Adaptive Fuzzy Output Feedback Control for Partial State Constrained Nonlinear Pure Feedback Systems

The adaptive fuzzy output feedback control problem for a class of pure feedback systems with partial state constraints is addressed in this paper. The fuzzy state observers are designed to estimate the unmeasured state while the fuzzy logic systems are used to approximate the unknown nonlinear functions. The proposed adaptive fuzzy output feedback controller can guarantee that the partial state constraints are not violated, and all closed-loop signals remain bounded by use of Barrier Lyapunov Functions (BLFs). A numerical example is presented to illustrate the effectiveness of the results in this paper.


Introduction
During the last decades, control design of nonlinear systems has attracted increasing interests.All kinds of control techniques have been proposed for both theoretical analysis and practical applications [1][2][3][4][5][6].Many practical systems are inherently nonlinear and subject to many forms of constraints such as saturation and physical stoppages.Violation of the constraints may degrade the control performance and even make the system unstable.Therefore, the constraints handling in control design has attracted considerable attention [6][7][8][9][10][11][12][13][14][15].There exist various techniques to tackle the constraints for nonlinear systems like nonlinear reference governor [7], invariance control [9], nonlinear model predictive control [11], etc.
Backstepping methodology is more effective in synthesis of robust and adaptive nonlinear controllers for various systems with parametric or dynamic nonlinearities and uncertainties [16][17][18][19][20][21][22].The work in [20] constructs an adaptive tracking controller by introducing an auxiliary integrator subsystem and using the improved backstepping method such that the closed-loop system has a unique solution that is globally bounded in probability.The concept of Barrier Lyapunov Function (BLF), which is developed via Control Lyapunov Function (CLF) [23] in backstepping design method, was first proposed in [24].The characteristic of BLF is that it will approach infinity whenever its arguments approach some limits.Transgression of constraints can be prevented through keeping BLF bounded in the closed-loop system.BLF-based backstepping control has been applied to many constrained nonlinear systems control synthesis [25][26][27][28][29]. Therein, [24,30,31] have solved the BLF-based control problem of strict feedback nonlinear systems.The work [27] investigates the output tracking control problem of constrained nonlinear switch systems.The work [29] deals with the problem of adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form.And the problem with respect to full state constraints is solved in [32,33].
As we all know, the adaptive fuzzy control has an automatic learning capability which can adjust the adaptive parameters to deal with the uncertainty.Using the approximation property, fuzzy logic systems (FLSs) have been employed to tackle unknown nonlinear systems [26,28,[34][35][36].A control for nonlinear sampled systems with the guaranteed suboptimal performance achieved robust tracking by using fuzzy disturbance observer approach [26].The work [34] studied an adaptive fuzzy dynamic surface control for nonlinear systems with fuzzy dead zone, unmodeled dynamics, dynamical disturbances, and unknown control gain functions.And the unknown system functions are approximated by the Takagi-Sugeno-type fuzzy logic systems.The work [36] addressed the adaptive control problem for nonlinear pure feedback systems based on fuzzy backstepping approach.Moreover, [28] constructed the approximation-based adaptive fuzzy tracking controller for non-strict-feedback stochastic nonlinear time-delay systems.As for the adaptive fuzzy observer design, to the best of the authors' knowledge, the existing results consider only the influence of full state constraints, and there is no further discussion about partial state constraints.There is seldom adaptive control method subject to partial state constraints and unmeasured state.
In this paper, we present an adaptive fuzzy backstepping tracking controller for a class of pure feedback nonlinear systems subject to unmeasured states and unknown nonlinear function.Firstly, the fuzzy state observers are designed to estimate the unmeasured state while the fuzzy logic systems are used to approximate the unknown nonlinear functions.Secondly, the proposed adaptive fuzzy output feedback controller can guarantee that the partial state constraints are not exceeded, and all closed-loop signals remain bounded with the using of BLF while the adaptive law for the estimations on uncertain parameters is constructed.A new coordinate transform is introduced during the process.Finally, a numerical example is given to validate our results presented in this paper.
The paper is organized as follows.Section 2 presents the problem formulation and some preliminaries.The main results are proposed in Section 3. Section 4 illustrates the effectiveness of the results by a numerical example.

Problem Statement and Preliminaries
where This paper is concerned with the problem of adaptive fuzzy output feedback control for system (1).Because of the existing unknown nonlinear functions, the fuzzy logic systems are employed to approximate the unknown nonlinear functions, and the fuzzy state observers are designed to handle the unmeasurable states.The following lemmas and assumptions related to backstepping design are given which will be used in the further analysis in the sequel.
Remark 7. The filtered signals x+1, and   are employed to avoid the so-called algebraic loop problem existing in [29,39] and to design the state observer and controller for nonlinear pure feedback systems.
Remark 8. Based on the statements in [26,36,38] Then (10) can be further rewritten into the following state space form: where  = [ ] .The vector  is chosen to make matrix  to be a strict Hurwitz matrix; i.e., for given a matrix  =  T > 0 there exists a matrix  =  T > 0 satisfying where  is the output of the system,   1 and   denote fuzzy sets, and  represents the number of fuzzy rules.By using the singleton fuzzifier, center average defuzzification and product inference [25,28], the final output can be expressed as follows: where ỹ = max ∈    ();    1 (  ) and    () stand for membership functions with respect to fuzzy sets   1 and   , respectively. Define as the basis function vector.The ideal constant weight vector is = [ỹ where   = [  1 ,   2 , . . .,    ] T is the center vector, and   is the width of the Gaussian function.
Then, the fuzzy output  in ( 15) is described as Accordingly, assume that the unknown nonlinear functions in (1) are approximated by the following fuzzy logic systems: where x+1, =   .The optimal weight vector  *  is defined as where Let   =   − x be an observer error vector.Then from ( 12) and ( 22), we have where  = [ 1 ,  2 , . . .,   ] T and  = [ 1 ,  2 , . . .,   ] T .Substituting ( 23) into ( 22), we have Consider the following Lyapunov function candidate: Computing the time derivative of  0 , one has According to Assumption 6, we get where Using Young's inequality and ( 28), we get To simplify the notation, let V 0 ≜ − 0 ‖‖ 2 +  0 , where

Main Results
In this section, we propose a generalised design to deal with partial state constraint and the adaptive fuzzy output feedback controller which based on the backstepping technique and fuzzy state observer will be proposed.To guarantee the system performance, the virtual control signals and adaptive laws are designed.A new design procedure is presented which may cover some results related to the full state constraint [37].To ensure that   remains in the constrained region, we give the feasibility conditions with respect to the design parameters and an initial state region, i.e., (0) ∈ Ω (0) , where Let the tracking error and the variables where  −1 is a virtual controller to be designed in Step .
The detailed design procedures are given below.
Step 1.According to ( 19), ( 21), (24), and ( 30), the derivative of  1 is calculated as follows: The Lyapunov function is defined as Then, substituting (33) into (34), one can have Design a virtual controller  1 , adaptive law θ 1 , and ε 1 as where θ1 is the estimation of  * 1 and θ =  *  − θ , ε1 =  * 1 − ε1 ,  1 ,  1 , and  are the positive constants to be designed, respectively.Substituting (36) into (35), it yields ( = 2, 3, . . .,   ).The derivative of   is calculated as follows: Choose the following Lyapunov function candidates: where w =  *  − ŵ .Similar to (35), one can have The virtual controller   , adaptive law θ  , and ε  are designed as where   and   are positive constants.Substituting (41) into (40), the following inequality can be obtained: (42) Step   + 1.The derivative of    +1 is calculated as follows: The Lyapunov function is chosen as Then, we have Design a virtual controller    +1 , adaptive law θ   +1 , and ε Substituting ( 46) into (45), the following inequality is obtained: j ( =   + 2, . . .,  − 1).The derivative of   is calculated as follows: The Lyapunov function is defined as Then, one can have Design a virtual controller  +1 , adaptive law θ +1 , and ε +1 as Mathematical Problems in Engineering 7 Substituting (51) into (50), the following inequality can be obtained: Step .The derivative of   is calculated as follows: Define the Lyapunov function as Then, one can have We choose the control law and the adaptive laws as Substituting ( 56) into (55), the following equality can be obtained: According to Young's inequality, we have Then, the derivative of   is given by Combined with the above analysis, we have come to the following conclusions.Theorem 9. Consider system (1).Assumptions 5 and 6 hold on the sets Ω   .For the virtual controller   ,  = 1, 2, . . .,  − 1, in (36), (41), and (51) and the actual controller  in (56) and the adaptive laws in (36), ( 41), (51), and (56), the following properties hold: (i) The proposed adaptive control scheme can guarantee that the tracking error converges to a bounded compact set (ii) All the signals in the closed-loop systems are bounded.
(iii) The partial state constraints are not violated.

Illustrative Example
In this section, we give an example to show how to apply the results proposed in this paper to investigate the stabilization of nonlinear pure feedback systems subject to partial state constraints.
Let us consider the following nonlinear systems: where the state constraints are | 1 | < 0.5; the reference signal is given as   = 0.1sin(/3) + 0.
The simulation is given in Figures 1-8. Figure 1 shows the trajectory of the state . Figure 2 is the swing curve of the control signal . Figure 3 is the state  1 which remains in the constraint region; it shows the trajectory of the state . Figure 4 stands for the variables of  1 and  2 , and these error variables can not violate their bounds.Figures 5 and 6 are used to illustrate the trajectories of the system states  1 ,  2 and the observer states x1 , x2 .Figures 7 and 8 show that all the signals in the closed-loop system are bounded.It is clear that the nonlinear pure feedback system subject to partial constraints under the output feedback law is bounded.From the simulation, we can conclude that the results proposed in Theorem 9 are very practicable in stability of nonlinear pure feedback systems with partial constraints.Meanwhile, it is a good tool in analyzing the stability problems of some      classes of nonlinear pure feedback systems in the presence of constraint.

Conclusions
In this paper, the tracking control problem of a class of nonlinear pure feedback systems subject to partial state constraints and adaptive fuzzy output feedback controls has been investigated by use of BLFs and backstepping.The output feedback control law, by which the stability of the closedloop system is guaranteed, is determined by constraints.Simulations show that the results obtained in this paper are very practicable in analyzing the stability of some classes of nonlinear pure feedback systems.