Prescribed Performance Fuzzy Adaptive Output-Feedback Control for Nonlinear Stochastic Systems

and Applied Analysis 3 appropriately determines the performance bounds of the error z i (t). To represent (3) by an equality form, we employ an error transformation as z i = μ i (t) Φ i (ζ i (t)) , ∀t ≥ 0, (4) where Φ i (ζ i ) = (δ imaxe ζi − δ imine i)/(e ζi + e i). Since the functionΦ i (ζ i ) is strictly monotonic increasing, its inverse function can be expressed as ζ i (t) = Φ −1 ( z i (t)


Introduction
In the past decade, control design and stability analysis on stochastic systems have received considerable attention, since stochastic modeling has come to play an important role in many real systems, including nuclear processes, thermal processes, chemical processes, biology, socioeconomics, and immunology [1][2][3][4].Especially, the investigations on the control design methods of nonlinear stochastic systems have received more attention in recent years based on backstepping technique.For example, the adaptive backstepping control problem has been investigated in [5] for a class of SISO strict-feedback stochastic systems by a risk-sensitive cost criterion.An output-feedback stabilization method has been proposed for a class of strict-feedback stochastic nonlinear systems by using the quartic Lyapunov function in [6].Two backstepping control design approaches have been developed for nonlinear stochastic systems with the Markovian switching in [7,8].By using a linear reduced-order state observer, several different output-feedback controllers have been developed for strict-feedback nonlinear stochastic systems with unmeasured states, such as tracking control [9], decentralized control [10], and time-delay systems [11].However, these proposed control methods are only suitable for those nonlinear stochastic systems with nonlinear dynamic models known exactly or with the unknown parameters appearing linearly with respect to known nonlinear functions.To cope with the problems that the nonlinear dynamic models are unknown or the system uncertainties are not linearly parameterized, the adaptive output-feedback control approaches have been proposed for a class of uncertain nonlinear stochastic systems by using neural networks in [12,13].The decentralized adaptive neural networks control methods have been developed in [14,15] for a class of uncertain large-scale nonlinear stochastic systems on the basis of [12,13].
Although the adaptive neural networks backstepping control approaches in [12][13][14][15] can solve the problem of the unmeasured states by designing a linear state observer, there is a limit; that is, uncertain terms are only the functions of the output of the controlled systems, not related to the other states variables.To solve this limit, some adaptive fuzzy output feedback control methods have been proposed for a class of nonlinear stochastic systems by designing nonlinear fuzzy state observers in [16][17][18].
It should be mentioned that the control methods [12][13][14][15][16][17][18] can only solve output-feedback stabilization problem and cannot solve the output feedback tracking control problem.
In addition, the tracking performance in the above control methods confined to converge to a small residual set, whose size depends on the design parameters and some unknown bounded terms; they cannot offer the guaranteed transient performance at time instants.As we know, the practical engineering often requires the proposed control scheme to satisfy certain quality of the performance indices, such as overshoot, convergence rate, and steady-state error.Prescribed performance issues are extremely challenging and difficult to be achieved, even in the case of the nonlinear behavior of the system in the presence of unknown uncertainties and external disturbances.More recently, a design solution called prescribed performance control for the problem has been proposed in [19] for a class of feedback linearization nonlinear systems and was extended to the class of nonlinear systems in [20].Its main idea is to introduce predefined performance bounds of the tracking errors and is able to adjust control performance indices.However, to the author's best knowledge, by far, the prescribed performance design methodology has not been applied to nonlinear strictfeedback systems with unknown functions and immeasurable states, which is important and more practical; thus, it has motivated us for this study.
In this paper, an adaptive fuzzy output-feedback control design with prescribed performance is developed for a class of uncertain SISO nonlinear stochastic systems with unmeasured states.With the help of fuzzy logic systems identifying the unknown nonlinear systems, a fuzzy adaptive observer is developed to estimate the immeasurable states.The backstepping control design technique based on predefined performance bounds is presented to design adaptive fuzzy output-feedback controller.It is shown that all the signals of the resulting closed-loop system are bounded in probability.Moreover, the tracking error converges to an adjustable neighborhood of the origin and remains within the prescribed performance bounds.Compared with the existing results, the main advantages of the proposed control scheme are as follows: (i) the restrictive assumption that all the states of the system be measured directly can be removed by designing a state observer; and (ii) by introducing predefined performance, the proposed adaptive control method can ensure that the tracking error converges to a predefined arbitrarily small residual set.
Our control objective is to design a stable output feedback control scheme for system (1) to ensure that all the signals are bounded in probability and that the system output () can track the given reference signal   () with the given prescribed performance bounds.
where ‖‖ denotes the 2-norm of a vector .

Prescribed
Performance.This section introduces preliminary knowledge on the prescribed performance concept reported in [20].According to [20], the prescribed performance is achieved by ensuring that each error   () evolves strictly within predefined decaying bounds as follows: where 1 ≤  ≤ ,   min and   max are design constants, and the performance functions   () are bounded and strictly positive decreasing smooth functions with the property lim  → ∞   () =  ,∞ ;  ,∞ > 0 are a constant.In this paper, the performance functions are chosen as the exponential form   () = ( ,0 −  ,∞ ) −   +  ,∞ , where   ,  ,0 , and  ,∞ are strictly positive constants,  ,0 >  ,∞ , and  ,0 =   (0) is selected such that −  min   (0) <   (0) <   max   (0) is satisfied.The constant  ,∞ denotes the maximum allowable size of   () at steady state that is adjustable to an arbitrary small value reflecting the resolution of the measurement device.The decreasing rate   represents a lower bound on the required speed of convergence of   ().Furthermore, the maximum overshoot of   () is prescribed less than max{  min   (0),   max   (0)}.Therefore, choosing the performance function   () and the constants   min ,   max appropriately determines the performance bounds of the error   ().
To represent (3) by an equality form, we employ an error transformation as where Since the function Φ  (  ) is strictly monotonic increasing, its inverse function can be expressed as with For the output-feedback control design of the nonlinear system, we design the following state transformation: And the transformation state dynamics is 2.3.Fuzzy Logic Systems.A fuzzy logic system (FLS) consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier.The knowledge base for FLS comprises a collection of fuzzy IF-THEN rules of the following form: Then     ,  = 1, 2, . . ., , where  = ( 1 , . . .,   )  and  are the FLS input and output, respectively.Fuzzy sets    and   are associated with the fuzzy functions     (  ) and    (), respectively. is the rule number of IF-THEN.
Through singleton function, center average defuzzification, and product inference [21], the FLS can be expressed as where   = max ∈    ().

Fuzzy State Observer Design
Since the states  2 , . . .,   in system (1) are not available for measurement, a state observer is to be established to estimate them in this section.
Rewrite (1) in the following form: where The vector  is chosen such that  is a Hurwitz matrix.Thus, given a positive definite matrix  =   > 0, there exists a positive definite matrix  =   > 0 satisfying By Lemma 4, we can assume that nonlinear terms   ( x ),  = 1, 2, . . .,  in (13) can be approximated by the following FLSs: Define the optimal parameter vectors  *  as where Ω  and   are bounded compact sets for   and x , respectively.Also, the fuzzy minimum approximation error   is defined as where   satisfies |  | ≤  *  , with  *  being a positive constant.The state observer for ( 13) is designed as where  = [1 ⋅ ⋅ ⋅ 0 ⋅ ⋅ ⋅ 0].

Adaptive Controller Design
In this section, an adaptive fuzzy output-feedback control scheme will be developed by using the above fuzzy state observer and the backstepping technique, and the stability of the closed-loop system will be given.
The controller design consists of step ; each step is based on the following change of coordinates: where  −1 is referred to as the intermediate control function, which will be designed later.
Step  (2 ≤  ≤  − 1).Similar to Step 1, we have where Choose intermediate control function   and adaptation law   as where   > 0,   > 0 and   > 0 are design parameters and   is the estimate of  *  , and Step .In the final design step, the actual control input  will be designed.Similar to Step  we have The controller  and adaptation law   are chosen as where   > 0,   > 0 and   > 0 are design parameters and   is the estimate of  *  .

Stability Analysis
where  max () is the largest eigenvalue of .Substituting (34)-( 35) into (33) gives where , and  min () is the minimal eigenvalue of .
From (49) and (50), we have Integrating (51) over [0, ], we get Taking expectation on (52), it follows that where (⋅) is probability expectation.The above inequality means that [()] is bounded by / in mean square.Thus, according to [12][13][14][15][16][17][18], it is concluded that all the signals of the closed-loop system are SGUUB in the sense of the four-moment.Moreover, it follows that the tracking errors and virtual tracking errors remain within the prescribed performance bounds for all time  ≥ 0.

Simulation Study
In this section, a simulation example is provided to evaluate the control performance of the proposed adaptive outputfeedback control method.Consider a stochastic system governed by the following form: where ẇ () is assumed to be a Gaussian white noise with zero mean and variance 1.0.The tracking reference signal is chosen as   () = sin().

Conclusion
In this paper, fuzzy adaptive output feedback tracking control problem has been investigated for a class of nonlinear stochastic systems in strict-feedback form.The addressed stochastic nonlinear systems contain unknown nonlinear  functions and without the measurements of the states.Fuzzy logic systems are used to identify the unknown nonlinear functions, and a fuzzy state filter observer has been designed for estimating the unmeasured states.By applying the backstepping recursive design technique and the predefined performance technique, a new robust fuzzy adaptive output-feedback control approach has been developed, and the stability of the closed-loop system has been proved.The main advantages of the proposed control approach are that it cannot only solve the state unmeasured problem of nonlinear stochastic systems, but can also guarantee that the tracking error converges to an adjustable neighborhood of the origin and remains within the prescribed performance bounds.Future research will be concentrated on an adaptive fuzzy output-feedback tracking control for multiinput and multioutput stochastic nonlinear systems with unmeasured states based on the results of [22,23] and this paper.

Figure 1 :
Figure 1: The curves of  (solid line) and   (dot line).

Figure 4 :
Figure 4: The curves of  1 and performance bounds.