Some Results And Their Applications In The Discrete Control Systems

The optimal control theory is the main component of mod-ern control theories, and its form and development establish thefoundation of the whole modern control theories. Time delay existsinevitably in active control systems. In theoretical analysis and con-trol design, even small time delay, it can lead the actuators inputtingenergy to the structures when no energy is required by the struc-tures, and this can cause degradation in control e?ciency or evenrender the structures unstable, so delay is the important problem tobe worth. Delay discrete systems are important for many applica-tions, especially to economic models, to qualitative and numericalaspects of optimization and control of various hereditary processes,to numerical solutions of control systems with distributed parame-ters, etc. As a result, the optimal control of delay discrete controlis very important.In this paper, we mainly study the discrete form of continuessystems with delays in state variables and the discrete form andstandard discrete form of time-variant linear systems with delays incontrol, then we deduce the necessary optimality conditions of themultiple-delays discrete control systems, at the same time, we givethe necessary optimality conditions of constant discrete system. Atthe end of this article, we give some application examples.In section 3 , we give the discrete form of continues systems with delays in state variables and the discrete form and standarddiscrete form of time-variant linear systemswith delays in control under the conditions ofτis integral multipleof period or un-integral multiple of period.In practice many control problems have delays in time. How-ever, generally speaking, it is di?cult to get analytical solutions ofcontinues time-delay systems. In section 4 , we obtain the necessaryoptim?ality conditions of discrete control systems (p)with two delays in state variables.Theorem 4. 1 For system(p), suppose sequence u(k)minimizesthe criterionJ[u], and x(k)is the optimal state sequence. Then, thereexists n-dimension vectorλ(k), and u(k),x(k),λ(k)(k = k0,k0 +1,...,k1 ? 1)satisfy the following conditions And on this basis, we deduce the necessary optimality condi-tions ?of the multiple-delays discrete control systems(p?)with multiple delays in state variables.定理4. 2 For system(p?), suppose sequence u(k)minimizes thecriterionJ[u], and x(k)is the optimal state sequence. Then, thereexists n-dimension vectorλ(k), and u(k),x(k),λ(k)(k = k0,k0 +1,...,k1 ? 1)satisfy the following conditions where A,Bare constant coe?cient matrices. By linear quadraticregular(LQR), we obtain optimal control laws under the criteriongiven by linear quadratic equations. At the end of this article, we give an application example of engineering mechanics and a simpleexample.