Optimal control of systems with delayed observation sharing patterns via input–output methods

Abstract

In this paper we present an input–output point of view of certain optimal control problems with constraints on the processing of the measurement data. In particular, we consider norm minimization optimal control problems under the so-called one-step delay observation sharing pattern. We present a Youla parametrization approach that leads to their solution by converting them to nonstandard, yet convex, model matching problems. This conversion is always possible whenever the part of the plant that relates controls to measurements possesses the same structure in its feedthrough term with the one imposed by the observation pattern on the feedthrough term of the controller, i.e., (block-)diagonal. When that is not the case, it amounts to the so-called non-classical information pattern problems. For the $\text{H}{}^{\mathrm{\infty }}$ case, using loop-shifting ideas, a simple sufficient condition is given under which the problem can be still converted to a convex, model matching problem. We also demonstrate that there are several nontrivial classes of problems satisfying this condition. Finally, we extend these ideas to the case of a N-step delay observation sharing pattern.

Introduction

Optimal control under decentralized information structures is a topic that, although it has been studied extensively over the last forty years or so, still remains a challenge to the control community. The early encounters with the problem date back in the fifties and early sixties under the framework of team theory (e.g., [14], [16].) Soon it was realized that, in general, optimal decision making is very difficult to obtain when decision makers have access to private information, but do not exchange their information [25]. Nonetheless, under particular decentralized information schemes such as the partially nested information structures [10] certain optimal control problems admit trackable solutions. Several results exist by now when exchange of information is allowed with a one-step time delay (which is a special case of the partially nested information structure.) To mention only a few we refer to [1], [2], [18] where LQG criteria are of interest, [7], [11], [12], [13], [21], [22] where linear exponential-quadratic Gaussian (LEQG) problems are considered and certain connections to minimax quadratic problems are furnished. The interested reader may further refer to [3] for further bibliographical information.

In this paper, in contrast to the state-space view-point of the works previously cited, we undertake an input–output approach to optimal control under the information scheme known as the one-step delay observation sharing pattern (e.g., [2]). Under this pattern, measurement information can be exchanged between the decision makers with a delay of one time step. In the paper we provide an approach for solving the ℓ¹, $\text{H}{}^{\mathrm{\infty }}$ and $\text{H}{}^2$ (or LQG) optimal disturbance rejection in the case of quasi-classical information exchange [4]. This is the case whenever the part of the plant that relates controls to measurements possesses the same structure in its feedthrough term with the one imposed by the observation pattern on the feedthrough term of the controller, i.e., (block-)diagonal. The key ingredient in this approach is the transformation of the decentralization constraints on the controller to linear constraints on the Youla parameter used to characterize all controllers. Hence, the resulting problems in the input–output setting are, although nonstandard, convex. These problems are of the same form as the ones appearing in optimal control of periodic systems when lifting techniques are employed [6], [24], and can be solved analogously by employing Duality, Nehari and Projection theorems respectively.

When the part of the plant that relates controls to measurements does not possess the same structure in its feedthrough term with the one imposed by the observation pattern on the feedthrough term of the controller, the information exchange is non-classical (e.g., [2]). The previous approach leads in general to nonconvex problems since the constraints on the Youla parameter are not linear any more. For the $\text{H}{}^{\mathrm{\infty }}$ case however, using loop-shifting ideas, a simple sufficient condition is given under which the problem can be still converted to a convex, model matching problem. We also demonstrate that there are nontrivial classes of problems satisfying this condition. Furthermore, we extend these ideas to the case of a N-step delay observation sharing pattern using lifting techniques.