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E-raamat: Lyapunov Matrix Equation in System Stability and Control

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The Lyapunov and Riccati equations are two of the fundamental equations of control and system theory, having special relevance for system identification, optimization, boundary value problems, power systems, signal processing, and communications. This study covers mathematical developments and applications while providing quick and easy references for solutions to engineering and mathematical problems. Examples of real-world systems are given throughout the text in order to demonstrate the effectiveness of the presented methods and algorithms. It should appeal to practicing engineers, theoreticians, applied mathematicians, and graduate students who seek a comprehensive view of the main results of the Lyapunov matrix equation. The text includes: techniques for solving and analysing the algebraic, differential, and difference Lyapunov matrix equations of continuous-time and discrete-time systems; summaries and references at the end of each chapter; examples of the use of the equation to solve real-world problems; and quick and easy references for the solutions to engineering and mathematical problems using the Lyapunov equation.
Part 1 Introduction: stability of linear systems; variance of linear
stochastic systems; quadratic performance measure; book organization. Part 2
Continuous algebraic Lyapunov equation: explicit solutions; solution sounds;
numerical solutions. Part 3 Discrete algebraic Lyapunov equation: explicit
solutions; bounds of solution's attributes; numerical solutions. Part 4
Differential and difference Lyapunov equation: explicit solutions; bounds of
solution's attributes; numerical solutions; singularly perturbed and weakly
coupled systems; coupled differential equations. Part 5 Algebraic Lyapunov
equation with small parameters: singularly perturbed continuous Lyapunov
equation; weakly coupled continuous Lyapunov equation; singularly perturbed
discrete systems; recursive methods for weakly coupled discrete systems. Part
6 Robustness and sensitivity of the Lyapunov equation: stability robustness;
sensitivity of algebraic Lyapunov equation. Part 7 Iterative methods and
parallel algorithms: Smith's algorithm; ADI iterative method; SOR iterative
method; parallel algorithms; parallel algorithms for coupled Lyapunov
equations. Part 8 Lyapunov iterations: Kleinman algorithm for Riccati
equation; Lyapunov iterations for jump linear systems; Lyapunov iterations
for Nash differential games; Lyapunov iterations for output feedback control.
Part 9 Concluding remarks: Sylvester equations; related topics; applications.
Appendix: matrix inequalities.