Optimizing Control of Timevarying Large-Scale Systems by Price Mechanism with Local Feedback

Abstract

In this paper we propose and investigate a certain optimizing control structure for large-scale systems which can be assumed to operate in steady state, but are time-varying. It means that we deal with systems which have fast dynamics when compared to the rate of change of external disturbances. It is assumed that the system is described in terms of subsystem input-output mappings which only approximate the unknown reality. This reflects the usual practical situation when the disturbances which affect the subsystems are not exactly known.The proposed control structure is organized in the form of a two-layer hierarchy: optimization layer and correction layer. For a given instant of time the task of the optimization layer is to solve, by using the Interaction Balance Method, the model-based optimization problem corresponding to the actual value of the disturbance estimate.The results are the value of price which is sent to the correction layer, and the value of control (set-point) which is applied to the system. Because of the difference between the disturbance and its estimate this model-optimal control is not optimal for the system. The task of the correction layer is to improve the performance. The correction layer uses local information feedback (interaction measurements) from the system and it is completely decentralized. This kind of control with local feedback has been first proposed in Pindeisen (1974) and its main properties in a stationary case have been investigated in Brdys and Ulanicki (1978), Brdys and Michalak (1978). The time - varying nature of the problem implies that the desired equilibrium point(Brdys and Ulanicki, 1978)cannot be achieved. Thus the correction layer iterative algorithm can only track the moving equilibrium point with certain accuracy, depending on the rate of disturbance changes.In the paper the bounds on this accuracy are given. Because the system is time-varying the optimization and the correction layers should repeat its action, appropriately to the disturbance changes. The central problem which is investigated in the paper is the following:for given bounds on the difference between the value of disturbance and its estimate and a bound on the rate of change of disturbances find the optimization and correction layers intervention instants such that the system overall performance obtained in the considered control structure tracks the optimal performance of the system with desired tracking accuracy.The results are obtained for general nonlinear case.