The matrices defined in 3. We also establish conditions that allow such a decomposition, and formulate the corresponding algorithm. Using the decomposition procedure for the discrete-time algebraic regulator Riccati equation presented in the previous section and the duality property between the optimal linear-quadratic filters and regulators, we will obtain an efficient decomposition scheme such that the slow Copyright 2001 by Marcel Dekker, Inc. The asymptotic expansion of the solution of a discrete periodic singularly perturbed linear-quadratic optimal control problem is constructed as series of non-negative integer powers of a small parameter. The 1986 edition of this book, reprinted here in its original form, provides the theoretical foundation for representative control applications. The completely decoupled filter in the new coordinates, driven by the system measurements and control inputs are -6. Iterative Solution of Nonlinear Equations on Several Variables, Academic Press, New York.
Maruyama -- Existence of periodic solutions for Kaldorian business fluctuations; D. The control robustness and efficiency are shown by computer simulations. The latter means that the problem under consideration is a singular optimal control problem. The solution of the regulator algebraic equation is used to find the optimal feedback gains. We carry out numerical experiments to illustrate the computational savings obtained for the upper and lower bound. Singularly Perturbed and Weakly Coupled Linear Control Systems — A Recursive Approach, Springer-Verlag, New York.
Discrete-Time Linear Optimal Control Systems 3. The following assumption is commonly used for the criterion penalty matrices. Sciffer -- An explicit non-expansive function whose subdifferential is the entire dual ball; G. By using the control optimality conditions, the solution of this problem is reduced to solution of the set of four equations: one Riccati-type matrix ordinary differential equation, two Riccati-type matrix partial differential equations of the first order, and one trivial scalar ordinary differential equation. Problem matrices A,B,Q, and R are given in Section 2. This book, Continuous Time Dynamical Systems: State Estimation and Optimal Control with Orthogonal Functions, considers different classes of systems with quadratic performance criteria.
It should be pointed out that some related problems still remain unsolved, especially corresponding problems in the discrete-time domain, and the optimization problems over a finite horizon. Copyright 2001 by Marcel Dekker, Inc. The slow and fast components affect each other by a regular O perturbation or by weak couplings. In that case we refer to Wilde and Kokotovic, 1973. Ten of his former doctoral students hold faculty positions at respected world universities. The corresponding steady state discrete-time results are either obtained or can be obtained by extending dual continuous-time results, except for the discrete-time high gain and cheap linear control problems and small measurement noise linear filtering problems, where meaningful problem formulations are still missing in the control literature. Even more, that theory has successfully extended the corresponding slow-fast decomposition of some classes of nonlinear singularly perturbed optimal control problems Fridman, 1999, 2000.
These systems, characterized by the presence of slow and fast variables, describe dynamics of many real physical systems such as aircraft, power systems, nuclear reactors, chemical reactors, electrical circuits, dc and induction motors, robots, large space flexible structures, synchronous machines, cars, and so on. The well-defined recursive numerical technique for the solution of nonlinear algebraic matrix equations, associated with the output feedback control problem of singularly perturbed systems has been developed. Thus, the new problem is a regular optimal control problem. The book studies a special class of linear control systems known as singularly perturbed systems. The extension to the multimodeling team problems Saksena and Basar, Copyright 2001 by Marcel Dekker, Inc.
Having obtained the solutions of 6. Asymptotics of integral manifold and decomposition of singularly perturbed systems of neutral type, Differential Equations, 26: 457-467. Conditions for complete separation of slow and fast regulator designs are formulated. Organized into independent chapters for easy access to the material, this text also contains several case studies, examples, exercises, computer assignments, and formulations of research problems to help instructors and students. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. In the era of an increased application of modern control theory results in real physical systems, this is a serious problem. Copyright 2001 by Marcel Dekker, Inc.
We first propose a combined control of both dynamics based on a stability analysis. Professor Gajic has delivered four plenary lectures at international conferences and presented close to one hundred fifty conference papers. Thus, the matrix A4 — K2C2 is stable for sufficiently small values of the small singular perturbation parameter e. Linear Optimal Control Systems, Wiley Interscience, New York. The concept of multimodeling was introduced to the control audience in Khalil and Kokotovic, 1978. These changes are often hard to predict and it is, therefore, fundamentally impossible to design for the worst case scenarios; such scenarios are not known due to the very combinatorial nature of the problem. In that direction, an extension of the results obtained is needed to cover various types of optimal control and filtering problems of linear singularly perturbed systems.
Integral Manifolds in Nonlinear Mechanics, Nauka, Moskow. In this chapter, we present a method for the exact decomposition of the optimal control associated with 4. The proposed method is successfully applied to the reducedorder design of optimal controllers for a hydro power plant Skataric and Gajic, 1992. Singular control has been extensively studied for the linear-quadratic regulator problem for achieving asymptotic stability of controlled linear systems with cheap control. It should be pointed that the recursive approach to slow-fast decomposition of the differential Riccati equation obtained in Grodt and Gajic, 1988 is very efficient for achieving a very high accuracy.
An algorithm for solving a general nonsymmetric algebraic Riccati equation was derived in Avramovic, 1979, see also Avramovic et al. This matrix has to be nonsingular in order to preserve the slowfast decomposition of the corresponding state-costate variables, that is, to keep the slow variables slow and the fast variable fast. The corresponding discretetime open-loop optimal control problem is considered in Section 3. Linear Optimal Control of Bilinear Systems: with Applications to Singular Perturbations and Weak Coupling, Springer Verlag, New York. Essential of Robust Control, Prentice Hall, Upper Saddle River, New Jersey.