Adaptive Fuzzy Control Of Non-affine Nonlinear System With Zero Dynamics

Non-affine nonlinear system can be divided into two classes. One is the system whose relative degree is equal to the system degree, the other is the system whose relative degree less than the system degree. The system whose relative degree less than the system degree have internal dynamics, which should be stable when the controller is designed. The introduction of zero dynamics can solve this problem easily. In this paper, adaptive fuzzy control algorithms are proposed for the systems that with stable zero dynamics and unstable zero dynamics separately.For the systems with stable zero dynamics, we assume that its zero dynamics is exponentially stable, and then we introduce an error signal, the convergence of which guarantees the convergence of the output tracking error. The input-output linearization is applied to design the controller. Fuzzy system is used to approximately cancel the modeling error. Lyapunov synthesis method is used to design the adaptive law. The stability proof shows that all the signals in the closed-loop system are bounded, and the tracking error can be adjusted using the control parameters. Meanwhile, we can prove the system internal dynamics is also stable. Therefore, we finish the design of the non-affine nonlinear systems with internal dynamics. Simulation results further illustrate the feasibility of the developed approaches.For the systems with unstable zero dynamics, we use backstepping method, which can change the subsystem with unstable zero dynamics into the system with stable zero dynamics, and design a tracking controller recursively for the whole system. At the final stage of backstepping, a controller is obtained by statically inverting a model and the modeling error is compensated by using a fuzzy system. Lyapunov synthesis method is used to design the adaptive law, and the Lyapunov function can be gotten from the backstepping. The stability proof shows that all the signals in the closed-loop system are bounded, and the tracking error can be adjusted using the control parameters. Meanwhile, we can prove the system internal dynamics is also stable. Therefore, we finish the design of the non-affine nonlinear systems with unstable zero dynamics. Simulation results further illustrate the feasibility of the developed approaches.