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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/22/3604 |
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| Summary: | 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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| ISSN: | 2227-7390 |