The Feedback Control for Distributed Systems

We study the problem of synthesis of lumped sources control for the objects with distributed parameters on the basis of continuous observation of phase state at given points of object. In the proposed approach the phase state space (phase space) is beforehand somehow partitioned at observable points into given subsets (zones). The synthesizing control actions therewith are taken from the class of piecewise constant functions. The current values of control actions are determined by the subset of phase space that contains the aggregate of current states of object at the observable points (in these states control actions take constant values). In the paper such synthesized control actions are called zone control actions. A technique to obtain optimal values of zone control actions with the use of smooth optimization methods is given. With this aim, the formulas of objective functional gradient in the space of zone control actions are obtained. Keywords—Feedback control, distributed systems, smooth optimization methods, lumped control synthesis. I. THE FORMULATION OF THE PROBLEM OF LUMPED CONTROL SYNTHESIS O illustrate the proposed approach to the study of the problem of control synthesis in distributed systems, we consider the problem of control of heat-exchanger process. Let us describe briefly the process. A liquid goes through the heat-exchanger. The temperature of the liquid at the entry point of heat-exchanger is ) ( 2 t ψ . The temperature of steam in the jacket is ) (t ν . The function ) (t ν is control parameter in the following problem: ] , 0 [ ], , 0 [ ), ( ) , ( ) , ( ) , ( 0 1 T t l x t av t x u a t x u a t x u x t ∈ ∈ + − − = (1) ). ( ) ( ) , 0 ( ), ( ) ( ) 0 , ( 2 2 1 1 t t t u x x x u Ψ ∈ = Ψ ∈ = ψ ψ (2) Here u(x,t) is the temperature of the liquid, ) ( ), ( 2 1 t x Ψ Ψ are given domains of initial and boundary conditions. Let us suppose that there are L sensors mounted at the heatexchanger at the points L j l x j ,..., 1 ], , 0 [ ~ = ∈ . The sensors perform on-line monitoring and reading-in of information about the temperature of liquid at these points into the process control system. This information is defined by the vector: Kamil Aida-zade is with the Institute of Cybernetics NAS of Azerbaijan Republic, Baku, Azerbaijan. Cemal Ardil is with the National Academy of Aviation, Baku, Azerbaijan. L j t u t x u j j ,..., 1 ), ( ~ ) , ~ ( = = , ] , [ 0 T t t ∈ The problem is to determine such control function U t ∈ ) ( ν with respect to current values of L j t u j ,..., 1 ), ( ~ = that minimizes the functional: ( ) ( ) ( ) ( ) [ ] ∫ ∫ − + = l T dx x u T x u dt t v v J 0 2 *