Iterative learning control for over-determined, under-determined, and ill conditioned systems (Q2702510)

The authors consider a controlled system represented by the operator \(F\) acting on Hilbert spaces \(U\) and \(Y\), that is \(F : U\to Y\), where \(U\) is the control space and \(Y\) is the space of observations or system outputs. The system evolution is given by \(y_k=F(u_k), k=0,1,\dots,\) where \(u_k\in U\) and \(y_k\in Y\) are the control input and the observed output of the system at the \(k\)-th cycle, respectively. Let \(y_d\) be the desired output of the system. The aim is to find a control \(u_d\) which is a solution of the operator equation \(y_d = F(u).\) This paper studies learning control for under-determined and over-determined systems. Discrete time systems whose operators are ill conditioned are also studied. For all these cases the authors recommend to use pseudoinverses or their approximations as a learning operator. In particular, it is shown that for over-determined systems, unless one knows the system exactly, the minimum possible output error is never achieved by a repetition invariant learning controller. Finally, the relation between several iterative learning control methods and Tikhonov-type regularization techniques is discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00062].