Topic
Riccati equation
About: Riccati equation is a research topic. Over the lifetime, 10428 publications have been published within this topic receiving 210015 citations. The topic is also known as: Riccati's differential equation.
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TL;DR: This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using Automatic Differentiation (AD), and shows that they can be seen as discretized methods for the and adjoint differential equation of the underlying control problem.
Abstract: This paper considers the numerical solution of optimal control problems based on ODEs. We assume that an explicit Runge-Kutta method is applied to integrate the state equation in the context of a recursive discretization approach. To compute the gradient of the cost function, one may employ Automatic Differentiation (AD). This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using AD. We show that they can be seen as discretization methods for the sensitivity and adjoint differential equation of the underlying control problem. Furthermore, we prove that the convergence rate of the scheme automatically derived for the sensitivity equation coincides with the convergence rate of the integration scheme for the state equation. Under mild additional assumptions on the coefficients of the integration scheme for the state equation, we show a similar result for the scheme automatically derived for the adjoint equation. Numerical results illustrate the presented theoretical results.
74 citations
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TL;DR: Numerical simulation with the nonlinear model of a spacecraft rendezvous system shows the effectiveness of the proposed gain scheduled approaches to the stabilization of linear systems with actuator saturation.
Abstract: This brief is concerned with gain scheduled approaches to the stabilization of linear systems with actuator saturation. For linear systems that are polynomially unstable, using the parametric Lyapunov equation-based and Riccati equation-based design, we propose gain scheduling approaches to increase the design parameter online so as to increase the convergence rates of the closed-loop systems. To apply the proposed gain scheduling approaches, only a scalar differential equation whose right-hand side is a function of the state vector is required to be integrated online. The closed-loop system is proven to be exponentially stable provided some parameters in the scheduling law are properly chosen. The established gain scheduling approaches are also extended to exponentially unstable linear systems with actuator saturation. As applications of the proposed dynamic gain scheduling approaches, the controller design of spacecraft rendezvous systems is revisited. Numerical simulation with the nonlinear model of a spacecraft rendezvous system shows the effectiveness of the proposed gain scheduling approaches.
74 citations
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TL;DR: In this paper, the authors considered a linear quadratic differential game in which the weighting on the minimizing control is allowed to approach zero and showed that if a certain minimum phase condition is satisfied then the value of the game will approach zero.
74 citations
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TL;DR: In this article, a group-theoretical approach to the Riccati equation has been developed, which is shown to be very useful for a better understanding of the properties of the equation.
Abstract: In this paper we develop some group-theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation, and we discuss some of its integrability conditions from a group-theoretical perspective. The nonlinear superposition principle also arises in a simple way.
74 citations
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TL;DR: In this paper, it was shown that the stabilizing solution P * to the algebraic Riccati equation in (A, B, C ) depends analytically on ( A, B, C ).
Abstract: The central result of this correspondence is Lemma 1.1, which states that the stabilizing solution P * to the algebraic Riccati equation in ( A, B, C ) depends analytically on ( A, B, C ). In the remainder of the correspondence, various control- and system-theoretic ramifications of the analyticity lemma are considered. An important consequence of Lemma 1.1 is that any smooth-state feedback control law for a finite-dimensional plant may be implemented using dynamic input-output compensators whose transfer functions depend smoothly on the plant transfer function.
74 citations