On the Alternatives of Lyapunov’s Direct Method in Adaptive Control Design

The prevailing methodology in designing adaptive controllers for strongly nonlinear systems is based on Lyapunov's PhD Thesis he defended in 1892 to study the stability of motion of systems for the solution of the equations of motion of which no closed form analytical solutions exist. The adaptive robot controllers developed in the nineties of the 20thcentury guarantee global (often asymptotic) stability of the controlled system by using his ingenious Direct Method that introduces a Lyapunov function for the behavior of which relatively simple numerical limitations have to be proved. Though for various problem classes typical Lyapunov function candidates are available, the application of this method requires far more knowledge than the implementation of some algorithm. Besides requiring creative designer's abilities, it often requires too much because it works with satisfactory conditionsinstead of necessary and satisfactoryones. To evade these difficulties, based on the firm mathematical background of constructing convergent iterative sequences by contractive maps in Banach spaces, an alternative of Lyapunov's technique was so introduced for digital controllers in 2008 that during one control cycle only one step of the required iteration was done. Besides its simplicity the main advantage of this approach was the possible evasion of complete state estimation that normally is required in the Lyapunov functionbased design. Though the convergence of the control sequence can be guaranteed only within a bounded basin, this approach seems to have considerable advantages. In the paper the current state of the art of this approach is briefly summarized.