On output feedback control of singularly perturbed systems

We treat the problem of robustness of output feedback controllers with respect to singular perturbations. Given a singularly perturbed control system whose boundary layer system is exponentially stable and whose reduced order system is exponentially stabilizable via a (possibly dynamical) output feedback controller, we present a sufficient condition which ensures that the system obtained by applying the same controller to the original full order singularly perturbed control system is exponentially stable for sufficiently small values of the perturbation parameter. This condition, which is less restrictive than those previously given in the literature, is shown to be always satisfied when the singular perturbation is due to the presence of fast actuators and/or sensors. Furthermore, we show explicitly that, in the linear time-invariant case, if this condition is not satisfied then there exists an output feedback controller which stabilizes the reduced order system but destabilizes the full order system.     

Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem

A system of equations that arises in a singularly perturbed optimal control problem is studied. We give conditions under which a formal asymptotic solution exists. This formal asymptotic solution consists of an outer expansion and left and right boundary-layer expansions. A feature of our procedure is that we do nota priori eliminate the control function from the problem. In particular, we construct a formal asymptotic expansion for the control directly. We apply our procedure to a Mayer-type problem. The paper concludes with a worked example.     

Approximate solution of singularly perturbed nonlinear pursuit-evasion games

A methodology to obtain an approximate solution of a singularly perturbed nonlinear differential game is presented. The outcome of the game with approximate strategies, defined as extended value, is related to the saddle-point value of the game. In an example of a simple pursuit-evasion game, it is shown that the proposed methodology leads to an easily implementable feedback form solution with fairly accurate results. This approach seems to be attractive for analyzing realistic air-combat models without solving a two-point boundary-value problem.This research was partially supported by AFSC Contract No. F-49620-79-6-0135. The authors are grateful to Prof. J. V. Breakwell for encouraging the approach taken in this research. Thanks are also due to Dr. S. Gutman and Dr. J. Lewin for their useful comments.     

Approximate solution of singularly perturbed nonlinear pursuit-evasion games

Several errors in Ref. 1 are corrected.     

Asymptotic method for solving the time-optimal control problem for a nonlinear singularly perturbed system

The time-optimal control problem for a nonlinear singularly perturbed system with multidimensional controls bounded in the Euclidean norm is considered. An algorithm for constructing asymptotic approximations to its solution is proposed. The main advantage of the algorithm is that the original optimal control problem decomposes into two unperturbed problems of lower dimensions.     

Parallel initial-value algorithms for singularly perturbed boundary-value problems

Initial-value methods for linear and semilinear singularly perturbed boundary-value problems are examined with a view to designing and implementing algorithms on parallel architectures. Practical experiments on a CRAY Y-MP 8/432 multiprocessor have been performed, showing the reliability and performance of several proposed parallel schemes.This work was supported by CNR, Rome, Italy (Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo, Sottoprogetto 1).The authors wish to thank Dr. A. Papini, who carried out most of the computations reported in this work.     

Feedback stabilization for perturbed linear systems *

In this paper, we consider the questions of stabilization of perturbed (or uncertain) linear systems in Hilbert space. Perturbations of the system operator represent the uncertainty in the modelling process and could be bounded or even unbounded. Sufficient conditions are presented that guarantee stabilizability of the perturbed system given that the nominal (unperturbed) system is stabilizable. In particular it is shown that for certain class of perturbations weak and strong stabilizability properties are preserved for the same state feedback control. Two numerical examples are given to illustrate our theory.     

Stochastic optimal linear feedback control systems using available measurements

The stochastic optimal control of linear systems with time-varying and partially observable parameters is synthesized under noisy measurements and a quadratic performance criterion. The structure of the regulator is given, and the optimal solution is reduced to a two-point boundary-value problem. Comments on the numerical solution by appropriate integration schemes is included.     

Optimal control of continuous-time deterministic systems with incomplete discrete feedback

Sufficient conditions for optimality are obtained for controls that depend on the current time and values of known functions of the state vector at finite points of the time interval. The equations for finding the required control laws are derived. An example is given for which an exact solution of the problem can be obtained.The author is grateful to Professor R. W. Rishel for his support during the preparation of the paper.     

Uniformly convergent scheme for two‐parameter singularly perturbed problems with non‐smooth data

A numerical scheme is constructed for the problems in which the diffusion and convection parameters (?₁ and ?₂ , respectively) both are small, and the convection and source terms have a jump discontinuity in the domain of consideration. Depending on the magnitude of the ratios , and two different cases have been considered separately. Through rigorous analysis, the theoretical error bounds on the singular and regular components of the solution are obtained separately, which shows that in both cases the method is convergent uniformly irrespective of the size of the parameters ?₁, ?₂ . Two test problems are included to validate the theoretical results.     

Power-series approximations for the optimal feedback control of noisy nonlinear systems

Power-series methods are developed for designing approximately optimal state regulators for a nonlinear system subject to white Gaussian random disturbances. The performance index of the control is an ensemble average of a quadratic form. A perfect observation of the system state is assumed. When the system nonlinearity is small and it is characterized by a polynomial function of the state, a definite method is presented to construct a suboptimal feedback control of a power-series form in a small nonlinearity parameter. If the variance of noise is small, an alternative method is also applicable which yields a suboptimal control in a power series with respect to a variance parameter. A simple one-dimensional problem is examined to make comparison between controls of the two different forms.     

Bounds of robustness for singularly perturbed feedback systems

The stability-bound problem of linear time-invariant singularlyperturbed feedback systems is studied in this paper. The systemand the input matrices are subject to structured parameter perturbations.An explicit expression of the bound on the singular perturbationparameter is derived such that a composite state-feedback controlrenders the closed-loop singularly perturbed system asymptoticallystable.     

A class of singularly perturbed,nonlinear, fixed-endpoint control problems

Singular perturbation techniques are applied to a class of nonlinear, fixed-endpoint control problems to decompose the full-order problem into three lower-order problems, namely, the reduced problem and the left and right boundary-layer problems. The boundary-layer problems are linear-quadratic and, contrary to previous singular perturbation works, the reduced problem has a simple formulation. The solutions of these lower-order problems are combined to yield an approximate solution to the full nonlinear problem. Based on the properties of the lower-order problems, the full problem is shown to possess an asymptotic series solution.This work was supported in part by the National Science Foundation under Grant No. ENG-47-20091 and in part by the US Air Force under Grant No. AFOSR-73-2570.The author acknowledges the helpful suggestions of Professor P. V. Kokotovic, University of Illinois, Urbana, Illinois.     

Optimal feedback control of bilinear systems

Feedback synthesis of optimal constrained controls for single-input bilinear systems is considered. Quadratic cost functionals (with and without quadratic control penalization) are modified by the inclusion of additional nonnegative state penalizing functions in the respective cost integrands. The latter functions are chosen so as to regularize the problems, in the sense that feedback solutions of particularly simple form are obtained. Finite and infinite time horizon problem formulations are treated, and associated aspects of feedback stabilization of bilinear systems are discussed.     

Invariance of asymptotic stability of perturbed linear systems on Hilbert space

The questions of stabilizability of structurally perturbed or uncertain linear systems in Hilbert space of the form are considered. The operatorA is assumed to be the infinitesimal generator of aC ₀-semigroup of contractionsT(t),t0, in a Hilbert spaceX;B is a bounded linear operator from another Hilbert spaceU toX; and {P(r),r } is a family of bounded or unbounded perturbations ofA inX, where is an arbitrary set, not necessarily carrying any topology. Sufficient conditions are presented that guarantee controllability and stabilizability of the perturbed system, given that the unperturbed system has similar properties. In particular, it is shown that, for certain classes of perturbations, weak and strong stabilizability properties are preserved for the same state feedback operator.This work was supported in part by the Natural Science and Engineering Research Council of Canada under Grant No. A7109.