State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear Systems
In this paper, a problem of random disturbance attenuation capabilities for linear time-invariant continuous systems, affected by random disturbances with bounded <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi...
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| author | Victor A. Boichenko Alexey A. Belov Olga G. Andrianova |
| author_facet | Victor A. Boichenko Alexey A. Belov Olga G. Andrianova |
| author_sort | Victor A. Boichenko |
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| description | In this paper, a problem of random disturbance attenuation capabilities for linear time-invariant continuous systems, affected by random disturbances with bounded <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy, is studied. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm defines a performance index of the system on the set of the aforementioned input signals. Two problems are considered. The first is a state-space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy analysis of linear systems, and the second is an optimal control design using the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm as an optimization objective. The state-space solution to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy analysis problem is derived from the representation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm in the frequency domain. The formulae of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm computation in the state space are presented in the form of coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations. Optimal control law is obtained using game theory and a saddle-point condition of optimality. The optimal state-feedback control, which minimizes the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm of the closed-loop system, is found from the solution of two algebraic Riccati equations, one Lyapunov equation, and the log determinant equation. |
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| spelling | doaj-art-b2b33da6744242c9a34bb5d17e9593e32025-08-20T02:04:54ZengMDPI AGMathematics2227-73902024-11-011222360410.3390/math12223604State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear SystemsVictor A. Boichenko0Alexey A. Belov1Olga G. Andrianova2V.A. Trapeznikov Institute of Control Sciences of RAS, Moscow 117997, RussiaV.A. Trapeznikov Institute of Control Sciences of RAS, Moscow 117997, RussiaV.A. Trapeznikov Institute of Control Sciences of RAS, Moscow 117997, RussiaIn this paper, a problem of random disturbance attenuation capabilities for linear time-invariant continuous systems, affected by random disturbances with bounded <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy, is studied. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm defines a performance index of the system on the set of the aforementioned input signals. Two problems are considered. The first is a state-space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy analysis of linear systems, and the second is an optimal control design using the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm as an optimization objective. The state-space solution to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy analysis problem is derived from the representation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm in the frequency domain. The formulae of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm computation in the state space are presented in the form of coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations. Optimal control law is obtained using game theory and a saddle-point condition of optimality. The optimal state-feedback control, which minimizes the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-entropy norm of the closed-loop system, is found from the solution of two algebraic Riccati equations, one Lyapunov equation, and the log determinant equation.https://www.mdpi.com/2227-7390/12/22/3604linear systemsspectral entropyoptimal controlrobust controlalgebraic Riccati equation |
| spellingShingle | Victor A. Boichenko Alexey A. Belov Olga G. Andrianova State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear Systems Mathematics linear systems spectral entropy optimal control robust control algebraic Riccati equation |
| title | State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear Systems |
| title_full | State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear Systems |
| title_fullStr | State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear Systems |
| title_full_unstemmed | State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear Systems |
| title_short | State-Space Solution to Spectral Entropy Analysis and Optimal State-Feedback Control for Continuous-Time Linear Systems |
| title_sort | state space solution to spectral entropy analysis and optimal state feedback control for continuous time linear systems |
| topic | linear systems spectral entropy optimal control robust control algebraic Riccati equation |
| url | https://www.mdpi.com/2227-7390/12/22/3604 |
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