Higher Order Nonlinear Systems With Uncertain Control Design And Stability Analysis

The problems of the control design for the so-called high-order nonlinear systems have been became a new direction in recent years. Because of the Jacobian linearization of the system at the origin has uncontrollable modes associated with eigenvalues on the open right-half plane, so the traditional method(such as feedback linearization, backstepping approach) is hardly applicable to the system, and high-order nonlinear systems have more extensive forms than general nonlinear systems, and furthermore, high-order nonlinear systems include a class of under-actuated, weakly coupled and unstable mechanical systems. Therefore, the investigations of control problems for high-order systems have theoretical and practical meanings. Based on the method of adding a power integrator, flexible adaptive technique and other techniques such as Young's inequality, we focus on the two control problems for a class of high-order uncertain nonlinear systems:(I) Adaptive state-feedback stabilization for a class of high-order uncertain nonlinear systems with unknown control coefficients and zero-dynamicsThis problem has been investigated under unmeasurable zero dynamics meet more spe-cial assumption. Restriction on zero dynamics is further relaxed in this paper. By using the method of adding a power integrator, a recursive design procedure is provided to achieve a continuous adaptive state-feedback controller, In the design procedures of the controller, to effectively deal with the unknown parameters coming from control coefficients and non-linearities, and according to the delicate definition of an appropriate unknown parameter, we only design one parameter updating law, thereby the dynamic order of the controller designed is minimum, and avoid the over-parametrization estimate phenomenon. By means of Lyapunov stability theory and Barbalat lemma, the controller designed guarantees that all the states of the whole closed-loop system are globally uniformly bounded, while the original system states globally asymptotically converge to zero. Finally, a simple example is given to illustrate the effectiveness of the control scheme.(II) Practical output tracking control for a class of high-order uncertain non-linear systems with unknown control coefficients and zero-dynamicsThis problem has been investigated under the assumption that the lower bounds of the unknown control coefficients are exactly known and unmeasurable zero dynamics meet more particular condition. Based on the idea of changing supply functions and continu- ous stabilization, Restriction on zero dynamics is further relaxed successfully, by using the backstepping approach, the method of adding a power integrator and flexible Algebra pro-cessing techniques, we successfully developed to achieve a continuous stabilizing tracking controller. The controller guarantees that all states of the whole closed-loop system are globally bounded, while the tracking error can be bounded by any given positive number after a finite time. Finally, we give a simple example to demonstrate the effectiveness of the control scheme.In a word, satisfactory solutions to control problems of high-order uncertain nonlinear systems not only guide the academic research of strict feedback nonlinear control systems, but also will enrich the design methods of traditional controllers.