抄録
According to the classical mechanics, dynamic systems have their own scalar function (V-function), in which the trajectories of the systems move, and have a functional ∫Ldt, where L is a Lagrangian. In our optimum control problem, first of all a plant is given and then a control law must be determined for the plant. The plant is also a dynamic system. Therefore it has a functional and a unique V-function.
In this paper, it is shown that when the plant is a linear system with constant coefficients, the functional is given by Js=∫∞t(x'Qsx+m2)dt and the V-function by Vs=x'Psx. If it is requied to design the plant to be optimum minimizing the performance index Jr=∫∞t(x'Qrx+u2)dt, then the system finally constructed by such a specification has also a functional Jf=∫∞t(x'Qfx+m2)dt and a V-function Vf=x'Pfx. Here Qf=Qr+Qs, Vf=Vr+Vs, and x' is the vector (x1, x2, …, xn) constructed from the states, in the canonical form of the dynamic equation. Also m=xn, u is the input of the plant and Vr is the minimum value of Jr.
The additivity of Q and V is an elegant result of this paper. Because of this property, the synthesis problems of the linear optimum systems with constant coefficients are reduced to an easy task, that is, only to solving an algebraic equation of order n.
Furthermore, this paper clarifies the relation between the modern control theory and the classical control systems theory.
