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Publisher Summary An adaptive control system must be able to perform the following three functions: provide continuous information about the present state of the system or identify the system; compare present system performance to the desired or optimum performance and make a decision as to how the system should be guided to achieve the optimum performance; and, finally, initiate a proper actuation so as to drive the control system to the optimum. These three principles of identification, decision, and actuation are inherent in any adaptive system. The advance of adaptive control theory is a natural evolution of the control technology. The impact on its development has many facets. Control engineers were confronted with problems in which the classical control techniques developed previously were proving very inadequate. This chapter highlights the problems associated with adaptive control systems. It describes one possible approach for solving adaptive control problems. These problems are formulated within the context of functional analysis. All the problems with least-squares criteria are basically identical with the operator-theoretic formulation. They can all be rewritten as the minimization of certain norms in specifically defined Hilbert spaces. This function-space approach for solving a class of control problems has been used extensively. The chapter discusses the interpretation of the well-known, generalized, least-squares error criterion as the minimization of a quadratic functional in Hilbert space. It also discusses minimum effort problems. The chapter discusses the least-squares estimation of a linear system weighting function matrix, the identification problem of a nonlinear system, and the optimal control of nonlinear systems.

The multivariable nonlinear system considered here is assumed to be characterized by a truncated functional power series. The system is to be identified under normal operating conditions. Least square error criterion is used to estimate the system weighting function matrix and a steepest descent method is employed for solving this problem. A recursive estimation scheme is devised for up-dating its estimation. An “Adjoint Space Approach”, which has the advartage of reducing the dimensionality of the identification problem is explored.