A high gain multiple time-scaling technique for global control of nonlinear systems

We consider a general high-gain scaling technique for global control of strict-feedback-like systems. Unlike previous results, the scaling utilizes arbitrary powers (instead of requiring successive powers) of the high gain parameter with the powers chosen to satisfy certain inequalities depending on system nonlinearities. The scaling induces a weak-cascading upper diagonal dominance (w-CUDD) structure on the dynamics. The analysis is based on our recent results on the w-CUDD property and uniform solvability of coupled state-dependent Lyapunov equations. The proposed scaling provides extensions in both state-feedback and output-feedback cases. The state-feedback problem is solved for a class of systems with certain ratios of nonlinear terms being polynomially bounded. The controller has a simple form being essentially linear with state-dependent dynamic gains and does not involve recursive computations. In the output-feedback case, the scaling technique is applied to the design of the observer which is then coupled with a backstepping controller. The results relax the assumption in our earlier papers on cascading dominance of upper diagonal terms. However, since the required upper diagonal cascading dominance in observer and controller contexts are dual, it is not possible to use a dual high-gain observer/controller in the proposed design preventing the bounds on uncertain functions from being of the more general form in our earlier work. A topic of further research is to examine the possibility of a scaling (perhaps utilizing more than one high gain parameter) that achieves bidirectional cascading dominance.