Almost Passivity and Simple Adaptive Control in Discrete‐Time Systems

Successful implementations of simple direct adaptive control techniques in various domains of application have been presented over the last two decades in the technical literature. The theoretical background concerning basic conditions needed for stability of the controller and the open questions relating the convergence of the adaptive gains have been clarified recently, yet only for the continuous-time algorithms. Apparently, asymptotic tracking in discrete time systems has been possible only with step input commands and the scope of the so called “almost strictly positive real (ASPR)” condition has also remained not clear. This paper expands the feasibility of discrete simple adaptive control methodology to include any desired input commands and almost all real-world systems. The proofs of stability are also rigorously revised to solve the ultimate adaptive gain values question that has remained open until now. Finally, a complex algebraic loop that seemed to be inherent in discrete ASPR systems and might prevent the use of passivity properties in discrete systems has also been eliminated.