We propose a terminal sliding mode control (SMC) law based on adaptive fuzzy-neural observer for nonaffine nonlinear uncertain system. First, a novel nonaffine nonlinear approximation algorithm is proposed for observer and controller design. Then, an adaptive fuzzy-neural observer is introduced to identify the simplified model and resolve the problem of the unavailability of the state variables. Moreover, based on the information of the adaptive observer, the terminal SMC law is designed. The Lyapunov synthesis approach is used to guarantee a global uniform ultimate boundedness property of the state estimation error and the asymptotic output tracking of the closed-loop control systems in spite of unknown uncertainties/disturbances, as well as all the other signals in the closed-loop system. Finally, using the designed terminal sliding mode controller, the simulation results on the dynamic model demonstrate the effectiveness of the proposed new control techniques.

Sliding mode control (SMC) is known to be a robust control scheme applicable for controlling uncertain systems. Great robustness is provided against various categories of uncertainties such as external disturbances and measurement errors [

Since fuzzy logic systems (FLSs) and neural networks (NNs) are universal function approximators [

In fact, most of the works in the fuzzy/neural control are dedicated to the control problem for the affine nonlinear systems, that is, systems characterized by inputs appearing linearly in the system input-output equation. Few results are accessible for nonaffine nonlinear systems in which the control input occurs in a nonlinear fashion. In this paper, a novel dynamic model approximation method is first proposed to approximate the nonaffine nonlinear dynamics, which is a solution that bridges the gap between affine and nonaffine control systems. Then we combine the FLSs and NNs, and adaptive techniques proposed an adaptive fuzzy-neural observer for nonaffine nonlinear systems. Because in many control problems, state variables may be partly unavailable. So an adaptive observer is designed, and it entails simultaneous estimation of parameters and unknown state variables. The update laws of fuzzy-neural network (FNN) parameters provide the Lyapunov stability for the closed-loop system and guarantee that all signals involved are globally uniformly ultimately bounded. Using terminal SMC, based on the adaptive FNN observer, the robust tracking controller is designed.

The rest of this paper is organized as follows. First, a brief descriptions of the fuzzy-neural network system. Main results include a novel dynamic model approximation technique, adaptive fuzzy-neural observer, and an SMC control algorithm which are proposed in Section

Figure

Functional link of a fuzzy-neural network structure.

Consider the nonaffine nonlinear system represented in the following normal form:

Observer-based model identification methods have been proposed in affine nonlinear systems in [

The problem of controlling the plants characterized by models that are nonaffine in the control input vector is a difficult one. For the tracking control especially, the linearization may result in the design of sufficiently accurate controllers in the case of stabilization around the operating point; in the case of tracking of desired trajectories, the problem becomes much more difficult because the linearized model is time-varying. Hence, there is a clear need for the development of systematic control design techniques for nonlinear models that are nonaffine in

For the nonaffine nonlinear model (

It is easy that (

Consider that

In Assumption

Convenient for the following statements,

By (

The traditional model simplification method is not global, mainly due to that the simplified model is fixed rather than time-varying model. It can be seen that (

The simplified model (

We approximate the functions

Using the FNN approximations, the dynamic equation of a fuzzy-neural observer that estimates the states in (

Defining the state and output estimate errors as

In order to construct the vector

Consider the observer system (

Consider the Lyapunov-function candidate

Define tracking error as

Consider terminal function

Define terminal function

Suppose that the control law is

Consider a Lyapunov-function candidate as follows:

Substituting the control law (

It follows from Assumption

In order to reduce chattering which is caused by discontinuous control signal, a continuous function vector is used to the controller design. In control law (

This section presents the simulation results of the proposed tracking controller to illustrate that the stability of the closed-loop system is guaranteed, and all signal involved are bounded. Consider the nonlinear system

The desired output is

System response with the proposed tracking controller.

System states and observer values.

Observer error of states.

We have carried out a systematic study on fuzzy-neural observer-based terminal SMC in this paper, based on an adaptive fuzzy-neural observer which is used to identify the model and estimate the states. So the information of mathematical model and states does not require to know, only using the measurable output. The proposed nonlinear tracking control scheme can guarantee the asymptotic output tracking of the closed-loop control systems in spite of unknown uncertainties/disturbances. Finally, simulation results are provided on a nonaffine nonlinear system to show the effective and advantages of the new control strategy.

This work was partially supported by the National Natural Science Foundation of China (61273171, 61034005, and 61174058), the National Aerospace Science Foundation of China (2011ZA52009, 2012ZA52017), PAPD, Funding of Jiangsu Innovation Program for Graduate Education (CXZZ11-0213), and funding for Outstanding Doctoral Dissertation in NUAA (BCXJ12-04).

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