Adaptive and robust controls for uncertain linear and nonlinear systems

The presence of various uncertainties in control systems is one of the main challenging issues to engineers since this problem in real applications gives rise to instability and degradation of performance. This dissertation addresses analysis and control design problems involving linear and nonlinear systems with structured and unstructured uncertainties.;In the first problem, the critical direction theory (CDT) an H∞ design is proposed as a systematic controller synthesis methodology for a linear system with parametric uncertainties. In the literature, a significant conservatism issue in this study of such uncertain systems with respect to robust control design has been reported. First, the mechanisms leading to conservatism in an earlier approach utilizing a constant overbounding uncertainty template are analyzed. Then, motivated by the desire to decrease the conservative uncertainty, a static weight approach based on the CDT is examined using a constant value to describe the uncertainties over all frequencies. This approach shows that the CDT combined with H∞ is effective and promising for less conservative stability and controller synthesis for a linear system with uncertainties.;The second is a study of the problem of the robustness of multi-input multi-output (MIMO) systems with parametric uncertainties. Two criteria that are used to assess the stability of MIMO systems are numerically evaluated. A numerical analysis provides an empirical demonstration of the relationship between the two stability margins.;The dissertation then proceeds to the study of design methods for systematic adaptive controls combined with robustness for uncertain nonlinear systems. The most common technique in designing a controller for a nonlinear system is to assume the nonlinear dynamics are exactly known, but in fact system nonlinearities are unknown in real applications. To overcome the uncertainties in nonlinear systems, a wavelet neural network (WNN) identifier with a sliding mode controller (SMC) is proposed. Then, a Robust Integral of the Sign of the Error (RISE) feedback control based on the WNN identifier is used to attenuate the effect of the wavelet network approximation error and exogenous disturbance. The proposed method ensures asymptotic tracking of a desired reference signal. A simulation of controlling memristor based chaotic system illustrates the effectiveness of the proposed approach and its performance in the presence of disturbances, and the result is supported through rigorous Lyapunov-based stability proofs.;Then, an adaptive robust tracking control scheme using a multi-layer neural network (NN) for a class of nonlinear dynamic systems with unknown time varying states is addressed. Typical adaptive NN backstepping controllers for uncertain nonlinear systems with time-delay give rise to computation complexity caused by the repeated derivatives of virtual controllers and nonlinear functions. Moreover, the combined techniques usually result in only uniformly ultimately bounded (UUB) stability due to the inherent NN approximation error. To begin with, we develop a modified desired compensation adaptation law (DCAL) formulation which avoids the “explosion of complexity” caused by the general NN backstepping scheme to compensate for nonlinear system uncertainties, bounded system disturbances, and unknown state time delays. Then, using a Lyapunov-Krasovskii (LK) functional, it is shown that the proposed controller renders the class of uncertain nonlinear time-delay systems asymptotically stable.;Finally, the effectiveness of the proposed multiple controller design methodologies is demonstrated for the 1:1 in-phase synchrony in the mutually coupled interneuron (MCI) network.