Reduction-Based Robustness Analysis of Linear Predictor Feedback for Distributed Input Delays

Abstract

Abstract-Lyapunov-Krasovskii approach is applied to parameter-and delay-robustness analysis of the feedback suggested by Manitius and Olbrot for a linear time-invariant system with distributed input delay. A functional is designed based on Artstein's system reduction technique. It depends on the norms of the reduction-transformed plant state and original actuator state. The functional is used to prove that the feedback is stabilizing when there is a slight mismatch in the system matrices and delay values between the plant and controller.Index Terms-Delay systems, predictive control for linear systems, robust control, Lyapunov methods. I. NOTATIONWe write M > 0 or M ≥ 0 to state that a symmetric real matrix M is positive definite or positive semidefinite, respectively. Also in this case λmin(M ) and λmax(M ) represent the minimal and maximal eigenvalues of M . Vector norms being used are x = √ x T x andwhere M > 0. Euclidean matrix norm is M . The symbol P C(T, X) stands for the space of piecewise continuous functions mappingGiven u ∈ P C(R, R r ), let ut be a function defined as ut(θ) = u(t + θ) for all θ ∈ [−h, 0). The constant h is specified below. II. INTRODUCTION A. The problemConsider the time-invariant systeṁn×r . For brevity, we will use Stieltjes integral notation and write the system under consideration aṡwhere h ≥ max{h1, h2, . . . , hN , hint},and χ is the Heaviside step function. The following control law was proposed for (3) in [1]:where F is a constant matrix. The feedback (5) is called a predictor feedback because it employs the plant's model (i.e., the matrices A and β) to, in a sense, predict the future state of the plant. Our goal is to investigate robustness of the feedback (5). In terms of (2), we are interested in:• parametric robustness (small uncertainty in A, Bi, and Bint);• delay-robustness (small uncertainty in hi and hint). B. Previous results overviewA range of methods is known to be suitable for analysis of linear systems of the form (3), (5). Let us separate them into those using Lyapunov-Krasovskii functional analysis and those doing otherwise.Most of the progress with non-Lyapunov techniques has been achieved in the area of systems with one discrete delay, e.g., delayrobustness of a predictive controller [2], [3], robustness with respect to a finite-sum implementation [4], [5], robustness of an adaptive controller in presence of a disturbance [6], and delay-robustness of a linear time-varying predictor feedback [7]. Furthermore, it has been shown in [8] that robustness with respect to a finite-sum implementation may be ensured by including a low-pass element in the control loop.Lyapunov-Krasovskii analysis of systems with one discrete delay was shown to succeed in proving delay-robustness of the predictor feedback [9] and robustness with respect to uncertain parameters [10]. Adaptive controllers were designed in [11], [12]. Recently, a predictor feedback for retarded [13] and neutral [14] systems with state delays and an input delay was proposed, the closed loop's exponential sta...

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