Transfer Function Approach to the Problems of Discrete-time Systems : H_∞-optimal Linear Control and Filtering
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The optimal linear regulator with state-dependent sampling is formulated and the existence of an optimal solution is proved. A search algorithm is developed which uses both the necessary conditions obtained from nonlinear programming theory and a sequential unconstrained minimization technique to compute the optimal control. An application is presented.
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The paper deals with a new approach for synthesis of time-optimal control for a class of linear systems. It is based on the decomposition of the time-optimal control problem into a class of decreasing order problems, and the properties and relations between problems within this class. First, the problems' state-space properties are analyzed, and then the optimal control is obtained by using a multi-step procedure avoiding the switching hyper surface description. The emphasis in this paper is on the optimal control synthesis stage of the approach proposed. A property of the considered class of problems is studied which enables development of a fast algorithm for synthesis of time-optimal control without using the switching hyper surface.
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The paper establishes a new procedure to obtain the solution for the discrete optimal tracking problem based on dynamic programming. The optimal control refers to a quadratic criterion with finite final time, regarding a perturbed discrete invariant linear system. The proposed algorithm can be easier implemented by comparison with other procedures.
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Several approximate dynamic programming (ADP) algorithms have been developed and demonstrated for the model-free control of continuous and discrete dynamical systems. However, their applicability to hybrid systems that involve both discrete and continuous state and control variables has yet to be demonstrated in the literature. This paper presents an ADP approach for hybrid systems (hybrid-ADP) that obtains the optimal control law and discrete action sequence via online learning. New recursive relationships for hybrid-ADP are presented for switched hybrid systems that are possibly nonlinear. In order to demonstrate the ability of the proposed ADP algorithm to converge to the optimal solution, the approach is demonstrated on a switched, linear hybrid system with a quadratic cost function, for which there exists an analytical solution. The results show that the ADP algorithm is capable of converging to the optimal switched control law, by minimizing the cost-to-go online, based on an observable state vector.
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The correspondence derives an algorithm for designing suboptimal dynamic controllers for linear discrete systems.
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An algorithmic framework is presented for the derivation of the explicit optimal control policy for continuous-time linear dynamic systems that involve constraints on the process inputs and outputs. The control actions are usually computed by regularly solving an on-line optimisation problem in the discrete-time space based on a set of measurements that specify the current process state. A way to derive the explicit optimal control law, thereby, eliminating the need for rigorous on-line computations has already been reported in the literature, but it is limited to discrete-time linear dynamic systems. The currently presented approach derives the optimal state-feedback control law off-line for a continuous-time dynamic plant representation. The control law is proved to be nonlinear piecewise differentiable with respect to the system state and does not require the repetitive solution of on-line optimisation problems. Hence, the on-line implementation is reduced to a sequence of function evaluations. The key advantages of the proposed algorithm are demonstrated via two illustrative examples.
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Abstract By the introduction of the shift transformation matrix, direct product matrix and summation matrix of the discrete Walsh series, the analysis of time-varying digital control systems is facilitated and the approximate solution of time-invariant digital optimal control problems is achieved of this study. The design algorithms of digital optimal control are based on the discrete variational principle combined with the idea of penalty functions to obtain the conveniently computational formulations for evaluating the optimal control and trajectory. Three examples are illustrated by using the discrete Walsh approach.
Matrix (chemical analysis)
Digital Control
Transformation matrix
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This paper considers the problem of designing optimal instantaneous output—feedback controllers for a class of discrete-time linear time-invariant systems with inaccessible state. The performance index is taken to be the expectation, with respect to the initial state, of the standard quadratic one for the discrete-time regulator. Necessary conditions for optimality are derived and existence of an optimal solution is proven. Implementation of the optimal controller requires the solution of two non-linear matrix algebraic equations, for which a computational algorithm is presented. The design is illustrated by a simple numerical example.
LTI system theory
Matrix (chemical analysis)
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