A SCHEME FOR OPTIMIZING CONSTRAINED PROPORTIONAL CONTROL PROBLEMS USING PENALTY FUNCTION METHOD

Abstract

The general continuous optimal proportional control problems constrained by ordinary di erential equations, delay di erential equations and two-dimensional proportional control problem with vector and matrix coe cients are considered. The general form of the problems are given in equations (1) 􀀀 (3), (4) 􀀀 (6) and (7) 􀀀 (9) respectively. Minimize J(x;w) = 1 2 Z T 0 (px2(t) + qw2(t))dt (1) Subject to x_ (t) = ax(t) + bw(t); t 2 [0; T] (2) x(0) = x0; w(t) = mx(t); (3) where p; q; a; b;m 2 R; p; q > 0 and m is the proportional control constant. Minimize J(x;w) = 1 2 Z T 0 (px2(t) + qw2(t))dt (4) Subject to x_ (t) = ax(t) + bw(t) + cx(t 􀀀 r); t 2 [0; T] (5) x(t) = h(t); t 2 [􀀀r; 0]; w(t) = mx(t); (6) where r is the delay term, h(t) is the state's initial condition, p; q; a; b;m 2 R; p; q > 0 and m is the proportional control constant. Minimize J(X;W) = 1 2 Z T 0 (XTPX +WTQW)dt (7) Subject to _X (t) = AX + BW; t 2 [0; T] (8) X(0) = X0; W = MX; (9) where X = X(t) 2 R2, W = W(t) 2 R2, P2 2;Q2 2 are symmetric positive de nite, A2 2;B2 2 are not necessarily symmetric positive de nite: M2 2 is a time-invariant proportional control constant. The optimization of proportional control problems with equality constraints is discussed. The discretization of the objective functions and the constraints is carried out using the Simpson's 3 8 Rule and the Fifth-Order Adams-Moulton Technique presented in equations (10) and (11) respectively. Z tn t0 f(t)dt = 3h 8 hf(t0) + 3 n􀀀2 X i=1;4;7;::: f(ti) + 3 n􀀀1 X i=2;5;8;::: f(ti) + 2 n􀀀3 X i=3;6;9;::: f(ti) + f(tn)i (10) i xi+4 =xi+3 + h 720h251f(ti+4; xi+4) + 646f(ti+3; xi+3) 􀀀 264f(ti+2; xi+2) + 106f(ti+1; xi+1) 􀀀 19f(ti; xi)i􀀀 n 160 h6f5( ) (11) With equality constraints in our research problems, the discretized form of the constrained proportional control problems with or without delay is converted to unconstrained problems using the conventional penalty function given as (x) = [hi(x)] = l Xi=1 k hi(x) k2 (12) With this formulation, the associated operators, which are amenable to the application of the Conjugate Gradient Method (CGM) for solving the problems are constructed. Seven real life examples that were solved by existing schemes and developed schemes are considered. The analysis of convergence of the results is carried out to demonstrate the e ectiveness, e ciency and accuracy of our new schemes over the existing ones. Hence, our results show that the new schemes are much closer to the analytical solutions than the existing schemes with superlinear convergence.