Optimal Regulator for Continuous-Time Descriptor System Using Generalized Riccati Equation
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This paper considers an LQ regulator problem for a continuous-time descriptor system: Ex=Ax+Bu. We first derive a generalized Riccati differential equation (GRDE) for the finite-horizon nonsingular (det E≠0) descriptor regulator problem. Then we show that for the singular case (det E=0), a solution of the GRDE can be used for computing an optimal control and the optimal cost. For the infinite-horizon case, a generalized algebraic Riccati equation (GARS) is derived as a steady-state version of the GRDE. Based on the generalized eigenproblem associated with the Hamiltonian equation, we present a method of computing admissible solutions to the GARE and optimal feedback gains. Numerical examples are included.Keywords:
Algebraic Riccati equation
Hamiltonian (control theory)
The matrix Riccati equation appears in many optimal control and filtering problems. In this paper the Riccati equation is studied from an algebraic point of view, and the results are applied on optimal control of linear time invariant systems with quadratic loss.
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The state dependent Riccati equation was originally developed for the continuous time systems. In the paper the optimality of a discrete time version of the state dependent Riccati equation is considered. The derivation of the optimal control strategy is based on the Hamiltonian optimal solution for the nonlinear optimal control problem. The new form of the discrete state dependent Riccati equation with a correction tensor is derived. The prediction of the future trajectory is used in the derivation.
Algebraic Riccati equation
Hamiltonian (control theory)
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Sufficient conditions are given for solutions of infinite-dimensional algebraic Riccati equations to be continuous in the uniform topology with respect to a parameter. The results are applied to three types of Riccati equations: the linear quadratic control Riccati equation, the positive-real Riccati equation and the bounded-real Riccati equation. For bounded generators we assume only exponential stabilizability and detectability, whereas for unbounded generators we assume an extra assumption on the generator.
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Algebraic equation
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Abstract In this paper we consider the linear-quadratic optimal regulator problem for continuous-time descriptor system by the dynamic programming approach. A generalized Riccati equation is derived by making use of the Hamilton-Jacobi equation for the descriptor system. Sufficient conditions for the existence of a solution of the generalized Riccati differential (algebraic) equation are studied. It is shown that the existence of the solution of the generalized Riccati differential (algebraic) equation depends on the solutions of a standard Riccati algebraic equation and a standard Riccati differential (algebraic) equation. As in the regular state-space system, the solution of the generalized Riccati equation is used in the construction of the optimal feedback gains.
Algebraic Riccati equation
Universal differential equation
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Algebraic differential equation
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Algebraic Riccati equation
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An approach to dealing with the linear-quadratic optimal control problems for the continuous-time descriptor systems is proposed. A Riccati equation with a new form is established through the use of the necessary conditions for the optimal control. Unlike the existing approaches, transformation to the Riccati equation is made directly to find its solution. The sufficient conditions for the existence of the unique solution of the Riccati equation are found. The solution of the Riccati equation is then used in the construction of the optimal control strategies. It is believed that the full-order Riccati equation derived here may have some relation to the dynamic programming theory for the descriptor system.< >
Algebraic Riccati equation
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Algebraic Riccati equation
Matrix difference equation
Matrix (chemical analysis)
Hamilton–Jacobi–Bellman equation
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Universal differential equation
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In this work, we introduce an approach to nonlinear model predictive based on the so-called state dependent Riccati equation, SDRE. In this approach, the model is first cast in a form similar to the linear state space representation. Then the algebraic Riccati equation is constructed based on a similarity with the linear quadratic regulator to obtain stable NMPC. The method requires the solution of the Riccati equation at each sampling period. Simulation results are quite encourageing.
Algebraic Riccati equation
Model Predictive Control
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The optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated. Both the case in which the terminal state is free and that in which the terminal state is constrained to be zero are treated. The integrand of the performance criterion is allowed to be fully quadratic in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem. Frequency-domain and time-domain conditions for the existence of solutions are derived. The algebraic Riccati equation is then examined, and a complete classification of all its solutions is presented. It is finally shown how the optimal control problems introduced in the beginning of the paper may be solved analytically via the algebraic Riccati equation.
Algebraic Riccati equation
Algebraic equation
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