Backstepping Control For Uncertain Nonlinear Systems

In recent years, the control problem of uncertain nonlinear systems becomes one of the research focuses. Particularly, the control theory for nonlinear systems with actuator failures or time delays has attracted increasing attention. It is well known that actuator failures are frequently encountered during operation of control systems. Also, time delay phenomenon widely exists. Up to now, many results have been achieved on adaptive control of linear systems with actuator failures. However, for nonlinear systems, adaptive actuator failure compensation has not been deeply investigated. On the other hand, despite the huge literature for linear time-delay systems, the control problem of nonlinear time-delay systems has been receiving relatively little attention until recently. Based on the current research findings, this dissertation uses the backstepping approach as a basic tool to address the problem of adaptive actuator failure compensation of uncertain nonlinear systems and uncertain nonlinear time-delay system control, respectively. The main contributions of the dissertation are as follows.1. A modified adaptive actuator failure compensation scheme is proposed for a class of multi-input and single-output nonlinear systems in the output-feedback form. A con-stant actuator failure model is considered. In the control design, a new parametric model with unknown plant parameters and actuator failure parameters is firstly established. Then, an adaptive compensation controller is constructed by utilizing the backstepping technique. The boundedness of all the closed-loop signals is guaranteed. The output tracking error is proved to converge to zero asymptotically. It is worth pointing out that, within the new model framework, the adaptive control design is programmed, and the stability analysis can be sim-plified greatly. Moreover, the lower-order filters can be exploited to estimate the unmeasured plant states. Thus, the resulting control scheme is easier to be implemented. Finally, the presented scheme is applied to a simplified dual-actuator ball-beam system for the actuator failure compensation study. Simulation results demonstrate the effectiveness of the proposed design method.2. The robust adaptive output-feedback control problem is addressed for a class of non-linear systems with unknown time-varying actuator failures. Additional un-modelled term in the actuator failure model is considered. A linearly parameterized model is proposed, where unknown plant parameters and actuator failure parameters are lumped up together and separated form known signals. Based on the new model, adaptive controller is recursively designed. The modified switching-?signals are incorporated in the adaptive laws to ensure stability and robustness. The boundedness of all the closed-loop signals is established. The desired control performance of the closed-loop system is guaranteed by appropriately choos- ing the design parameters. Finally, a dual-actuator ball-beam system with state-dependent nonlinearities is used to illustrate the effectiveness of the proposed scheme.3. For a class of nonlinear systems with unknown high-frequency gain signs and param-eterizable time-varying actuator failures, an adaptive output-feedback failure compensation scheme is proposed based on the backstepping technique. The Nussbaum gain approach is introduced to remove the restrictive assumption on the high-frequency gain signs. In view of unknown high-frequency gain signs, some constant variables are redefined, the para-metric model with actuator failure information is reestablished, and the stability analysis is retreated. The boundedness of all the closed-loop signals and the asymptotic output track-ing are guaranteed in spite of the unknown actuator failures. Finally, the proposed control scheme is applied to a two-axis positioning stage system with redundant actuators. The effectiveness of the control algorithm is verified by the simulation studies.4. The problem of output-feedback tracking control is addressed for a class of single-input and single-output nonlinear systems in the strict-feedback form, which are subjected to both uncertain delay-related functions and external disturbances. A reduced-order observer is firstly introduced to provide the estimates of the unmeasured states. The advantage of exploiting the lower-dimension observer is two-fold. On the one hand, the resulting closed-loop system is easier to be implemented. On the other hand, since the design parameters are determined by a linear matrix inequality (LMI), the cost involved by the computation of the LMI may be much less. Then, an output-feedback controller is recursively designed based on the backstepping method. An appropriate Lyapunov-Krasovskii functional is constructed to compensate for all delayed state perturbation functions. All the signals in the closed-loop system are proved to be bounded. The desired tracking performance is guaranteed by suitably choosing the design parameters. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed control algorithm.5. Based on the backstepping approach, the problem of adaptive stabilization for a class of nonlinear time-delay systems in triangular structure is addressed. Both the parameters of the system to be controlled and the upper bounds of the time delays and their deriva-tives are assumed to be unknown, which are estimated online by designing adaptive laws. The assumption on the delay-related nonlinearities is further relaxed. To achieve the con-trol objective under such weakened condition, a lemma on function factorization is firstly introduced. Then, an appropriate Lyapunov-Krasovskii functional is constructed and adap-tive controller is designed recursively. To counteract the influence of the parameter estimates on stability of the closed-loop system, from the second step of the backstepping procedure on, two terms are added in each virtual control function. It is worth pointing out that these additional functions should be selected carefully such that the backstepping design can be performed smoothly. It is shown that all the closed-loop signals are bounded, while the plant states converge to zero asymptotically. Simulation studies are provided to demonstrate the effectiveness of the proposed control strategy.6. An adaptive backstepping tracking control scheme is proposed for a class of non-linear state time-varying delay systems, which are subject to parametric uncertainties and external disturbances. Tuning function method is exploited to construct the control law and adaptive laws recursively. This method not only effectively overcome the difficulty in the choice of additional terms in virtual control functions in the last chapter, but also make adap-tive control design more programmed. Unknown time-varying delays are compensated for by using appropriate Lyapunov-Krasovskii functional. The bounds of the time delays and their derivatives are also assumed to be unknown. Two set of tuning functions are designed to guarantee the boundedness of all the closed-loop signals. The tracking performance is ad-justed by choosing suitable design parameters. Simulation results illustrate the effectiveness of the proposed design procedure.