Discretizing continuous-time controllers for infinite-dimensional linear systems∗

We investigate a fundamental question about the discretization of continuous-time controllers. We consider a linear feedback system that works in continuous time, and has satisfactory performance. We want to replace the controller with a combination of a discrete-time system (a digital processor), an analog-to-digital converter (a filter with a sampler) and a digital-to-analog converter (a hold device). The problem is to determine whether it is possible to achieve closed-loop performance arbitrarily close to the performance of the original feedback system. The performance is represented by the input-output maps of the closed-loop system, and the distance is measured in the operator norm. We show that arbitrarily close performance can be achieved, by choosing a sufficiently small sampling period and an appropriate controller structure, if the original controller is strictly proper. We begin with a description of the problem treated in the paper and its context. Our terminology and notation are mostly adopted from the book of Chen and Francis [3] and the papers of Kannai and Weiss [7] and Weiss and Curtain [12]. ∗This research was initiated during a visit of the 3rd author to the University of Exeter, supported in part by the University of Exeter Research Fund. We first introduce our terminology about transfer functions, which is fairly standard. A Cp×m-valued analytic function G is said to be a well-posed transfer function if it is bounded and analytic on some right half-plane in C. Such a function defines an input-output operator from Lloc([0,∞),C ) to Lloc([0,∞),C ) via the formula ŷ(s) = G(s)û(s), where û and ŷ are the Laplace transforms of the input u and the output y. This operator is time-invariant (i.e., right shift-invariant) and hence causal. In this paper we represent linear and timeinvariant (LTI) systems by their transfer functions, i.e., we ignore their state space. By some abuse of notation, we use the same symbol to denote both the transfer function and the corresponding input-output operator. Finitedimensional systems have rational transfer functions, but in this paper we shall never assume that the transfer functions we deal with are rational, because it would not simplify our arguments. A transfer function G is called strictly proper if lim|s|→∞ ‖G(s)‖ = 0. The space H∞(C+,C) consists of bounded analytic Cp×m-valued functions defined on the right halfplane C+. This is a Banach space with the norm ‖G‖ = sups∈C+ ‖G(s)‖, where ‖G(s)‖ denotes the greatest singular value of G(s). For transfer functions G ∈ H∞(C+,C), the corresponding input-output operator is in L(L2([0,∞),C), L2([0,∞),C)). In this case the norm of G as an input-output operator is equal to the H∞-norm of G. Such transfer functions are called stable. The transfer function G is called exponentially stable if it is bounded and analytic on a right half-plane which strictly includes C+. In other words, there exists a δ > 0 such that the function Gδ defined by Gδ(s) = G(s − δ) is in H∞(C+,C). Our terminology and conventions for discrete-time transfer functions are very similar. A Cp×m-valued discrete-time well-posed transfer function is bounded and analytic on the complement of some disk centered at 0. Such a transfer function defines a time-invariant operator from C-valued sequences to C-valued sequences, via the z-transform. Again, we use the same symbol to denote the transfer function and the corresponding operator. The stable transfer functions are those in H∞(Dc,Cp×m), where D denotes the complement of the unit disk.