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eISSN: 2574-8092

International Robotics & Automation Journal

Research Article Volume 4 Issue 3

Output control of a single-airscrew helicopter’s longitudinal motion spectrum

Nikolay E Zubov, Vladimir N Ryabchenko, Igor V Sorokin

Moscow, Bauman Moscow State Technical University, Russia

Correspondence: Nikolay E Zubov, Moscow, Bauman Moscow State Technical University, Russia, Tel 79637637687

Received: August 17, 2017 | Published: May 7, 2018

Citation: Zubov NE, Ryabchenko VN, Sorokin IV. Output control of a single-airscrew helicopter’s longitudinal motion spectrum. Int Rob Auto J. 2018;4(3):157-162. DOI: 10.15406/iratj.2018.04.00114

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Abstract

The problem of stabilization law synthesis of a single-airscrew helicopter’s longitudinal motion for lack of information about the vertical speed of its motion is analytically solved. The solution is based on the method of output control synthesis of the MIMO-system motion spectrum to be used as the basis of an especially designed multilevel decomposition of the dynamic system model in state space.

Keywords: single-airscrew helicopter, longitudinal motion, linear MIMO-system, stabilization, output feedback, motion spectrum, Control laws synthesis, multilevel decomposition

Introduction

The physical principle of a single-airscrew helicopter’s spatial motion provides the opportunity to categorize it as a complex multidimensional dynamic system (control object). At the same time, it is possible to mark out the following problems in less detail, which can be attributed to the control of helicopters as multidimensional dynamic systems.1,2

  1. The problem of stabilization, or the problem of stabilizing control law synthesis, i.e. determination of a feedback (of a controller), which provides stability for a disturbed motion. In searching for this kind of problem solution, as a rule, modal control methods or pole control methods are used.3–6
  2. The closed-loop system decoupling problem, i.e. determination of a controller’s coefficients that provide decoupling of the control object’s subsystems. The group of methods used is based on the analysis and usage of the reference system operators’ kernels.7,8
  3. The problem of making a set of stabilizing control laws, i.e. obtaining all, or almost all, control laws (algorithms), which provide stability for the closed-loop system, if the solution to the first problem, from those listed above, is not the only one. As a rule, the methods used are based on the solutions to Diophantine equations and the subsequent parameterization (for instance, the Youla-Kuĉera parametrization).7,9–12

It is possible to find other, not so commonly used, statements and solutions to the problem. For instance, there is a statement and a solution to the problem when stability and decoupling of the closed-loop controlled system, and also specified placement of the invariant system’s zeros, i.e. complex frequencies, on which the closed-loop system “locks” completely or partially, can be provided simultaneously. The strengthened type of decoupling, when it is necessary to provide not only the block-diagonal form of the closed-loop system’s operator, but also specific placement of the invariant system’s zeros on the complex plane, belongs to it. Many of the practically important solutions to the problems listed above are not accompanied by published methods on how to obtain them. The analytical solutions have to fill this gap. In this paper for the first time the analytical solution to the problem of a single-airscrew helicopter’s stabilization in the vertical plane with the specified placement of poles and the incomplete measurement vector of motion parameters is presented. We use the linearized model of longitudinal motion of a single-airscrew helicopter (SH) having the following form:13

( V ˙ x V ˙ y ω ˙ z υ ˙ )=( a V x V x a V x V y a V x ω z a V x υ a V y V x a V y V y a V y ω z a V y υ a ω z V x a ω z V y a ω z ω z 0 0 0 1 0 )( V x V y ω z υ )+( b V x u x b V x u gp b V y u x b V y u gp b ω z u x b ω z u gp 0 0 )( u x u gp ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpu0dh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaajuaGba qcLbsafaqabeabbaaaaKqbagaajugibiqadAfagaGaaOWaaSbaaKqb GeaajugWaiaadIhaaKqbagqaaaqaaKqzGeGabmOvayaacaGcdaWgaa qcfasaaKqzadGaamyEaaqcfayabaaabaqcLbsacuaHjpWDgaGaaKqb aoaaBaaajuaibaqcLbmacaWG6baajuaibeaaaKqbagaajugibiqbew 8a1zaacaaaaaqcfaOaayjkaiaawMcaaKqzGeGaeyypa0Jcdaqadaqc fayaaKqzGeqbaeqabqabaaaaaKqbagaajugibiaadggajuaGdaqhaa qcfasaaKqzadGaamOvaKqbaoaaBaaajuaibaqcLbmacaWG4baajuai beaaaeaajugWaiaadAfajuaGdaWgaaqcfasaaKqzadGaamiEaaqcfa sabaaaaaqcfayaaKqzGeGaamyyaKqbaoaaDaaajuaibaqcLbmacaWG wbqcfa4aaSbaaKqbGeaajugWaiaadIhaaKqbGeqaaaqaaKqzadGaam OvaKqbaoaaBaaajuaibaqcLbmacaWG5baajuaibeaaaaaajuaGbaqc LbsacaWGHbqcfa4aa0baaKqbGeaajugWaiaadAfajuaGdaWgaaqcfa saaKqzadGaamiEaaqcfasabaaabaqcLbmacqaHjpWDjuaGdaWgaaqc fasaaKqzadGaamOEaaqcfasabaaaaaqcfayaaKqzGeGaamyyaKqbao aaDaaajuaibaqcLbmacaWGwbqcfa4aaSbaaKqbGeaajugWaiaadIha aKqbGeqaaaqaaKqzadGaeqyXduhaaaqcfayaaKqzGeGaamyyaKqbao aaDaaajuaibaqcLbmacaWGwbqcfa4aaSbaaKqbGeaajugWaiaadMha aKqbGeqaaaqaaKqzadGaamOvaKqbaoaaBaaajuaibaqcLbmacaWG4b aajuaibeaaaaaajuaGbaqcLbsacaWGHbqcfa4aa0baaKqbGeaajugW aiaadAfajuaGdaWgaaqcfasaaKqzadGaamyEaaqcfasabaaabaqcLb macaWGwbqcfa4aaSbaaKqbGeaajugWaiaadMhaaKqbGeqaaaaaaKqb agaajugibiaadggajuaGdaqhaaqcfasaaKqzadGaamOvaKqbaoaaBa aajuaibaqcLbmacaWG5baajuaibeaaaeaajugWaiabeM8a3Lqbaoaa BaaajuaibaqcLbmacaWG6baajuaibeaaaaaajuaGbaqcLbsacaWGHb qcfa4aa0baaKqbGeaajugWaiaadAfajuaGdaWgaaqcfasaaKqzadGa amyEaaqcfasabaaabaqcLbmacqaHfpqDaaaajuaGbaqcLbsacaWGHb qcfa4aa0baaKqbGeaajugWaiabeM8a3LqbaoaaBaaajuaibaqcLbma caWG6baajuaibeaaaeaajugWaiaadAfajuaGdaWgaaqcfasaaKqzad GaamiEaaqcfasabaaaaaqcfayaaKqzGeGaamyyaKqbaoaaDaaajuai baqcLbmacqaHjpWDjuaGdaWgaaqcfasaaKqzadGaamOEaaqcfasaba aabaqcLbmacaWGwbqcfa4aaSbaaKqbGeaajugWaiaadMhaaKqbGeqa aaaaaKqbagaajugibiaadggajuaGdaqhaaqcfasaaKqzadGaeqyYdC xcfa4aaSbaaKqbGeaajugWaiaadQhaaKqbGeqaaaqaaKqzadGaeqyY dCxcfa4aaSbaaKqbGeaajugWaiaadQhaaKqbGeqaaaaaaKqbagaaju gibiaaicdaaKqbagaajugibiaaicdaaKqbagaajugibiaaicdaaKqb agaajugibiaaigdaaKqbagaajugibiaaicdaaaaajuaGcaGLOaGaay zkaaGcdaqadaqcfayaaKqzGeqbaeqabqqaaaaajuaGbaqcLbsacaWG wbGcdaWgaaqcfasaaKqzadGaamiEaaqcfayabaaabaqcLbsacaWGwb GcdaWgaaqcfasaaKqzadGaamyEaaqcfayabaaabaqcLbsacqaHjpWD kmaaBaaajuaibaqcLbmacaWG6baajuaGbeaaaeaajugibiabew8a1b aaaKqbakaawIcacaGLPaaajugibiabgUcaROWaaeWaaKqbagaajugi buaabeqaeiaaaaqcfayaaKqzGeGaamOyaKqbaoaaDaaajuaibaqcLb macaWGwbqcfa4aaSbaaKqbGeaajugWaiaadIhaaKqbGeqaaaqaaKqz adGaamyDaKqbaoaaBaaajuaibaqcLbmacaWG4baajuaibeaaaaaaju aGbaqcLbsacaWGIbqcfa4aa0baaKqbGeaajugWaiaadAfajuaGdaWg aaqcfasaaKqzadGaamiEaaqcfasabaaabaqcLbmacaWG1bqcfa4aaS baaKqbGeaajugWaiaabEgacaqGWbaajuaibeaaaaaajuaGbaqcLbsa caWGIbqcfa4aa0baaKqbGeaajugWaiaadAfajuaGdaWgaaqcfasaaK qzadGaamyEaaqcfasabaaabaqcLbmacaWG1bqcfa4aaSbaaKqbGeaa jugWaiaadIhaaKqbGeqaaaaaaKqbagaajugibiaadkgajuaGdaqhaa qcfasaaKqzadGaamOvaKqbaoaaBaaajuaibaqcLbmacaWG5baajuai beaaaeaajugWaiaadwhajuaGdaWgaaqcfasaaKqzadGaae4zaiaabc haaKqbGeqaaaaaaKqbagaajugibiaadkgajuaGdaqhaaqcfasaaKqz adGaeqyYdCxcfa4aaSbaaKqbGeaajugWaiaadQhaaKqbGeqaaaqaaK qzadGaamyDaKqbaoaaBaaajuaibaqcLbmacaWG4baajuaibeaaaaaa juaGbaqcLbsacaWGIbqcfa4aa0baaKqbGeaajugWaiabeM8a3Lqbao aaBaaajuaibaqcLbmacaWG6baajuaibeaaaeaajugWaiaadwhajuaG daWgaaqcfasaaKqzadGaae4zaiaabchaaKqbGeqaaaaaaKqbagaaju gibiaaicdaaKqbagaajugibiaaicdaaaaajuaGcaGLOaGaayzkaaGc daqadaqcfayaaKqzGeqbaeqabiqaaaqcfayaaKqzGeGaamyDaKqbao aaBaaajuaibaqcLbmacaWG4baajuaibeaaaKqbagaajugibiaadwha kmaaBaaajuaibaqcLbmacaqGNbGaaeiCaaqcfayabaaaaaGaayjkai aawMcaaKqzGeGaaiOlaaaa@7088@  (1)
Here V x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGwbGcdaWgaaqcfasaaKqzadGaamiEaaqcfaya baaaaa@4098@  – deviation from specified value of the longitudinal speed; V y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGwbWcdaWgaaqcfasaaKqzadGaamyEaaqcfasa baaaaa@403A@  – deviation from specified value of the vertical speed; ω z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaaHjpWcdaWgaaqcfasaaKqzadGaamOEaaqcfasa baaaaa@40B2@  – deviation from specified value of the pitch angular velocity; υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHfp qDaaa@39A7@ – deviation from specified value of the angle of pitch; u x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbb f9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq =He9q8qqQ8frFve9Fve9Ff0dmeaabaqaceGabiGaamqabaqaaeaaea aakeaajugibiaadwhajuaGdaWgaaqcfasaaKqzadGaamiEaaqcfaya baaaaa@3FCF@  – deviation angle of a main rotor’s cone in the longitudinal direction; u gp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWG1bGcdaWgaaqcfasaaKqzadGaae4zaiaabcha aKqbagqaaaaa@4197@  – general pitch of a main rotor; , a V x V x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbb f9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq =He9q8qqQ8frFve9Fve9Ff0dmeaabaqaceGabiGaamqabaqaaeaaea aakeaajugibiaadggajuaGdaqhaaqcfasaaKqzadGaamOvaKqbaoaa BaaajuaibaqcLbmacaWG4baajuaibeaaaeaajugWaiaadAfajuaGda WgaaqcfasaaKqzadGaamiEaaqcfasabaaaaaaa@4781@ , a V x V y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadIhaaKqbGeqaaaqaaKqzadGaamOvaSWaaSbaaK qbGeaajugWaiaadMhaaKqbGeqaaaaaaaa@4765@ , a V x ω z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadIhaaKqbGeqaaaqaaKqzadGaaqyYdSWaaSbaaK qbGeaajugWaiaadQhaaKqbGeqaaaaaaaa@47DD@ , a V x υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadIhaaKqbGeqaaaqaaKqzadGaaqyXdaaaaaa@4524@ , a V y V x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadMhaaKqbGeqaaaqaaKqzadGaamOvaSWaaSbaaK qbGeaajugWaiaadIhaaKqbGeqaaaaaaaa@4765@ , a V y V y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadMhaaKqbGeqaaaqaaKqzadGaamOvaSWaaSbaaK qbGeaajugWaiaadMhaaKqbGeqaaaaaaaa@4766@ , a V y ω z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadMhaaKqbGeqaaaqaaKqzadGaaqyYdSWaaSbaaK qbGeaajugWaiaadQhaaKqbGeqaaaaaaaa@47DE@ , a V y υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadMhaaKqbGeqaaaqaaKqzadGaaqyXdaaaaaa@4525@ , a ω z V x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaaqyYdSWaaSba aKqbGeaajugWaiaadQhaaKqbGeqaaaqaaKqzadGaamOvaSWaaSbaaK qbGeaajugWaiaadIhaaKqbGeqaaaaaaaa@47DD@ , a ω z V y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaaqyYdSWaaSba aKqbGeaajugWaiaadQhaaKqbGeqaaaqaaKqzadGaamOvaSWaaSbaaK qbGeaajugWaiaadMhaaKqbGeqaaaaaaaa@47DE@ , a ω z ω z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaqhaaqcfasaaKqzadGaaqyYdSWaaSba aKqbGeaajugWaiaadQhaaKqbGeqaaaqaaKqzadGaaqyYdSWaaSbaaK qbGeaajugWaiaadQhaaKqbGeqaaaaaaaa@4856@ , b V x u x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGIbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadIhaaKqbGeqaaaqaaKqzadGaamyDaSWaaSbaaK qbGeaajugWaiaadIhaaKqbGeqaaaaaaaa@4784@ , b V x u gp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGIbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadIhaaKqbGeqaaaqaaKqzadGaamyDaSWaaSbaaK qbGeaajugWaiaabEgacaqGWbaajuaibeaaaaaaaa@4864@ , b V y u x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGIbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadMhaaKqbGeqaaaqaaKqzadGaamyDaSWaaSbaaK qbGeaajugWaiaadIhaaKqbGeqaaaaaaaa@4785@ , b V y u gp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGIbWcdaqhaaqcfasaaKqzadGaamOvaSWaaSba aKqbGeaajugWaiaadMhaaKqbGeqaaaqaaKqzadGaamyDaSWaaSbaaK qbGeaajugWaiaabEgacaqGWbaajuaibeaaaaaaaa@4865@ , b ω z u x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGIbWcdaqhaaqcfasaaKqzadGaaqyYdSWaaSba aKqbGeaajugWaiaadQhaaKqbGeqaaaqaaKqzadGaamyDaSWaaSbaaK qbGeaajugWaiaadIhaaKqbGeqaaaaaaaa@47FD@ , b ω z u x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGIbWcdaqhaaqcfasaaKqzadGaaqyYdSWaaSba aKqbGeaajugWaiaadQhaaKqbGeqaaaqaaKqzadGaamyDaSWaaSbaaK qbGeaajugWaiaadIhaaKqbGeqaaaaaaaa@47FD@  – linearization coefficients.8,14
We use the following notation
x= ( V x V y ω z υ ) T ,u= ( u x u gp ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWG4bGaeyypa0JcdaqadaqcfayaaKqzGeqbaeqa beabaaaajuaGbaqcLbsacqWIZwIvcaWGwbWcdaWgaaqcfasaaKqzad GaamiEaaqcfasabaaajuaGbaqcLbsacqWIZwIvcaWGwbGcdaWgaaqc fasaaKqzadGaamyEaaqcfayabaaabaqcLbsacqWIZwIvcaaHjpWcda WgaaqcfasaaKqzadGaamOEaaqcfasabaaajuaGbaqcLbsacqWIZwIv caaHfpaaaaqcfaOaayjkaiaawMcaaSWaaWbaaKqbGeqabaqcLbmaca qGubaaaKqzGeGaaiilaiaaykW7caaMc8UaamyDaiabg2da9OWaaeWa aKqbagaajugibuaabeqabiaaaKqbagaajugibiabloBjwjaadwhakm aaBaaajuaibaqcLbmacaWG4baajuaGbeaaaeaajugibiabloBjwjaa dwhakmaaBaaajuaibaqcLbmacaqGNbGaaeiCaaqcfayabaaaaaGaay jkaiaawMcaaOWaaWbaaKqbagqajuaibaqcLbmacaqGubaaaaaa@7631@ ,
and also
a 11 = a V x V x , a 12 = a V x V y , a 13 = a V x ω z , a 14 = a V x υ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaWgaaqcKvay=haajugWaiaaigdacaaI XaaajqwbG9FabaqcLbsacqGH9aqpcaWGHbWcdaqhaaqcKvay=haaju gWaiaadAfalmaaBaaajqwbG9FaaKqzadGaamiEaaqcKvay=hqaaaqa aKqzadGaamOvaSWaaSbaaKazfa2=baqcLbmacaWG4baajqwbG9Faba aaaKqzGeGaaiilaiaaykW7caaMc8UaaGPaVlaadggakmaaBaaajuai baqcLbmacaaIXaGaaGOmaaqcfayabaqcLbsacqGH9aqpcaWGHbWcda qhaaqcfasaaKqzadGaamOvaSWaaSbaaKqbGeaajugWaiaadIhaaKqb GeqaaaqaaKqzadGaamOvaSWaaSbaaKqbGeaajugWaiaadMhaaKqbGe qaaaaajugibiaacYcacaaMc8UaaGPaVlaadggakmaaBaaajuaibaqc LbmacaaIXaGaaG4maaqcfayabaqcLbsacqGH9aqpcaWGHbWcdaqhaa qcfasaaKqzadGaamOvaSWaaSbaaKqbGeaajugWaiaadIhaaKqbGeqa aaqaaKqzadGaaqyYdSWaaSbaaKqbGeaajugWaiaadQhaaKqbGeqaaa aajugibiaacYcacaaMc8UaaGPaVlaadggakmaaBaaajuaibaqcLbma caaIXaGaaGinaaqcfayabaqcLbsacqGH9aqpcaWGHbWcdaqhaaqcfa saaKqzadGaamOvaSWaaSbaaKqbGeaajugWaiaadIhaaKqbGeqaaaqa aKqzadGaaqyXdaaajugibiaacYcacaaMc8oaaa@9A51@ a 21 = a V y V x , a 22 = a V y V y , a 23 = a V y ω z , a 24 = a V y υ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGHbWcdaWgaaqcfasaaKqzadGaaGOmaiaaigda aKqbGeqaaKqzGeGaeyypa0JaamyyaSWaa0baaKqbGeaajugWaiaadA falmaaBaaajuaibaqcLbmacaWG5baajuaibeaaaeaajugWaiaadAfa lmaaBaaajuaibaqcLbmacaWG4baajuaibeaaaaqcLbsacaGGSaGaaG PaVlaaykW7caaMc8UaamyyaOWaaSbaaKqbGeaajugWaiaaikdacaaI YaaajuaGbeaajugibiabg2da9iaadggalmaaDaaajuaibaqcLbmaca WGwbWcdaWgaaqcfasaaKqzadGaamyEaaqcfasabaaabaqcLbmacaWG wbWcdaWgaaqcfasaaKqzadGaamyEaaqcfasabaaaaKqzGeGaaiilai aaykW7caaMc8UaaGPaVlaadggalmaaBaaajuaibaqcLbmacaaIYaGa aG4maaqcfasabaqcLbsacqGH9aqpcaWGHbWcdaqhaaqcfasaaKqzad GaamOvaSWaaSbaaKqbGeaajugWaiaadMhaaKqbGeqaaaqaaKqzadGa aqyYdSWaaSbaaKqbGeaajugWaiaadQhaaKqbGeqaaaaajugibiaacY cacaaMc8UaaGPaVlaaykW7caWGHbGcdaWgaaqcfasaaKqzadGaaGOm aiaaisdaaKqbagqaaKqzGeGaeyypa0JaamyyaSWaa0baaKqbGeaaju gWaiaadAfalmaaBaaajuaibaqcLbmacaWG5baajuaibeaaaeaajugW aiaaew8aaaqcLbsacaGGSaaaaa@9010@ a 31 = a ω z V x , a 32 = a ω z V y , a 33 = a ω z ω z , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaaMc8UaaGPaVlaadggalmaaBaaajuaibaqcLbma caaIZaGaaGymaaqcfasabaqcLbsacqGH9aqpcaWGHbWcdaqhaaqcfa saaKqzadGaaqyYdSWaaSbaaKqbGeaajugWaiaadQhaaKqbGeqaaaqa aKqzadGaamOvaSWaaSbaaKqbGeaajugWaiaadIhaaKqbGeqaaaaaju gibiaacYcacaaMc8UaaGPaVlaaykW7caWGHbWcdaWgaaqcfasaaKqz adGaaG4maiaaikdaaKqbGeqaaKqzGeGaeyypa0JaamyyaSWaa0baaK qbGeaajugWaiaaeM8almaaBaaajuaibaqcLbmacaWG6baajuaibeaa aeaajugWaiaadAfalmaaBaaajuaibaqcLbmacaWG5baajuaibeaaaa qcLbsacaGGSaGaaGPaVlaaykW7caaMc8UaamyyaSWaaSbaaKqbGeaa jugWaiaaiodacaaIZaaajuaibeaajugibiabg2da9iaadggalmaaDa aajuaibaqcLbmacaaHjpWcdaWgaaqcfasaaKqzadGaamOEaaqcfasa baaabaqcLbmacaaHjpWcdaWgaaqcfasaaKqzadGaamOEaaqcfasaba aaaKqzGeGaaiilaaaa@7FCE@
b 11 = b V x u x , b 12 = b V x u gp , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGIb GcdaWgaaqcfasaaKqzadGaaGymaiaaigdaaKqbagqaaKqzGeGaeyyp a0JaamOyaSWaa0baaKqbGeaajugWaiaadAfalmaaBaaajuaibaqcLb macaWG4baajuaibeaaaeaajugWaiaadwhalmaaBaaajuaibaqcLbma caWG4baajuaibeaaaaqcLbsacaGGSaGaaGPaVlaaykW7caaMc8Uaam OyaOWaaSbaaKqbGeaajugWaiaaigdacaaIYaaajuaGbeaajugibiab g2da9iaadkgalmaaDaaajuaibaqcLbmacaWGwbWcdaWgaaqcfasaaK qzadGaamiEaaqcfasabaaabaqcLbmacaWG1bWcdaWgaaqcfasaaKqz adGaae4zaiaabchaaKqbGeqaaaaajugibiaacYcacaaMc8oaaa@632B@
b 21 = b V y u x , b 22 = b V y u gp , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqeguuD JXwAKbqeeuuDJXwAKbsr4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGabiqaciaadeqaaeaaba abaaGcbaqcLbsacaWGIbGcdaWgaaqcfasaaKqzadGaaGOmaiaaigda aKqbagqaaKqzGeGaeyypa0JaamOyaSWaa0baaKqbGeaajugWaiaadA falmaaBaaajuaibaqcLbmacaWG5baajuaibeaaaeaajugWaiaadwha lmaaBaaajuaibaqcLbmacaWG4baajuaibeaaaaqcLbsacaGGSaGaaG PaVlaaykW7caaMc8UaamOyaOWaaSbaaKqbGeaajugWaiaaikdacaaI YaaajuaGbeaajugibiabg2da9iaadkgalmaaDaaajuaibaqcLbmaca WGwbWcdaWgaaqcfasaaKqzadGaamyEaaqcfasabaaabaqcLbmacaWG 1bWcdaWgaaqcfasaaKqzadGaae4zaiaabchaaKqbGeqaaaaajugibi aacYcaaaa@666F@
b 31 = b ω z u x , b 32 = b ω z u gp , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaMc8 UaaGPaVlaadkgakmaaBaaajuaibaqcLbmacaaIZaGaaGymaaqcfaya baqcLbsacqGH9aqpcaWGIbWcdaqhaaqcfasaaKqzadGaeqyYdC3cda WgaaqcfasaaKqzadGaamOEaaqcfasabaaabaqcLbmacaWG1bWcdaWg aaqcfasaaKqzadGaamiEaaqcfasabaaaaKqzGeGaaiilaiaaykW7ca aMc8UaaGPaVlaadkgakmaaBaaajuaibaqcLbmacaaIZaGaaGOmaaqc fayabaqcLbsacqGH9aqpcaWGIbWcdaqhaaqcfasaaKqzadGaeqyYdC 3cdaWgaaqcfasaaKqzadGaamOEaaqcfasabaaabaqcLbmacaWG1bWc daWgaaqcfasaaSWaaSbaaKqbGeaajugWaiaabEgacaqGWbaajuaibe aaaeqaaaaajugibiaacYcaaaa@66FC@


Then instead of expression (1) we will obtain the following linearized model of the SH longitudinal motion:
  x ˙ =( a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 0 0 0 1 0 )x+( b 11 b 12 b 21 b 22 b 31 b 32 0 0 )u. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiqa=HhagaGaaiabg2da9O WaaeWaaKqbagaajugibuaabeqaeqaaaaaajuaGbaqcLbsacaWGHbWc daWgaaqcfasaaKqzadGaaGymaiaaigdaaKqbGeqaaaqcfayaaKqzGe GaamyyaOWaaSbaaKqbGeaajugWaiaaigdacaaIYaaajuaGbeaaaeaa jugibiaadggakmaaBaaajuaibaqcLbmacaaIXaGaaG4maaqcfayaba aabaqcLbsacaWGHbGcdaWgaaqcfasaaKqzadGaaGymaiaaisdaaKqb agqaaaqaaKqzGeGaamyyaSWaaSbaaKqbGeaajugWaiaaikdacaaIXa aajuaibeaaaKqbagaajugibiaadggakmaaBaaajuaibaqcLbmacaaI YaGaaGOmaaqcfayabaaabaqcLbsacaWGHbGcdaWgaaqcfasaaKqzad GaaGOmaiaaiodaaKqbagqaaaqaaKqzGeGaamyyaOWaaSbaaKqbGeaa jugWaiaaikdacaaI0aaajuaGbeaaaeaajugibiaadggakmaaBaaaju aibaqcLbmacaaIZaGaaGymaaqcfayabaaabaqcLbsacaWGHbGcdaWg aaqcfasaaKqzadGaaG4maiaaikdaaKqbagqaaaqaaKqzGeGaamyyaO WaaSbaaKqbGeaajugWaiaaiodacaaIZaaajuaGbeaaaeaajugibiaa icdaaKqbagaajugibiaaicdaaKqbagaajugibiaaicdaaKqbagaaju gibiaaigdaaKqbagaajugibiaaicdaaaaajuaGcaGLOaGaayzkaaqc LbsacaWF4bGaey4kaSIcdaqadaqcfayaaKqzGeqbaeqabqGaaaaaju aGbaqcLbsacaWGIbGcdaWgaaqcfasaaKqzadGaaGymaiaaigdaaKqb agqaaaqaaKqzGeGaamOyaOWaaSbaaKqbGeaajugWaiaaigdacaaIYa aajuaGbeaaaeaajugibiaadkgakmaaBaaajuaibaqcLbmacaaIYaGa aGymaaqcfayabaaabaqcLbsacaWGIbGcdaWgaaqcfasaaKqzadGaaG OmaiaaikdaaKqbagqaaaqaaKqzGeGaamOyaOWaaSbaaKqbGeaajugW aiaaiodacaaIXaaajuaGbeaaaeaajugibiaadkgalmaaBaaajuaiba qcLbmacaaIZaGaaGOmaaqcfasabaaajuaGbaqcLbsacaaIWaaajuaG baqcLbsacaaIWaaaaaqcfaOaayjkaiaawMcaaKqzGeGaa8xDaiaac6 caaaa@A224@ (2)
Hereafter it will be considered that information about vertical speed of the SH motion is not available. As a result the system output vector can be expressed as
y= ( Δ V x Δ ω z Δυ ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbsaca WF5bGaeyypa0JcdaqadaqcfayaaKqzGeqbaeqabeWaaaqcfayaaKqz GeGaeuiLdqKaamOvaOWaaSbaaKqbGeaajugWaiaadIhaaKqbagqaaa qaaKqzGeGaeuiLdqKaeqyYdCNcdaWgaaqcfasaaKqzadGaamOEaaqc fayabaaabaqcLbsacqqHuoarcqaHfpqDaaaajuaGcaGLOaGaayzkaa GcdaahaaqcfayabKqbGeaajugWaiaabsfaaaaaaa@5125@ .
Taking into consideration the assumptions made, the initial mathematical model can be written in the form of a dynamic MIMO-system of the “input – state – output” type:
  x ˙ =Ax+Βu,y=Cx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiqa=HhagaGaaiaa=1daca WFbbGaa8hEaiaa=TcacaWFsoGaa8xDaiaa=XcacaaMc8UaaGPaVlaa ykW7caaMc8Uaa8xEaiaa=1dacaWFdbGaa8hEaaaa@42B3@ , (3)
Where x n ,n=4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=HhacqGHiiIZcqWIDe sOkmaaCaaajuaGbeqcfasaaKqzadGaamOBaaaajugibiaacYcacaaM c8UaaGPaVlaaykW7caaMc8UaamOBaiabg2da9iaaisdaaaa@434A@  is the state vector; u r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbsaca WF1bGaeyicI4SaeSyhHe6cdaahaaqcfasabeaajugWaiaadkhaaaaa aa@3E53@  is the input vector (control vector), where r=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb Gaeyypa0JaaGOmaaaa@391F@ ; y m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieqajugibiaa=LhacqGHiiIZcqWIDe sOkmaaCaaajuaGbeqcfasaaKqzadGaamyBaaaaaaa@3929@  is the output vector (measurement vector), where m=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaad2gacqGH9aqpcaaIZaaaaa@34DE@ . Here MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqWIDe sOaaa@37D6@  is the set of real numbers.
In the equations (3) the corresponding matrices can be written as
A=( a 11 a 12 a 13 a 14 a 21 a 22 a 14 a 24 a 31 a 32 a 33 0 0 0 1 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=feacqGH9aqpkmaabm aajuaGbaqcLbsafaqabeabeaaaaaqcfayaaKqzGeGaamyyaSWaaSba aKqbGeaajugWaiaaigdacaaIXaaajuaibeaaaKqbagaajugibiaadg gakmaaBaaajuaibaqcLbmacaaIXaGaaGOmaaqcfayabaaabaqcLbsa caWGHbGcdaWgaaqcfasaaKqzadGaaGymaiaaiodaaKqbagqaaaqaaK qzGeGaamyyaOWaaSbaaKqbGeaajugWaiaaigdacaaI0aaajuaGbeaa aeaajugibiaadggalmaaBaaajuaibaqcLbmacaaIYaGaaGymaaqcfa sabaaajuaGbaqcLbsacaWGHbGcdaWgaaqcfasaaKqzadGaaGOmaiaa ikdaaKqbagqaaaqaaKqzGeGaamyyaOWaaSbaaKqbGeaajugWaiaaig dacaaI0aaajuaGbeaaaeaajugibiaadggakmaaBaaajuaibaqcLbma caaIYaGaaGinaaqcfayabaaabaqcLbsacaWGHbGcdaWgaaqcfayaaK qzGeGaaG4maiaaigdaaKqbagqaaaqaaKqzGeGaamyyaOWaaSbaaKqb GeaajugWaiaaiodacaaIYaaajuaGbeaaaeaajugibiaadggakmaaBa aajuaibaqcLbmacaaIZaGaaG4maaqcfayabaaabaqcLbsacaaIWaaa juaGbaqcLbsacaaIWaaajuaGbaqcLbsacaaIWaaajuaGbaqcLbsaca aIXaaajuaGbaqcLbsacaaIWaaaaaqcfaOaayjkaiaawMcaaaaa@77C6@ ,   B=( b 11 b 12 b 21 b 22 b 31 b 32 0 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=jeacqGH9aqpkmaabm aajuaGbaqcLbsafaqabeabcaaaaKqbagaajugibiaadkgalmaaBaaa juaibaqcLbmacaaIXaGaaGymaaqcfasabaaajuaGbaqcLbsacaWGIb WcdaWgaaqcfasaaKqzadGaaGymaiaaikdaaKqbGeqaaaqcfayaaKqz GeGaamOyaSWaaSbaaKqbGeaajugWaiaaikdacaaIXaaajuaibeaaaK qbagaajugibiaadkgakmaaBaaajuaibaqcLbmacaaIYaGaaGOmaaqc fayabaaabaqcLbsacaWGIbGcdaWgaaqcfasaaKqzadGaaG4maiaaig daaKqbagqaaaqaaKqzGeGaamOyaOWaaSbaaKqbGeaajugWaiaaioda caaIYaaajuaGbeaaaeaajugibiaaicdaaKqbagaajugibiaaicdaaa aajuaGcaGLOaGaayzkaaqcLbsacaGGSaGaaGPaVdaa@5C65@ . C=( 1 0 0 0 0 0 1 0 0 0 0 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=neacqGH9aqpkmaabm aajuaGbaqcLbsafaqabeWaeaaaaKqbagaajugibiaaigdaaKqbagaa jugibiaaicdaaKqbagaajugibiaaicdaaKqbagaajugibiaaicdaaK qbagaajugibiaaicdaaKqbagaajugibiaaicdaaKqbagaajugibiaa igdaaKqbagaajugibiaaicdaaKqbagaajugibiaaicdaaKqbagaaju gibiaaicdaaKqbagaajugibiaaicdaaKqbagaajugibiaaigdaaaaa juaGcaGLOaGaayzkaaaaaa@4D6E@ (4)
If, as a control law(3), to suggest the expression of the following form:
  u=Fy=FCx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=vhacaWF9aGaa8Nrai aa=LhacaWF9aGaa8Nraiaa=neacaWF4baaaa@38E9@ , (5)
Where F r×m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=zeacqGHiiIZcqWIDe sOkmaaCaaajuaGbeqcfasaaKqzadGaamOCaiabgEna0kaad2gaaaaa aa@3C05@  is a matrix of the output controller, then, in accordance with2 for the system under consideration (3) – (5) a case of the dynamic MIMO-system output control will take place.
Hereafter we assume that matrix B 4×2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=jeacqGHiiIZcqWIDe sOkmaaCaaajuaGbeqcfasaaKqzadGaaGinaiabgEna0kaaikdaaaaa aa@3B92@  (4) has a full rank ( rankB=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbsaca WFYbGaa8xyaiaa=5gacaWFRbGaaGPaVlaa=jeacaWF9aGaa8Nmaaaa @3DDD@  in this case), or that its equivalent matrix B T B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=jeakmaaCaaajuaGbe qcfasaaKqzadGaaeivaaaajugibiaa=jeaaaa@3736@  is invertible, i.e. det( B T B )0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiGacsgacaGGLbGaaiiDaOWaae WaaKqbagaaieGajugibiaa=jeakmaaCaaajuaGbeqcfasaaKqzadGa aeivaaaajugibiaa=jeaaKqbakaawIcacaGLPaaajugibiabgcMi5k aaicdaaaa@404F@ . Let us now consider the matrix spectrum A 4×4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=feacqGHiiIZcqWIDe sOlmaaCaaajuaibeqaaKqzadGaaGinaiabgEna0kaaisdaaaaaaa@3B06@  (5). It will be understood as a set of matrix A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiqcLbsaca WFbbaaaa@3732@  eigen values (poles), eig( A )={ λ i :det( λ i I 4 A )=0,i= 1,4 ¯ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaabwgacaqGPbGaae4zaOWaae WaaKqbagaaieGajugibiaa=feaaKqbakaawIcacaGLPaaajugibiab g2da9OWaaiWaaKqbagaajugibiabeU7aSTWaaSbaaKqbGeaajugWai aadMgaaKqbGeqaaKqzGeGaeyicI4SaeSOaHmQaaiOoaiaaykW7caaM c8UaaGPaVlGacsgacaGGLbGaaiiDaOWaaeWaaKqbagaajugibiabeU 7aSTWaaSbaaKqbGeaajugWaiaadMgaaKqbGeqaaKqzGeGaa8xsaSWa aSbaaKqbGeaajugWaiaaisdaaKqbGeqaaKqzGeGaeyOeI0Iaa8xqaa qcfaOaayjkaiaawMcaaKqzGeGaeyypa0JaaGimaiaacYcacaaMc8Ua aGPaVlaadMgacqGH9aqpkmaanaaajuaGbaqcLbsacaaIXaGaaiilai aaisdaaaaajuaGcaGL7bGaayzFaaaaaa@6764@ . Here I 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbsaca WFjbGcdaWgaaqcfasaaKqzadGaa8hnaaqcfayabaaaaa@3B7D@ – identity matrix of size 4×4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaaisdacqGHxdaTcaaI0aaaaa@35BD@ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiablkqiJcaa@3380@  – the set of complex numbers. Let Λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHBo ataaa@37FB@  be the given spectrum of the matrix A+BFC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=feacaWFRaGaa8Nqai aa=zeacaWFdbaaaa@35EE@  of the corresponding close-loop controlled system, i.e.
  Λ={ λ 1 , λ 2 , λ 3 , λ 4 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiabfU5amjabg2da9OWaaiWaaK qbagaajugibiqbeU7aSzaataWcdaWgaaqcfasaaKqzadGaaGymaaqc fasabaqcLbsacaqGSaGaaGPaVlaaykW7cuaH7oaBgaWeaSWaaSbaaK qbGeaajugWaiaaikdaaKqbGeqaaKqzGeGaaiilaiaaykW7caaMc8Ua fq4UdWMbambakmaaBaaajuaibaqcLbmacaaIZaaajuaGbeaajugibi aacYcacaaMc8UaaGPaVlqbeU7aSzaataGcdaWgaaqcfasaaKqzadGa aGinaaqcfayabaaacaGL7bGaayzFaaaaaa@56BA@ . (6)
It is required to determine (i.e. synthesize) explicitly the controller matrix FÎ R 2×3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=zeacaWFodGaa8NuaS WaaWbaaKqbGeqabaqcLbmacaWFYaGaa831aiaa=ndaaaaaaa@3963@ , such that the equality Λ=eig( A+BFC ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHBo atcqGH9aqpcaqGLbGaaeyAaiaabEgakmaabmaajuaGbaqcLbsacaWG bbGaey4kaSIaamOqaiaadAeacaWGdbaajuaGcaGLOaGaayzkaaaaaa@4459@ should be satisfied exactly. The complexity of this problem is a necessity for obtaining a solution in explicit analytical form, since A,B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8xqai aa=XcacaaMc8Uaa8Nqaaaa@399C@  matrices in (2) have a piecewise constant form. This paper is dedicated to solving of the problem.

Decomposition of a dynamic system
As a first step of the given problem solution we will consider the multilevel decomposition of the SH model suggested.15–17 Since in this case the inequality mr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaKqzGe GaeyyzImRaamOCaaaa@3B8F@  (i.e. the number of system’s outputs is greater than the number of its inputs) is implemented, then, in general, not taking into consideration specific numerical values for m and r, we consider the multilevel decomposition of system (3) – (5) of the following form: –zero decomposition level
A 0 =A, B 0 =B, C 0 =C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=fealmaaBaaajuaiba qcLbmacaWFWaaajuaibeaajugibiaa=1dacaWFbbGaa8hlaiaaykW7 caaMc8UaaGPaVlaaykW7caWFcbGcdaWgaaqcfasaaKqzadGaa8hmaa qcfayabaqcLbsacaWF9aGaa8Nqaiaa=XcacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaa=neakmaaBaaajuaibaqcLbmacaWFWaaajuaGbe aajugibiaa=1dacaWFdbaaaa@51DB@ (7)
kth decomposition level ( k= 1,M ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadUgacqGH9aqpkmaanaaaju aGbaqcLbsacaaIXaGaaiilaiaaykW7caaMc8Uaamytaaaaaaa@3AAB@ , where M=ceil(n/r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaad2eacqGH9aqpcaqGJbGaae yzaiaabMgacaqGSbGaaGPaVlaacIcacaWGUbGaai4laiaadkhacaGG Paaaaa@3D2C@ , ceil(*) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaabogacaqGLbGaaeyAaiaabY gacaaMc8UaaiikaiaacQcacaGGPaaaaa@3965@ –is the operation of rounding the number« * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacQcaaaa@3699@ »upwards)
A k = B k1 A k1 B k1 + , B k = B k1 A k1 B k1 , C k = C k1 A k1 B k1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadgeakmaaBaaajuaibaqcLb macaWGRbaajuaGbeaajugibiabg2da9iaadkeajuaGdaqhaaqcfasa aKqzadGaam4AaiabgkHiTiaaigdaaKqbGeaajugWaiabgwQiEbaaju gibiaadgeakmaaBaaajuaibaqcLbmacaWGRbGaeyOeI0IaaGymaaqc fayabaqcLbsacaWGcbGcdaqhaaqcfasaaKqzadGaam4AaiabgkHiTi aaigdaaKqbGeaajugWaiabgwQiEjabgUcaRaaajugibiaacYcacaaM c8UaaGPaVlaadkeakmaaBaaajuaibaqcLbmacaWGRbaajuaGbeaaju gibiabg2da9iaadkeakmaaDaaajuaibaqcLbmacaWGRbGaeyOeI0Ia aGymaaqcfasaaKqzadGaeyyPI4faaOGaamyqamaaBaaajuaibaqcLb macaWGRbGaeyOeI0IaaGymaaqcfasabaGccaWGcbWaaSbaaKqbGeaa jugWaiaadUgacqGHsislcaaIXaaajuaibeaajugibiaacYcacaaMc8 UaaGPaVlaadoeakmaaBaaajuaibaqcLbmacaWGRbaajuaibeaajugi biabg2da9iaadoeakmaaBaaajuaibaqcLbmacaWGRbGaeyOeI0IaaG ymaaqcfayabaGccaWGbbWaaSbaaKqbGeaajugWaiaadUgacqGHsisl caaIXaaajuaibeaajugibiaadkeakmaaDaaajuaibaqcLbmacaWGRb GaeyOeI0IaaGymaaqcfasaaKqzadGaeyyPI4Laey4kaScaaaaa@8AB0@ . (8)
Equations (7), (8) for a set of indices k= 0,M ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadUgacqGH9aqpkmaanaaaju aGbaqcLbsacaaIWaGaaiilaiaaykW7caaMc8Uaamytaaaaaaa@3AAA@  involve the matrices with the following properties:
( B k B k + ) 1 =( B k + B k ), B k B k =0, B k + B k = I r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqcfayaaKqzGeqbaeqabeGaaW qcfayaaKqzGeGaamOqaOWaaSbaaKqbGeaajugWaiaadUgaaKqbagqa aaqaaKqzGeGaamOqaSWaa0baaKqbGeaajugWaiaadUgaaKqbGeaaju gWaiabgwQiEjaabUcaaaaaaaqcfaOaayjkaiaawMcaaOWaaWbaaKqb agqajuaibaqcLbmacqGHsislcaaIXaaaaKqzGeGaeyypa0Jcdaqada qcfayaaKqzGeqbaeqabiqamaqcfayaaKqzGeGaamOqaOWaa0baaKqb GeaajugWaiaadUgaaKqbGeaajugWaiabgUcaRaaaaKqbagaajugibi aadkeakmaaDaaajuaibaqcLbmacaWGRbaajuaibaqcLbmacqGHLkIx aaaaaaqcfaOaayjkaiaawMcaaKqzGeGaaiilaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaamOqaOWaa0baaKqbGeaajugWaiaadUgaaKqb GeaajugWaiabgwQiEbaajugibiaadkeakmaaBaaajuaibaqcLbmaca WGRbaajuaGbeaajugibiabg2da9iaaicdacaGGSaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWGcbGcdaqhaaqcfasaaKqzadGaam4Aaa qcfasaaKqzadGaey4kaScaaKqzGeGaamOqaOWaaSbaaKqbGeaajugW aiaadUgaaKqbagqaaKqzGeGaeyypa0JaamysaOWaaSbaaKqbGeaaju gWaiaadkhaaKqbGeqaaaaa@8541@ ,
( C k C k ) 1 =( C k + C k + ), C k C k =0, C k C k + = I m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqcfayaaKqzGeqbaeqabiqama qcfayaaKqzGeGaam4qaSWaaSbaaKqbGeaajugWaiaadUgaaKqbGeqa aaqcfayaaKqzGeGaam4qaSWaa0baaKqbGeaajugWaiaadUgaaKqbGe aajugWaiabgwQiEbaaaaaajuaGcaGLOaGaayzkaaGcdaahaaqcfaya bKqbGeaajugWaiabgkHiTiaaigdaaaqcLbsacqGH9aqpkmaabmaaju aGbaqcLbsafaqabeqacaadjuaGbaqcLbsacaWGdbWcdaqhaaqcfasa aKqzadGaam4AaaqcfasaaKqzadGaey4kaScaaaqcfayaaKqzGeGaam 4qaSWaa0baaKqbGeaajugWaiaadUgaaKqbGeaajugWaiabgwQiEjaa bUcaaaaaaaqcfaOaayjkaiaawMcaaKqzGeGaaiilaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8Uaam4qaOWaaSbaaKqbagaajugWaiaadUga aSqabaqcLbsacaWGdbWcdaqhaaqcfasaaKqzadGaam4AaaqcfasaaK qzadGaeyyPI4faaKqzGeGaeyypa0JaaGimaiaacYcacaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaadoeakmaaBaaajuaibaqcLbmacaWGRb aajuaGbeaajugibiaadoealmaaDaaajuaibaqcLbmacaWGRbaajuai baqcLbmacqGHRaWkaaqcLbsacqGH9aqpcaWGjbGcdaWgaaqcfasaaK qzadGaamyBaaqcfayabaaaaa@85B4@ ,
Where the superscript « MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiabgwQiEbaa@33FB@ » denotes orthogonal annihilators (divisors of zero), and the superscript « + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8urps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiabgUcaRaaa@334C@ » denotes the Moore-Penrose pseudo inverse matrices.13,16–19 Also, we consider the recurrence formulae of controllers for the spectrum control on the corresponding decomposition levels, written down in reverse order:
– -th decomposition level
F M =( Φ M B M + B M + A M ) C M + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadAeakmaaBaaajuaibaqcLb macaWGnbaajuaGbeaajugibiabg2da9OWaaeWaaKqbagaajugibiab fA6agPWaaSbaaKqbGeaajugWaiaad2eaaKqbagqaaKqzGeGaamOqaS Waa0baaKqbGeaajugWaiaad2eaaKqbGeaajugWaiabgUcaRaaajugi biabgkHiTiaadkealmaaDaaajuaibaqcLbmacaWGnbaajuaibaqcLb macqGHRaWkaaGaamyqaOWaaSbaaKqbGeaajugWaiaad2eaaKqbagqa aaGaayjkaiaawMcaaKqzGeGaam4qaSWaa0baaKqbGeaajugWaiaad2 eaaKqbGeaajugWaiabgUcaRaaaaaa@54E9@ ,(9)
k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36C7@ -th decomposition level ( k= 0,M1 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadUgacqGH9aqpkmaanaaaju aGbaqcLbsacaaIWaGaaiilaiaaykW7caaMc8UaamytaiabgkHiTiaa igdaaaaaaa@3C52@ )
F k =( Φ k B k B k A k ) C k + , B k = B k + F k+1 C k+1 B k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadAeakmaaBaaajuaibaqcLb macaWGRbaajuaGbeaajugibiabg2da9OWaaeWaaKqbagaajugibiab fA6agPWaaSbaaKqbGeaajugWaiaadUgaaKqbagqaaOGaamOqaSWaa0 baaKqbGeaajugWaiaadUgaaKqbGeaajugWaiabgkHiTaaajugibiab gkHiTiaadkeakmaaDaaajuaibaqcLbmacaWGRbaajuaibaqcLbmacq GHsislaaqcLbsacaWGbbGcdaWgaaqcfasaaKqzadGaam4Aaaqcfaya baaacaGLOaGaayzkaaGccaWGdbWaa0baaKqbGeaajugWaiaadUgaaK qbGeaajugWaiabgUcaRaaajugibiaacYcacaaMc8UaaGPaVlaaykW7 caWGcbGcdaqhaaqcfasaaKqzadGaam4AaaqcfasaaKqzadGaeyOeI0 caaKqzGeGaeyypa0JaamOqaOWaa0baaKqbGeaajugWaiaadUgaaKqb GeaajugWaiabgUcaRaaajugibiabgkHiTiaadAeakmaaBaaajuaGba qcLbsacaWGRbGaey4kaSIaaGymaaqcfayabaGccaWGdbWcdaWgaaqc fayaaKqzadGaam4AaiabgUcaRiaaigdaaKqbagqaaKqzGeGaamOqaO Waa0baaKqbGeaajugWaiaadUgaaKqbGeaajugWaiabgwQiEbaaaaa@7B00@ . (10)

The multilevel decomposition procedure considered is then implemented.

III. Algorithm for synthesis of the MIMO-system output control
The following statement is true that has been proven13 and given here with an allowance for replacement of matrix annihilators, which are not possessed of a property of orthogonality, i.e., replacement with matrices, for which, in general, not necessarily the equality B k B k + = I nr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb WcdaqhaaqcfasaaKqzadGaam4AaaqcfasaaKqzadGaeyyPI4faaKqz GeGaamOqaSWaa0baaKqbGeaajugWaiaadUgaaKqbGeaajugWaiabgw QiEjaabUcaaaqcLbsacqGH9aqpcaWGjbGcdaWgaaqcfasaaKqzadGa amOBaiabgkHiTiaadkhaaKqbagqaaaaa@4B8B@  is implemented.

Theorem1. Let mr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpu0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaeyyzImRaamOCaaaa@3B90@ , and the following matrices exist and are pair wise completely controllable:
Then, there exists a nonempty set of matrices G k T = B k A k C k ( B k C k ) + , H k T = ( B k C k ) ,k= 0,M ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadEealmaaDaaajuaibaqcLb macaWGRbaajuaibaqcLbmacaqGubaaaKqzGeGaeyypa0JaamOqaSWa a0baaKqbGeaajugWaiaadUgaaKqbGeaajugWaiabgkHiTaaajugibi aadgeajuaGdaWgaaqcfasaaKqzadGaam4AaaqcfayabaqcLbsacaWG dbWcdaqhaaqcfasaaKqzadGaam4AaaqcfasaaKqzadGaeyyPI4faaK qbaoaabmaabaqcLbsacaWGcbWcdaqhaaqcfasaaKqzadGaam4Aaaqc fasaaKqzadGaeyOeI0caaKqzGeGaam4qaSWaa0baaKqbGeaajugWai aadUgaaKqbGeaajugWaiabgwQiEbaaaKqbakaawIcacaGLPaaadaah aaqabKqbGeaajugWaiabgUcaRaaajugibiaacYcacaaMc8UaaGPaVl aadIealmaaDaaajuaibaqcLbmacaWGRbaajuaibaqcLbmacaqGubaa aKqzGeGaeyypa0tcfa4aaeWaaeaajugibiaadkealmaaDaaajuaiba qcLbmacaWGRbaajuaibaqcLbmacqGHsislaaqcLbsacaWGdbWcdaqh aaqcfasaaKqzadGaam4AaaqcfasaaKqzadGaeyyPI4faaaqcfaOaay jkaiaawMcaamaaCaaabeqcfasaaKqzadGaeyyPI4faaKqzGeGaaiil aiaaykW7caaMc8UaaGPaVlaadUgacqGH9aqpjuaGdaqdaaqaaKqzGe GaaGimaiaacYcacaaMc8UaaGPaVlaad2eaaaGaaiilaaaa@89C6@ K i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadUealmaaBaaajuaGbaqcLb macaWGPbaajuaGbeaaaaa@365E@ , i= 0,M ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadMgacqGH9aqpkmaanaaaju aGbaqcLbsacaaIWaGaaiilaiaaykW7caaMc8Uaamytaaaaaaa@3AA8@ such that
Φ i = G i + K i T H i =( B i A i C i ) ( B i C i ) + + K i T ( B i C i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiabfA6agLqbaoaaBaaabaqcLb macaWGPbaajuaGbeaajugibiabg2da9iaadEeajuaGdaWgaaqaaKqz adGaamyAaaqcfayabaqcLbsacqGHRaWkcaWGlbqcfa4aa0baaeaaju gWaiaadMgaaKqbagaajugWaiaabsfaaaqcLbsacaWGibqcfa4aaSba aeaajugWaiaadMgaaKqbagqaaKqzGeGaeyypa0JcdaqadaqcfayaaK qzGeGaamOqaKqbaoaaDaaabaqcLbmacaWGPbaajuaGbaqcLbmacqGH sislaaqcLbsacaWGbbqcfa4aaSbaaeaajugWaiaadMgaaKqbagqaaK qzGeGaam4qaKqbaoaaDaaabaqcLbmacaWGPbaajuaGbaqcLbmacqGH LkIxaaaajuaGcaGLOaGaayzkaaGcdaqadaqcfayaaKqzGeGaamOqaK qbaoaaDaaabaqcLbmacaWGPbaajuaGbaqcLbmacqGHsislaaqcLbsa caWGdbqcfa4aa0baaeaajugWaiaadMgaaKqbagaajugWaiabgwQiEb aaaKqbakaawIcacaGLPaaakmaaCaaajuaGbeqaaKqzadGaey4kaSca aKqzGeGaey4kaSIaam4saKqbaoaaDaaabaqcLbmacaWGPbaajuaGba qcLbmacaqGubaaaOWaaeWaaKqbagaajugibiaadkeajuaGdaqhaaqa aKqzadGaamyAaaqcfayaaKqzadGaeyOeI0caaKqzGeGaam4qaKqbao aaDaaabaqcLbmacaWGPbaajuaGbaqcLbmacqGHLkIxaaaajuaGcaGL OaGaayzkaaWaaWbaaeqabaqcLbmacqGHLkIxaaaaaa@920B@ , (11) and(9), (10) satisfy the equalities of spectra
eig( A k + B k F k C k )= i=k M eig( Φ i ) ,k= 1,M ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaabwgacaqGPbGaae4zaOWaae WaaKqbagaajugibiaadgeakmaaBaaajuaGbaqcLbmacaWGRbaajuaG beaajugibiabgUcaRiaadkeakmaaBaaajuaGbaqcLbmacaWGRbaaju aGbeaajugibiaadAeakmaaBaaajuaGbaqcLbmacaWGRbaajuaGbeaa jugibiaadoeakmaaBaaajuaGbaqcLbmacaWGRbaajuaGbeaaaiaawI cacaGLPaaajugibiabg2da9OWaambCaKqbagaajugibiaabwgacaqG PbGaae4zaiaabIcacqqHMoGrlmaaBaaajuaGbaqcLbmacaWGPbaaju aGbeaajugibiaabMcaaKqbagaajugibiaadMgacqGH9aqpcaWGRbaa juaGbaqcLbsacaWGnbaacqWIQisvaiaacYcacaaMc8UaaGPaVlaayk W7caWGRbGaeyypa0JcdaqdaaqcfayaaKqzGeGaaGymaiaacYcacaaM c8UaaGPaVlaad2eaaaaaaa@6C95@ ,
A 0 =A, B 0 =B, C 0 =C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadgealmaaBaaajuaibaqcLb macaaIWaaajuaibeaajugibiabg2da9iaadgeacaGGSaGaaGPaVlaa ykW7caaMc8UaaGPaVlaadkealmaaBaaajuaibaqcLbmacaaIWaaaju aibeaajugibiabg2da9iaadkeacaGGSaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWGdbGcdaWgaaqcfasaaKqzadGaaGimaaqcfayaba qcLbsacqGH9aqpcaWGdbaaaa@5284@ ,
A k = C k-1 ^ A k-1 C k-1 ^+ , B k = B k-1 A k-1 C k-1 ^+ , C k = C k-1 A k-1 C k-1 ^+ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacPqpw0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=feajuaGdaWgaaqcfa saaKqzadGaa83AaaqcfasabaqcLbsacaWF9aGaa83qaKqbaoaaDaaa juaibaqcLbmacaWFRbGaa8xlaiaa=fdaaKqbGeaajugWaiaa=5faaa qcLbsacaWFbbqcfa4aaSbaaKqbGeaajugWaiaa=TgacaWFTaGaa8xm aaqcfasabaqcLbsacaWFdbqcfa4aa0baaKqbGeaajugWaiaa=Tgaca WFTaGaa8xmaaqcfasaaKqzadGaa8Nxaiaa=TcaaaqcLbsacaWFSaGa aGPaVlaaykW7caWFcbGcdaWgaaqcfasaaKqzadGaa83Aaaqcfayaba qcLbsacaWF9aGaa8NqaOWaaSbaaKqbGeaajugWaiaa=TgacaWFTaGa a8xmaaqcfayabaqcLbsacaWFbbqcfa4aaSbaaKqbGeaajugWaiaa=T gacaWFTaGaa8xmaaqcfasabaqcLbsacaWFdbqcfa4aa0baaKqbGeaa jugWaiaa=TgacaWFTaGaa8xmaaqcfasaaKqzadGaa8Nxaiaa=Tcaaa qcLbsacaWFSaGaaGPaVlaaykW7caWFdbGcdaWgaaqcfasaaKqzadGa a83AaaqcfayabaqcLbsacaWF9aGaa83qaKqbaoaaBaaajuaibaqcLb macaWFRbGaa8xlaiaa=fdaaKqbGeqaaKqzGeGaa8xqaKqbaoaaBaaa juaibaqcLbmacaWFRbGaa8xlaiaa=fdaaKqbGeqaaKqzGeGaa83qaK qbaoaaDaaajuaibaqcLbmacaWFRbGaa8xlaiaa=fdaaKqbGeaajugW aiaa=5facaWFRaaaaaaa@895C@ , k= 1,N ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaaykW7caWGRbGaeyypa0Jcda qdaaqcfayaaKqzGeGaaGymaiaacYcacaaMc8UaaGPaVlaad6eaaaaa aa@3C36@ moreover, the following matrices exist and are pair wise completely controllable:
G k = ( B k ^ C k + ) + B k ^ A k C k + , H k = ( B k ^ C k + ) ^ ,k= 0,N ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=DealmaaBaaajuaGba qcLbmacaWFRbaajuaGbeaajugibiaa=1dakmaabmaajuaGbaqcLbsa caWFcbWcdaqhaaqcfayaaKqzadGaa83AaaqcfayaaKqzadGaa8Nxaa aajugibiaa=nealmaaDaaajuaGbaqcLbmacaWFRbaajuaGbaqcLbma caWFRaaaaaqcfaOaayjkaiaawMcaaOWaaWbaaKqbagqabaqcLbmaca WFRaaaaKqzGeGaa8NqaSWaa0baaKqbagaajugWaiaa=TgaaKqbagaa jugWaiaa=5faaaqcLbsacaWFbbGcdaWgaaqcfayaaKqzadGaa83Aaa qcfayabaqcLbsacaWFdbWcdaqhaaqcfayaaKqzadGaa83Aaaqcfaya aKqzadGaa83kaaaajugibiaa=XcacaaMc8UaaGPaVlaaykW7caWFib WcdaWgaaqcfayaaKqzadGaa83AaaqcfayabaqcLbsacaWF9aGcdaqa daqcfayaaKqzGeGaa8NqaSWaa0baaKqbagaajugWaiaa=TgaaKqbag aajugWaiaa=5faaaqcLbsacaWFdbWcdaqhaaqcfayaaKqzadGaa83A aaqcfayaaKqzadGaa83kaaaaaKqbakaawIcacaGLPaaalmaaCaaaju aGbeqaaKqzadGaa8Nxaaaajugibiaa=XcacaaMc8UaaGPaVlaaykW7 caWFRbGaa8xpaOWaa0aaaKqbagaajugibiaa=bdacaWFSaGaaGPaVl aaykW7caWFobaaaaaa@8862@ .           
Then, there exists a nonempty set of matrices L i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=XeakmaaBaaaleaaju gWaiaa=LgaaSqabaaaaa@3559@ , i= 0,N ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadMgacqGH9aqpkmaanaaaju aGbaqcLbsacaaIWaGaaiilaiaaykW7caaMc8UaamOtaaaaaaa@3AA9@ , such that Ψ i = G i + H i L i T = ( B i ^ C i + ) + B i ^ A i C i + + ( B i ^ C i + ) ^ L i T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=H6almaaBaaajuaGba qcLbmacaWFPbaajuaGbeaajugibiaa=1dacaWFhbWcdaWgaaqcfaya aKqzadGaa8xAaaqcfayabaqcLbsacaWFRaGaa8hsaOWaaSbaaKqbag aajugWaiaa=LgaaKqbagqaaKqzGeGaa8htaSWaa0baaKqbagaajugW aiaa=LgaaKqbagaajugWaiaa=rfaaaqcLbsacaWF9aGcdaqadaqcfa yaaKqzGeGaa8NqaSWaa0baaKqbagaajugWaiaa=LgaaKqbagaajugW aiaa=5faaaqcLbsacaWFdbWcdaqhaaqcfayaaKqzadGaa8xAaaqcfa yaaKqzadGaa83kaaaaaKqbakaawIcacaGLPaaakmaaCaaajuaGbeqa aKqzadGaa83kaaaajugibiaa=jealmaaDaaajuaGbaqcLbmacaWFPb aajuaGbaqcLbmacaWFEbaaaKqzGeGaa8xqaOWaaSbaaKqbagaajugW aiaa=LgaaKqbagqaaKqzGeGaa83qaSWaa0baaKqbagaajugWaiaa=L gaaKqbagaajugWaiaa=TcaaaqcLbsacaWFRaGcdaqadaqcfayaaKqz GeGaa8NqaSWaa0baaKqbagaajugWaiaa=LgaaKqbagaajugWaiaa=5 faaaqcLbsacaWFdbWcdaqhaaqcfayaaKqzadGaa8xAaaqcfayaaKqz adGaa83kaaaaaKqbakaawIcacaGLPaaalmaaCaaajuaGbeqaaKqzad Gaa8Nxaaaajugibiaa=XealmaaDaaajuaGbaqcLbmacaWFPbaajuaG baqcLbmacaWFubaaaaaa@88F2@  and, for F N = B M + ( C M + Ψ M - A M C M + ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=zealmaaBaaajuaGba qcLbmacaWFobaajuaGbeaajugibiaa=1dacaWFcbWcdaqhaaqcfaya aKqzadGaa8xtaaqcfayaaKqzadGaa83kaaaakmaabmaajuaGbaqcLb sacaWFdbWcdaqhaaqcfayaaKqzadGaa8xtaaqcfayaaKqzadGaa83k aaaajugibiaa=H6almaaBaaajuaGbaqcLbmacaWFnbaajuaGbeaaju gibiaa=1cacaWFbbGcdaWgaaqcfayaaKqzadGaa8xtaaqcfayabaqc LbsacaWFdbWcdaqhaaqcfayaaKqzadGaa8xtaaqcfayaaKqzadGaa8 3kaaaaaKqbakaawIcacaGLPaaaaaa@5741@ , F k = B k + ( C k - Ψ k - A k C k - ), C k - = C k + - C k ^T B k+1 F k+1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=zeakmaaBaaajuaGba qcLbmacaWFRbaajuaGbeaajugibiaa=1dacaWFcbWcdaqhaaqcfaya aKqzadGaa83AaaqcfayaaKqzadGaa83kaaaakmaabmaajuaGbaqcLb sacaWFdbWcdaqhaaqcfayaaKqzadGaa83AaaqcfayaaKqzadGaa8xl aaaajugibiaa=H6akmaaBaaajuaGbaqcLbmacaWFRbaajuaGbeaaju gibiaa=1cacaWFbbWcdaWgaaqcfayaaKqzadGaa83Aaaqcfayabaqc LbsacaWFdbWcdaqhaaqcfayaaKqzadGaa83AaaqcfayaaKqzadGaa8 xlaaaaaKqbakaawIcacaGLPaaajugibiaa=XcacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaa=nealmaaDaaajuaGbaqcLbmacaWFRbaaju aGbaqcLbmacaWFTaaaaKqzGeGaa8xpaiaa=nealmaaDaaajuaGbaqc LbmacaWFRbaajuaGbaqcLbmacaWFRaaaaKqzGeGaa8xlaiaa=nealm aaDaaajuaGbaqcLbmacaWFRbaajuaGbaqcLbmacaWFEbGaa8hvaaaa jugibiaa=jeakmaaBaaajuaGbaqcLbmacaWFRbGaa83kaiaa=fdaaK qbagqaaKqzGeGaa8NraOWaaSbaaKqbagaajugWaiaa=TgacaWFRaGa a8xmaaqcfayabaqcLbsacaWFSaaaaa@83E3@ k= 1,N-1 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=TgacaWF9aGcdaqdaa qaaKqzGeGaa8xmaiaa=XcacaaMc8UaaGPaVlaa=5eacaWFTaGaa8xm aaaaaaa@3B2A@  it holds that
eig( A k + B k F k C k )= i=k N eig( Ψ i ) ,k= 1,N ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=vgacaWFPbGaa83zaO WaaeWaaKqbagaajugibiaa=fealmaaBaaajuaGbaqcLbmacaWFRbaa juaGbeaajugibiaa=TcacaWFcbWcdaWgaaqcfayaaKqzadGaa83Aaa qcfayabaqcLbsacaWFgbGcdaWgaaqcfayaaKqzadGaa83Aaaqcfaya baqcLbsacaWFdbGcdaWgaaqcfayaaKqzadGaa83Aaaqcfayabaaaca GLOaGaayzkaaqcLbsacaWF9aGcdaWeWbqcfayaaKqzGeGaa8xzaiaa =LgacaWFNbGaa8hkaiaa=H6almaaBaaajuaGbaqcLbmacaWFPbaaju aGbeaajugibiaa=LcaaKqbagaajugibiaa=LgacaWF9aGaa83Aaaqc fayaaKqzGeGaa8NtaaGaeSOkIufacaWFSaGaaGPaVlaaykW7caaMc8 Uaa83Aaiaa=1dakmaanaaajuaGbaqcLbsacaWFXaGaa8hlaiaaykW7 caaMc8Uaa8Ntaaaaaaa@6AEF@ , eig( A 0 + B 0 F 0 C 0 )= i=0 N eig( Ψ i ) =Λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=vgacaWFPbGaa83zaO WaaeWaaKqbagaajugibiaa=feakmaaBaaajuaGbaqcLbmacaWFWaaa juaGbeaajugibiaa=TcacaWFcbWcdaWgaaqcfayaaKqzadGaa8hmaa qcfayabaqcLbsacaWFgbGcdaWgaaqcfayaaKqzadGaa8hmaaqcfaya baqcLbsacaWFdbGcdaWgaaqcfayaaKqzadGaa8hmaaqcfayabaaaca GLOaGaayzkaaqcLbsacaWF9aGcdaWeWbqcfayaaKqzGeGaa8xzaiaa =LgacaWFNbGaa8hkaiaa=H6akmaaBaaajuaGbaqcLbmacaWFPbaaju aGbeaajugibiaa=LcaaKqbagaajugibiaa=LgacaWF9aGaa8hmaaqc fayaaKqzGeGaa8NtaaGaeSOkIufacaWF9aGaa83Mdaaa@5E2F@ . It should be noted that here, as in the algorithms described,13,16–19 only semi orthogonal and pseudo inverse matrices are used in the transformations, which at least do not reduce the condition number of the equations. Moreover, this approach does not impose restrictions in the form of the differentiation between the algebraic and geometric multiplicities of the elements of the spectrum to be assigned; there are also no restrictions on the size of the problem. This is confirmed by simulation, which shows a high relative accuracy of spectrum control and the practical absence of restrictions on the size of system (3).

Analytical synthesis of a single-airscrew helicopter’s longitudinal motion control
In accordance with the problem statement, it is required to find explicitly a formula of controller Fin the control law (5) that can be expressed in this case as:
( u x u gp )=FC( V x V y ω z υ )=F( V x ω z υ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqcfayaaKqzGeqbaeqabiqaaa qcfayaaGqaaKqzGeGaa8xDaSWaaSbaaKqbagaajugWaiaa=HhaaKqb agqaaaqaaKqzGeGaa8xDaOWaaSbaaKqbagaajugWaiaa=DgacaWFWb aajuaGbeaaaaaacaGLOaGaayzkaaqcLbsacaWF9aGaa8Nraiaa=nea kmaabmaajuaGbaqcLbsafaqabeabbaaaaKqbagaajugibiaa=zfakm aaBaaajuaGbaqcLbmacaWF4baajuaGbeaaaeaajugibiaa=zfakmaa BaaajuaGbaqcLbmacaWF5baajuaGbeaaaeaajugibiaa=L8akmaaBa aajuaGbaqcLbmacaWF6baajuaGbeaaaeaajugibiaa=v8aaaaajuaG caGLOaGaayzkaaqcLbsacaWF9aGaa8NraOWaaeWaaKqbagaajugibu aabeqadeaaaKqbagaajugibiaa=zfakmaaBaaajuaGbaqcLbmacaWF 4baajuaGbeaaaeaajugibiaa=L8akmaaBaaajuaGbaqcLbmacaWF6b aajuaGbeaaaeaajugibiaa=v8aaaaajuaGcaGLOaGaayzkaaaaaa@6914@ ,
And provides for the close-loop system «HS + control system» of a specified earlier spectrum(6). We perform for the system (2) with matrices(3), (4) – the multilevel decomposition described in Section 1, which has in this case two eig( A 0 + B 0 F 0 C 0 )= i=0 M eig( Φ i ) =Λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=vgacaWFPbGaa83zaO WaaeWaaKqbagaajugibiaa=fealmaaBaaajuaGbaqcLbmacaWFWaaa juaGbeaajugibiaa=TcacaWFcbGcdaWgaaqcfayaaKqzadGaa8hmaa qcfayabaqcLbsacaWFgbWcdaWgaaqcfayaaKqzadGaa8hmaaqcfaya baqcLbsacaWFdbGcdaWgaaqcfayaaKqzadGaa8hmaaqcfayabaaaca GLOaGaayzkaaqcLbsacaWF9aGcdaWeWbqcfayaaKqzGeGaa8xzaiaa =LgacaWFNbGaa8hkaiaa=z6almaaBaaajuaGbaqcLbmacaWFPbaaju aGbeaajugibiaa=LcaaKqbagaajugibiaa=LgacaWF9aGaa8hmaaqc fayaaKqzGeGaa8xtaaGaeSOkIufacaWF9aGaa83Mdaaa@5E2E@ . The condition mr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaeyyzImRaamOCaaaa@3A15@  in Theorem 1 is not restrictive; it is introduced to indicate that, in the present case, F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiqcLbsaca WFgbaaaa@3737@  matrix from (5) is conventionally considered as a matrix of controller (i.e. the number of inputs is less than the number of outputs). For the case mr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb GaeyizImQaamOCaaaa@3A04@ , Theorem 1 has a dual formulation, and matrix F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiqcLbsaca WFgbaaaa@3737@  is replaced with the observer matrix L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiqcLbsaca WFmbaaaa@373D@  (the number of inputs is greater than the number of outputs).
Theorem 2. Let mr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaad2gacqGHKjYOcaWGYbaaaa@35C7@ , N=ceil(n/m) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaad6eacqGH9aqpcaqGJbGaae yzaiaabMgacaqGSbGaaGPaVlaacIcacaWGUbGaai4laiaad2gacaGG Paaaaa@3D28@ , and the following decomposition of system (3) hold: decomposition levels ( M=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaad2eacqGH9aqpcaaIXaaaaa@34BD@ ): zero level (7) and first level (8). Therefore, we will have
B 0 =( l 11 l 11 1 0 0 0 0 1 ), B 0 + =( l 11 l 1 0 l 12 l 1 0 l 1 0 0 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=jealmaaDaaajuaGba qcLbmacaaIWaaajuaGbaqcLbmacqGHLkIxaaqcLbsacqGH9aqpkmaa bmaajuaGbaqcLbsafaqabeGaeaaaaKqbagaajugibiaadYgakmaaBa aajuaGbaqcLbmacaaIXaGaaGymaaqcfayabaaabaqcLbsacaWGSbGc daWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaaqaaKqzGeGaaG ymaaqcfayaaKqzGeGaaGimaaqcfayaaKqzGeGaaGimaaqcfayaaKqz GeGaaGimaaqcfayaaKqzGeGaaGimaaqcfayaaKqzGeGaaGymaaaaaK qbakaawIcacaGLPaaajugibiaacYcacaaMc8UaaGPaVlaaykW7caWF cbWcdaqhaaqcfayaaKqzadGaaGimaaqcfayaaKqzadGaeyyPI4Laey 4kaScaaKqzGeGaeyypa0JcdaqadaqcfayaaKqzGeqbaeqabqGaaaaa juaGbaqcLbsacaWGSbGcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaK qbagqaaKqzGeGaamiBaSWaaWbaaKqbagqabaqcLbmacqGHsislcaaI XaaaaaqcfayaaKqzGeGaaGimaaqcfayaaKqzGeGaamiBaSWaaSbaaK qbagaajugWaiaaigdacaaIYaaajuaGbeaajugibiaadYgalmaaCaaa juaGbeqaaKqzadGaeyOeI0IaaGymaaaaaKqbagaajugibiaaicdaaK qbagaajugibiaadYgalmaaCaaajuaGbeqaaKqzadGaeyOeI0IaaGym aaaaaKqbagaajugibiaaicdaaKqbagaajugibiaaicdaaKqbagaaju gibiaaigdaaaaajuaGcaGLOaGaayzkaaaaaa@8B39@ , (12)
A 1 =( a 11 1 a 12 1 a 21 1 0 ), B 1 =( b 11 1 b 12 1 b 21 1 b 22 1 ), C 1 =( c 11 1 c 12 1 c 21 1 c 22 1 c 31 1 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieaajugibiaa=fealmaaBaaajuaGba qcLbmacaWFXaaajuaGbeaajugibiabg2da9OWaaeWacKqbagaajugi buaabeqaciaaaKqbagaajugibiaadggalmaaDaaajuaGbaqcLbmaca aIXaGaaGymaaqcfayaaKqzadGaaGymaaaaaKqbagaajugibiaadgga lmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaaGymaa aaaKqbagaajugibiaadggalmaaDaaajuaGbaqcLbmacaaIYaGaaGym aaqcfayaaKqzadGaaGymaaaaaKqbagaajugibiaaicdaaaaajuaGca GLOaGaayzkaaqcLbsacaGGSaGaaGPaVlaaykW7caaMc8UaaGPaVlaa =jealmaaBaaajuaGbaqcLbmacaWFXaaajuaGbeaajugibiabg2da9O WaaeWacKqbagaajugibuaabeqaciaaaKqbagaajugibiaadkgalmaa DaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqzadGaaGymaaaaaK qbagaajugibiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqc fayaaKqzadGaaGymaaaaaKqbagaajugibiaadkgalmaaDaaajuaGba qcLbmacaaIYaGaaGymaaqcfayaaKqzadGaaGymaaaaaKqbagaajugi biaadkgalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqzad GaaGymaaaaaaaajuaGcaGLOaGaayzkaaqcLbsacaGGSaGaaGPaVlaa ykW7caaMc8Uaa83qaOWaaSbaaKqbagaajugWaiaa=fdaaKqbagqaaK qzGeGaeyypa0JcdaqadaqcfayaaKqzGeqbaeqabmGaaaqcfayaaKqz GeGaam4yaSWaa0baaKqbagaajugWaiaaigdacaaIXaaajuaGbaqcLb macaaIXaaaaaqcfayaaKqzGeGaam4yaSWaa0baaKqbagaajugWaiaa igdacaaIYaaajuaGbaqcLbmacaaIXaaaaaqcfayaaKqzGeGaam4yaS Waa0baaKqbagaajugWaiaaikdacaaIXaaajuaGbaqcLbmacaaIXaaa aaqcfayaaKqzGeGaam4yaSWaa0baaKqbagaajugWaiaaikdacaaIYa aajuaGbaqcLbmacaaIXaaaaaqcfayaaKqzGeGaam4yaSWaa0baaKqb agaajugWaiaaiodacaaIXaaajuaGbaqcLbmacaaIXaaaaaqcfayaaK qzGeGaaGimaaaaaKqbakaawIcacaGLPaaaaaa@BBE1@ ,(13)
Where, taking in view the following notation l * = b 11 b 22 b 12 b 21 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadYgakmaaCaaajuaGbeqaaK qzadGaaiOkaaaajugibiabg2da9iaadkgalmaaBaaajuaGbaqcLbma caaIXaGaaGymaaqcfayabaqcLbsacaWGIbWcdaWgaaqcfayaaKqzad GaaGOmaiaaikdaaKqbagqaaKqzGeGaeyOeI0IaamOyaOWaaSbaaKqb agaajugWaiaaigdacaaIYaaajuaGbeaajugibiaadkgakmaaBaaaju aGbaqcLbmacaaIYaGaaGymaaqcfayabaqcLbsacaaMc8oaaa@4F47@ , the matrices’ elements in(12), (13) is equal to:
l 11 = b 21 b 32 b 22 b 31 l * , l 12 = ( b 11 b 32 b 12 b 31 ) l * , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadYgalmaaBaaajuaGbaqcLb macaaIXaGaaGymaaqcfayabaqcLbsacqGH9aqpkmaalaaajuaGbaqc LbsacaWGIbWcdaWgaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagqaaK qzGeGaamOyaOWaaSbaaKqbagaajugWaiaaiodacaaIYaaajuaGbeaa jugibiabgkHiTiaadkgakmaaBaaajuaGbaqcLbmacaaIYaGaaGOmaa qcfayabaqcLbsacaWGIbWcdaWgaaqcfayaaKqzadGaaG4maiaaigda aKqbagqaaaqaaKqzGeGaamiBaOWaaWbaaKqbagqabaqcLbmacaGGQa aaaaaajugibiaacYcacaaMc8UaaGPaVlaaykW7caWGSbGcdaWgaaqc fayaaKqzadGaaGymaiaaikdaaKqbagqaaKqzGeGaeyypa0JcdaWcaa qcfayaaKqzGeGaeyOeI0IaaiikaiaadkgakmaaBaaajuaGbaqcLbma caaIXaGaaGymaaqcfayabaqcLbsacaWGIbGcdaWgaaqcfayaaKqzad GaaG4maiaaikdaaKqbagqaaKqzGeGaeyOeI0IaamOyaOWaaSbaaKqb agaajugWaiaaigdacaaIYaaajuaGbeaajugibiaadkgalmaaBaaaju aGbaqcLbmacaaIZaGaaGymaaqcfayabaqcLbsacaGGPaaajuaGbaqc LbsacaWGSbWcdaahaaqcfayabeaajugWaiaacQcaaaaaaKqzGeGaai ilaaaa@7FFF@ ,
l= l 11 2 + l 12 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaaykW7caaMc8UaamiBaiabg2 da9iaadYgalmaaDaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqz adGaaGOmaaaajugibiabgUcaRiaadYgalmaaDaaajuaGbaqcLbmaca aIXaGaaGOmaaqcfayaaKqzadGaaGOmaaaajugibiabgUcaRiaaigda aaa@4864@
a 11 1 l= a 33 + a 13 l 11 + a 23 l 12 + a 31 l 11 + a 32 l 12 + a 11 l 11 2 + + a 22 l 12 2 + a 12 l 11 l 12 + a 21 l 11 l 12 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakqaabeqaaKqzGeGaamyyaSWaa0baaKqbag aajugWaiaaigdacaaIXaaajuaGbaqcLbmacaaIXaaaaKqzGeGaamiB aiabg2da9iaadggakmaaBaaajuaGbaqcLbmacaaIZaGaaG4maaqcfa yabaqcLbsacqGHRaWkcaWGHbWcdaWgaaqcfayaaKqzadGaaGymaiaa iodaaKqbagqaaKqzGeGaamiBaOWaaSbaaKqbagaajugWaiaaigdaca aIXaaajuaGbeaajugibiabgUcaRiaadggakmaaBaaajuaGbaqcLbma caaIYaGaaG4maaqcfayabaqcLbsacaWGSbGcdaWgaaqcfayaaKqzad GaaGymaiaaikdaaKqbagqaaKqzGeGaey4kaSIaamyyaSWaaSbaaKqb agaajugWaiaaiodacaaIXaaajuaGbeaajugibiaadYgakmaaBaaaju aGbaqcLbmacaaIXaGaaGymaaqcfayabaqcLbsacqGHRaWkcaWGHbGc daWgaaqcfayaaKqzadGaaG4maiaaikdaaKqbagqaaKqzGeGaamiBaO WaaSbaaKqbagaajugWaiaaigdacaaIYaaajuaGbeaajugibiabgUca RiaadggakmaaBaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayabaqcLb sacaWGSbWcdaqhaaqcfayaaKqzadGaaGymaiaaigdaaKqbagaajugW aiaaikdaaaqcLbsacqGHRaWkaOqaaKqzGeGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaey4kaS IaamyyaOWaaSbaaKqbagaajugWaiaaikdacaaIYaaajuaGbeaajugi biaadYgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaKqzad GaaGOmaaaajugibiabgUcaRiaadggakmaaBaaajuaGbaqcLbmacaaI XaGaaGOmaaqcfayabaqcLbsacaWGSbWcdaWgaaqcfayaaKqzadGaaG ymaiaaigdaaKqbagqaaKqzGeGaamiBaOWaaSbaaKqbagaajugWaiaa igdacaaIYaaajuaGbeaajugibiabgUcaRiaadggakmaaBaaajuaGba qcLbmacaaIYaGaaGymaaqcfayabaqcLbsacaWGSbWcdaWgaaqcfaya aKqzadGaaGymaiaaigdaaKqbagqaaKqzGeGaamiBaOWaaSbaaKqbag aajugWaiaaigdacaaIYaaajuaGbeaajugibiaacYcacaaMc8oaaaa@CC7F@
a 12 1 = a 13 l 11 + a 24 l 12 , a 21 1 = l 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadggalmaaDaaajuaGbaqcLb macaaIXaGaaGOmaaqcfayaaKqzadGaaGymaaaajugibiabg2da9iaa dggakmaaBaaajuaGbaqcLbmacaaIXaGaaG4maaqcfayabaqcLbsaca WGSbWcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaKqzGeGa ey4kaSIaamyyaOWaaSbaaKqbagaajugWaiaaikdacaaI0aaajuaGbe aajugibiaadYgakmaaBaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfaya baqcLbsacaGGSaGaaGPaVlaaykW7caWGHbWcdaqhaaqcfayaaKqzad GaaGOmaiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacqGH9aqpcaWG SbWcdaahaaqcfayabeaajugWaiabgkHiTiaaigdaaaqcLbsacaGGSa GaaGPaVdaa@645F@
b 21 1 = b 31 , b 22 1 = b 32 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaGymaaqcfayaaKqzadGaaGymaaaajugibiabg2da9iaa dkgakmaaBaaajuaGbaqcLbmacaaIZaGaaGymaaqcfayabaqcLbsaca GGSaGaaGPaVlaadkgalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqc fayaaKqzadGaaGymaaaajugibiabg2da9iaadkgakmaaBaaajuaGba qcLbmacaaIZaGaaGOmaaqcfayabaqcLbsacaGGSaaaaa@5082@
b 11 1 = b 11 ( a 31 + a 11 l 11 + a 21 l 12 )+ b 21 ( a 32 + a 12 l 11 + a 22 l 12 )+ + b 31 ( a 33 + a 13 l 11 + a 23 l 12 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakqaabeqaaKqzGeGaamOyaSWaa0baaKqbag aajugWaiaaigdacaaIXaaajuaGbaqcLbmacaaIXaaaaKqzGeGaeyyp a0JaamOyaSWaaSbaaKqbagaajugWaiaaigdacaaIXaaajuaGbeaaju gibiaacIcacaWGHbWcdaWgaaqcfayaaKqzadGaaG4maiaaigdaaKqb agqaaKqzGeGaey4kaSIaamyyaSWaaSbaaKqbagaajugWaiaaigdaca aIXaaajuaGbeaajugibiaadYgalmaaBaaajuaGbaqcLbmacaaIXaGa aGymaaqcfayabaqcLbsacqGHRaWkcaWGHbGcdaWgaaqcfayaaKqzad GaaGOmaiaaigdaaKqbagqaaKqzGeGaamiBaOWaaSbaaKqbagaajugW aiaaigdacaaIYaaajuaGbeaajugibiaacMcacqGHRaWkcaWGIbGcda WgaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagqaaKqzGeGaaiikaiaa dggakmaaBaaajuaGbaqcLbmacaaIZaGaaGOmaaqcfayabaqcLbsacq GHRaWkcaWGHbGcdaWgaaqcfayaaKqzadGaaGymaiaaikdaaKqbagqa aKqzGeGaamiBaOWaaSbaaKqbagaajugWaiaaigdacaaIXaaajuaGbe aajugibiabgUcaRiaadggakmaaBaaajuaGbaqcLbmacaaIYaGaaGOm aaqcfayabaqcLbsacaWGSbGcdaWgaaqcfayaaKqzadGaaGymaiaaik daaKqbagqaaKqzGeGaaiykaiabgUcaRaGcbaqcLbsacaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaey4kaS IaamOyaOWaaSbaaKqbagaajugWaiaaiodacaaIXaaajuaGbeaajugi biaacIcacaWGHbGcdaWgaaqcfayaaKqzadGaaG4maiaaiodaaKqbag qaaKqzGeGaey4kaSIaamyyaOWaaSbaaKqbagaajugWaiaaigdacaaI ZaaajuaGbeaajugibiaadYgalmaaBaaajuaGbaqcLbmacaaIXaGaaG ymaaqcfayabaqcLbsacqGHRaWkcaWGHbGcdaWgaaqcfayaaKqzadGa aGOmaiaaiodaaKqbagqaaKqzGeGaamiBaOWaaSbaaKqbagaajugWai aaigdacaaIYaaajuaGbeaajugibiaacMcacaGGSaaaaaa@CF7D@
b 12 1 = b 12 ( a 31 + a 11 l 11 + a 21 l 12 )+ b 22 ( a 32 + a 12 l 11 + a 22 l 12 )+ + b 32 ( a 33 + a 13 l 11 + a 23 l 12 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakqaabeqaaKqzGeGaamOyaSWaa0baaKqbag aajugWaiaaigdacaaIYaaajuaGbaqcLbmacaaIXaaaaKqzGeGaeyyp a0JaamOyaSWaaSbaaKqbagaajugWaiaaigdacaaIYaaajuaGbeaaju gibiaacIcacaWGHbGcdaWgaaqcfayaaKqzadGaaG4maiaaigdaaKqb agqaaKqzGeGaey4kaSIaamyyaOWaaSbaaKqbagaajugWaiaaigdaca aIXaaajuaGbeaajugibiaadYgalmaaBaaajuaGbaqcLbmacaaIXaGa aGymaaqcfayabaqcLbsacqGHRaWkcaWGHbGcdaWgaaqcfayaaKqzad GaaGOmaiaaigdaaKqbagqaaKqzGeGaamiBaOWaaSbaaKqbagaajugW aiaaigdacaaIYaaajuaGbeaajugibiaacMcacqGHRaWkcaWGIbWcda WgaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagqaaKqzGeGaaiikaiaa dggalmaaBaaajuaGbaqcLbmacaaIZaGaaGOmaaqcfayabaqcLbsacq GHRaWkcaWGHbGcdaWgaaqcfayaaKqzadGaaGymaiaaikdaaKqbagqa aKqzGeGaamiBaOWaaSbaaKqbagaajugWaiaaigdacaaIXaaajuaGbe aajugibiabgUcaRiaadggalmaaBaaajuaGbaqcLbmacaaIYaGaaGOm aaqcfayabaqcLbsacaWGSbWcdaWgaaqcfayaaKqzadGaaGymaiaaik daaKqbagqaaKqzGeGaaiykaiabgUcaRaGcbaqcLbsacaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWkcaWGIb GcdaWgaaqcfayaaKqzadGaaG4maiaaikdaaKqbagqaaKqzGeGaaiik aiaadggalmaaBaaajuaGbaqcLbmacaaIZaGaaG4maaqcfayabaqcLb sacqGHRaWkcaWGHbGcdaWgaaqcfayaaKqzadGaaGymaiaaiodaaKqb agqaaKqzGeGaamiBaSWaaSbaaKqbagaajugWaiaaigdacaaIXaaaju aGbeaajugibiabgUcaRiaadggakmaaBaaajuaGbaqcLbmacaaIYaGa aG4maaqcfayabaqcLbsacaWGSbGcdaWgaaqcfayaaKqzadGaaGymai aaikdaaKqbagqaaKqzGeGaaiykaiaacYcaaaaa@CDF9@ c 11 1 =( a 13 + a 11 l 11 + a 12 l 12 ) l 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadogalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaaGymaaaajugibiabg2da9iaa cIcacaWGHbGcdaWgaaqcfayaaKqzadGaaGymaiaaiodaaKqbagqaaK qzGeGaey4kaSIaamyyaOWaaSbaaKqbagaajugWaiaaigdacaaIXaaa juaGbeaajugibiaadYgalmaaBaaajuaGbaqcLbmacaaIXaGaaGymaa qcfayabaqcLbsacqGHRaWkcaWGHbWcdaWgaaqcfayaaKqzadGaaGym aiaaikdaaKqbagqaaKqzGeGaamiBaOWaaSbaaKqbagaajugWaiaaig dacaaIYaaajuaGbeaajugibiaacMcacaWGSbGcdaahaaqcfayabeaa jugWaiabgkHiTiaaigdaaaqcLbsacaGGSaGaaGPaVdaa@5FE0@
c 12 1 = a 13 , c 21 1 =( a 33 + a 31 l 11 + a 32 l 12 ) l 1 , c 22 1 =0, c 31 1 = l 1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadogalmaaDaaajuaGbaqcLb macaaIXaGaaGOmaaqcfayaaKqzadGaaGymaaaajugibiabg2da9iaa dggakmaaBaaajuaGbaqcLbmacaaIXaGaaG4maaqcfayabaqcLbsaca GGSaGaaGPaVlaaykW7caWGJbWcdaqhaaqcfayaaKqzadGaaGOmaiaa igdaaKqbagaajugWaiaaigdaaaqcLbsacqGH9aqpcaGGOaGaamyyaS WaaSbaaKqbagaajugWaiaaiodacaaIZaaajuaGbeaajugibiabgUca RiaadggalmaaBaaajuaGbaqcLbmacaaIZaGaaGymaaqcfayabaqcLb sacaWGSbWcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaKqz GeGaey4kaSIaamyyaOWaaSbaaKqbagaajugWaiaaiodacaaIYaaaju aGbeaajugibiaadYgakmaaBaaajuaGbaqcLbmacaaIXaGaaGOmaaqc fayabaqcLbsacaGGPaGaamiBaOWaaWbaaKqbagqabaqcLbmacqGHsi slcaaIXaaaaKqzGeGaaiilaiaaykW7caaMc8Uaam4yaSWaa0baaKqb agaajugWaiaaikdacaaIYaaajuaGbaqcLbmacaaIXaaaaKqzGeGaey ypa0JaaGimaiaacYcacaaMc8UaaGPaVlaadogalmaaDaaajuaGbaqc LbmacaaIZaGaaGymaaqcfayaaKqzadGaaGymaaaajugibiabg2da9i aadYgakmaaCaaajuaGbeqaaKqzadGaeyOeI0IaaGymaaaajugibiaa c6caaaa@8DDB@ To check the controllability conditions in Theorem 1, we calculate the matrices:
C 0 T =( 0 1 0 0 ), B 0 + =( b 11 0+ b 12 0+ b 14 0+ 0 b 21 0+ b 22 0+ b 23 0+ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=nealmaaDaaajuaGba qcLbmacaaIWaaajuaGbaqcLbmacqGHLkIxcaqGubaaaKqzGeGaeyyp a0JcdaqadaqcfayaaKqzGeqbaeqabeabaaaajuaGbaqcLbsacaaIWa aajuaGbaqcLbsacaaIXaaajuaGbaqcLbsacaaIWaaajuaGbaqcLbsa caaIWaaaaaqcfaOaayjkaiaawMcaaKqzGeGaaiilaiaaykW7caaMc8 Uaa8NqaSWaa0baaKqbagaajugWaiaaicdaaKqbagaajugWaiabgUca Raaajugibiabg2da9OWaaeWaaKqbagaajugibuaabeqacqaaaaqcfa yaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaigdacaaIXaaajuaG baqcLbmacaaIWaGaey4kaScaaaqcfayaaKqzGeGaamOyaSWaa0baaK qbagaajugWaiaaigdacaaIYaaajuaGbaqcLbmacaaIWaGaey4kaSca aaqcfayaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaigdacaaI0a aajuaGbaqcLbmacaaIWaGaey4kaScaaaqcfayaaKqzGeGaaGimaaqc fayaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIXaaaju aGbaqcLbmacaaIWaGaey4kaScaaaqcfayaaKqzGeGaamOyaSWaa0ba aKqbagaajugWaiaaikdacaaIYaaajuaGbaqcLbmacaaIWaGaey4kaS caaaqcfayaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaikdacaaI ZaaajuaGbaqcLbmacaaIWaGaey4kaScaaaqcfayaaKqzGeGaaGimaa aaaKqbakaawIcacaGLPaaaaaa@8D7C@ , in which, in presence of the introduced parameter
b 0+ = b 11 2 b 22 2 + b 11 2 b 32 2 2 b 11 b 12 b 21 b 22 2 b 11 b 12 b 31 b 32 + + b 12 2 b 21 2 + b 12 2 b 31 2 + b 21 2 b 32 2 2 b 21 b 22 b 31 b 32 + b 22 2 b 31 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakqaabeqaaKqzGeGaamOyaOWaaWbaaKqbag qabaqcLbmacaaIWaGaey4kaScaaKqzGeGaeyypa0JaamOyaSWaa0ba aKqbagaajugWaiaaigdacaaIXaaajuaGbaqcLbmacaaIYaaaaKqzGe GaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIYaaajuaGbaqcLbma caaIYaaaaKqzGeGaey4kaSIaamOyaSWaa0baaKqbagaajugWaiaaig dacaaIXaaajuaGbaqcLbmacaaIYaaaaKqzGeGaamOyaSWaa0baaKqb agaajugWaiaaiodacaaIYaaajuaGbaqcLbmacaaIYaaaaKqzGeGaey OeI0IaaGOmaiaadkgalmaaBaaajuaGbaqcLbmacaaIXaGaaGymaaqc fayabaqcLbsacaWGIbWcdaWgaaqcfayaaKqzadGaaGymaiaaikdaaK qbagqaaKqzGeGaamOyaOWaaSbaaKqbagaajugWaiaaikdacaaIXaaa juaGbeaajugibiaadkgakmaaBaaajuaGbaqcLbmacaaIYaGaaGOmaa qcfayabaqcLbsacqGHsislcaaIYaGaamOyaOWaaSbaaKqbagaajugW aiaaigdacaaIXaaajuaGbeaajugibiaadkgakmaaBaaajuaGbaqcLb macaaIXaGaaGOmaaqcfayabaqcLbsacaWGIbGcdaWgaaqcfayaaKqz adGaaG4maiaaigdaaKqbagqaaKqzGeGaamOyaSWaaSbaaKqbagaaju gWaiaaiodacaaIYaaajuaGbeaajugibiabgUcaRaGcbaqcLbsacaaM c8UaaGPaVlaaykW7cqGHRaWkcaWGIbWcdaqhaaqcfayaaKqzadGaaG ymaiaaikdaaKqbagaajugWaiaaikdaaaqcLbsacaWGIbWcdaqhaaqc fayaaKqzadGaaGOmaiaaigdaaKqbagaajugWaiaaikdaaaqcLbsacq GHRaWkcaWGIbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaa jugWaiaaikdaaaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGaaG4mai aaigdaaKqbagaajugWaiaaikdaaaqcLbsacqGHRaWkcaWGIbWcdaqh aaqcfayaaKqzadGaaGOmaiaaigdaaKqbagaajugWaiaaikdaaaqcLb sacaWGIbWcdaqhaaqcfayaaKqzadGaaG4maiaaikdaaKqbagaajugW aiaaikdaaaqcLbsacqGHsislcaaIYaGaamOyaOWaaSbaaKqbagaaju gWaiaaikdacaaIXaaajuaGbeaajugibiaadkgakmaaBaaajuaGbaqc LbmacaaIYaGaaGOmaaqcfayabaqcLbsacaWGIbGcdaWgaaqcfayaaK qzadGaaG4maiaaigdaaKqbagqaaKqzGeGaamOyaOWaaSbaaKqbagaa jugWaiaaiodacaaIYaaajuaGbeaajugibiabgUcaRiaadkgalmaaDa aajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqzadGaaGOmaaaajugi biaadkgalmaaDaaajuaGbaqcLbmacaaIZaGaaGymaaqcfayaaKqzad GaaGOmaaaajugibiaacYcacaaMc8UaaGPaVdaaaa@E3C6@
Corresponding elements can be expressed as
b 11 0+ = b 11 b 22 2 b 12 b 21 b 22 + b 11 b 32 2 b 12 b 31 b 32 b 0+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaaGimaiabgUcaRaaajugibiab g2da9OWaaSaaaKqbagaajugibiaadkgakmaaBaaajuaGbaqcLbmaca aIXaGaaGymaaqcfayabaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGOmaiaaikdaaKqbagaajugWaiaaikdaaaqcLbsacqGHsislcaWGIb GcdaWgaaqcfayaaKqzadGaaGymaiaaikdaaKqbagqaaKqzGeGaamOy aOWaaSbaaKqbagaajugWaiaaikdacaaIXaaajuaGbeaajugibiaadk gakmaaBaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfayabaqcLbsacqGH RaWkcaWGIbGcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaK qzGeGaamOyaSWaa0baaKqbagaajugWaiaaiodacaaIYaaajuaGbaqc LbmacaaIYaaaaKqzGeGaeyOeI0IaamOyaOWaaSbaaKqbagaajugWai aaigdacaaIYaaajuaGbeaajugibiaadkgalmaaBaaajuaGbaqcLbma caaIZaGaaGymaaqcfayabaqcLbsacaWGIbGcdaWgaaqcfayaaKqzad GaaG4maiaaikdaaKqbagqaaaqaaKqzGeGaamOyaOWaaWbaaKqbagqa baqcLbmacaaIWaGaey4kaScaaaaajugibiaacYcaaaa@7EAB@ b 12 0+ = b 21 b 12 2 b 11 b 22 b 12 + b 21 b 32 2 b 22 b 31 b 32 b 0+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGOmaaqcfayaaKqzadGaaGimaiabgUcaRaaajugibiab g2da9OWaaSaaaKqbagaajugibiaadkgalmaaBaaajuaGbaqcLbmaca aIYaGaaGymaaqcfayabaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGymaiaaikdaaKqbagaajugWaiaaikdaaaqcLbsacqGHsislcaWGIb GcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaKqzGeGaamOy aOWaaSbaaKqbagaajugWaiaaikdacaaIYaaajuaGbeaajugibiaadk gakmaaBaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayabaqcLbsacqGH RaWkcaWGIbWcdaWgaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagqaaK qzGeGaamOyaSWaa0baaKqbagaajugWaiaaiodacaaIYaaajuaGbaqc LbmacaaIYaaaaKqzGeGaeyOeI0IaamOyaOWaaSbaaKqbagaajugWai aaikdacaaIYaaajuaGbeaajugibiaadkgakmaaBaaajuaGbaqcLbma caaIZaGaaGymaaqcfayabaqcLbsacaWGIbGcdaWgaaqcfayaaKqzad GaaG4maiaaikdaaKqbagqaaaqaaKqzGeGaamOyaOWaaWbaaKqbagqa baqcLbmacaaIWaGaey4kaScaaaaajugibiaacYcaaaa@7EAE@
b 13 0+ = b 31 b 12 2 b 11 b 32 b 12 + b 31 b 22 2 b 21 b 32 b 22 b 0+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaG4maaqcfayaaKqzadGaaGimaiabgUcaRaaajugibiab g2da9OWaaSaaaKqbagaajugibiaadkgakmaaBaaajuaGbaqcLbmaca aIZaGaaGymaaqcfayabaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGymaiaaikdaaKqbagaajugWaiaaikdaaaqcLbsacqGHsislcaWGIb WcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaKqzGeGaamOy aOWaaSbaaKqbagaajugWaiaaiodacaaIYaaajuaGbeaajugibiaadk gakmaaBaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayabaqcLbsacqGH RaWkcaWGIbWcdaWgaaqcfayaaKqzadGaaG4maiaaigdaaKqbagqaaK qzGeGaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIYaaajuaGbaqc LbmacaaIYaaaaKqzGeGaeyOeI0IaamOyaSWaaSbaaKqbagaajugWai aaikdacaaIXaaajuaGbeaajugibiaadkgakmaaBaaajuaGbaqcLbma caaIZaGaaGOmaaqcfayabaqcLbsacaWGIbWcdaWgaaqcfayaaKqzad GaaGOmaiaaikdaaKqbagqaaaqaaKqzGeGaamOyaSWaaWbaaKqbagqa baqcLbmacaaIWaGaey4kaScaaaaajugibiaacYcaaaa@7EB3@
b 21 0+ = b 12 b 21 2 b 11 b 22 b 21 + b 12 b 31 2 b 11 b 32 b 31 b 0+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaGymaaqcfayaaKqzadGaaGimaiabgUcaRaaajugibiab g2da9OWaaSaaaKqbagaajugibiaadkgakmaaBaaajuaGbaqcLbmaca aIXaGaaGOmaaqcfayabaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGOmaiaaigdaaKqbagaajugWaiaaikdaaaqcLbsacqGHsislcaWGIb GcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaKqzGeGaamOy aSWaaSbaaKqbagaajugWaiaaikdacaaIYaaajuaGbeaajugibiaadk galmaaBaaajuaGbaqcLbmacaaIYaGaaGymaaqcfayabaqcLbsacqGH RaWkcaWGIbGcdaWgaaqcfayaaKqzadGaaGymaiaaikdaaKqbagqaaK qzGeGaamOyaSWaa0baaKqbagaajugWaiaaiodacaaIXaaajuaGbaqc LbmacaaIYaaaaKqzGeGaeyOeI0IaamOyaSWaaSbaaKqbagaajugWai aaigdacaaIXaaajuaGbeaajugibiaadkgalmaaBaaajuaGbaqcLbma caaIZaGaaGOmaaqcfayabaqcLbsacaWGIbGcdaWgaaqcfayaaKqzad GaaG4maiaaigdaaKqbagqaaaqaaKqzGeGaamOyaOWaaWbaaKqbagqa baqcLbmacaaIWaGaey4kaScaaaaajugibiaacYcacaaMc8oaaa@8038@
b 22 0+ = b 22 b 11 2 b 12 b 21 b 11 + b 22 b 31 2 b 21 b 32 b 31 b 0+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaGOmaaqcfayaaKqzadGaaGimaiabgUcaRaaajugibiab g2da9OWaaSaaaKqbagaajugibiaadkgalmaaBaaajuaGbaqcLbmaca aIYaGaaGOmaaqcfayabaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGymaiaaigdaaKqbagaajugWaiaaikdaaaqcLbsacqGHsislcaWGIb GcdaWgaaqcfayaaKqzadGaaGymaiaaikdaaKqbagqaaKqzGeGaamOy aSWaaSbaaKqbagaajugWaiaaikdacaaIXaaajuaGbeaajugibiaadk galmaaBaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayabaqcLbsacqGH RaWkcaWGIbGcdaWgaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagqaaK qzGeGaamOyaSWaa0baaKqbagaajugWaiaaiodacaaIXaaajuaGbaqc LbmacaaIYaaaaKqzGeGaeyOeI0IaamOyaOWaaSbaaKqbagaajugWai aaikdacaaIXaaajuaGbeaajugibiaadkgakmaaBaaajuaGbaqcLbma caaIZaGaaGOmaaqcfayabaqcLbsacaWGIbGcdaWgaaqcfayaaKqzad GaaG4maiaaigdaaKqbagqaaaqaaKqzGeGaamOyaOWaaWbaaKqbagqa baqcLbmacaaIWaGaey4kaScaaaaajugibiaacYcacaaMc8oaaa@8039@
b 23 0+ = b 32 b 11 2 b 12 b 31 b 11 + b 32 b 21 2 b 22 b 31 b 21 b 0+ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaG4maaqcfayaaKqzadGaaGimaiabgUcaRaaajugibiab g2da9OWaaSaaaKqbagaajugibiaadkgakmaaBaaajuaGbaqcLbmaca aIZaGaaGOmaaqcfayabaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGymaiaaigdaaKqbagaajugWaiaaikdaaaqcLbsacqGHsislcaWGIb GcdaWgaaqcfayaaKqzadGaaGymaiaaikdaaKqbagqaaKqzGeGaamOy aOWaaSbaaKqbagaajugWaiaaiodacaaIXaaajuaGbeaajugibiaadk galmaaBaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayabaqcLbsacqGH RaWkcaWGIbGcdaWgaaqcfayaaKqzadGaaG4maiaaikdaaKqbagqaaK qzGeGaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIXaaajuaGbaqc LbmacaaIYaaaaKqzGeGaeyOeI0IaamOyaSWaaSbaaKqbagaajugWai aaikdacaaIYaaajuaGbeaajugibiaadkgakmaaBaaajuaGbaqcLbma caaIZaGaaGymaaqcfayabaqcLbsacaWGIbGcdaWgaaqcfayaaKqzad GaaGOmaiaaigdaaKqbagqaaaqaaKqzGeGaamOyaOWaaWbaaKqbagqa baqcLbmacaaIWaGaey4kaScaaaaajugibiaac6caaaa@7EB1@

Besides, the following equations take place:
H 0 = ( B 0 + C 0 ) T = ( b 22 0+ b 12 0+ 1 ) T , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=HealmaaBaaajuaGba qcLbmacaaIWaaajuaGbeaajugibiabg2da9OWaaeWaaKqbagaajugi biaa=jealmaaDaaajuaGbaqcLbmacaaIWaaajuaGbaqcLbmacqGHRa WkaaqcLbsacaWFdbWcdaqhaaqcfayaaKqzadGaaGimaaqcfayaaKqz adGaeyyPI4faaaqcfaOaayjkaiaawMcaaOWaaWbaaKqbagqabaqcLb macqGHLkIxcaqGubaaaKqzGeGaeyypa0JcdaqadaqcfayaaKqzGeqb aeqabeGaaaqcfayaaKqzGeGaeyOeI0IcdaWcaaqcfayaaKqzGeGaam OyaSWaa0baaKqbagaajugWaiaaikdacaaIYaaajuaGbaqcLbmacaaI WaGaey4kaScaaaqcfayaaKqzGeGaamOyaSWaa0baaKqbagaajugWai aaigdacaaIYaaajuaGbaqcLbmacaaIWaGaey4kaScaaaaaaKqbagaa jugibiaaigdaaaaajuaGcaGLOaGaayzkaaGcdaahaaqcfayabeaaju gWaiaabsfaaaqcLbsacaGGSaaaaa@6B2A@
G 0 = ( B 0 C 0 ) +T ( B 0 A 0 C 0 ) T = =( b 12 0+ ( a 12 b 11 0+ + a 22 b 12 0+ + a 32 b 13 0+ ) ( b 12 0+ ) 2 + ( b 22 0+ ) 2 b 12 0+ ( a 12 b 21 0+ + a 22 b 22 0+ + a 32 b 23 0+ ) ( b 12 0+ ) 2 + ( b 22 0+ ) 2 b 22 0+ ( a 12 b 11 0+ + a 22 b 12 0+ + a 32 b 13 0+ ) ( b 12 0+ ) 2 + ( b 22 0+ ) 2 b 22 0+ ( a 12 b 21 0+ + a 22 b 22 0+ + a 32 b 23 0+ ) ( b 12 0+ ) 2 + ( b 22 0+ ) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacOqpw0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakqaabeqaaKqzGeGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8ocbiGaa83raKqbao aaBaaabaqcLbmacaaIWaaajuaGbeaajugibiabg2da9OWaaeWaaKqb agaajugibiaa=jeajuaGdaqhaaqaaKqzadGaaGimaaqcfayaaKqzad GaeyOeI0caaKqzGeGaa83qaKqbaoaaDaaabaqcLbmacaaIWaaajuaG baqcLbmacqGHLkIxaaaajuaGcaGLOaGaayzkaaGcdaahaaqcfayabe aajugWaiabgUcaRiaabsfaaaGcdaqadaqcfayaaKqzGeGaa8NqaKqb aoaaDaaabaqcLbmacaaIWaaajuaGbaqcLbmacqGHsislaaqcLbsaca WFbbqcfa4aaSbaaeaajugWaiaaicdaaKqbagqaaKqzGeGaa83qaKqb aoaaDaaabaqcLbmacaaIWaaajuaGbaqcLbmacqGHLkIxaaaajuaGca GLOaGaayzkaaGcdaahaaqcfayabeaajugWaiaabsfaaaqcLbsacqGH 9aqpaOqaaKqzGeGaeyypa0JcdaqadaqcfayaaKqzGeqbaeqabiGaaa qcfayaaOWaaSaaaKqbagaajugibiaadkgajuaGdaqhaaqaaKqzadGa aGymaiaaikdaaKqbagaajugWaiaaicdacqGHRaWkaaqcLbsacaGGOa GaamyyaKqbaoaaBaaabaqcLbmacaaIXaGaaGOmaaqcfayabaqcLbsa caWGIbqcfa4aa0baaeaajugWaiaaigdacaaIXaaajuaGbaqcLbmaca aIWaGaey4kaScaaKqzGeGaey4kaSIaamyyaKqbaoaaBaaabaqcLbma caaIYaGaaGOmaaqcfayabaqcLbsacaWGIbqcfa4aa0baaeaajugWai aaigdacaaIYaaajuaGbaqcLbmacaaIWaGaey4kaScaaKqzGeGaey4k aSIaamyyaOWaaSbaaKqbagaajugWaiaaiodacaaIYaaajuaGbeaaju gibiaadkgajuaGdaqhaaqaaKqzadGaaGymaiaaiodaaKqbagaajugW aiaaicdacqGHRaWkaaqcLbsacaGGPaaajuaGbaqcLbsacaGGOaGaam OyaKqbaoaaDaaabaqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaaGim aiabgUcaRaaajugibiaacMcajuaGdaahaaqabeaajugWaiaaikdaaa qcLbsacqGHRaWkcaGGOaGaamOyaKqbaoaaDaaabaqcLbmacaaIYaGa aGOmaaqcfayaaKqzadGaaGimaiabgUcaRaaajugibiaacMcakmaaCa aajuaGbeqaaKqzadGaaGOmaaaaaaaajuaGbaGcdaWcaaqcfayaaKqz GeGaamOyaKqbaoaaDaaabaqcLbmacaaIXaGaaGOmaaqcfayaaKqzad GaaGimaiabgUcaRaaajugibiaacIcacaWGHbqcfa4aaSbaaeaajugW aiaaigdacaaIYaaajuaGbeaajugibiaadkgajuaGdaqhaaqaaKqzad GaaGOmaiaaigdaaKqbagaajugWaiaaicdacqGHRaWkaaqcLbsacqGH RaWkcaWGHbGcdaWgaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagqaaK qzGeGaamOyaKqbaoaaDaaabaqcLbmacaaIYaGaaGOmaaqcfayaaKqz adGaaGimaiabgUcaRaaajugibiabgUcaRiaadggakmaaBaaajuaGba qcLbmacaaIZaGaaGOmaaqcfayabaqcLbsacaWGIbqcfa4aa0baaeaa jugWaiaaikdacaaIZaaajuaGbaqcLbmacaaIWaGaey4kaScaaKqzGe GaaiykaaqcfayaaKqzGeGaaiikaiaadkgajuaGdaqhaaqaaKqzadGa aGymaiaaikdaaKqbagaajugWaiaaicdacqGHRaWkaaqcLbsacaGGPa qcfa4aaWbaaeqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaiikaiaa dkgajuaGdaqhaaqaaKqzadGaaGOmaiaaikdaaKqbagaajugWaiaaic dacqGHRaWkaaqcLbsacaGGPaGcdaahaaqcfayabeaajugWaiaaikda aaaaaaqcfayaaOWaaSaaaKqbagaajugibiaadkgajuaGdaqhaaqaaK qzadGaaGOmaiaaikdaaKqbagaajugWaiaaicdacqGHRaWkaaqcLbsa caGGOaGaamyyaKqbaoaaBaaabaqcLbmacaaIXaGaaGOmaaqcfayaba qcLbsacaWGIbqcfa4aa0baaeaajugWaiaaigdacaaIXaaajuaGbaqc LbmacaaIWaGaey4kaScaaKqzGeGaey4kaSIaamyyaOWaaSbaaKqbag aajugWaiaaikdacaaIYaaajuaGbeaajugibiaadkgajuaGdaqhaaqa aKqzadGaaGymaiaaikdaaKqbagaajugWaiaaicdacqGHRaWkaaqcLb sacqGHRaWkcaWGHbGcdaWgaaqcfayaaKqzadGaaG4maiaaikdaaKqb agqaaKqzGeGaamOyaKqbaoaaDaaabaqcLbmacaaIXaGaaG4maaqcfa yaaKqzadGaaGimaiabgUcaRaaajugibiaacMcaaKqbagaajugibiaa cIcacaWGIbqcfa4aa0baaeaajugWaiaaigdacaaIYaaajuaGbaqcLb macaaIWaGaey4kaScaaKqzGeGaaiykaKqbaoaaCaaabeqaaKqzadGa aGOmaaaajugibiabgUcaRiaacIcacaWGIbqcfa4aa0baaeaajugWai aaikdacaaIYaaajuaGbaqcLbmacaaIWaGaey4kaScaaKqzGeGaaiyk aKqbaoaaCaaabeqaaKqzadGaaGOmaaaaaaaajuaGbaGcdaWcaaqcfa yaaKqzGeGaamOyaKqbaoaaDaaabaqcLbmacaaIYaGaaGOmaaqcfaya aKqzadGaaGimaiabgUcaRaaajugibiaacIcacaWGHbqcfa4aaSbaae aajugWaiaaigdacaaIYaaajuaGbeaajugibiaadkgajuaGdaqhaaqa aKqzadGaaGOmaiaaigdaaKqbagaajugWaiaaicdacqGHRaWkaaqcLb sacqGHRaWkcaWGHbGcdaWgaaqcfayaaKqzadGaaGOmaiaaikdaaKqb agqaaKqzGeGaamOyaKqbaoaaDaaabaqcLbmacaaIYaGaaGOmaaqcfa yaaKqzadGaaGimaiabgUcaRaaajugibiabgUcaRiaadggakmaaBaaa juaGbaqcLbmacaaIZaGaaGOmaaqcfayabaqcLbsacaWGIbqcfa4aa0 baaeaajugWaiaaikdacaaIZaaajuaGbaqcLbmacaaIWaGaey4kaSca aKqzGeGaaiykaaqcfayaaKqzGeGaaiikaiaadkgajuaGdaqhaaqaaK qzadGaaGymaiaaikdaaKqbagaajugWaiaaicdacqGHRaWkaaqcLbsa caGGPaGcdaahaaqcfayabeaajugWaiaaikdaaaqcLbsacqGHRaWkca GGOaGaamOyaKqbaoaaDaaabaqcLbmacaaIYaGaaGOmaaqcfayaaKqz adGaaGimaiabgUcaRaaajugibiaacMcajuaGdaahaaqabeaajugWai aaikdaaaaaaaaaaKqbakaawIcacaGLPaaaaaaa@E096@ (14)
C 1 T =0, B 1 + =( b 11 1+ b 12 1+ b 21 1+ b 22 1+ ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=nealmaaDaaajuaGba qcLbmacaaIXaaajuaGbaqcLbmacqGHLkIxcaqGubaaaKqzGeGaeyyp a0JaaGimaiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caWFcbWcdaqhaaqcfayaaKqzadGaaGymaaqcfayaaKqzadGaey4k aScaaKqzGeGaeyypa0JcdaqadaqcfayaaKqzGeqbaeqabiGaaaqcfa yaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaigdacaaIXaaajuaG baqcLbmacaaIXaGaey4kaScaaaqcfayaaKqzGeGaamOyaSWaa0baaK qbagaajugWaiaaigdacaaIYaaajuaGbaqcLbmacaaIXaGaey4kaSca aaqcfayaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIXa aajuaGbaqcLbmacaaIXaGaey4kaScaaaqcfayaaKqzGeGaamOyaSWa a0baaKqbagaajugWaiaaikdacaaIYaaajuaGbaqcLbmacaaIXaGaey 4kaScaaaaaaKqbakaawIcacaGLPaaajugibiaacYcacaaMc8oaaa@76C7@
H 1 = ( B 1 + C 1 T ) T = I 2 , H 1 + = ( H 1 ) + = I 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=HealmaaBaaajuaGba qcLbmacaaIXaaajuaGbeaajugibiabg2da9OWaaeWaaeaajugibiaa =jealmaaDaaabaqcLbmacaaIXaaaleaajugWaiabgUcaRaaajugibi aa=nealmaaDaaabaqcLbmacaaIXaaaleaajugWaiabgwQiEjaabsfa aaaakiaawIcacaGLPaaadaahaaWcbeqaaKqzadGaeyyPI4Laaeivaa aajugibiabg2da9iaa=LeakmaaBaaaleaajugWaiaaikdaaSqabaqc LbsacaGGSaGaaGPaVlaaykW7caaMc8Uaa8hsaSWaa0baaeaajugWai aaigdaaSqaaKqzadGaey4kaScaaKqzGeGaeyypa0JcdaqadaqaaKqz GeGaa8hsaSWaaSbaaKqbagaajugWaiaaigdaaKqbagqaaaGccaGLOa GaayzkaaWaaWbaaSqabeaajugWaiabgUcaRaaajugibiabg2da9iaa =LeakmaaBaaaleaajugWaiaaikdaaSqabaaaaa@67D6@ , (15)
G 1 = ( B 1 + C 1 T ) +T ( B 1 + A 1 C 1 T ) T =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=DealmaaBaaajuaGba qcLbmacaaIXaaajuaGbeaajugibiabg2da9OWaaeWaaKqbagaajugi biaa=jealmaaDaaajuaGbaqcLbmacaaIXaaajuaGbaqcLbmacqGHRa WkaaqcLbsacaWFdbWcdaqhaaqcfayaaKqzadGaaGymaaqcfayaaKqz adGaeyyPI4LaaeivaaaaaKqbakaawIcacaGLPaaakmaaCaaajuaGbe qaaKqzadGaey4kaSIaaeivaaaakmaabmaajuaGbaqcLbsacaWFcbWc daqhaaqcfayaaKqzadGaaGymaaqcfayaaKqzadGaey4kaScaaKqzGe Gaa8xqaSWaaSbaaKqbagaajugWaiaaigdaaKqbagqaaKqzGeGaa83q aSWaa0baaKqbagaajugWaiaaigdaaKqbagaajugWaiabgwQiEjaabs faaaaajuaGcaGLOaGaayzkaaWcdaahaaqcfayabeaajugWaiaabsfa aaqcLbsacqGH9aqpcaaIWaaaaa@67F0@ ,
Where, for brevity, the following notation is used:
h 1 + = b 11 1 b 22 1 b 12 1 b 21 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadIgalmaaDaaajuaGbaqcLb macaaIXaaajuaGbaqcLbmacqGHRaWkaaqcLbsacqGH9aqpcaWGIbWc daqhaaqcfayaaKqzadGaaGymaiaaigdaaKqbagaajugWaiaaigdaaa qcLbsacaWGIbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagaa jugWaiaaigdaaaqcLbsacqGHsislcaWGIbWcdaqhaaqcfayaaKqzad GaaGymaiaaikdaaKqbagaajugWaiaaigdaaaqcLbsacaWGIbWcdaqh aaqcfayaaKqzadGaaGOmaiaaigdaaKqbagaajugWaiaaigdaaaaaaa@577F@ , b 11 1+ = b 22 1 / h 1 + , b 12 1+ = b 12 1 / h 1 + , b 21 1+ = b 21 1 / h 1 + , b 22 1+ = b 11 1 / h 1 + . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiab g2da9iaadkgalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaK qzadGaaGymaaaajugibiaac+cacaWGObWcdaqhaaqcfayaaKqzadGa aGymaaqcfayaaKqzadGaey4kaScaaKqzGeGaaiilaiaaykW7caaMc8 UaaGPaVlaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfaya aKqzadGaaGymaiabgUcaRaaajugibiabg2da9iaaykW7cqGHsislca WGIbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaa igdaaaqcLbsacaGGVaGaamiAaSWaa0baaKqbagaajugWaiaaigdaaK qbagaajugWaiabgUcaRaaajugibiaacYcacaaMc8UaaGPaVlaadkga lmaaDaaajuaGbaqcLbmacaaIYaGaaGymaaqcfayaaKqzadGaaGymai abgUcaRaaajugibiabg2da9iabgkHiTiaadkgalmaaDaaajuaGbaqc LbmacaaIYaGaaGymaaqcfayaaKqzadGaaGymaaaajugibiaac+caca WGObWcdaqhaaqcfayaaKqzadGaaGymaaqcfayaaKqzadGaey4kaSca aKqzGeGaaiilaiaaykW7caaMc8UaamOyaSWaa0baaKqbagaajugWai aaikdacaaIYaaajuaGbaqcLbmacaaIXaGaey4kaScaaKqzGeGaeyyp a0JaamOyaSWaa0baaKqbagaajugWaiaaigdacaaIXaaajuaGbaqcLb macaaIXaaaaKqzGeGaai4laiaadIgalmaaDaaajuaGbaqcLbmacaaI XaaajuaGbaqcLbmacqGHRaWkaaqcLbsacaGGUaaaaa@A2E9@  For the zero and first decomposition levels, we calculate the ranks of the following block matrices ( H 0 G 0 H 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqcfayaaKqzGeqbaeqabeGaaW qcfayaaGqacKqzGeGaa8hsaSWaaSbaaKqbagaajugWaiaaicdaaKqb agqaaaqaaKqzGeGaa83raOWaaSbaaKqbagaajugWaiaaicdaaKqbag qaaKqzGeGaa8hsaOWaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaaa aiaawIcacaGLPaaaaaa@4287@ , ( H 1 G 1 H 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqcfayaaKqzGeqbaeqabeGaaW qcfayaaGqacKqzGeGaa8hsaSWaaSbaaKqbagaajugWaiaaigdaaKqb agqaaaqaaKqzGeGaa83raSWaaSbaaKqbagaajugWaiaaigdaaKqbag qaaKqzGeGaa8hsaSWaaSbaaKqbagaajugWaiaaigdaaKqbagqaaaaa aiaawIcacaGLPaaaaaa@428C@  as a result we will obtain: rank( H 0 G 0 H 0 )=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaabkhacaqGHbGaaeOBaiaabU gakmaabmaajuaGbaqcLbsafaqabeqacaadjuaGbaacbiqcLbsacaWF ibWcdaWgaaqcfayaaKqzadGaaGimaaqcfayabaaabaqcLbsacaWFhb GcdaWgaaqcfayaaKqzadGaaGimaaqcfayabaqcLbsacaWFibWcdaWg aaqcfayaaKqzadGaaGimaaqcfayabaaaaaGaayjkaiaawMcaaKqzGe Gaeyypa0JaaGOmaaaa@492A@ , rank( H 1 G 1 H 1 )=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaabkhacaqGHbGaaeOBaiaabU gakmaabmaajuaGbaqcLbsafaqabeqacaadjuaGbaacbiqcLbsacaWF ibGcdaWgaaqcfayaaKqzadGaaGymaaqcfayabaaabaqcLbsacaWFhb WcdaWgaaqcfayaaKqzadGaaGymaaqcfayabaqcLbsacaWFibWcdaWg aaqcfayaaKqzadGaaGymaaqcfayabaaaaaGaayjkaiaawMcaaKqzGe Gaeyypa0JaaGOmaaaa@492D@ , this corresponds to the number of “independent” inputs r=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkhacqGH9aqpcaaIYaaaaa@34E3@ . Therefore, each level of decomposition satisfies the controllability condition in Theorem 1. According to the form of controllers(9), (10) we define a matrix whose Eigen values will be assigned to the first decomposition level. Since matrices (15) are invertible, we may choose any matrix Ф 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYd Ni=xH8yiVeYlH8siW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqadeGa ciqaaiaabeqaaeaabaWaaaGcbaacbaqcLbsacaWFKqWcdaWgaaqcfa yaaKqzadGaaGymaaqcfayabaaaaa@36B9@  that has the prescribed spectrum for the first decomposition level. For simplicity, we set this matrix diagonal Ф 1 =diag( λ 1 , λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaabscblmaaBaaajuaGbaqcLb macaaIXaaajuaGbeaajugibiabg2da9iaabsgacaqGPbGaaeyyaiaa bEgakmaabmaajuaGbaqcLbsacuaH7oaBgaWeaSWaaSbaaKqbagaaju gWaiaaigdaaKqbagqaaKqzGeGaaiilaiaaykW7caaMc8Uafq4UdWMb ambakmaaBaaajuaGbaqcLbmacaaIYaaajuaGbeaaaiaawIcacaGLPa aaaaa@4C3E@ , and calculate the pseudo inverse matrix
C 1 + =( c 11 1+ c 12 1+ c 13 1+ c 21 1+ c 22 1+ c 23 1+ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=nealmaaDaaajuaGba qcLbmacaaIXaaajuaGbaqcLbmacqGHRaWkaaqcLbsacqGH9aqpkmaa bmaajuaGbaqcLbsafaqabeGadaaajuaGbaqcLbsacaWGJbWcdaqhaa qcfayaaKqzadGaaGymaiaaigdaaKqbagaajugWaiaaigdacqGHRaWk aaaajuaGbaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaGymaiaaik daaKqbagaajugWaiaaigdacqGHRaWkaaaajuaGbaqcLbsacaWGJbWc daqhaaqcfayaaKqzadGaaGymaiaaiodaaKqbagaajugWaiaaigdacq GHRaWkaaaajuaGbaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaGOm aiaaigdaaKqbagaajugWaiaaigdacqGHRaWkaaaajuaGbaqcLbsaca WGJbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagaajugWaiaa igdacqGHRaWkaaaajuaGbaqcLbsacaWGJbWcdaqhaaqcfayaaKqzad GaaGOmaiaaiodaaKqbagaajugWaiaaigdacqGHRaWkaaaaaaqcfaOa ayjkaiaawMcaaaaa@7197@ .
Here, again for brevity, we introduce the notation
c 1+ = ( c 11 1 c 22 1 ) 2 2 c 11 1 c 12 1 c 21 1 c 22 1 + ( c 12 1 c 21 1 ) 2 + ( c 12 1 c 31 1 ) 2 + ( c 22 1 c 31 1 ) 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadogalmaaCaaameqajuaGba qcLbmacaaIXaGaey4kaScaaKqzGeGaeyypa0Jaaiikaiaadogalmaa DaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqzadGaaGymaaaaju gibiaadogalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqz adGaaGymaaaajugibiaacMcakmaaCaaajuaGbeqaaKqzadGaaGOmaa aajugibiabgkHiTiaaikdacaWGJbWcdaqhaaqcfayaaKqzadGaaGym aiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfa yaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaaigdaaaqcLbsacaWG JbWcdaqhaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagaajugWaiaaig daaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqb agaajugWaiaaigdaaaqcLbsacqGHRaWkcaGGOaGaam4yaSWaa0baaK qbagaajugWaiaaigdacaaIYaaajuaGbaqcLbmacaaIXaaaaKqzGeGa am4yaSWaa0baaKqbagaajugWaiaaikdacaaIXaaajuaGbaqcLbmaca aIXaaaaKqzGeGaaiykaSWaaWbaaKqbagqabaqcLbmacaaIYaaaaKqz GeGaey4kaSIaaiikaiaadogalmaaDaaajuaGbaqcLbmacaaIXaGaaG OmaaqcfayaaKqzadGaaGymaaaajugibiaadogalmaaDaaajuaGbaqc LbmacaaIZaGaaGymaaqcfayaaKqzadGaaGymaaaajugibiaacMcakm aaCaaajuaGbeqaaKqzadGaaGOmaaaajugibiabgUcaRiaacIcacaWG JbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagaajugWaiaaig daaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaG4maiaaigdaaKqb agaajugWaiaaigdaaaqcLbsacaGGPaGcdaahaaqcfayabeaajugWai aaikdaaaqcLbsacaGGSaaaaa@A713@ c 11 1+ = c 22 1 ( c 11 1 c 22 1 c 12 1 c 21 1 )/ c 1+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadogalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiab g2da9iaadogalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaK qzadGaaGymaaaajugibiaacIcacaWGJbWcdaqhaaqcfayaaKqzadGa aGymaiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaa qcfayaaKqzadGaaGOmaiaaikdaaKqbagaajugWaiaaigdaaaqcLbsa cqGHsislcaWGJbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbag aajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaGOm aiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaGGPaGaai4laiaado galmaaCaaajuaGbeqaaKqzadGaaGymaiabgUcaRaaajugibiaacYca aaa@688B@ c 12 1+ = c 12 1 ( c 11 1 c 22 1 c 12 1 c 21 1 )/ c 1+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadogalmaaDaaabaqcLbmaca aIXaGaaGOmaaWcbaqcLbmacaaIXaGaey4kaScaaKqzGeGaeyypa0Ja eyOeI0Iaam4yaSWaa0baaeaajugWaiaaigdacaaIYaaaleaajugWai aaigdaaaqcLbsacaGGOaGaam4yaSWaa0baaeaajugWaiaaigdacaaI XaaaleaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqaaKqzadGaaG OmaiaaikdaaSqaaKqzadGaaGymaaaajugibiabgkHiTiaadogalmaa DaaabaqcLbmacaaIXaGaaGOmaaWcbaqcLbmacaaIXaaaaKqzGeGaam 4yaSWaa0baaeaajugWaiaaikdacaaIXaaaleaajugWaiaaigdaaaqc LbsacaGGPaGaai4laiaadogakmaaCaaaleqabaqcLbmacaaIXaGaey 4kaScaaKqzGeGaaiilaaaa@628E@
c 13 1+ = c 31 1 ( c 12 1 c 12 1 + c 22 1 c 22 1 )/ c 1+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadogalmaaDaaajuaGbaqcLb macaaIXaGaaG4maaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiab g2da9iaadogalmaaDaaajuaGbaqcLbmacaaIZaGaaGymaaqcfayaaK qzadGaaGymaaaajugibiaacIcacaWGJbWcdaqhaaqcfayaaKqzadGa aGymaiaaikdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaa qcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaaigdaaaqcLbsa cqGHRaWkcaWGJbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbag aajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaGOm aiaaikdaaKqbagaajugWaiaaigdaaaqcLbsacaGGPaGaai4laiaado galmaaCaaabeqaaKqzadGaaGymaiabgUcaRaaajugibiaacYcaaaa@67F6@ c 21 1+ =( c 12 1 c 21 1 c 21 1 c 11 1 c 21 1 c 22 1 + c 12 1 c 31 1 c 31 1 )/ c 1+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadogalmaaDaaajuaGbaqcLb macaaIYaGaaGymaaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiab g2da9iaacIcacaWGJbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaK qbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGa aGOmaiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaa qcfayaaKqzadGaaGOmaiaaigdaaKqbagaajugWaiaaigdaaaqcLbsa cqGHsislcaWGJbWcdaqhaaqcfayaaKqzadGaaGymaiaaigdaaKqbag aajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaGOm aiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfa yaaKqzadGaaGOmaiaaikdaaKqbagaajugWaiaaigdaaaqcLbsacqGH RaWkcaWGJbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaaju gWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaG4maiaa igdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaK qzadGaaG4maiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaGGPaGa ai4laiaadogalmaaCaaajuaGbeqaaKqzadGaaGymaiabgUcaRaaaju gibiaacYcaaaa@86A7@
c 22 1+ =( c 22 1 c 11 1 c 11 1 c 12 1 c 21 1 c 11 1 + c 22 1 c 31 1 c 31 1 )/ c 1+ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadogalmaaDaaajuaGbaqcLb macaaIYaGaaGOmaaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiab g2da9iaacIcacaWGJbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaK qbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGa aGymaiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaa qcfayaaKqzadGaaGymaiaaigdaaKqbagaajugWaiaaigdaaaqcLbsa cqGHsislcaWGJbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbag aajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaGOm aiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfa yaaKqzadGaaGymaiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacqGH RaWkcaWGJbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagaaju gWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaKqzadGaaG4maiaa igdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaK qzadGaaG4maiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaGGPaGa ai4laiaadogalmaaCaaajuaGbeqaaKqzadGaaGymaiabgUcaRaaaju gibiaacYcaaaa@86A7@
c 23 1+ = c 31 1 ( c 11 1 c 12 1 c 21 1 c 22 1 )/ c 1+ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadogalmaaDaaajuaGbaqcLb macaaIYaGaaG4maaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiab g2da9iabgkHiTiaadogalmaaDaaajuaGbaqcLbmacaaIZaGaaGymaa qcfayaaKqzadGaaGymaaaajugibiaacIcacaWGJbWcdaqhaaqcfaya aKqzadGaaGymaiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacaWGJb WcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaaigda aaqcLbsacqGHsislcaWGJbWcdaqhaaqcfayaaKqzadGaaGOmaiaaig daaKqbagaajugWaiaaigdaaaqcLbsacaWGJbWcdaqhaaqcfayaaKqz adGaaGOmaiaaikdaaKqbagaajugWaiaaigdaaaqcLbsacaGGPaGaai 4laiaadogalmaaCaaajuaGbeqaaKqzadGaaGymaiabgUcaRaaajugi biaac6caaaa@697D@
Based on equation(9), the first decomposition level yields the following formula for the controller:
F 1 = B 1 + ( Φ 1 A 1 ) C 1 + =( f 11 1 f 12 1 f 13 1 f 21 1 f 22 1 f 23 1 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=zealmaaBaaajuaGba qcLbmacaaIXaaajuaGbeaajugibiabg2da9iaa=jealmaaDaaajuaG baqcLbmacaaIXaaajuaGbaqcLbmacqGHRaWkaaGcdaqadaqcfayaaK qzGeGaeuOPdy0cdaWgaaqcfayaaKqzadGaaGymaaqcfayabaqcLbsa cqGHsislcaWFbbWcdaWgaaqcfayaaKqzadGaaGymaaqcfayabaaaca GLOaGaayzkaaqcLbsacaWFdbWcdaqhaaqcfayaaKqzadGaaGymaaqc fayaaKqzadGaey4kaScaaKqzGeGaeyypa0JcdaqadaqcfayaaKqzGe qbaeqabiWaaaqcfayaaKqzGeGaamOzaSWaa0baaKqbagaajugWaiaa igdacaaIXaaajuaGbaqcLbmacaaIXaaaaaqcfayaaKqzGeGaamOzaS Waa0baaKqbagaajugWaiaaigdacaaIYaaajuaGbaqcLbmacaaIXaaa aaqcfayaaKqzGeGaamOzaSWaa0baaKqbagaajugWaiaaigdacaaIZa aajuaGbaqcLbmacaaIXaaaaaqcfayaaKqzGeGaamOzaSWaa0baaKqb agaajugWaiaaikdacaaIXaaajuaGbaqcLbmacaaIXaaaaaqcfayaaK qzGeGaamOzaSWaa0baaKqbagaajugWaiaaikdacaaIYaaajuaGbaqc LbmacaaIXaaaaaqcfayaaKqzGeGaamOzaSWaa0baaKqbagaajugWai aaikdacaaIZaaajuaGbaqcLbmacaaIXaaaaaaaaKqbakaawIcacaGL PaaajugibiaacYcaaaa@8687@
Where the following notation is used:
d 1 = a 12 1 b 11 1+ b 12 1+ λ 1 , d 2 = a 11 1 b 11 1+ + a 21 1 b 12 1+ b 11 1+ λ 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadsgalmaaBaaajuaGbaqcLb macaaIXaaajuaGbeaajugibiabg2da9iaadggalmaaDaaajuaGbaqc LbmacaaIXaGaaGOmaaqcfayaaKqzadGaaGymaaaajugibiaadkgalm aaDaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqzadGaaGymaiab gUcaRaaajugibiabgkHiTiaadkgalmaaDaaajuaGbaqcLbmacaaIXa GaaGOmaaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiqbeU7aSzaa taGcdaWgaaqcfayaaKqzadGaaGymaaqcfayabaqcLbsacaGGSaGaaG PaVlaaykW7caWGKbWcdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaqc LbsacqGH9aqpcaWGHbWcdaqhaaqcfayaaKqzadGaaGymaiaaigdaaK qbagaajugWaiaaigdaaaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGymaiaaigdaaKqbagaajugWaiaaigdacqGHRaWkaaqcLbsacqGHRa WkcaWGHbWcdaqhaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagaajugW aiaaigdaaaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGaaGymaiaaik daaKqbagaajugWaiaaigdacqGHRaWkaaqcLbsacqGHsislcaWGIbWc daqhaaqcfayaaKqzadGaaGymaiaaigdaaKqbagaajugWaiaaigdacq GHRaWkaaqcLbsacuaH7oaBgaWeaSWaaSbaaKqbagaajugWaiaaigda aKqbagqaaKqzGeGaaiilaaaa@8EA0@ d 3 = a 12 1 b 21 1+ b 22 1+ λ 2 , d 4 = a 11 1 b 21 1+ + a 21 1 b 22 1+ b 11 1+ λ 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadsgalmaaBaaajuaGbaqcLb macaaIZaaajuaGbeaajugibiabg2da9iaadggalmaaDaaajuaGbaqc LbmacaaIXaGaaGOmaaqcfayaaKqzadGaaGymaaaajugibiaadkgalm aaDaaajuaGbaqcLbmacaaIYaGaaGymaaqcfayaaKqzadGaaGymaiab gUcaRaaajugibiabgkHiTiaadkgalmaaDaaajuaGbaqcLbmacaaIYa GaaGOmaaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiqbeU7aSzaa taWcdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaqcLbsacaGGSaGaaG PaVlaaykW7caWGKbGcdaWgaaqcfayaaKqzadGaaGinaaqcfayabaqc LbsacqGH9aqpcaWGHbWcdaqhaaqcfayaaKqzadGaaGymaiaaigdaaK qbagaajugWaiaaigdaaaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGOmaiaaigdaaKqbagaajugWaiaaigdacqGHRaWkaaqcLbsacqGHRa WkcaWGHbWcdaqhaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagaajugW aiaaigdaaaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGaaGOmaiaaik daaKqbagaajugWaiaaigdacqGHRaWkaaqcLbsacqGHsislcaWGIbWc daqhaaqcfayaaKqzadGaaGymaiaaigdaaKqbagaajugWaiaaigdacq GHRaWkaaqcLbsacuaH7oaBgaWeaOWaaSbaaKqbagaajugWaiaaikda aKqbagqaaKqzGeGaaiilaaaa@8EA9@
and where the components are:
f 11 1 = c 21 1+ d 1 c 11 1+ d 2 , f 12 1 = c 22 1+ d 1 c 12 1+ d 2 , f 13 1 = c 23 1+ d 1 c 13 1+ d 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadAgalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaaGymaaaajugibiabg2da9iab gkHiTiaadogalmaaDaaajuaGbaqcLbmacaaIYaGaaGymaaqcfayaaK qzadGaaGymaiabgUcaRaaajugibiaadsgalmaaBaaajuaGbaqcLbma caaIXaaajuaGbeaajugibiabgkHiTiaadogalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiaa dsgalmaaDaaajuaGbaqcLbmacaaIYaaajuaGbaaaaKqzGeGaaiilai aaykW7caaMc8UaamOzaSWaa0baaKqbagaajugWaiaaigdacaaIYaaa juaGbaqcLbmacaaIXaaaaKqzGeGaeyypa0JaeyOeI0Iaam4yaSWaa0 baaKqbagaajugWaiaaikdacaaIYaaajuaGbaqcLbmacaaIXaGaey4k aScaaKqzGeGaamizaSWaaSbaaKqbagaajugWaiaaigdaaKqbagqaaK qzGeGaeyOeI0Iaam4yaSWaa0baaKqbagaajugWaiaaigdacaaIYaaa juaGbaqcLbmacaaIXaGaey4kaScaaKqzGeGaamizaSWaa0baaKqbag aajugWaiaaikdaaKqbagaaaaqcLbsacaGGSaGaaGPaVlaaykW7caWG MbWcdaqhaaqcfayaaKqzadGaaGymaiaaiodaaKqbagaajugWaiaaig daaaqcLbsacqGH9aqpcqGHsislcaWGJbWcdaqhaaqcfayaaKqzadGa aGOmaiaaiodaaKqbagaajugWaiaaigdacqGHRaWkaaqcLbsacaWGKb WcdaWgaaqcfayaaKqzadGaaGymaaqcfayabaqcLbsacqGHsislcaWG JbWcdaqhaaqcfayaaKqzadGaaGymaiaaiodaaKqbagaajugWaiaaig dacqGHRaWkaaqcLbsacaWGKbWcdaqhaaqcfayaaKqzadGaaGOmaaqc fayaaaaajugibiaacYcacaaMc8oaaa@A7A2@ f 21 1 = c 21 1+ d 3 c 11 1+ d 4 , f 22 1 = c 22 1+ d 3 c 12 1+ d 4 , f 23 1 = c 23 1+ d 3 c 13 1+ d 4 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadAgalmaaDaaajuaGbaqcLb macaaIYaGaaGymaaqcfayaaKqzadGaaGymaaaajugibiabg2da9iab gkHiTiaadogalmaaDaaajuaGbaqcLbmacaaIYaGaaGymaaqcfayaaK qzadGaaGymaiabgUcaRaaajugibiaadsgakmaaBaaajuaGbaqcLbma caaIZaaajuaGbeaajugibiabgkHiTiaadogalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiaa dsgakmaaBaaajuaGbaqcLbmacaaI0aaajuaGbeaajugibiaacYcaca aMc8UaaGPaVlaadAgalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqc fayaaKqzadGaaGymaaaajugibiabg2da9iabgkHiTiaadogalmaaDa aajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqzadGaaGymaiabgUca RaaajugibiaadsgakmaaBaaajuaGbaqcLbmacaaIZaaajuaGbeaaju gibiabgkHiTiaadogalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqc fayaaKqzadGaaGymaiabgUcaRaaajugibiaadsgakmaaBaaajuaGba qcLbmacaaI0aaajuaGbeaajugibiaacYcacaaMc8UaaGPaVlaadAga lmaaDaaajuaGbaqcLbmacaaIYaGaaG4maaqcfayaaKqzadGaaGymaa aajugibiabg2da9iabgkHiTiaadogalmaaDaaajuaGbaqcLbmacaaI YaGaaG4maaqcfayaaKqzadGaaGymaiabgUcaRaaajugibiaadsgalm aaBaaajuaGbaqcLbmacaaIZaaajuaGbeaajugibiabgkHiTiaadoga lmaaDaaajuaGbaqcLbmacaaIXaGaaG4maaqcfayaaKqzadGaaGymai abgUcaRaaajugibiaadsgakmaaBaaajuaGbaqcLbmacaaI0aaajuaG beaajugibiaac6cacaaMc8oaaa@A7AB@ To calculate matrix B 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=jealmaaDaaajuaGba qcLbmacaaIWaaajuaGbaqcLbmacqGHsislaaaaaa@3843@  that is needed for determining the zero level controller, we use the second formula in(10). As a result, we obtain the expression
B 0 = B 0 + F 1 С 1 B 0 =( b 11 m b 12 m b 13 m b 14 m b 21 m b 22 m b 23 m b 24 m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=jealmaaDaaajuaGba qcLbmacaaIWaaajuaGbaqcLbmacqGHsislaaqcLbsacqGH9aqpcaWF cbWcdaqhaaqcfayaaKqzadGaaGimaaqcfayaaKqzadGaey4kaScaaK qzGeGaeyOeI0Iaa8NraSWaaSbaaKqbagaajugWaiaaigdaaKqbagqa aKqzGeGaa8xieSWaaSbaaKqbagaajugWaiaaigdaaKqbagqaaKqzGe Gaa8NqaSWaa0baaKqbagaajugWaiaaicdaaKqbagaajugWaiabgwQi Ebaajugibiabg2da9OWaaeWaaKqbagaajugibuaabeqacqaaaaqcfa yaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaigdacaaIXaaajuaG baqcLbmacaWGTbaaaaqcfayaaKqzGeGaamOyaSWaa0baaKqbagaaju gWaiaaigdacaaIYaaajuaGbaqcLbmacaWGTbaaaaqcfayaaKqzGeGa amOyaSWaa0baaKqbagaajugWaiaaigdacaaIZaaajuaGbaqcLbmaca WGTbaaaaqcfayaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaigda caaI0aaajuaGbaqcLbmacaWGTbaaaaqcfayaaKqzGeGaamOyaSWaa0 baaKqbagaajugWaiaaikdacaaIXaaajuaGbaqcLbmacaWGTbaaaaqc fayaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIYaaaju aGbaqcLbmacaWGTbaaaaqcfayaaKqzGeGaamOyaSWaa0baaKqbagaa jugWaiaaikdacaaIZaaajuaGbaqcLbmacaWGTbaaaaqcfayaaKqzGe GaamOyaSWaa0baaKqbagaajugWaiaaikdacaaI0aaajuaGbaqcLbma caWGTbaaaaaaaKqbakaawIcacaGLPaaaaaa@9699@ .
Here we denote linear combinations:
b 11 m = b 11 0+ l 11 ( c 11 1 f 11 1 + c 21 1 f 12 1 + c 31 1 f 13 1 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaamyBaaaajugibiabg2da9iaa dkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqzadGaaG imaiabgUcaRaaajugibiabgkHiTiaadYgakmaaBaaajuaGbaqcLbma caaIXaGaaGymaaqcfayabaqcLbsacaGGOaGaam4yaSWaa0baaKqbag aajugWaiaaigdacaaIXaaajuaGbaqcLbmacaaIXaaaaKqzGeGaamOz aSWaa0baaKqbagaajugWaiaaigdacaaIXaaajuaGbaqcLbmacaaIXa aaaKqzGeGaey4kaSIaam4yaSWaa0baaKqbagaajugWaiaaikdacaaI XaaajuaGbaqcLbmacaaIXaaaaKqzGeGaamOzaSWaa0baaKqbagaaju gWaiaaigdacaaIYaaajuaGbaqcLbmacaaIXaaaaKqzGeGaey4kaSIa am4yaSWaa0baaKqbagaajugWaiaaiodacaaIXaaajuaGbaqcLbmaca aIXaaaaKqzGeGaamOzaSWaa0baaKqbagaajugWaiaaigdacaaIZaaa juaGbaqcLbmacaaIXaaaaKqzGeGaaiykaiaacYcaaaa@78E1@
b 12 m = b 12 0+ l 12 ( c 11 1 f 11 1 + c 21 1 f 12 1 + c 31 1 f 13 1 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGOmaaqcfayaaKqzadGaamyBaaaajugibiabg2da9iaa dkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaaG imaiabgUcaRaaajugibiabgkHiTiaadYgalmaaBaaajuaGbaqcLbma caaIXaGaaGOmaaqcfayabaqcLbsacaGGOaGaam4yaSWaa0baaKqbag aajugWaiaaigdacaaIXaaajuaGbaqcLbmacaaIXaaaaKqzGeGaamOz aSWaa0baaKqbagaajugWaiaaigdacaaIXaaajuaGbaqcLbmacaaIXa aaaKqzGeGaey4kaSIaam4yaSWaa0baaKqbagaajugWaiaaikdacaaI XaaajuaGbaqcLbmacaaIXaaaaKqzGeGaamOzaSWaa0baaKqbagaaju gWaiaaigdacaaIYaaajuaGbaqcLbmacaaIXaaaaKqzGeGaey4kaSIa am4yaSWaa0baaKqbagaajugWaiaaiodacaaIXaaajuaGbaqcLbmaca aIXaaaaKqzGeGaamOzaSWaa0baaKqbagaajugWaiaaigdacaaIZaaa juaGbaqcLbmacaaIXaaaaKqzGeGaaiykaiaacYcaaaa@78E5@
b 13 m = b 13 0+ c 11 1 f 11 1 c 21 1 f 12 1 c 31 1 f 13 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaG4maaqcfayaaKqzadGaamyBaaaajugibiabg2da9iaa dkgalmaaDaaajuaGbaqcLbmacaaIXaGaaG4maaqcfayaaKqzadGaaG imaiabgUcaRaaajugibiabgkHiTiaadogalmaaDaaajuaGbaqcLbma caaIXaGaaGymaaqcfayaaKqzadGaaGymaaaajugibiaadAgalmaaDa aajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqzadGaaGymaaaajugi biabgkHiTiaadogalmaaDaaajuaGbaqcLbmacaaIYaGaaGymaaqcfa yaaKqzadGaaGymaaaajugibiaadAgalmaaDaaajuaGbaqcLbmacaaI XaGaaGOmaaqcfayaaKqzadGaaGymaaaajugibiabgkHiTiaadogalm aaDaaajuaGbaqcLbmacaaIZaGaaGymaaqcfayaaKqzadGaaGymaaaa jugibiaadAgalmaaDaaajuaGbaqcLbmacaaIXaGaaG4maaqcfayaaK qzadGaaGymaaaajugibiaacYcacaaMc8oaaa@73C2@
b 14 m = c 12 1 f 11 1 c 22 1 f 12 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGinaaqcfayaaKqzadGaamyBaaaajugibiabg2da9iab gkHiTiaadogalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaK qzadGaaGymaaaajugibiaadAgalmaaDaaajuaGbaqcLbmacaaIXaGa aGymaaqcfayaaKqzadGaaGymaaaajugibiabgkHiTiaadogalmaaDa aajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqzadGaaGymaaaajugi biaadAgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaKqzad GaaGymaaaajugibiaacYcaaaa@5A7D@
b 21 m = b 21 0+ l 11 ( c 11 1 f 21 1 + c 21 1 f 22 1 + c 31 1 f 23 1 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaGymaaqcfayaaKqzadGaamyBaaaajugibiabg2da9iaa dkgalmaaDaaajuaGbaqcLbmacaaIYaGaaGymaaqcfayaaKqzadGaaG imaiabgUcaRaaajugibiabgkHiTiaadYgalmaaBaaajuaGbaqcLbma caaIXaGaaGymaaqcfayabaqcLbsacaGGOaGaam4yaSWaa0baaKqbag aajugWaiaaigdacaaIXaaajuaGbaqcLbmacaaIXaaaaKqzGeGaamOz aSWaa0baaKqbagaajugWaiaaikdacaaIXaaajuaGbaqcLbmacaaIXa aaaKqzGeGaey4kaSIaam4yaSWaa0baaKqbagaajugWaiaaikdacaaI XaaajuaGbaqcLbmacaaIXaaaaKqzGeGaamOzaSWaa0baaKqbagaaju gWaiaaikdacaaIYaaajuaGbaqcLbmacaaIXaaaaKqzGeGaey4kaSIa am4yaSWaa0baaKqbagaajugWaiaaiodacaaIXaaajuaGbaqcLbmaca aIXaaaaKqzGeGaamOzaSWaa0baaKqbagaajugWaiaaikdacaaIZaaa juaGbaqcLbmacaaIXaaaaKqzGeGaaiykaiaacYcaaaa@78E7@
b 22 m = b 22 0+ l 12 ( c 11 1 f 21 1 + c 21 1 f 22 1 + c 31 1 f 23 1 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaGOmaaqcfayaaKqzadGaamyBaaaajugibiabg2da9iaa dkgalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqzadGaaG imaiabgUcaRaaajugibiabgkHiTiaadYgakmaaBaaajuaGbaqcLbma caaIXaGaaGOmaaqcfayabaqcLbsacaGGOaGaam4yaSWaa0baaKqbag aajugWaiaaigdacaaIXaaajuaGbaqcLbmacaaIXaaaaKqzGeGaamOz aSWaa0baaKqbagaajugWaiaaikdacaaIXaaajuaGbaqcLbmacaaIXa aaaKqzGeGaey4kaSIaam4yaSWaa0baaKqbagaajugWaiaaikdacaaI XaaajuaGbaqcLbmacaaIXaaaaKqzGeGaamOzaSWaa0baaKqbagaaju gWaiaaikdacaaIYaaajuaGbaqcLbmacaaIXaaaaKqzGeGaey4kaSIa am4yaSWaa0baaKqbagaajugWaiaaiodacaaIXaaajuaGbaqcLbmaca aIXaaaaKqzGeGaamOzaSWaa0baaKqbagaajugWaiaaikdacaaIZaaa juaGbaqcLbmacaaIXaaaaKqzGeGaaiykaiaaykW7caGGSaGaaGPaVd aa@7BFF@
b 23 m = b 23 0+ c 11 1 f 21 1 c 21 1 f 22 1 c 31 1 f 23 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaG4maaqcfayaaKqzadGaamyBaaaajugibiabg2da9iaa dkgalmaaDaaajuaGbaqcLbmacaaIYaGaaG4maaqcfayaaKqzadGaaG imaiabgUcaRaaajugibiabgkHiTiaadogalmaaDaaajuaGbaqcLbma caaIXaGaaGymaaqcfayaaKqzadGaaGymaaaajugibiaadAgalmaaDa aajuaGbaqcLbmacaaIYaGaaGymaaqcfayaaKqzadGaaGymaaaajugi biabgkHiTiaadogalmaaDaaajuaGbaqcLbmacaaIYaGaaGymaaqcfa yaaKqzadGaaGymaaaajugibiaadAgalmaaDaaajuaGbaqcLbmacaaI YaGaaGOmaaqcfayaaKqzadGaaGymaaaajugibiabgkHiTiaadogalm aaDaaajuaGbaqcLbmacaaIZaGaaGymaaqcfayaaKqzadGaaGymaaaa jugibiaadAgalmaaDaaajuaGbaqcLbmacaaIYaGaaG4maaqcfayaaK qzadGaaGymaaaajugibiaacYcaaaa@723C@
b 24 m = c 12 1 f 21 1 c 22 1 f 22 1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaGinaaqcfayaaKqzadGaamyBaaaajugibiabg2da9iaa ykW7cqGHsislcaWGJbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaK qbagaajugWaiaaigdaaaqcLbsacaWGMbWcdaqhaaqcfayaaKqzadGa aGOmaiaaigdaaKqbagaajugWaiaaigdaaaqcLbsacqGHsislcaWGJb WcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagaajugWaiaaigda aaqcLbsacaWGMbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbag aajugWaiaaigdaaaqcLbsacaGGUaaaaa@5C0D@
According to Theorem 1, we complete the system of the zero decomposition level using (7). For this purpose we use equation (14) and, applying new notation, we obtain
G 0 =( a 11 a a 12 a a 21 a a 22 a ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=DealmaaBaaajuaGba qcLbmacaaIWaaajuaGbeaajugibiabg2da9OWaaeWaaKqbagaajugi buaabeqaciaaaKqbagaajugibiaadggalmaaDaaajuaGbaqcLbmaca aIXaGaaGymaaqcfayaaKqzadGaamyyaaaaaKqbagaajugibiaadgga lmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaamyyaa aaaKqbagaajugibiaadggalmaaDaaajuaGbaqcLbmacaaIYaGaaGym aaqcfayaaKqzadGaamyyaaaaaKqbagaajugibiaadggalmaaDaaaju aGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqzadGaamyyaaaaaaaajuaG caGLOaGaayzkaaaaaa@5B23@ ,
Where we denote
a 11 a = b 12 0+ d 1 / d , a 12 a = b 12 0+ d 2 / d , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadggalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaamyyaaaajugibiabg2da9iaa dkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaaG imaiabgUcaRaaajugibiaadsgalmaaBaaajuaGbaqcLbmacaaIXaaa juaGbeaajugibiaac+cacaWGKbWcdaWgaaqcfayaaaqabaqcLbsaca GGSaGaaGPaVlaaykW7caaMc8UaamyyaSWaa0baaKqbagaajugWaiaa igdacaaIYaaajuaGbaqcLbmacaWGHbaaaKqzGeGaeyypa0JaamOyaS Waa0baaKqbagaajugWaiaaigdacaaIYaaajuaGbaqcLbmacaaIWaGa ey4kaScaaKqzGeGaamizaOWaaSbaaKqbagaajugWaiaaikdaaKqbag qaaKqzGeGaai4laiaadsgalmaaBaaajuaGbaaabeaajugibiaacYca aaa@689B@
a 21 a = b 22 0+ d 1 / d , a 22 a = b 22 0+ d 2 / d , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadggalmaaDaaajuaGbaqcLb macaaIYaGaaGymaaqcfayaaKqzadGaamyyaaaajugibiabg2da9iaa dkgalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqzadGaaG imaiabgUcaRaaajugibiaadsgalmaaBaaajuaGbaqcLbmacaaIXaaa juaGbeaajugibiaac+cacaWGKbWcdaWgaaqcfayaaaqabaqcLbsaca GGSaGaaGPaVlaaykW7caWGHbWcdaqhaaqcfayaaKqzadGaaGOmaiaa ikdaaKqbagaajugWaiaadggaaaqcLbsacqGH9aqpcaWGIbWcdaqhaa qcfayaaKqzadGaaGOmaiaaikdaaKqbagaajugWaiaaicdacqGHRaWk aaqcLbsacaWGKbWcdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaqcLb sacaGGVaGaamizaSWaaSbaaKqbagaaaeqaaKqzGeGaaiilaaaa@6715@
d= ( b 12 0+ ) 2 + ( b 22 0+ ) 2 , d 1 = a 12 b 11 m + a 22 b 12 m + a 32 b 13 m , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadsgacqGH9aqpcaGGOaGaam OyaSWaa0baaKqbagaajugWaiaaigdacaaIYaaajuaGbaqcLbmacaaI WaGaey4kaScaaKqzGeGaaiykaSWaaWbaaKqbagqabaqcLbmacaaIYa aaaKqzGeGaey4kaSIaaiikaiaadkgalmaaDaaajuaGbaqcLbmacaaI YaGaaGOmaaqcfayaaKqzadGaaGimaiabgUcaRaaajugibiaacMcalm aaCaaajuaGbeqaaKqzadGaaGOmaaaajugibiaacYcacaaMc8UaaGPa VlaadsgalmaaBaaajuaGbaqcLbmacaaIXaaajuaGbeaajugibiabg2 da9iaadggakmaaBaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayabaqc LbsacaWGIbWcdaqhaaqcfayaaKqzadGaaGymaiaaigdaaKqbagaaju gWaiaad2gaaaqcLbsacqGHRaWkcaWGHbGcdaWgaaqcfayaaKqzadGa aGOmaiaaikdaaKqbagqaaKqzGeGaamOyaSWaa0baaKqbagaajugWai aaigdacaaIYaaajuaGbaqcLbmacaWGTbaaaKqzGeGaey4kaSIaamyy aSWaaSbaaKqbagaajugWaiaaiodacaaIYaaajuaGbeaajugibiaadk galmaaDaaajuaGbaqcLbmacaaIXaGaaG4maaqcfayaaKqzadGaamyB aaaajugibiaacYcacaaMc8oaaa@8298@
d 2 = a 12 b 21 m + a 22 b 22 m + a 32 b 23 m . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadsgalmaaBaaajuaGbaqcLb macaaIYaaajuaGbeaajugibiabg2da9iaadggakmaaBaaajuaGbaqc LbmacaaIXaGaaGOmaaqcfayabaqcLbsacaWGIbWcdaqhaaqcfayaaK qzadGaaGOmaiaaigdaaKqbagaajugWaiaad2gaaaqcLbsacqGHRaWk caWGHbWcdaWgaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagqaaKqzGe GaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIYaaajuaGbaqcLbma caWGTbaaaKqzGeGaey4kaSIaamyyaSWaaSbaaKqbagaajugWaiaaio dacaaIYaaajuaGbeaajugibiaadkgalmaaDaaajuaGbaqcLbmacaaI YaGaaG4maaqcfayaaKqzadGaamyBaaaajugibiaac6caaaa@6106@
Now, we should determine Ф 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbfv3ySLgzGueE0jxyaibaieYd Ni=xH8yiVeYlH8siW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pw e9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqadeGa ciqaaiaabeqaaeaabaWaaaGcbaqcLbsacaqGKqWcdaWgaaqcfayaaK qzadGaaGimaaqcfayabaaaaa@36B1@  for the zero decomposition level. For this purpose, we decompose the matrices H 0 , G 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=HealmaaBaaajuaGba qcLbmacaaIWaaajuaGbeaajugibiaacYcacaaMc8UaaGPaVlaa=Dea lmaaBaaajuaGbaqcLbmacaaIWaaajuaGbeaaaaa@3E7A@  of the zero level into two sublevels and calculate the corresponding matrices. We obtain
( H 0 ) 0 = H 0 , ( G 0 ) 0 = G 0 , ( H 0 ) 0 =( b 12 m / b 22 m 1 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadiqcfayaaGqacKqzGeGaa8hsaS WaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaGaayjkaiaawMcaaSWa aSbaaKqbagaajugWaiaaicdaaKqbagqaaKqzGeGaeyypa0Jaa8hsaS WaaSbaaKqbagaajugWaiaaicdaaKqbagqaaKqzGeGaaiilaiaaykW7 caaMc8UcdaqadiqcfayaaKqzGeGaa83raSWaaSbaaKqbagaajugWai aaicdaaKqbagqaaaGaayjkaiaawMcaaOWaaSbaaKqbagaajugWaiaa icdaaKqbagqaaKqzGeGaeyypa0Jaa83raSWaaSbaaKqbagaajugWai aaicdaaKqbagqaaKqzGeGaaiilaiaaykW7caaMc8Ucdaqadiqcfaya aKqzGeGaa8hsaSWaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaGaay jkaiaawMcaaSWaa0baaKqbagaajugWaiaaicdaaKqbagaajugWaiab gwQiEbaajugibiabg2da9OWaaeWaaKqbagaajugibuaabeqabiaaaK qbagaajugibiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqc fayaaKqzadGaamyBaaaajugibiaac+cacaWGIbWcdaqhaaqcfayaaK qzadGaaGOmaiaaikdaaKqbagaajugWaiaad2gaaaaajuaGbaqcLbsa caaIXaaaaaqcfaOaayjkaiaawMcaaKqzGeGaaiilaiaaykW7aaa@7F9B@
( H 0 ) 0 + =( b 12 m b 22 m / b m ( b 12 m ) 2 / b m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadiqcfayaaGqacKqzGeGaa8hsaS WaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaGaayjkaiaawMcaaSWa a0baaKqbagaajugWaiaaicdaaKqbagaajugWaiabgwQiEjabgUcaRa aajugibiabg2da9OWaaeWaaKqbagaajugibuaabeqaceaaaKqbagaa jugibiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaK qzadGaamyBaaaajugibiaadkgalmaaDaaajuaGbaqcLbmacaaIYaGa aGOmaaqcfayaaKqzadGaamyBaaaajugibiaac+cacaWGIbWcdaqhaa qcfayaaaqaaKqzadGaamyBaaaaaKqbagaajugibiaacIcacaWGIbWc daqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaad2gaaa qcLbsacaGGPaGcdaahaaqcfayabeaajugWaiaaikdaaaqcLbsacaGG VaGaamOyaSWaa0baaKqbagaaaeaajugWaiaad2gaaaaaaaqcfaOaay jkaiaawMcaaaaa@6A55@ ,
( H 0 ) 0 + =( b 12 m b 22 m / b m ( b 12 m ) 2 / b m ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadiqcfayaaGqacKqzGeGaa8hsaS WaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaGaayjkaiaawMcaaSWa a0baaKqbagaajugWaiaaicdaaKqbagaajugWaiabgUcaRaaajugibi abg2da9OWaaeWaaKqbagaajugibuaabeqabiaaaKqbagaajugibiab gkHiTiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaK qzadGaamyBaaaajugibiaadkgalmaaDaaajuaGbaqcLbmacaaIYaGa aGOmaaqcfayaaKqzadGaamyBaaaajugibiaac+cacaWGIbWcdaqhaa qcfayaaaqaaKqzadGaamyBaaaaaKqbagaajugibiaacIcacaWGIbWc daqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaad2gaaa qcLbsacaGGPaGcdaahaaqcfayabeaajugWaiaaikdaaaqcLbsacaGG VaGaamOyaSWaa0baaKqbagaaaeaajugWaiaad2gaaaaaaaqcfaOaay jkaiaawMcaaKqzGeGaaiOlaaaa@6AD2@
Here b m = ( b 12 m ) 2 + ( b 22 m ) 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaaaba qcLbmacaWGTbaaaKqzGeGaeyypa0JaaiikaiaadkgalmaaDaaajuaG baqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaamyBaaaajugibiaacM cakmaaCaaajuaGbeqaaKqzadGaaGOmaaaajugibiabgUcaRiaacIca caWGIbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagaajugWai aad2gaaaqcLbsacaGGPaWcdaahaaqcfayabeaajugWaiaaikdaaaqc LbsacaGGSaaaaa@5134@  and, in addition,
( G 0 ) 1 = ( H 0 ) 0 G 0 ( H 0 ) 0 = a 1 aa = = ( b 22 m ) 2 ( a 22 a + a 12 a b 12 m / b 22 m )/ b m + + b 12 m b 22 m ( a 21 a + a 11 a b 12 m / b 22 m )/ b m , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakqaabeqaamaabmGajuaGbaacbiqcLbsaca WFhbWcdaWgaaqcfayaaKqzadGaaGimaaqcfayabaaacaGLOaGaayzk aaGcdaWgaaqcfayaaKqzadGaaGymaaqcfayabaqcLbsacqGH9aqpkm aabmGajuaGbaqcLbsacaWFibGcdaWgaaqcfayaaKqzadGaaGimaaqc fayabaaacaGLOaGaayzkaaWcdaqhaaqcfayaaKqzadGaaGimaaqcfa yaaaaakiaa=DealmaaBaaajuaGbaqcLbmacaaIWaaajuaGbeaakmaa bmGajuaGbaqcLbsacaWFibGcdaWgaaqcfayaaKqzadGaaGimaaqcfa yabaaacaGLOaGaayzkaaWcdaqhaaqcfayaaKqzadGaaGimaaqcfaya aKqzadGaeyyPI4faaKqzGeGaeyypa0JaamyyaSWaa0baaKqbagaaju gWaiaaigdaaKqbagaajugWaiaadggacaWGHbaaaKqzGeGaeyypa0da juaGbaqcLbsacqGH9aqpcaGGOaGaamOyaSWaa0baaKqbagaajugWai aaikdacaaIYaaajuaGbaqcLbmacaWGTbaaaKqzGeGaaiykaOWaaWba aKqbagqabaqcLbmacaaIYaaaaKqzGeGaaiikaiaadggalmaaDaaaju aGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqzadGaamyyaaaajugibiab gUcaRiaadggalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaK qzadGaamyyaaaajugibiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGa aGOmaaqcfayaaKqzadGaamyBaaaajugibiaac+cacaWGIbWcdaqhaa qcfayaaKqzadGaaGOmaiaaikdaaKqbagaajugWaiaad2gaaaqcLbsa caGGPaGaai4laiaadkgalmaaCaaajuaGbeqaaKqzadGaamyBaaaaju gibiabgUcaRaGcbaqcLbsacaaMc8Uaey4kaSIaamOyaSWaa0baaKqb agaajugWaiaaigdacaaIYaaajuaGbaqcLbmacaWGTbaaaKqzGeGaam OyaSWaa0baaKqbagaajugWaiaaikdacaaIYaaajuaGbaqcLbmacaWG TbaaaKqzGeGaaiikaiaadggalmaaDaaajuaGbaqcLbmacaaIYaGaaG ymaaqcfayaaKqzadGaamyyaaaajugibiabgUcaRiaadggalmaaDaaa juaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqzadGaamyyaaaajugibi aadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGa amyBaaaajugibiaac+cacaWGIbWcdaqhaaqcfayaaKqzadGaaGOmai aaikdaaKqbagaajugWaiaad2gaaaqcLbsacaGGPaGaai4laiaadkga kmaaCaaajuaGbeqaaKqzadGaamyBaaaajugibiaacYcacaaMc8oaaa a@D116@
( H 0 ) 1 = ( H 0 ) 0 G 0 H 0 = b 11 bb MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadiqcfayaaGqacKqzGeGaa8hsaS WaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaGaayjkaiaawMcaaSWa aSbaaKqbagaajugWaiaaigdaaKqbagqaaKqzGeGaeyypa0Jcdaqadi qcfayaaKqzGeGaa8hsaOWaaSbaaKqbagaajugWaiaaicdaaKqbagqa aaGaayjkaiaawMcaaSWaa0baaKqbagaajugWaiaaicdaaKqbagaaju gWaiabgwQiEbaajugibiaa=DealmaaBaaajuaGbaqcLbmacaaIWaaa juaGbeaajugibiaa=HeakmaaBaaajuaGbaqcLbmacaaIWaaajuaGbe aajugibiabg2da9iaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGym aaqcfayaaKqzadGaamOyaiaadkgaaaaaaa@5C39@ .
Besides,
b 11 bb = a 22 a ( b 22 m ( a 21 a + a 11 a b 12 m / b 22 m )/ b 12 m + a 12 a b 12 m / b 22 m . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaamOyaiaadkgaaaqcLbsacqGH 9aqpcaWGHbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagaaju gWaiaadggaaaqcLbsacqGHsislcaGGOaGaamOyaSWaa0baaKqbagaa jugWaiaaikdacaaIYaaajuaGbaqcLbmacaWGTbaaaKqzGeGaaiikai aadggalmaaDaaajuaGbaqcLbmacaaIYaGaaGymaaqcfayaaKqzadGa amyyaaaajugibiabgUcaRiaadggalmaaDaaajuaGbaqcLbmacaaIXa GaaGymaaqcfayaaKqzadGaamyyaaaajugibiaadkgalmaaDaaajuaG baqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaamyBaaaajugibiaac+ cacaWGIbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagaajugW aiaad2gaaaqcLbsacaGGPaGaai4laiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGOmaaqcfayaaKqzadGaamyBaaaajugibiabgUcaRiaa dggalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaam yyaaaajugibiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqc fayaaKqzadGaamyBaaaajugibiaac+cacaWGIbWcdaqhaaqcfayaaK qzadGaaGOmaiaaikdaaKqbagaajugWaiaad2gaaaqcLbsacaGGUaaa aa@8E06@ (16)
Using the values of b 11 bb MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaamOyaiaadkgaaaaaaa@39F9@  (16), we then find the matrix (scalar, in this case) ( H 0 ) 1 + = b p11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadiqcfayaaGqacKqzGeGaa8hsaS WaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaGaayjkaiaawMcaaSWa a0baaKqbagaajugWaiaaigdaaKqbagaajugWaiabgUcaRaaajugibi abg2da9iaadkgalmaaDaaajuaGbaqcLbmacaWGWbGaaGymaiaaigda aKqbagaaaaaaaa@44E6@ , where b p11 =1/ b 11 bb MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadkgalmaaDaaajuaGbaqcLb macaWGWbGaaGymaiaaigdaaKqbagaaaaqcLbsacqGH9aqpkmaalyaa juaGbaqcLbsacaaIXaaajuaGbaqcLbsacaWGIbWcdaqhaaqcfayaaK qzadGaaGymaiaaigdaaKqbagaajugWaiaadkgacaWGIbaaaaaaaaa@446C@ .
Let us now assign the Eigen value as a scalar matrix ( Ф 0 ) 1 = λ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeGabaaiemaabmGajuaGbaqcLbsacaqGKq GcdaWgaaqcfayaaKqzadGaaGimaaqcfayabaaacaGLOaGaayzkaaWc daWgaaqcfayaaKqzadGaaGymaaqcfayabaqcLbsacqGH9aqpcuaH7o aBgaWeaSWaaSbaaKqbagaajugWaiaaiodaaKqbagqaaaaa@420A@  and calculate the matrix of feedback coefficients for the first sublevel of the zero decomposition level. We obtain k 1 = b p11 ( a 1 aa λ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadUgalmaaBaaajuaGbaqcLb macaaIXaaajuaGbeaajugibiabg2da9iabgkHiTiaadkgalmaaBaaa juaGbaqcLbmacaWGWbGaaGymaiaaigdaaKqbagqaaOWaaeWaaKqbag aajugibiaadggalmaaDaaajuaGbaqcLbmacaaIXaaajuaGbaqcLbma caWGHbGaamyyaaaajugibiabgkHiTiqbeU7aSzaataGcdaWgaaqcfa yaaKqzadGaaG4maaqcfayabaaacaGLOaGaayzkaaaaaa@4ED2@ . Next, we calculate the matrix
( H 0 ) 0 = ( H 0 ) 0 + k 1 ( H 0 ) 0 =( h 11 0 h 12 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadiqcfayaaGqacKqzGeGaa8hsaS WaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaGaayjkaiaawMcaaSWa a0baaKqbagaajugWaiaaicdaaKqbagaajugWaiabgkHiTaaajugibi abg2da9OWaaeWacKqbagaajugibiaa=HeakmaaBaaajuaGbaqcLbma caaIWaaajuaGbeaaaiaawIcacaGLPaaalmaaDaaajuaGbaqcLbmaca aIWaaajuaGbaqcLbmacqGHRaWkaaqcLbsacqGHsislcaWGRbWcdaWg aaqcfayaaKqzadGaaGymaaqcfayabaGcdaqadiqcfayaaKqzGeGaa8 hsaOWaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaGaayjkaiaawMca aSWaa0baaKqbagaajugWaiaaicdaaKqbagaajugWaiabgwQiEbaaju gibiabg2da9OWaaeWaaKqbagaajugibuaabeqabiaaaKqbagaajugi biaadIgalmaaDaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqzad GaaGimaiabgkHiTaaaaKqbagaajugibiaadIgalmaaDaaajuaGbaqc LbmacaaIXaGaaGOmaaqcfayaaKqzadGaaGimaiabgkHiTaaaaaaaju aGcaGLOaGaayzkaaqcLbsacaGGSaaaaa@7521@
Where h 11 0 = b p11 b 12 m (( a 1 aa λ 3 )/ b 22 m + b 12 m b 22 m / b m , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadIgalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaaGimaiabgkHiTaaajugibiab g2da9iaadkgakmaaBaaajuaGbaqcLbmacaWGWbGaaGymaiaaigdaaK qbagqaaKqzGeGaamOyaSWaa0baaKqbagaajugWaiaaigdacaaIYaaa juaGbaqcLbmacaWGTbaaaKqzGeGaaiikaiaacIcacaWGHbWcdaqhaa qcfayaaKqzadGaaGymaaqcfayaaKqzadGaamyyaiaadggaaaqcLbsa cqGHsislcuaH7oaBgaWeaOWaaSbaaKqbagaajugWaiaaiodaaKqbag qaaKqzGeGaaiykaiaac+cacaWGIbWcdaqhaaqcfayaaKqzadGaaGOm aiaaikdaaKqbagaajugWaiaad2gaaaqcLbsacqGHRaWkcaWGIbWcda qhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaad2gaaaqc LbsacaWGIbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbagaaju gWaiaad2gaaaqcLbsacaGGVaGaamOyaSWaa0baaKqbagaaaeaajugW aiaad2gaaaqcLbsacaGGSaGaaGPaVdaa@78CB@ h 12 0 = b p11 ( a 1 aa λ 3 )+ ( b 12 m ) 2 / b m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadIgalmaaDaaajuaGbaqcLb macaaIXaGaaGOmaaqcfayaaKqzadGaaGimaiabgkHiTaaajugibiab g2da9iaadkgakmaaBaaajuaGbaqcLbmacaWGWbGaaGymaiaaigdaaK qbagqaaKqzGeGaaiikaiaadggalmaaDaaajuaGbaqcLbmacaaIXaaa juaGbaqcLbmacaWGHbGaamyyaaaajugibiabgkHiTiqbeU7aSzaata GcdaWgaaqcfayaaKqzadGaaG4maaqcfayabaqcLbsacaGGPaGaey4k aSIaaiikaiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfa yaaKqzadGaamyBaaaajugibiaacMcalmaaCaaajuaGbeqaaKqzadGa aGOmaaaajugibiaac+cacaWGIbWcdaqhaaqcfayaaaqaaKqzadGaam yBaaaaaaa@62A2@ , and specify again the matrix of Eigen values of the zero sublevel of the zero decomposition level by ( Ф 0 ) 0 = λ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadiqcfayaaKqzGeGaaeijeSWaaS baaKqbagaajugWaiaaicdaaKqbagqaaaGaayjkaiaawMcaaOWaaSba aKqbagaajugWaiaaicdaaKqbagqaaKqzGeGaeyypa0Jafq4UdWMbam balmaaBaaajuaGbaqcLbmacaaI0aaajuaGbeaaaaa@41E3@ .
Finally, we find matrix K 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=TealmaaBaaajuaGba qcLbmacaaIWaaajuaGbeaaaaa@3630@  by the rule
K 0 = ( Ф 0 ) 0 ( H 0 ) 0 ( H 0 ) 0 G 0 =( K 11 K 12 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=TealmaaBaaajuaGba qcLbmacaaIWaaajuaGbeaajugibiabg2da9OWaaeWacKqbagaajugi biaabscbkmaaBaaajuaGbaacbaqcLbmacaGFWaaajuaGbeaaaiaawI cacaGLPaaalmaaBaaajuaGbaqcLbmacaaIWaaajuaGbeaakmaabmGa juaGbaqcLbsacaWFibWcdaWgaaqcfayaaKqzadGaaGimaaqcfayaba aacaGLOaGaayzkaaWcdaqhaaqcfayaaKqzadGaaGimaaqcfayaaKqz adGaeyOeI0caaKqzGeGaeyOeI0IcdaqadiqcfayaaKqzGeGaa8hsaS WaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaGaayjkaiaawMcaaSWa a0baaKqbagaajugWaiaaicdaaKqbagaajugWaiabgkHiTaaajugibi aa=DealmaaBaaajuaGbaqcLbmacaaIWaaajuaGbeaajugibiabg2da 9OWaaeWaaKqbagaajugibuaabeqabiaaaKqbagaajugibiaadUeakm aaBaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayabaaabaqcLbsacaWG lbWcdaWgaaqcfayaaKqzadGaaGymaiaaikdaaKqbagqaaaaaaiaawI cacaGLPaaaaaa@6EE7@ ,
Where in the case under examination
K 11 = a 11 a (( b 12 m b 22 m / b m b p11 b 12 m ( a 1 aa λ 3 )/ b 22 m ) a 21 a ( b p11 ( a 1 aa λ 3 ) + ( b 12 m ) 2 / b m ) λ 4 ( b 12 m b 22 m / b m b p11 b 12 m ( a 1 aa λ 3 )/ b 22 m ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakqaabeqaaKqzGeGaam4saSWaaSbaaKqbag aajugWaiaaigdacaaIXaaajuaGbeaajugibiabg2da9iaadggalmaa DaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqzadGaamyyaaaaju gibiaacIcacaGGOaGaamOyaSWaa0baaKqbagaajugWaiaaigdacaaI YaaajuaGbaqcLbmacaWGTbaaaKqzGeGaamOyaSWaa0baaKqbagaaju gWaiaaikdacaaIYaaajuaGbaqcLbmacaWGTbaaaKqzGeGaai4laiaa dkgalmaaDaaajuaGbaaabaqcLbmacaWGTbaaaKqzGeGaeyOeI0Iaam OyaSWaa0baaKqbagaajugWaiaadchacaaIXaGaaGymaaqcfayaaaaa jugibiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayaaK qzadGaamyBaaaajugibiaacIcacaWGHbWcdaqhaaqcfayaaKqzadGa aGymaaqcfayaaKqzadGaamyyaiaadggaaaqcLbsacqGHsislcuaH7o aBgaWeaSWaaSbaaKqbagaajugWaiaaiodaaKqbagqaaKqzGeGaaiyk aiaac+cacaWGIbWcdaqhaaqcfayaaKqzadGaaGOmaiaaikdaaKqbag aajugWaiaad2gaaaqcLbsacaGGPaGaeyOeI0IaamyyaSWaa0baaKqb agaajugWaiaaikdacaaIXaaajuaGbaqcLbmacaWGHbaaaKqzGeGaai ikaiaadkgalmaaDaaajuaGbaqcLbmacaWGWbGaaGymaiaaigdaaKqb agaaaaqcLbsacaGGOaGaamyyaSWaa0baaKqbagaajugWaiaaigdaaK qbagaajugWaiaadggacaWGHbaaaKqzGeGaeyOeI0Iafq4UdWMbamba lmaaBaaajuaGbaqcLbmacaaIZaaajuaGbeaajugibiaacMcaaOqaaK qzGeGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab gUcaRiaacIcacaWGIbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaK qbagaajugWaiaad2gaaaqcLbsacaGGPaWcdaqhaaqcfayaaaqaaKqz adGaaGOmaaaajugibiaac+cacaWGIbWcdaqhaaqcfayaaaqaaKqzad GaamyBaaaajugibiaacMcacqGHsislcuaH7oaBgaWeaOWaaSbaaKqb agaajugWaiaaisdaaKqbagqaaKqzGeGaaiikaiaadkgalmaaDaaaju aGbaqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaamyBaaaajugibiaa dkgalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaaKqzadGaam yBaaaajugibiaac+cacaWGIbWcdaqhaaqcfayaaaqaaKqzadGaamyB aaaajugibiabgkHiTiaadkgalmaaDaaajuaGbaqcLbmacaWGWbGaaG ymaiaaigdaaKqbagaaaaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGymaiaaikdaaKqbagaajugWaiaad2gaaaqcLbsacaGGOaGaamyyaS Waa0baaKqbagaajugWaiaaigdaaKqbagaajugWaiaadggacaWGHbaa aKqzGeGaeyOeI0Iafq4UdWMbambakmaaBaaajuaGbaqcLbmacaaIZa aajuaGbeaajugibiaacMcacaGGVaGaamOyaSWaa0baaKqbagaajugW aiaaikdacaaIYaaajuaGbaqcLbmacaWGTbaaaKqzGeGaaiykaiaacY caaaaa@FC74@
K 12 = λ 4 ( b p11 ( a 1 aa λ 3 )+ ( b 12 m ) 2 / b m ) a 22 a ( b p11 ( a 1 aa λ 3 )+ + ( b 12 m ) 2 / b m )+ a 12 a (( b 12 m b 22 m / b m b p11 b 12 m ( a 1 aa λ 3 )/ b 22 m ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakqaabeqaaKqzGeGaam4saOWaaSbaaKqbag aajugWaiaaigdacaaIYaaajuaGbeaajugibiabg2da9iqbeU7aSzaa taWcdaWgaaqcfayaaKqzadGaaGinaaqcfayabaqcLbsacaGGOaGaam OyaSWaa0baaKqbagaajugWaiaadchacaaIXaGaaGymaaqcfayaaaaa jugibiaacIcacaWGHbWcdaqhaaqcfayaaKqzadGaaGymaaqcfayaaK qzadGaamyyaiaadggaaaqcLbsacqGHsislcuaH7oaBgaWeaSWaaSba aKqbagaajugWaiaaiodaaKqbagqaaKqzGeGaaiykaiabgUcaRiaacI cacaWGIbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugW aiaad2gaaaqcLbsacaGGPaWcdaqhaaqcfayaaaqaaKqzadGaaGOmaa aajugibiaac+cacaWGIbWcdaqhaaqcfayaaaqaaKqzadGaamyBaaaa jugibiaacMcacqGHsislcaWGHbWcdaqhaaqcfayaaKqzadGaaGOmai aaikdaaKqbagaajugWaiaadggaaaqcLbsacaGGOaGaamOyaSWaa0ba aKqbagaajugWaiaadchacaaIXaGaaGymaaqcfayaaaaajugibiaacI cacaWGHbWcdaqhaaqcfayaaKqzadGaaGymaaqcfayaaKqzadGaamyy aiaadggaaaqcLbsacqGHsislcuaH7oaBgaWeaSWaaSbaaKqbagaaju gWaiaaiodaaKqbagqaaKqzGeGaaiykaiabgUcaRaGcbaqcLbsacaaM c8UaaGPaVlabgUcaRiaacIcacaWGIbWcdaqhaaqcfayaaKqzadGaaG ymaiaaikdaaKqbagaajugWaiaad2gaaaqcLbsacaGGPaWcdaqhaaqc fayaaaqaaKqzadGaaGOmaaaajugibiaac+cacaWGIbWcdaqhaaqcfa yaaaqaaKqzadGaamyBaaaajugibiaacMcacqGHRaWkcaWGHbWcdaqh aaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaadggaaaqcLb sacaGGOaGaaiikaiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGOm aaqcfayaaKqzadGaamyBaaaajugibiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaGOmaaqcfayaaKqzadGaamyBaaaajugibiaac+cacaWG IbWcdaqhaaqcfayaaaqaaKqzadGaamyBaaaajugibiabgkHiTiaadk galmaaDaaajuaGbaqcLbmacaWGWbGaaGymaiaaigdaaKqbagaaaaqc LbsacaWGIbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaaju gWaiaad2gaaaqcLbsacaGGOaGaamyyaSWaa0baaKqbagaajugWaiaa igdaaKqbagaajugWaiaadggacaWGHbaaaKqzGeGaeyOeI0Iafq4UdW MbambalmaaBaaajuaGbaqcLbmacaaIZaaajuaGbeaajugibiaacMca caGGVaGaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIYaaajuaGba qcLbmacaWGTbaaaKqzGeGaaiykaiaac6caaaaa@E2DB@ As a result, we obtain using equation(11), the matrix Ф 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGKq GcdaWgaaqcfayaaKqzadGaaGimaaqcfayabaaaaa@3A3F@ , whose eigen values λ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiqbeU7aSzaataGcdaWgaaqcfa yaaKqzadGaaG4maaqcfayabaaaaa@3729@ , λ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiqbeU7aSzaataWcdaWgaaqcfa yaaKqzadGaaGinaaqcfayabaaaaa@372B@ , are ensured by the output controller, namely,
Ф 0 =( a 11 a b 22 m b 12 m K 11 a 21 a + K 11 a 12 a b 22 m b 12 m K 12 a 22 a + K 12 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaabscblmaaBaaajuaGbaqcLb macaaIWaaajuaGbeaajugibiabg2da9OWaaeWaaKqbagaajugibuaa beqaciaaaKqbagaajugibiaadggalmaaDaaajuaGbaqcLbmacaaIXa GaaGymaaqcfayaaKqzadGaamyyaaaajugibiabgkHiTOWaaSaaaKqb agaajugibiaadkgalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfa yaaKqzadGaamyBaaaaaKqbagaajugibiaadkgalmaaDaaajuaGbaqc LbmacaaIXaGaaGOmaaqcfayaaKqzadGaamyBaaaaaaqcLbsacaWGlb WcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaaqaaKqzGeGa amyyaSWaa0baaKqbagaajugWaiaaikdacaaIXaaajuaGbaqcLbmaca WGHbaaaKqzGeGaey4kaSIaam4saSWaaSbaaKqbagaajugWaiaaigda caaIXaaajuaGbeaaaeaajugibiaadggalmaaDaaajuaGbaqcLbmaca aIXaGaaGOmaaqcfayaaKqzadGaamyyaaaajugibiabgkHiTOWaaSaa aKqbagaajugibiaadkgalmaaDaaajuaGbaqcLbmacaaIYaGaaGOmaa qcfayaaKqzadGaamyBaaaaaKqbagaajugibiaadkgalmaaDaaajuaG baqcLbmacaaIXaGaaGOmaaqcfayaaKqzadGaamyBaaaaaaqcLbsaca WGlbWcdaWgaaqcfayaaKqzadGaaGymaiaaikdaaKqbagqaaaqaaKqz GeGaamyyaSWaa0baaKGbagaajugWaiaaikdacaaIYaaajyaGbaqcLb macaWGHbaaaKqzGeGaey4kaSIaam4saOWaaSbaaKqbagaajugWaiaa igdacaaIYaaajuaGbeaaaaaacaGLOaGaayzkaaqcLbsacaGGUaaaaa@946D@
Further calculations, which were described, for instance, 13,17–19 finally yield the following formula for the SH output controller vector:
F=( f 11 f 12 f 13 f 21 f 22 f 23 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaieGajugibiaa=zeaieWacaGF9aGcda qadaqcfayaaKqzGeqbaeqabiWaaaqcfayaaKqzGeGaamOzaOWaaSba aKqbagaajugWaiaaigdacaaIXaaajuaGbeaaaeaajugibiaadAgakm aaBaaajuaGbaqcLbmacaaIXaGaaGOmaaqcfayabaaabaqcLbsacaWG MbWcdaWgaaqcfayaaKqzadGaaGymaiaaiodaaKqbagqaaaqaaKqzGe GaamOzaSWaaSbaaKqbagaajugWaiaaikdacaaIXaaajuaGbeaaaeaa jugibiaadAgakmaaBaaajuaGbaqcLbmacaaIYaGaaGOmaaqcfayaba aabaqcLbsacaWGMbGcdaWgaaqcfayaaKqzadGaaGOmaiaaiodaaKqb agqaaaaaaiaawIcacaGLPaaaaaa@577E@ . (17)
Here elements of the following notation:
d 11 = a 11 a b 22 m b 12 m K 11 , d 12 = a 21 a + K 11 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadsgakmaaBaaajuaGbaqcLb macaaIXaGaaGymaaqcfayabaqcLbsacqGH9aqpcaWGHbWcdaqhaaqc fayaaKqzadGaaGymaiaaigdaaKqbagaajugWaiaadggaaaqcLbsacq GHsislkmaalaaajuaGbaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGOmaiaaikdaaKqbagaajugWaiaad2gaaaaajuaGbaqcLbsacaWGIb WcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaad2ga aaaaaKqzGeGaam4saOWaaSbaaKqbagaajugWaiaaigdacaaIXaaaju aGbeaajugibiaacYcacaaMc8UaaGPaVlaaykW7caWGKbGcdaWgaaqc fayaaKqzadGaaGymaiaaikdaaKqbagqaaKqzGeGaeyypa0JaamyyaS Waa0baaKqbagaajugWaiaaikdacaaIXaaajuaGbaqcLbmacaWGHbaa aKqzGeGaey4kaSIaam4saOWaaSbaaKqbagaajugWaiaaigdacaaIXa aajuaGbeaajugibiaacYcaaaa@711B@
d 21 = a 12 a b 22 m b 12 m K 12 , d 22 = a 22 a + K 12 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadsgakmaaBaaajuaGbaqcLb macaaIYaGaaGymaaqcfayabaqcLbsacqGH9aqpcaWGHbWcdaqhaaqc fayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaadggaaaqcLbsacq GHsislkmaalaaajuaGbaqcLbsacaWGIbWcdaqhaaqcfayaaKqzadGa aGOmaiaaikdaaKqbagaajugWaiaad2gaaaaajuaGbaqcLbsacaWGIb WcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaad2ga aaaaaKqzGeGaam4saSWaaSbaaKqbagaajugWaiaaigdacaaIYaaaju aGbeaajugibiaacYcacaaMc8UaaGPaVlaaykW7caWGKbWcdaWgaaqc fayaaKqzadGaaGOmaiaaikdaaKqbagqaaKqzGeGaeyypa0JaamyyaS Waa0baaKqbagaajugWaiaaikdacaaIYaaajuaGbaqcLbmacaWGHbaa aKqzGeGaey4kaSIaam4saOWaaSbaaKqbagaajugWaiaaigdacaaIYa aajuaGbeaajugibiaacYcaaaa@7123@ Can be expressed as
f 11 = b 11 m d 11 b 12 m a 21 b 13 m a 31 b 11 m a 11 + b 21 m d 12 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeGabaaXcKqzGeGaamOzaSWaaSbaaKqbag aajugWaiaaigdacaaIXaaajuaGbeaajugibiabg2da9iaadkgalmaa DaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqzadGaamyBaaaaju gibiaadsgakmaaBaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayabaqc LbsacqGHsislcaWGIbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaK qbagaajugWaiaad2gaaaqcLbsacaWGHbGcdaWgaaqcfayaaKqzadGa aGOmaiaaigdaaKqbagqaaKqzGeGaeyOeI0IaamOyaSWaa0baaKqbag aajugWaiaaigdacaaIZaaajuaGbaqcLbmacaWGTbaaaKqzGeGaamyy aSWaaSbaaKqbagaajugWaiaaiodacaaIXaaajuaGbeaajugibiabgk HiTiaadkgalmaaDaaajuaGbaqcLbmacaaIXaGaaGymaaqcfayaaKqz adGaamyBaaaajugibiaadggakmaaBaaajuaGbaqcLbmacaaIXaGaaG ymaaqcfayabaqcLbsacqGHRaWkcaWGIbWcdaqhaaqcfayaaKqzadGa aGOmaiaaigdaaKqbagaajugWaiaad2gaaaqcLbsacaWGKbWcdaWgaa qcfayaaKqzadGaaGymaiaaikdaaKqbagqaaKqzGeGaaiilaaaa@7E36@
f 12 = b 13 m d 11 b 13 m a 33 b 14 m + b 23 m d 12 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadAgakmaaBaaajuaGbaqcLb macaaIXaGaaGOmaaqcfayabaqcLbsacqGH9aqpcaWGIbWcdaqhaaqc fayaaKqzadGaaGymaiaaiodaaKqbagaajugWaiaad2gaaaqcLbsaca WGKbGcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaKqzGeGa eyOeI0IaamOyaSWaa0baaKqbagaajugWaiaaigdacaaIZaaajuaGba qcLbmacaWGTbaaaKqzGeGaamyyaOWaaSbaaKqbagaajugWaiaaioda caaIZaaajuaGbeaajugibiabgkHiTiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGinaaqcfayaaKqzadGaamyBaaaajugibiabgUcaRiaa dkgalmaaDaaajuaGbaqcLbmacaaIYaGaaG4maaqcfayaaKqzadGaam yBaaaajugibiaadsgalmaaBaaajuaGbaqcLbmacaaIXaGaaGOmaaqc fayabaqcLbsacaGGSaaaaa@6A43@
f 13 = b 14 m d 11 b 13 m a 34 b 11 m a 13 b 12 m a 24 + b 24 m d 12 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadAgakmaaBaaajuaGbaqcLb macaaIXaGaaG4maaqcfayabaqcLbsacqGH9aqpcaWGIbWcdaqhaaqc fayaaKqzadGaaGymaiaaisdaaKqbagaajugWaiaad2gaaaqcLbsaca WGKbWcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaKqbagqaaKqzGeGa eyOeI0IaamOyaSWaa0baaKqbagaajugWaiaaigdacaaIZaaajuaGba qcLbmacaWGTbaaaKqzGeGaamyyaSWaaSbaaKqbagaajugWaiaaioda caaI0aaajuaGbeaajugibiabgkHiTiaadkgalmaaDaaajuaGbaqcLb macaaIXaGaaGymaaqcfayaaKqzadGaamyBaaaajugibiaadggakmaa BaaajuaGbaqcLbmacaaIXaGaaG4maaqcfayabaqcLbsacqGHsislca WGIbWcdaqhaaqcfayaaKqzadGaaGymaiaaikdaaKqbagaajugWaiaa d2gaaaqcLbsacaWGHbWcdaWgaaqcfayaaKqzadGaaGOmaiaaisdaaK qbagqaaKqzGeGaey4kaSIaamOyaSWaa0baaKqbagaajugWaiaaikda caaI0aaajuaGbaqcLbmacaWGTbaaaKqzGeGaamizaSWaaSbaaKqbag aajugWaiaaigdacaaIYaaajuaGbeaajugibiaacYcaaaa@7D7E@
f 21 = b 11 m d 21 b 22 m a 21 b 23 m a 31 b 21 m a 11 + b 21 m d 22 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadAgakmaaBaaajuaGbaqcLb macaaIYaGaaGymaaqcfayabaqcLbsacqGH9aqpcaWGIbWcdaqhaaqc fayaaKqzadGaaGymaiaaigdaaKqbagaajugWaiaad2gaaaqcLbsaca WGKbGcdaWgaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagqaaKqzGeGa eyOeI0IaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIYaaajuaGba qcLbmacaWGTbaaaKqzGeGaamyyaSWaaSbaaKqbagaajugWaiaaikda caaIXaaajuaGbeaajugibiabgkHiTiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaG4maaqcfayaaKqzadGaamyBaaaajugibiaadggalmaa BaaajuaGbaqcLbmacaaIZaGaaGymaaqcfayabaqcLbsacqGHsislca WGIbWcdaqhaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagaajugWaiaa d2gaaaqcLbsacaWGHbGcdaWgaaqcfayaaKqzadGaaGymaiaaigdaaK qbagqaaKqzGeGaey4kaSIaamOyaSWaa0baaKqbagaajugWaiaaikda caaIXaaajuaGbaqcLbmacaWGTbaaaKqzGeGaamizaSWaaSbaaKqbag aajugWaiaaikdacaaIYaaajuaGbeaajugibiaacYcaaaa@7D73@
f 22 = b 13 m d 21 b 23 m a 33 b 24 m + b 23 m d 22 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadAgakmaaBaaajuaGbaqcLb macaaIYaGaaGOmaaqcfayabaqcLbsacqGH9aqpcaWGIbWcdaqhaaqc fayaaKqzadGaaGymaiaaiodaaKqbagaajugWaiaad2gaaaqcLbsaca WGKbWcdaWgaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagqaaKqzGeGa eyOeI0IaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIZaaajuaGba qcLbmacaWGTbaaaKqzGeGaamyyaSWaaSbaaKqbagaajugWaiaaioda caaIZaaajuaGbeaajugibiabgkHiTiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaGinaaqcfayaaKqzadGaamyBaaaajugibiabgUcaRiaa dkgalmaaDaaajuaGbaqcLbmacaaIYaGaaG4maaqcfayaaKqzadGaam yBaaaajugibiaadsgakmaaBaaajuaGbaqcLbmacaaIYaGaaGOmaaqc fayabaqcLbsacaGGSaaaaa@6A49@
f 23 = b 14 m d 21 b 23 m a 34 b 22 m a 24 b 21 m a 13 + b 24 m d 22 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaadAgalmaaBaaajuaGbaqcLb macaaIYaGaaG4maaqcfayabaqcLbsacqGH9aqpcaWGIbWcdaqhaaqc fayaaKqzadGaaGymaiaaisdaaKqbagaajugWaiaad2gaaaqcLbsaca WGKbWcdaWgaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagqaaKqzGeGa eyOeI0IaamOyaSWaa0baaKqbagaajugWaiaaikdacaaIZaaajuaGba qcLbmacaWGTbaaaKqzGeGaamyyaSWaaSbaaKqbagaajugWaiaaioda caaI0aaajuaGbeaajugibiabgkHiTiaadkgalmaaDaaajuaGbaqcLb macaaIYaGaaGOmaaqcfayaaKqzadGaamyBaaaajugibiaadggakmaa BaaajuaGbaqcLbmacaaIYaGaaGinaaqcfayabaqcLbsacqGHsislca WGIbWcdaqhaaqcfayaaKqzadGaaGOmaiaaigdaaKqbagaajugWaiaa d2gaaaqcLbsacaWGHbWcdaWgaaqcfayaaKqzadGaaGymaiaaiodaaK qbagqaaKqzGeGaey4kaSIaamOyaSWaa0baaKqbagaajugWaiaaikda caaI0aaajuaGbaqcLbmacaWGTbaaaKqzGeGaamizaOWaaSbaaKqbag aajugWaiaaikdacaaIYaaajuaGbeaajugibiaac6caaaa@7D86@

The synthesized controller (the SH control system laws) ensures exactly the specified spectrum (6) for controlled longitudinal motion of the helicopter. This assertion can be directly checked with the help of appropriate analytical calculations. For this purpose it is sufficient to make use of the package Symbolic Toolbox Matlab; namely, one can use the eig instruction to calculate the eigen values of the A+BFСmatrix, where matrices can be expressed as(4), and the F matrixis determined by (17).
Numerical analysis
Suppose that we want the closed-loop system “SH + control system” to have the following multiple real spectrum
Λ={ 1.5,1.5,1.5,1.5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiabfU5amjabg2da9OWaaiWaaK qbagaajugibiabgkHiTiaaigdacaGGUaGaaGynaiaacYcacaaMc8Ua aGPaVlabgkHiTiaaigdacaGGUaGaaGynaiaacYcacaaMc8UaaGPaVl abgkHiTiaaigdacaGGUaGaaGynaiaacYcacaaMc8UaaGPaVlabgkHi TiaaigdacaGGUaGaaGynaaqcfaOaay5Eaiaaw2haaaaa@5040@ . (18)
Let use for the hypothetical SH the A, B and C (4) matrices having the following numerical values:
A=( 0.0598 0.0233 0.1095 0.1703 0.0268 0.8899 1.2091 0.0135 1.5158 2.3207 2.3464 0 0 0 1 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiqcLbsaca WFbbGaeyypa0JcdaqadaqcfayaaKqzGeqbaeqabqabaaaaaKqbagaa jugibiabgkHiTiaaicdacaGGUaGaaGimaiaaiwdacaaI5aGaaGioaa qcfayaaKqzGeGaeyOeI0IaaGimaiaac6cacaaIWaGaaGOmaiaaioda caaIZaaajuaGbaqcLbsacaaIWaGaaiOlaiaaigdacaaIWaGaaGyoai aaiwdaaKqbagaajugibiabgkHiTiaaicdacaGGUaGaaGymaiaaiEda caaIWaGaaG4maaqcfayaaKqzGeGaeyOeI0IaaGimaiaac6cacaaIWa GaaGOmaiaaiAdacaaI4aaajuaGbaqcLbsacqGHsislcaaIWaGaaiOl aiaaiIdacaaI4aGaaGyoaiaaiMdaaKqbagaajugibiabgkHiTiaaig dacaGGUaGaaGOmaiaaicdacaaI5aGaaGymaaqcfayaaKqzGeGaeyOe I0IaaGimaiaac6cacaaIWaGaaGymaiaaiodacaaI1aaajuaGbaqcLb sacaaIXaGaaiOlaiaaiwdacaaIXaGaaGynaiaaiIdaaKqbagaajugi biabgkHiTiaaikdacaGGUaGaaG4maiaaikdacaaIWaGaaG4naaqcfa yaaKqzGeGaeyOeI0IaaGOmaiaac6cacaaIZaGaaGinaiaaiAdacaaI 0aaajuaGbaqcLbsacaaIWaaajuaGbaqcLbsacaaIWaaajuaGbaqcLb sacaaIWaaajuaGbaqcLbsacaaIXaaajuaGbaqcLbsacaaIWaaaaaqc faOaayjkaiaawMcaaKqzGeGaaiilaaaa@8AEE@
B=( 0.3050 0.0228 1.2817 2.0890 28.178 22.7255 0 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiqcLbsaca WFcbGaeyypa0JcdaqadaqcfayaaKqzGeqbaeqabqGaaaaajuaGbaqc LbsacqGHsislcaaIWaGaaiOlaiaaiodacaaIWaGaaGynaiaaicdaaK qbagaajugibiabgkHiTiaaicdacaGGUaGaaGimaiaaikdacaaIYaGa aGioaaqcfayaaKqzGeGaeyOeI0IaaGymaiaac6cacaaIYaGaaGioai aaigdacaaI3aaajuaGbaqcLbsacaaIYaGaaiOlaiaaicdacaaI4aGa aGyoaiaaicdaaKqbagaajugibiaaikdacaaI4aGaaiOlaiaaigdaca aI3aGaaGioaaqcfayaaKqzGeGaaGOmaiaaikdacaGGUaGaaG4naiaa ikdacaaI1aGaaGynaaqcfayaaKqzGeGaaGimaaqcfayaaKqzGeGaaG imaaaaaKqbakaawIcacaGLPaaaaaa@63CD@
With use of equation (17) for the controller we obtain the numerical value for the matrixF. As a result we have
F=( 3.6163 1.8622 3.4996 2.7936 2.2309 3.6237 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiqcLbsaca WFgbGaeyypa0JcdaqadaqcfayaaKqzGeqbaeqabiWaaaqcfayaaKqz GeGaaG4maiaac6cacaaI2aGaaGymaiaaiAdacaaIZaaajuaGbaqcLb sacqGHsislcaaIXaGaaiOlaiaaiIdacaaI2aGaaGOmaiaaikdaaKqb agaajugibiabgkHiTiaaiodacaGGUaGaaGinaiaaiMdacaaI5aGaaG OnaaqcfayaaKqzGeGaeyOeI0IaaGOmaiaac6cacaaI3aGaaGyoaiaa iodacaaI2aaajuaGbaqcLbsacaaIYaGaaiOlaiaaikdacaaIZaGaaG imaiaaiMdaaKqbagaajugibiaaiodacaGGUaGaaGOnaiaaikdacaaI ZaGaaG4naaaaaKqbakaawIcacaGLPaaaaaa@5F80@ .
The computation of Eigen values of the matrix A+BFC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbsaca WFbbGaa83kaiaa=jeacaWFgbGaa83qaaaa@3A2A@  yields the set
eig( A+BFC ) ={ 1.5,1.5151,1.4924±0.0131i } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaajugibiaabwgacaqGPbGaae4zaOWaae WaaKqbagaaieGajugibiaa=feacqGHRaWkcaWFcbGaa8Nraiaa=nea aKqbakaawIcacaGLPaaaieWajugibiaa+bcacaGF9aGcdaGadaqcfa yaaKqzGeGaeyOeI0IaaGymaiaac6cacaaI1aGaaiilaiaaykW7caaM c8UaeyOeI0IaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaIXaGaai ilaiaaykW7caaMc8UaeyOeI0IaaGymaiaac6cacaaI0aGaaGyoaiaa ikdacaaI0aGaeyySaeRaaGimaiaac6cacaaIWaGaaGymaiaaiodaca aIXaGaamyAaaqcfaOaay5Eaiaaw2haaaaa@5E7D@ , Which lies quite close to the specified values (18). Note that to implement the sets (18) using the method5 is not possible.20,21

Conclusion

The problem of a hypothetical single-airscrew helicopter’s longitudinal motion stabilization for lack of information about the vertical speed of its motion has been analytically solved. The solution is based on the method of the output signal control synthesis that provides a specified spectrum of the MIMO-system’s motion, presented 13,16–19.

Acknowledgements

My Institute’s (Bauman Moscow State Technical University) representative is fully aware of this submission.

Conflict of interest

The author declares there is no conflict of interest.

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