Parameter estimation aspects in adaptive control

This paper provides a solution to the convergence of the parameter estimates to their true values in an ideal adaptive regulation context. The key design feature consists in the use of an asymptotically vanishing internally generated exciting sequence. 1. Introduction IN THIS paper, the following question is addressed: given a known order linear time invariant system, how does one design an adaptive regulator which ensures the convergence of both the parameter estimates error and the system output to zero? The first objective (i.e. the convergence of the parameter estimates to their true values) can be obtained using an external persistently exciting signal as in Elliott et al. regulation of the system output to zero) cannot be obtained with such an approach. This drawback can be removed using an asymptotically vanishing probing signal. This issue has recently been investigated in Chen and Guo (1987) and Canudas de Wit (1987) using an external asymptotically vanishing sequence. Such an approach requires extra-prior knowledge about the system to be controlled. In the present paper, we propose an answer to the above stated question. The main features of the underlying solution are: (1) It is carried out irrespective of both the parameter estimator and the control law; and (2) It involves an asymptotically vanishing exciting signal. Such a vanishing property is a consequence of the ideal framework we are concerned with. Of particular interest, the considered probing signal is internally generated, properly exploiting the parameter estimator properties. Intuitively, the basic idea of this paper is then to build and to use an internally generated exciting signal that tends to zero sufficiently slowly to ensure the parameter convergence to the true parameters. The paper is organized as follows. In Section 2 the control problem statement is given. The adaptive regulator is