Eric S. Hamby†, Yeo-Chow Juan‡, Pierre T. Kabamba†,     †Aerospace Engineering Department, The University of Michigan, Ann Arbor, Michigan 48109-2118; ‡Ford Motor Company, 20000 Rotunda Drive, Dearborn, Michigan 48121

I Introduction

The classical approach to sampled-data control assumes that the continuous-time control input is generated from the discrete-time output of the digital controller using pre specified digital-to-analog conversion devices – typically zero-order or first-order hold [1]. Recently, however, a new approach to sampled-data control has been introduced [2–4]. This method is called “Generalized Sampled-Data Hold Function Control” (GSHF), and its original feature is to consider the hold function as a design parameter. GSHF control has been investigated for finite dimensional, continuous time systems [2, 3, 5–7], finite dimensional, discrete time systems [8, 9], and infinite dimensional, continuous time systems [10–13]. In addition, the advantages and disadvantages of GSHF control, compared to classical sampled-data control, have been documented. Roughly speaking, GSHF control appears capable of handling structured uncertainties [7, 14], but may provide little robustness in the face of unstructured uncertainties [15–17].

This paper presents solutions to two classes of optimal design problems in GSHF control: regulation and tracking for an analog plant in a standard 4-block configuration. The distinguishing feature of these optimization problems is that the performance index explicitly penalizes the intersampling behavior of the closed loop system, instead of penalizing only its discrete-time behavior1. This is accomplished by using penalty indices that are time integrals of a quadratic function of the state vector and control input of the analog plant. The optimization problem then becomes a standard finite-horizon optimal control problem where the “control input” is the hold function. Existence and uniqueness of an optimal hold function are proven under the assumption of a “fixed monodromy,” that is, the discrete-time behavior of the closed-loop system is specified.

The specific problems we solve in regulation (Section II) are the GSHF control versions of the well-known LQ and LQG regulators. These results are a minor generalization of the results in [19] because they are formulated for a plant in a standard 4-block configuration. We also treat the question: “When can we expect the optimal hold function to be close to a zero-order hold?” Specifically, we give a sufficient condition under which the optimal hold function is exactly a zero-order hold, and the answer follows by continuity.

We then consider (Section III) the problem of causing the output of an analog system to asymptotically track a reference signal at the sampling times. This reference signal is itself assumed to be the output of a linear time-invariant system that is observable, but not controllable. After augmenting the dynamics of the plant-controller system with the reference model, we apply the results of Section II. The difficulty is that this augmented system is uncontrollable due to the (possibly unstable) reference model, even though the original plant is controllable. We must therefore make the dynamics of the reference model unobservable through the regulation error. We give necessary and sufficient conditions for this, which are shown to satisfy the Internal Model Principle [20]. We then optimize the hold function with respect to a criterion that reflects the intersample output tracking error and the control energy.

In Section IV, we illustrate several features of optimal GSHF control through examples, by comparison with zero-order hold conversion. We show that optimizing the hold function may yield substantial gain, as measured by the performance index. We identify instances where the optimal hold function can be expected to be close to a zero-order hold. We also show that, sometimes, optimizing the hold function may not yield much improvement, even though the optimal hold function is far from a zero-order hold. Finally, we show that optimal GSHF control can be used to achieve ripple-free deadbeat.

We use the following standard notation: superscript T denotes matrix transpose; E[·] expected value; tr(·) denotes the trace of a matrix; δ(·) denotes the Kronecker symbol in both the discrete-time and continuous-time case; Ip denotes the identity matrix of order p. Let and denote the columns of X as xi, i = 1, …, n, i.e., X = [x1, …, xn]. The vec operator on X is defined as vec(X) = col[x1, x2, …, xn], where vec(X) ∈.

II Optimal Hold Functions for Sampled-Data Regulation