离散非线性系统P型迭代控制的收敛性

On Ensuring Convergence While Removing Restrictionson Arimoto′s Learning Theory for Discrete Systems

收敛性是迭代控制中最重要的问题之一。本文给出并证明了当被控系统的状态方程为一般非线性离散方程时，采用Ｐ型算法在开环与闭环两种情形下迭代控制收敛的充分条件和必要条件。其结果表明迭代控制的收敛性条件与状态方程的具体形式无关，因而与线性情形时有同样的结果。并且证明了给出的条件比已有的结果放宽了。

In the study of control problems such as the locus of a robot′s arm, we find it desirable to remove the four restrictions, to be named later, encountered when applying Arimoto′s learning theory . The convergence of general nonlinear discrete systems under both openloop and closedloop algorithm is studied in this paper. We consider the following discrete system:x n+1 (i)=f(i,x n(i),u n(i)) y n(i)=g(i,x n(i))+Du n(i)(1) Our learning algorithm is given byu n+1 (i)=u n(i)+Γe n(i)(4) Or u n+1 (i)=u n(i)+Γe n+1 (i)(5) So far research results on system described by eq.(1), based on Arimoto′s learning theory, suffers from four restrictions: (1) Only some special nonlinear state equations are considered. Usually, the function f is requried to be linear with respect to the input u . (2) Convergence conditions given in terms of matrix norm are unnecessarily strong. (3) There is no proof about the convergence of the discrete system under closedloop algorithm even in the case of the linear discrete system. (4) The initial state error is always set to zero. The above four restrictions were completely overcome in this paper. We obtained the following two convergence theorems. Theorem 1 Suppose that functions f and g are continuous, and the initial states x n (0) satisfy that lim n→∞(x n+1 (0)-x n(0))=0. Then as long as the spectral radius of the matrix I-ΓD is less than one, the system described by eq.(1) is convergent under the algorithm described by eq.(4) Theorem 2 Suppose that the functions f and g are continuous, and the initial states x n(0) satisfy that lim n→∞(x n+1 (0)-x n(0))=0. Then as long as the spectral radius of the matrix (I+ΓD) -1 is less than one, the system (1) is convergent under the algorithm described by eq.(5).