Robust Adaptive Nonlinear Control System Designs

We have developed a powerful and flexible paradigm for dynamic high-gain based design of controllers and observers for various classes of nonlinear systems. The design technique is applicable in both the state-feedback and the output-feedback cases. The technique utilizes a state scaling generated through an appropriately designed dynamics driven by the measured outputs of the system. The resulting controller and observer are algebraically simple requiring no recursive computations and the associated Lyapunov functions have a simple scaled quadratic form. The stability analysis is based on the solution of a pair of coupled matrix Lyapunov equations. Necessary and sufficient conditions for the solvability of the coupled Lyapunov equations have been obtained in our recent results. The approach provides a unified design procedure applicable to both lower triangular (feedback) and upper triangular (feedforward) systems and also to some classes of polynomially bounded systems without requiring any triangularity in the system structure. The controller provides strong robustness properties and allows coupling with a dynamic high-gain observer whose design is dual to that of the controller to obtain output-feedback stabilization and tracking results. This represents the first output-feedback result for feedforward systems. The controller and observer designs in the case of feedforward systems are strongly parallel to the designs in the case of strict-feedback systems suggesting that the proposed technique could allow further extensions for feedforward systems along various lines that have been hitherto investigated only for strict-feedback systems. In both, the strict feedback and the feedforward cases, a greater generality and complexity of bounds on uncertain functions in the system does not increase the complexity of the control law, the observer, and the Lyapunov function, but is instead handled through the dynamics of the scaling parameter. Furthermore, the scaling parameter dynamics can be designed to provide robustness to various perturbations including unknown parameters, additive disturbances, inverse dynamics, and appended Input-to-State Stable (ISS) dynamics. A generalized scaling technique utilizing arbitrary powers of the scaling parameter has been developed to weaken the assumptions on the system and to extend the results to non-triangular systems. Current research effort on this topic is focussed on extending the results in the following directions: 1) systems with ISS appended dynamics considering general interconnections (dependent on all states) 2) adaptive control without a priori bounds on unknown parameters 3) decentralized control for large-scale systems with each subsystem control input designed using the dynamic high-gain scaling based technique 4) disturbance attenuation 5) relaxation of assumptions by considering more general scaling patterns/ multiple scalings and more general scaling dynamics.