Optimal control of systems with delayed observation sharing patterns via input–output methods☆
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, and (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 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.
Section snippets
Problem definition
The standard block diagram for the disturbance rejection problem is depicted in Fig. 1. In this figure, P denotes some fixed linear time invariant (LTI) causal plant, C denotes the compensator, and the signals , and u are defined as follows: w, exogenous disturbance; z, signals to be regulated; y, measured plant output; and u, control inputs to the plant. In what follows we will assume that both P and C are LTI systems; we comment on this restriction on C later. Furthermore, we assume that
Problem solution
We consider separately two cases. In the first case it is assumed that the feedthrough term D22 is block diagonal which corresponds to a quasi-classical information pattern. This case may naturally arise in large scale systems where u1 and y1 are generated and measured, respectively in a station that is remotely located from the station where u2 and y2 are generated and measured, respectively.
The second case is when D22 is not block diagonal which corresponds to a non-classical information
Optimal control with N-step delay observation sharing patterns
In the wake of the approach in the previous section, we consider herein the control with a N-step delay observation pattern. In particular, let the instant k be expressed as k=mN+i where m, i are integers with 0⩽i<N. The information pattern considered here is that u1(k) is constrained to depend on the data {YmN−1,y1(mN),y1(mN+1),…,y1(k)} and similarly u2(k) is constrained to depend on the data {YmN−1,y2(mN),y2(mN+1),…,y2(k)} where, as in Section 2, YmN−1={y1(0),y2(0),…,y1(mN−1),y2(mN−1)}.
Conclusions
We presented an input–output point of view for norm minimization optimal control problems under the one-step delay observation sharing pattern. In the case where 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 problem is convex and a procedure that leads to its solution was given. It was also documented in this
References (26)
- et al.
Team decision for linear continuous-time systems
IEEE Trans. Automat. Control
(1980) Two-criteria LQG decision problems with one-step delay observation sharing pattern
Inform. Control
(1978)- T. Başar, R. Bansal, The theory of teams: a selective annotated bibliography, in: Differential Games and Applications,...
- et al.
Concepts and methods in multiperson coordination and control, in: Optimization and Control of Dynamic Operational Research Models
(1982) - et al.
Rejection of persistent bounded disturbancesnonlinear controllers
Syst. Control Lett.
(1992) - et al.
Optimal and robust controllers for periodic and multirate systems
IEEE Trans. Automat. Control
(1992) - et al.
Centralized and decentralized solutions to the linear exponential-Gaussian problem
IEEE Trans. Automat. Control
(1994) A Course in H∞ Control Theory
(1987)- et al.
A constructive algorithm for sensitivity optimization of periodic systems
SIAM J. Control Optim.
(1987) - et al.
Team decision theory and information structures in optimal control problems—parts I and II
IEEE Trans. Automat. Control
(1972)
The dynamic linear exponential Gaussian team problem
IEEE Trans. Automat. Control
Static team problems—Part IIaffine control laws, projections, algorithms and the LEGT problem
IEEE Trans. Automat. Control
Static team problems—Part Isufficient conditions and the exponential cost criterion
IEEE Trans. Automat. Control
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2013, SIAM Journal on Control and OptimizationDistributed control and filtering for industrial systems
2012, Distributed Control and Filtering for Industrial SystemsOptimal disturbance accommodation with limited model information
2012, Proceedings of the American Control Conference
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This work was supported in part by ONR grant N00014-95-1-0948/N00014-97-1-0153 and National Science Foundation Grant CCR 00-85917 ITR.