A prescribed performance fuzzy adaptive output-feedback control approach is proposed for a class of single-input and single-output nonlinear stochastic systems with unmeasured states. Fuzzy logic systems are used to identify the unknown nonlinear system, and a fuzzy state observer is designed for estimating the unmeasured states. Based on the backstepping recursive design technique and the predefined performance technique, a new fuzzy adaptive output-feedback control method is developed. It is shown that all the signals of the resulting closed-loop system are bounded in probability and the tracking error remains an adjustable neighborhood of the origin with the prescribed performance bounds. A simulation example is provided to show the effectiveness of the proposed approach.
1. 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 [1–4]. 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 [12–15] 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 [16–18].
It should be mentioned that the control methods [12–18] 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 strict-feedback 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.
2. System Descriptions and Preliminaries 2.1. Nonlinear System Descriptions
Consider the following SISO strict-feedback nonlinear stochastic system:
(1)dx1=(x2+f1(x1)+d1(t))dt+g1(x)dw,dx2=(x3+f2(x-2)+d2(t))dt+g2(x)dw,⋮dxn-1=(xn+fn-1(x-n-1)+dn-1(t))dt+gn-1(x)dw,dxn=(u+fn(x-n)+dn(t))dt+gn(x)dw,y=x1,
where x-i=[x1,x2,…,xi]T∈Ri, i=1,2,…,n(x=x_n) is the state vector; u∈R and y∈R are the control input and system output, respectively. fi(x-i) and gi(x)i=1,2,…,n are unknown continuous nonlinear functions, and di(t), i=1,2,…,n is the external disturbance. w∈R is an independent standard Wiener process defined on a complete probability space with the incremental covariance E{dw·dwjT}=σ(t)σ(t)Tdt.
In this paper, the states xi(i≥2) are assumed not to be available for measurement.
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 y(t) can track the given reference signal yd(t) with the given prescribed performance bounds.
Assumption 1.
The external disturbances di(t) are bounded; that is, |di(t)|≤di* with di* being an unknown constant.
Assumption 2 (see [17]).
Assume that functions fi(·) satisfy the global Lipschitz condition; that is, there exist known constants mi, i=1,2,…,n such that for all X1,X2∈Ri, the following inequalities hold:
(2)|fi(X1)-fi(X2)|≤mi∥X1-X2∥,
where ∥X∥ denotes the 2-norm of a vector X.
Assumption 3 (see [9]).
The disturbance covariance gTσσTg=σ-σ-T is bounded, where g=[g1,…,gn]T.
2.2. 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 zi(t) evolves strictly within predefined decaying bounds as follows:
(3)-δiminμi(t)<zi(t)<δimaxμi(t),∀t≥0,
where 1≤i≤n, δimin and δimax are design constants, and the performance functions μi(t) are bounded and strictly positive decreasing smooth functions with the property limt→∞μi(t)=μi,∞; μi,∞>0 are a constant. In this paper, the performance functions are chosen as the exponential form μi(t)=(μi,0-μi,∞)e-nit+μi,∞, where ni, μi,0, and μi,∞ are strictly positive constants, μi,0>μi,∞, and μi,0=μi(0) is selected such that -δiminμi(0)<zi(0)<δimaxμi(0) is satisfied. The constant μi,∞ denotes the maximum allowable size of zi(t) at steady state that is adjustable to an arbitrary small value reflecting the resolution of the measurement device. The decreasing rate ni represents a lower bound on the required speed of convergence of zi(t). Furthermore, the maximum overshoot of zi(t) is prescribed less than max{δiminμi(0),δimaxμi(0)}. Therefore, choosing the performance function μi(t) and the constants δimin, δimax appropriately determines the performance bounds of the error zi(t).
To represent (3) by an equality form, we employ an error transformation as
(4)zi=μi(t)Φi(ζi(t)),∀t≥0,
where Φi(ζi)=(δimaxeζi-δimine-ζi)/(eζi+e-ζi).
Since the function Φi(ζi) is strictly monotonic increasing, its inverse function can be expressed as
(5)ζi(t)=Φ-1(zi(t)μi(t))=12lnΦi-δiminδimax-Φi,ζ˙i(t)=pi(z˙i-μ˙iziμi)
with pi=(1/2μi)[(1/(Φi+δimin))-(1/(Φi-δimax))].
For the output-feedback control design of the nonlinear system, we design the following state transformation:
(6)zi(t)=ζi(t)-12lnδiminδimax.
And the transformation state dynamics is
(7)z˙i(t)=pi(z˙i-μ˙iziμi).
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:
(8)Rl:Ifx1isF1l,x2isF2l,…,xnisFnl,ThenyisGl,l=1,2,…,N,
where x=(x1,…,xn)T and y are the FLS input and output, respectively. Fuzzy sets Fil and Gl are associated with the fuzzy functions μFil(xi) and μGl(y), respectively. N is the rule number of IF-THEN.
Through singleton function, center average defuzzification, and product inference [21], the FLS can be expressed as
(9)y(x)=∑l=1Ny-l∏i=1nμFil(xi)∑l=1N[∏i=1nμFil(xi)],
where y-l=maxy∈RμGl(y).
Define the fuzzy basis functions as
(10)φl=∏i=1nμFil(xi)∑l=1N(∏i=1nμFil(xi)).
Denoting θT=[y-1,y-2,…,y-N]=[θ1,θ2,…,θN] and φ(x)=[φ1(x),…,φN(x)]T, then FLS (9) can be rewritten as
(11)y(x)=θTφ(x).
Lemma 4 (see [21]).
Let f(x) be a continuous function defined on a compact set Ω. Then for any constant ε>0, there exists a FLS (11) such as
(12)supx∈Ω|f(x)-θTφ(x)|≤ε.
3. Fuzzy State Observer Design
Since the states x2,…,xn 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:
(13)dx-n=(Ax-n+Ky+∑i=1nBi[fi(x-i)+di(t)]+Bu)dt+g(x)dw=(Ax-n+Ky+∑i=1nBi[fi(x-^i)+Δfi+di(t)]+Bu)dt+g(x)dw,
where x-^i=(x^1,x^2,…,x^i)T is the estimate of x-i=(x1,x2,…,xi)T, A=[-k1⋮I-kn0⋯0], K=[k1,k2,…,kn]T, Bi=[0⋯1⋯0]T, B=[0⋯0⋯1]T, Δfi=fi(x-i)-fi(x-^i), g(x)=[g1(x),…,gn(x)]T.
The vector K is chosen such that A is a Hurwitz matrix. Thus, given a positive definite matrix Q=QT>0, there exists a positive definite matrix P=PT>0 satisfying
(14)ATP+PA=-2Q.
By Lemma 4, we can assume that nonlinear terms fi(x-^i), i=1,2,…,n in (13) can be approximated by the following FLSs:
(15)f^i(x-^iθi)=θiTφi(x-^i).
Define the optimal parameter vectors θi* as
(16)θi*=argminθi∈Ωi[supx-^i∈Ui|f^i(x-^i∣θi)-fi(x-^i)|],
where Ωi and Ui are bounded compact sets for θi and x-^i, respectively. Also, the fuzzy minimum approximation error εi is defined as
(17)εi=fi(x-^i)-f^i(x-^i∣θi*),
where εi satisfies |εi|≤εi*, with εi* being a positive constant.
The state observer for (13) is designed as
(18)x-^˙n=Ax-^n+Ky+∑i=1nBi[f^i(x-^i∣θi)]+Bu,y^=Cx-^n,
where C=[1⋯0⋯0].
4. 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 n; each step is based on the following change of coordinates:
(19)z1=y-yd,zi=x^i-αi-1,(i=2,…,n),
where αi-1 is referred to as the intermediate control function, which will be designed later.
Step 1.
From (1), (7), and (19), according to Itô’s differentiation rule, we can obtain
(20)dz1=p1(x2+f1(x1)+d1-y˙d-μ˙1z1μ1)dt+p1g1(x)dw=p1(μ˙1z1μ1z2+α1+e2+θ1Tφ1(x^1)+θ~1Tφ1(x^1)+ε1+d1-y˙d+Δf1-μ˙1z1μ1)dt+p1g1(x)dw.
Choose the intermediate control function α1 and the adaptation law for θ1 as follows:
(21)α1=-c1z1p1-94z1p11/3-34z1p13-z13p1-θ1Tφ1(x^1)+y˙d+μ˙1z1μ1,(22)θ˙1=η1z13p1φ1(x^1)-σ1θ1,
where c1>0, σ1>0 and η1>0 are design parameters and θ1 is the estimate of θ1*.
Step i(2≤i≤n-1). Similar to Step 1, we have
(23)dzi=[∑l=1i-1∂αl-1∂yd(l-1)ydl-∂α1∂yy˙-μ˙iziμipi(∑l=1i-1∂αl-1∂yd(l-1)ydl-∂α1∂yy˙-μ˙iziμizi+1+αi+kie1+θiTφ(x-^i)hhhhh-∑l=1i-1∂αl-1∂x^lx^˙l-∑l=1i-1∂αl-1∂θlθ˙lhhhhh-∑l=1i-1∂αl-1∂yd(l-1)ydl-∂α1∂yy˙-μ˙iziμi)]dt-pi∂αi-1∂yg1(x)dw=[-Hi-μ˙iziμi)pi(-μ˙iziμizi+1+αi+kie1+θiTφ(x-^i)hhhhh+θ~iTφ(x-^i)-θ~iTφ(x-^i)-∂α1∂yhhhhh×[θ~1Tφ1(x^1)+e2+ε1+Δf1+d1]hhhhh-Hi-μ˙iziμi)]dt-pi∂αi-1∂yg1(x)dw,
where
(24)Hi=-∑l=1i-1∂αl-1∂x^l[x^l+1+θlTφl(x-^l)+kle1]-∂α1∂θ1θ˙1-∂α1∂yd(l-1)y˙d-∂α1∂y[x^2+θ1Tφ1(x^1)].
Choose intermediate control function αi and adaptation law θi as
(25)αi=-cizipi-kie1-H-i-θiTφi(x-^i)-34zipi1/3-14zi+μ˙iziμi,(26)θ˙i=ηizi3piφi(x-^i)-σiθi,
where ci>0, σi>0 and ηi>0 are design parameters and θi is the estimate of θi*, and
(27)H-i=Hi+(∂αi-1∂x1)2zi3pi+34zipi3(∂αi-1∂y)4+12zi3pi+14zi3pi(∂2αi-1∂y2)2.
Step n. In the final design step, the actual control input u will be designed. Similar to Step i we have
(28)dzn=pnμ˙nzn(t)μn(u+kne1+θnTφn(x-^n)-α˙n-1-μ˙nzn(t)μn)dt-pn∂αn-1∂yg1(x)dw.
The controller u and adaptation law θn are chosen as
(29)u=-cnznpn-kne1-θnTφn(x-^n)-H-n-14zn+μ˙nznμn,(30)θ˙n=ηnzn3pnφn(x-^n)-σnθn,
where cn>0, σn>0 and ηi>0 are design parameters and θn is the estimate of θn*.
5. Stability Analysis
Consider the total Lyapunov candidate functions V as the sum of local Lyapunov candidate functions V0 and Vi, namely, V=V0+Vi, with V0=(1/2)eTPe, and Vi=∑i=1n((1/4)zi4+(1/2ηi)θ~iTθ~i), where e=x--x-^ is the observer error vector, ηi is positive design constant, and θ~i=θi*-θi.
Theorem 5.
For the stochastic nonlinear system (1), if Assumptions 1–3 are satisfied, the controller (29) with the state observer (18), together with the intermediate control functions (21) and (25), and adaptation laws (22), (26), and (30) can guarantee that all signals in the closed-loop system are semiglobally uniformly ultimately bounded in probability, and the tracking error remains in a neighborhood of the origin within the prescribed performance bounds for all t≥0.
Proof.
The infinitesimal generator of V is
(31)ℓV=ℓV0+ℓVi.
From (13) and (18), we have the observer error equation
(32)de=(Ae+∑i=1nBi[fi(x-^i)-f^i(x-^i∣θi)hhhhhhhhhhh+Δfi+di(x-i)-f^i(x-iθi)]∑i=1nBi)dt+g(x)dw=(Ae+∑i=1nBi[εi+θ~iTφi(x-^i)hhhhhhhhhhh+Δfi+diεi+θ~iTφi]∑i=1nBi)dt+g(x)dw=(+∑i=1nBiθ~iTφi(x-i)Ae+ε+d+Δf+∑i=1nBiθ~iTφi(x-^i))dt+g(x)dw,
where Δf=[Δf1,…,Δfn]T, ε=[ε1,…,εn]T, d=[d1,…,dn]T, θ~i=θi*-θi.
The infinitesimal generator of V0 along with (32) is
(33)ℓV0≤-λmin(Q)∥e∥2+eTP(ε+d+Δf)+∑i=1neTPBiθ~iTφi(x-^i)+Tr[σgTPgσT].
By Young’s inequality, Assumptions 1–3, we have
(34)eTP(d+ε+Δf)≤32∥e∥2+12∥P∥2∥ε*∥2+12∥P∥2∥d*∥2+12∥P∥2∥Δf∥2≤(32+12∥P∥2∑i=1nmi2)∥e∥2+12∥P∥2∥ε*∥2+12∥P∥2∥d*∥2,Tr[σgTPgσT]≤12∥P∥2+12|σ-σ-T|2,
where ε*=[ε1*,…,εn*]T, d*=[d1*,…,dn*]T.
Note that φiT(x-^i)φi(x-^i)≤1; by Young’s inequality, we have
(35)eTP∑i=1nBiθ~iTφi(x-^i)≤14eTPPTe+∑i=1nθ~iTφi(x-^i)φiT(x-^i)θ~i≤14λmax2(P)∥e∥2+∑i=1nθ~iTθ~i,
where λmax(P) is the largest eigenvalue of P.
Substituting (34)-(35) into (33) gives
(36)V˙0≤-q0∥e∥2+∑i=1nθ~iTθ~i+λ0,
where q0=λmin(Q)-((3/2)+(1/2)∥P∥2∑i=1nmi2+(1/4)λmax2(P)), λ0=(1/2)∥P∥2∥ε*∥2+(1/2)∥P∥2∥d*∥2+(1/2)∥P∥2+(1/2)|σ-σ-T|2, and λmin(Q) is the minimal eigenvalue of Q.
From (19), (20), (23), and (28) we have
(37)z13z˙1=z13p1(x2+f1(x1)+d1-y˙d-μ˙1z1μ1)+32z12p12g1TσσTg1=z13p1(μ˙1z1μ1z2+α1+e2+θ1Tφ1(x^1)+θ~1Tφ1(x^1)hhhhhhh+ε1+d1-y˙d+Δf1-μ˙1z1μ1)+32z12p12g1TσσTg1,zi3z˙i=zi3pi(∂2αi-1∂y2g1TσσTg1-μ˙iziμizi+1+αi+kie1-∂α1∂yhhhhhhh×[θ~1Tφ1(x^1)+e2+Δf1+d1+ε1]hhhhhhh+θiTφi(x-^i)+θ~iTφi(x-^i)-θ~iTφi(x-^i)-Hihhhhhhh-12∂2αi-1∂y2g1TσσTg1-μ˙iziμi)+32pi2zi2(∂αi-1∂y)2g1TσσTg1,zn3z˙n=zn3pnμ˙nzn(t)μn(θnTφn(x-^n)+u+14zn+kne1-H-n-μ˙nzn(t)μn)+θ~nTφn(x-^n)-θ~nTφn(x-^n)+32pn2zn2(∂αn-1∂y)2g1TσσTg1.
By Young’s inequality and Assumptions 1–3, we have
(38)z13p1z2+z13p1e2+z13p1ε1+z13p1d1+z13p1Δf1≤34z14p14/3+14z24+12z16p12+12∥e∥2+34z14p14/3+14ε1*4+34z14p14/3+14d1*4+12z16p12+12Δf12≤94z14p14/3+z16p12+14z24+14ε1*4+14d1*4+12∥e∥2+12∥e∥2m12,(39)32z12p12g1TσσTg1≤34z14p14+34|σ-σ-T|2,(40)zi3pizi+1-zi3piθ~iTφi(x-^i)≤34zi4pi4/3+14zi+14+12zi6pi2+12θ~iTθ~i,(41)-zi3pi∂αi-1∂x1e2≤12∥e∥2+12(∂αi-1∂x1)2zi6pi2,(42)-zi3pi∂αi-1∂x1θ~1Tφ1(x^1)≤12θ~1Tθ~1+12(∂αi-1∂x1)2zi6pi2,(43)-zi3pi∂αi-1∂x1[ε1+Δf1+d1]≤32(∂αi-1∂x1)2zi6pi2+12ε1*2+12d1*2+m122∥e∥2,(44)-12zi3pi∂2αi-1∂y2g1TσσTg1+32pi2zi2(∂αi-1∂y)2g1TσσTg1≤14zi6pi2(∂2αi-1∂y2)2+14|σ-σ-|2+34pi4zi4(∂αi-1∂y)4+34|σ-σ-T|2.
From (21)-(22), (25)-(26), (29)-(30), and (41)–(44), we have
(45)ℓV≤-qn∥e∥2-∑i=1ncizi4pi2+∑i=1nσiηiθ~iTθi+2∑i=1nθ~iTθ~i+n-12θ~1Tθ~1+λn,
where qn=q0-(n/2)-(nm12/2), λn=λ0+(n/2)ε12+(n/2)d1*2+n|σ-σ-|2.
Note that
(46)∑i=1nσiηiθ~iTθi≤-12∑i=1nσiηiθ~iTθ~i+12∑i=1nσiηiθi*Tθi*.
Substituting the above inequality into (52) gives
(47)ℓV≤-qn∥e∥2-∑i=1ncizi4pi2-∑i=2n(σi2ηi-2)θ~iTθ~i-(σ12η1-n+32)θ~1Tθ~1+λ,
where D=∑i=1n(σi/2ηi)θi*Tθi*+λn. Let qn>0, ci>0, (σi/2ηi)>1, and define
(48)C=min{2qnλmin(P),4cipi2,(i=1,…,n),σ1-(n+3)η1,2(σi-4ηi),(i=2,…,n)2qnλmin}.
Then (47) can be written as
(49)ℓV≤-CV+D.
Multiplying V by eCt and by Itô formula leads to
(50)d(eCtV)=eCt(CV+ℓV)dt+eCtΩ1dw,
where Ω1=(∂V/∂z1)g1(x)-∑i=2n(∂V/∂zi)(∂αi-1/∂y)g1(x)+(∂V/∂e)g(x).
From (49) and (50), we have
(51)d(eCtV)≤eCtDdt+eCtΩ1dw.
Integrating (51) over [0,T], we get
(52)V(T)≤eCTV(0)+DC+eCT∫0TeCsΩ1dw(s).
Taking expectation on (52), it follows that
(53)E[V(T)]≤EV(0)e-CT+DC,
where E(·) is probability expectation.
The above inequality means that E[V(T)] is bounded by D/C in mean square. Thus, according to [12–18], 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 t≥0.
6. Simulation Study
In this section, a simulation example is provided to evaluate the control performance of the proposed adaptive output-feedback control method.
Consider a stochastic system governed by the following form:
(54)dx1=[x2+f1(x1)]dt+g1(x)dw,dx2=[f2(x1,x2)]dt+u+g2(x)dw,y=x1,
where f1(x1)=sin(x12), f2(x1,x2)=x1sin(x22)-x1e0.5x1, g1(x)=sin(x1)/(1+0.5cos(x2)), g2(x)=x1x2/(1+(x1x2)2). w˙(t) is assumed to be a Gaussian white noise with zero mean and variance 1.0. The tracking reference signal is chosen as yd(t)=sin(t).
Choose fuzzy membership functions as
(55)μFil(x^i)=exp[-(x^i-3+l)216],l=1,2,3,4,5.
Construct the FLSs f^i(x-^i∣θi)=θiTφi(x-^i) to appreciate the unknown nonlinear functions fi(·), i=1,2.
Choose the design parameters and performance functions as k1=0.8, k2=10, c1=0.01, c2=1, η1=η2=0.01, μ1,0=2, μ1,∞=0.5, n1=0.5, σ1=σ2=0.01, δ1min=0.01, δ1max=0.02, and μ1(t)=1.5e-0.5t+0.5.
The initial conditions are chosen as follows: x1(0)=0, x2(0)=0.1, x^1(0)=0, x^2(0)=-0.1, θ1T(0)=[0,0,-0.1,0,0], and θ2T(0)=[0,0,0,-0.1,0].
Applying the control method in this paper to control (54), the simulation results are shown by Figures 1–4, where Figure 1 expresses the curves of the output yand tracking signal yd; Figure 2 expresses the curves of the observer error e1 and e2; Figure 3 expresses the curve of the control input u. Figure 4 express the curve the tracking error of the proposed control method. Figure 4 reveals that the evolution of the proposed adaptive controller remains within the prescribed performance bounds for all t≥0; that is, the prescribed performance is satisfied.
The curves of y (solid line) and yd (dot line).
The curves of e1 (solid line) and e2 (dot line).
The curve of u.
The curves of z1 and performance bounds.
7. 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.
Conflict of Interests
None of the authors of the paper have declared any conflict of interests.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 61374113, 61074014, and 61203008) and Liaoning Innovative Research Team in University (LT2012013).
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