Topic

# Riccati equation

About: Riccati equation is a(n) research topic. Over the lifetime, 10428 publication(s) have been published within this topic receiving 210015 citation(s). The topic is also known as: Riccati's differential equation.

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TL;DR: In this article, simple state-space formulas are derived for all controllers solving the following standard H/sub infinity / problem: for a given number gamma > 0, find all controllers such that the H/ sub infinity / norm of the closed-loop transfer function is (strictly) less than gamma.

Abstract: Simple state-space formulas are derived for all controllers solving the following standard H/sub infinity / problem: For a given number gamma >0, find all controllers such that the H/sub infinity / norm of the closed-loop transfer function is (strictly) less than gamma . It is known that a controller exists if and only if the unique stabilizing solutions to two algebraic Riccati equations are positive definite and the spectral radius of their product is less than gamma /sup 2/. Under these conditions, a parameterization of all controllers solving the problem is given as a linear fractional transformation (LFT) on a contractive, stable, free parameter. The state dimension of the coefficient matrix for the LFT, constructed using the two Riccati solutions, equals that of the plant and has a separation structure reminiscent of classical LQG (i.e. H/sub 2/) theory. This paper is intended to be of tutorial value, so a standard H/sub 2/ solution is developed in parallel. >

5,130 citations

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01 Jun 1979TL;DR: In this article, an augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems, with step-by-step explanations that show clearly how to make practical use of the material.

Abstract: This augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems. It explores linear optimal control theory from an engineering viewpoint, with step-by-step explanations that show clearly how to make practical use of the material. The three-part treatment begins with the basic theory of the linear regulator/tracker for time-invariant and time-varying systems. The Hamilton-Jacobi equation is introduced using the Principle of Optimality, and the infinite-time problem is considered. The second part outlines the engineering properties of the regulator. Topics include degree of stability, phase and gain margin, tolerance of time delay, effect of nonlinearities, asymptotic properties, and various sensitivity problems. The third section explores state estimation and robust controller design using state-estimate feedback. Numerous examples emphasize the issues related to consistent and accurate system design. Key topics include loop-recovery techniques, frequency shaping, and controller reduction, for both scalar and multivariable systems. Self-contained appendixes cover matrix theory, linear systems, the Pontryagin minimum principle, Lyapunov stability, and the Riccati equation. Newly added to this Dover edition is a complete solutions manual for the problems appearing at the conclusion of each section.

3,137 citations

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Fudan University

^{1}TL;DR: In this article, an extended tanh-function method is proposed for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way, and the key idea of this method is to take full advantage of a Riccati equation involving a parameter and use its solutions to replace the tanh function.

Abstract: An extended tanh-function method is proposed for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The key idea of this method is to take full advantages of a Riccati equation involving a parameter and use its solutions to replace the tanh function in the tanh-function method. It is quite interesting that the sign of the parameter can be used to exactly judge the numbers and types of these travelling wave solutions. In addition, by introducing appropriate transformations, it is shown that the extended tanh-function method still is applicable to nonlinear PDEs whose balancing numbers may be any nonzero real numbers. Some illustrative equations are investigated by this means and new travelling wave solutions are found.

1,687 citations

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17 Aug 2010TL;DR: The Fokker-Planck equation as mentioned in this paper describes the evolution of conditional probability density for given initial states for a Markov process, which satisfies the Ito stochastic differential equation.

Abstract: In 1984, H. Risken authored a book (H. Risken, The Fokker-Planck Equation: Methods of Solution, Applications, Springer-Verlag, Berlin, New York) discussing the Fokker-Planck equation for one variable, several variables, methods of solution and its applications, especially dealing with laser statistics. There has been a considerable progress on the topic as well as the topic has received greater clarity. For these reasons, it seems worthwhile again to summarize previous as well as recent developments, spread in literature, on the topic. The Fokker-Planck equation describes the evolution of conditional probability density for given initial states for a Markov process, which satisfies the Ito stochastic differential equation. The structure of the Fokker-Planck equation for the vector case is

1,472 citations

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01 Jan 1995

TL;DR: Geometric theory: the complex case 8.

Abstract: 1. Preliminaries from the theory of matrices 2. Indefinite scalar products 3. Skew-symmetric scalar products 4. Matrix theory and control 5. Linear matrix equations 6. Rational matrix functions 7. Geometric theory: the complex case 8. Geometric theory: the real case 9. Constructive existence and comparison theorems 10. Hermitian solutions and factorizations of rational matrix functions 11. Perturbation theory 12. Geometric theory for the discrete algebraic Riccati equation 13. Constructive existence and comparison theorems 14. Perturbation theory for discrete algebraic Riccati equations 15. Discrete algebraic Riccati equations and matrix pencils 16. Linear-quadratic regulator problems 17. The discrete Kalman filter 18. The total least squares technique 19. Canonical factorization 20. Hoo control problems 21. Contractive rational matrix functions 22. The matrix sign function 23. Structured stability radius Bibliography List of notations Index

1,417 citations