Adaptive robust control of a class of nonlinear systems in semi‐strict feedback form with non‐uniformly detectable unmeasured internal states

Abstract

This paper proposes a novel control method for a special class of nonlinear systems in semi-strict feedback form. The main characteristic of this class of systems is that the unmeasured internal states are non-uniformly detectable, which means that no observer for these states can be designed to make the observation error exponentially converge to zero. In view of this, a projection-based adaptive robust control law is developed in this paper for this kind of system. This method uses a projection-type adaptation algorithm for the estimation of both the unknown parameters and the internal states. Robust feedback term is synthesized to make the system robust to uncertain nonlinearities and disturbances. Although the estimation error for both the unknown parameters and the internal states may not converge to zero, the tracking error of the closed-loop system is proved to converge to zero asymptotically if the system has only parametric uncertainties. Furthermore, it is theoretically proved that all the signals are bounded, and the control algorithm is robust to bounded disturbances and uncertain nonlinearities with guaranteed output tracking transient performance and steady-state accuracy in general. The class of system considered here has wide engineering applications, and a practical example-control of mechanical systems with dynamic friction-is used as a case study. Simulation results are obtained to demonstrate the applicability of the proposed control methodology. The deterministic robust controllers are able to guarantee transient performance and final tracking accuracy in the presence of various kinds of uncertainties. However, some problems like switching [1] or infinite-gain [3] feedback will happen, which are undesirable for industrial application. In contrast, the adaptive controllers [4,5] are able to achieve asymptotic tracking in the presence of parametric uncertainties without using discontinuous or infinite-gain feedback. However, this approach may result in unstable closed-loop system in the presence of external disturbances. To remedy, a modification method called robust adaptive control (RAC) [4] has been developed to robust the system. But some trade offs have to be made, since the property of asymptotic tracking may be lost using this technique. In [6,7], an adaptive robust control (ARC) algorithm has been proposed, which incorporates the design methods of DRC and AC effectively. The resulting ARC controllers have the advantages of both DRC and AC while overcoming their practical limitations. The proposed ARC algorithm has been successfully applied to various systems such as electro-mechanical systems [8,9] and electro-hydraulic systems [10].Besides parametric uncertainties and uncertain nonlinearities, some systems may be further subjected to dynamic uncertainties. This kind of system has exogenous dynamic systems whose states cannot be measured. The control of this kind of system has received more and more attention in the recent years not only because there are few previous results available,...

A Piecewise Envelope Approach to H∞ Control of Nonlinear Systems

“…Based on this proposition, the computation of the piecewise envelope can be reformulated as a quadratic optimization problem with the discrete performance index (6) and constraint (9), which can be solved effectively using quadratic program method. The above idea is summarized as the following algorithm.…”

“…However, when applied to the UNS with model uncertainty, the closed-loop system may not be stable due to the model uncertainty and disturbance. To deal with this problem, [8][9][10] incorporated the robust control theory (Lyapunov-based control and sliding mode control) in the traditional adaptive control theory, namely the adaptive robust control (ARC), to guarantee the tracking accuracy and robust stability together. However, the ARC design method is still subject to the same limitations of the Lyapunov-based control and the sliding mode control methods when dealing with nonlinear systems.…”