The problem of fuzzy-based direct adaptive tracking control is considered for a class of pure-feedback stochastic nonlinear systems. During the controller design, fuzzy logic systems are used to approximate the packaged unknown nonlinearities, and then a novel direct adaptive controller is constructed via backstepping technique. It is shown that the proposed controller guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. The main advantages lie in that the proposed controller structure is simpler and only one adaptive parameter needs to be updated online. Simulation results are used to illustrate the effectiveness of the proposed approach.
1. Introduction
During the past decades, many control methods have been developed to control design of nonlinear systems, such as adaptive control [1–3], backstepping control [4], and fault tolerant control [5–9]. Particularly, backstepping-based adaptive control has been an effective tool to deal with the control problem of nonlinear strict-feedback systems without satisfying matching condition. So far, many interesting results have been obtained for deterministic nonlinear systems in [4, 10–21] and for stochastic cases in [22–35]. In the aforementioned papers, however, the existing results required that the nonlinear functions be in the affine forms; that is, systems are characterized by input appearing linearly in the system state equation.
Pure-feedback nonlinear systems, which have no affine appearance of the state variables that can be used as virtual control and the actual control input, stand for a more representative form than strict-feedback systems. Many practical systems are in nonaffine structure, such as biochemical process [4] and mechanical systems [36]. Therefore, the study on stability analysis and controller synthesis for pure-feedback nonlinear system is important both in theory and in practice applications [37–42]. By combining adaptive neural control and backstepping, in [37, 38], a class of pure-feedback systems was investigated, which contained only partial nonaffine functions and at least a control variable or virtual control signal existed in affine form. In [39], an “ISS-modular” approach combined with the small-gain theorem was presented for adaptive neural control of completely nonaffine pure-feedback systems. Recently, in [41], an adaptive neural tracking control scheme was presented for a class of nonaffine pure-feedback systems with multiple unknown state time-varying delays. On the other hand, it is well known that stochastic disturbance often exists in practical systems and is usually a source of instability of control systems. So, the consideration of the control design for pure-feedback nonlinear systems with stochastic disturbance is a meaningful issue and has attracted increasing attention in the control community in recent years [43–45]. In [43], the problem of adaptive fuzzy control for pure-feedback stochastic nonlinear systems has been reported. Then, Yu et al. [44] presented an adaptive neural controller to guarantee the four-moment boundedness for a class of uncertain nonaffine stochastic nonlinear system with time-varying delays. However, the considered systems in [43–45] are of a special kind of pure-feedback stochastic nonlinear systems in which only the last equation was viewed as nonaffine equation.
Motivated by the above observations, we will develop a novel fuzzy-based direct adaptive tracking control scheme for a class of pure-feedback stochastic nonlinear systems. The presented controller guarantees that all the signals in the closed-loop system remain bounded in probability and the tracking error converges to a small neighborhood around the origin in the sense of mean quartic value. The main contributions of this paper lie in that the structure of the proposed controller is simpler and only one adaptive parameter needs to be updated online. As a result, the computational burden is significantly alleviated and the control scheme may be more implemented in practice.
The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. An adaptive fuzzy tracking control scheme is presented in Section 3. The simulation example is given in Section 4, and it is followed by Section 5 which concludes the work.
2. Problem Formulation and Preliminaries
To introduce some useful conceptions and lemmas, consider the following stochastic system:
(1)dx=f(x,t)dt+h(x,t)dw,∀x∈Rn,
where x∈Rn is state vector, w is a r-dimensional independent standard Brownian motion defined on the complete probability space (Ω,F,{Ft}t≥0,P) with Ω being a sample space, F being a σ-field, {Ft}t≥0 being a filtration, and P being a probability measure, and f:Rn×R+→Rn, h:Rn×R+→Rn×r are locally Lipschitz functions in x and satisfy f(0,t)=0, h(0,t)=0.
Definition 1 (see [25]).
For any given V(x,t)∈C2,1, associated with the stochastic differential equation (1), define the differential operator L as follows:
(2)LV=∂V∂t+∂V∂xf+12Tr{hT∂2V∂x2h},
where Tr(A) is the trace of A.
Remark 2.
The term (1/2)Tr{hT(∂2V/(∂x2)h} is called Itô correction term, in which the second-order differential ∂2V/∂x2 makes the controller design much more difficult than that of the deterministic system.
Definition 3 (see [46]).
The trajectory {x(t), t≥0} of stochastic system (1) is said to be semiglobally uniformly ultimately bounded in pth moment, if, for some compact set Ω∈Rn and any initial state x0=x(t0), there exist a constant ɛ>0 and a time constant T=T(ɛ,x0) such that E(|x(t)|p)<ɛ, for all t>t0+T. Particularly, when p=2, it is usually called semiglobally uniformly ultimately bounded in mean square.
Lemma 4 (see [46]).
Suppose that there exist a C2,1 function V(x,t):Rn×R+→R+, two constants c1>0 and c2>0, and class K∞-functions α¯1 and α¯2 such that
(3)α¯1(|x|)≤V(x,t)≤α¯2(|x|),LV(x,t)≤-c1V(x,t)+c2,
for all x∈Rn and t>t0. Then, there is a unique strong solution of system (1) for each x0∈Rn and it satisfies
(4)E[V(x,t)]≤V(x0)e-c1t+c2c1,∀t>t0.
In this paper, we consider a class of pure-feedback stochastic nonlinear systems described by
(5)dxi=fi(x¯i,xi+1)dt+hiT(x¯i)dw,1≤i≤n-1,dxn=fn(x¯n,u)dt+hnT(x¯n)dw,y=x1,
where x and w are defined in (1), u∈R and y∈R represent the control input and the system output, and respectively, x¯i=[x1,x2,…,xi]T∈Ri, fi(·):Ri+1→R, hi(·):Ri→Rr (i=1,2,…,n) are unknown smooth nonlinear functions.
The control objective is to design a fuzzy-based adaptive tracking control law u for system (5) such that the system output y follows a desired reference signal yd in the sense of mean quartic value, and all signals in the closed-loop system remain bounded in probability. To this end, define y¯d(i)=[yd,yd(1),…,yd(i)]T, i=1,2,…,n, where yd(i) denotes the ith time derivative of yd.
For the system (5), define
(6)gi(x¯i,xi+1)=∂fi(x¯i,xi+1)∂xi+1,i=1,2,…,n,
where xn+1=u.
Assumption 5 (see [42]).
The signs of gi(x¯i,xi+1), i=1,2,…,n, are known, and there exist unknown constants bm and bM such that for 1≤i≤n,
(7)0<bm≤|gi(x¯i,xi+1)|≤bM<∞,∀(x¯i,xi+1)∈Ri×R.
Assumption 6 (see [17]).
The reference signal yd(t) and its nth order derivatives are continuous and bounded.
In this note, fuzzy logic system will be used to approximate a continuous function f(x) defined on some compact sets. Adopt the singleton fuzzifier, the product inference, and the center-average defuzzifier to deduce the following fuzzy rules:
: IF x1 is F1i and x2 is F2i and … and xn is Fni.
Then y is Bi (i=1,2,…,N),
where x=[x1,…,xn]T∈Rn and y∈R are the input and output of the fuzzy system, respectively. Fij and Bi are fuzzy sets in R. Since the strategy of singleton fuzzification, center-average defuzzification, and product inference is used, the output of the fuzzy system can be formulated as
(8)y(x)=∑j=1NW¯j∏i=1nμFij(xi)∑j=1N[∏i=1nμFij(xi)],
where W¯j is the point at which fuzzy membership function μBj(W¯j) achieves its maximum value, which is assumed to be 1. Let
(9)sj(x)=∏i=1nμFij(xi)∑j=1N[∏i=1nμFij(xi)],S(x)=[s1(x),…,sN(x)]T, and W=[W¯1,…,W¯N]T. Then, the fuzzy logic system can be rewritten as
(10)y(x)=WTS(x).
If all memberships are chosen as Gaussian functions, the lemma below holds.
Lemma 7 (see [47]).
Let f(x) be a continuous function defined on a compact set Ω. Then, for any given constant ɛ>0, there exists a fuzzy logic system (10) such that
(11)supx∈Ω|f(x)-WTS(x)|≤ɛ.
The following lemma will be used in this note.
Lemma 8 (Young’s inequality [23]).
For ∀(x,y)∈R2, the following inequality holds:
(12)xy≤ɛpp|x|p+1qɛq|y|q,
where ɛ>0, p>1, q>1, and (p-1)(q-1)=1.
3. Adaptive Fuzzy Control Design
In this section, a fuzzy-based adaptive tracking control scheme is proposed for the system (5). The backstepping design procedure contains n steps and is developed based on the coordinate transformation zi=xi-αi-1, i=1,2,…,n, where α0=yd. The virtual control signal and the adaption law will be constructed in the following forms:
(13)αi=-12ai2zi3θ^,(14)θ^˙=∑i=1nσ2ai2zi6-γθ^,
where ai (i=1,2,…,n), σ, and γ are positive design parameters and Z1=[x1,yd,y˙d]T∈ΩZ1⊂R3, Zi=[x¯iT,θ^,y¯d(i)T]T∈ΩZi⊂R2i+2(i=2,…,n).θ^ is the estimation of unknown constant θ which is specified as
(15)θ=max{bM2bm∥Wi∥2;i=1,2,…,n}.
Specially, αn is the actual control input u(t).
For simplicity, in the following, the time variable t and the state vector x¯i are omitted from the corresponding functions, and let Si(Zi)=Si.
Step 1.
Since z1=x1-yd, the error dynamic satisfies
(16)dz1=(f1(x1,x2)-y˙d)dt+h1Tdw.
Consider a Lyapunov function candidate as
(17)V1=14z14+bm2σθ~2,
where θ~=θ-θ^ is the parameter error. By (2) and the completion of squares, one has
(18)LV1≤z13(f1(x1,x2)-y˙d)+3z122h1Th1-bmσθ~θ^˙≤z13(f1(x1,x2)-y˙d+34l1-2z1∥h1∥4)+34l12-bmσθ~θ^˙,
where l1 is a constant. Define a new function
(19)w1=-y˙d+32z1+k1z1+34l1-2z1∥h1∥4,
then (18) can be rewritten as
(20)LV1≤z13(f1(x1,x2)+w1)-(k1+32)z14+34l12-bmσθ~θ^˙.
Based on Assumption 5 and the fact of ∂w1/∂x2=0, one has
(21)∂[f1(x1,x2)+w1]∂x2≥bm>0.
According to Lemma 1 in [37], for each value of x1 and w1, there exists a smooth ideal control input x2=α¯1(x1,w1) such that
(22)f1(x1,α¯1)+w1=0.
Applying mean value theorem [48], there exists μ1 (0<μ1<1) which makes
(23)f1(x1,x2)=f1(x1,α¯1)+gμ1(x2-α¯1),
where gμ1:=g1(x1,xμ1)=(∂f1(x1,x2)/∂x2)|x2=xμ1, xμ1=μ1x2+(1-μ1)α¯1.
Apparently, Assumption 5 on g1(x1,x2) is still valid for gμ1. Substituting (23) into (20) and applying the result (22) and z2=x2-α1 results in
(24)LV1≤z13gμ1(x2-α¯1)-(k1+32)z14+34l12-bmσθ~θ^˙≤z13gμ1z2+z13gμ1(α1-α¯1)-(k1+32)z14+34l12-bmσθ~θ^˙.
Since α¯1 contains the unknown function h1, α¯1 cannot be directly implemented in practice. According to Lemma 7, there exists a fuzzy logic system W1TS1(Z1) which can model the unknown function α¯1 over a compact set ΩZ1⊂R3, such that, for any given ɛ1>0,
(25)α¯1=W1TS1(Z1)+δ1(Z1),|δ1(Z1)|≤ɛ1,
where δ1(Z1) is approximation error. Then, based on Young’s inequality, it follows that
(26)-z13gμ1α¯1≤bmz162a12∥W1∥2bmS1TS1+a122+34z14+bM44ɛ14≤bm2a12z16θ+12a12+34z14+14bM4ɛ14,
where the unknown constant θ is defined in (15) and the property of S1TS1≤1 is used in (26). By choosing virtual control signal α1 in (13) with i=1, and using the property of θ^≥0 and Assumption 5, the following result holds:
(27)z13gμ1α1≤-bm2a12z16θ^.
Substituting (26) and (27) into (20) and using Young’s inequality to the term z13gμ1z2 produces
(28)LV1≤-k1z14+14bM4z24+ρ1+bmσθ~(σ2a12z16-θ^˙),
with ρ1=(1/2)a12+(1/4)bM4ɛ14+(3/4)l12.
Step 2.
Based on the coordinate transformation z2=x2-α1 and Itô formula, one has
(29)dz2=(f2(x¯2,x3)-ℓα1)dt+(h2-∂α1∂x1h1)Tdw
with
(30)ℓα1=∂α1∂x1f1(x1,x2)+∑j=01∂α1∂yd(j)yd(j+1)+∂α1∂θ^θ^˙+12∂2α1∂x12h1Th1.
Take the following Lyapunov function:
(31)V2=V1+14z24.
Then, by using a similar procedure as Step 1, it follows that
(32)LV2=LV1+z23(f2(x¯2,x3)-ℓα1)+32z22(h2-∂α1∂x1h1)T(h2-∂α1∂x1h1).
It is noticed that
(33)32z22∥h2-∂α1∂x1h1∥2≤34l22z24∥h2-∂α1∂x1h1∥4+3l224,
where l2 is a constant.
Furthermore, it can be verified by substituting (28) and (33) into (32) that
(34)LV2≤-k1z14+bmσθ~(σ2a12z16-θ^˙)+z23(f2(x¯2,x3)-ℓα1+3z24l22∥h2-∂α1∂x1h1∥4+14bM4z2∥h2-∂α1∂x1h1∥4)+ρ1+34l22.
From the definition of θ^˙ in (14), we have
(35)∂α1∂θ^θ^˙=∂α1∂θ^(∑j=12σ2aj2zj6-γθ^)+∂α1∂θ^∑j=3nσ2aj2zj6.
By combining (30), (34), and (35), the following result holds:
(36)LV2≤-k1z14+ρ1+z23(f2(x¯2,x3)+w2)-(k2+32)z24+bmσθ~(σ2a12z16-θ^˙)-∂α1∂θ^z23∑j=3nσ2aj2zj6+34l22,
where
(37)w2=bM44z2-∂α1∂x1f2(x1,x2)-∑j=01∂α1∂yd(j)yd(j+1)-12∂2α1∂x12h1Th1+34l2-2z2∥h2-∂α1∂x1h1∥4-∂α1∂θ^(∑j=12σ2aj2zj6-σθ^)+(k2+32)z2.
It can be seen from (37) that ∂w2/∂x3=0. Then, from Assumption 5, we have
(38)∂[f2(x¯2,x3)+w2]∂x3≥bm>0.
According to Lemma 1 in [37], there exists a smooth ideal control input x3=α¯2(x¯2,w2) such that
(39)f2(x¯2,α¯2)+w2=0.
By applying mean value theorem [48], there exists μ2 (0<μ2<1) such that
(40)f2(x¯2,x3)=f2(x¯2,α¯2)+gμ2(x3-α¯2),
where gμ2=(∂f2(x¯2,x3)/∂x3)|x3=xμ2,xμ2=μ2x3+(1-μ2)α¯2. Assumption 5 on g2(x¯2,x3) is still valid for gμ2.
Substituting (39) and (40) into (36) produces
(41)LV2≤-k1z14+ρ1+z23gμ2z3+z23gμ2(α2-α¯2)-(k2+32)z24+bmσθ~(σ2a12z16-θ^˙)-∂α1∂θ^z23∑j=3nσ2aj2zj6+34l22,
where z3=x3-α2.
Further, fuzzy logic system W2TS(Z2) is used to approximate the desired unknown virtual signal α¯i over a compact set ΩZ2⊂R6 such that, for any given positive constant ɛi, α¯2 can be shown as
(42)α¯2=W2TS2(Z2)+δ2(Z2),|δ2(Z2)|≤ɛ2,
with δ2(Z2) being the approximation error.
Then, constructing the virtual control signal α2 in (13) and repeating the same methods used in (26) and (27), one has
(43)-z23gμ2α¯2≤bm2a22z26θ+12a22+34z24+14bM4ɛ24,z23gμ2α2≤-bm2a22z26θ^.
By taking (43) into account and using Young’s inequality to the term z23gμ2z3, (41) can be rewritten as
(44)LV2≤∑j=12(-kjzj4+ρj)+bmθ~σ(∑j=12σ2aj2zj6-θ^˙)-∂α1∂θ^z23∑j=3nσ2aj2zj6+14bM4z34,
where ρj=(1/2)aj2+(1/4)bM4ɛj4+(3/4)lj2,j=1,2.
Remark 9.
The adaptive law θ^˙ defined in (14) is a function of all the error variables. So, unlike the conventional approximation-based adaptive control schemes, (∂α1/∂θ^)θ^˙ in (30) cannot be approximated directly by fuzzy logic system W2TS2(Z2). To solve this problem, in (35), (∂α1/∂θ^)θ^˙ is divided into two parts. The first term in the right hand side of (35) is contained in α¯2 to be modeled by W2TS2(Z2), and the last term in (35) which is a function of the latter error variables, namely, zj, j=3,4,…,n, will be dealt with in the later design steps. This design idea will be repeated at the following steps.
Step i (3≤i≤n-1). From zi=xi-αi-1 and Itô formula, we have
(45)dzi=(fi(x¯i,xi+1)-ℓαi-1)dt+(hi-∑j=1i-1∂αi-1∂xjhj)Tdw,
where
(46)ℓαi-1=∑j=1i-1∂αi-1∂xjfj(x¯j,xj+1)+∑j=0i-1∂αi-1∂yd(j)yd(j+1)+∂αi-1∂θ^θ^˙+12∑p,q=1i-1∂2αi-1∂xp∂xqhqThp.
Choose a Lyapunov function as
(47)Vi=Vi-1+14zi4.
Then, it follows that
(48)LVi=LVi-1+zi3(fi(x¯i,xi+1)-ℓαi-1)+32zi2(hi-∑j=1i-1∂αi-1∂xjhj)T×(hi-∑j=1i-1∂αi-1∂xjhj),
where the term LVi-1 in (48) can be obtained in the following form by straightforward derivations as those in former steps:
(49)LVi-1≤∑j=1i-1(-kjzj4+ρj)+bmθ~σ(∑j=1i-1σ2aj2zj6-θ^˙)-∑m=1i-2∂αm∂θ^zm+13∑j=inσ2aj2zj6+bM44zi4,
where ρj=(1/2)aj2+(1/4)bM4ɛj4+(3/4)lj2.
According to the definition of θ^˙ in (14), one has
(50)∂αi-1∂θ^θ^˙=∂αi-1∂θ^(∑j=1iσ2aj2zj6-γθ^)+∂αi-1∂θ^∑j=i+1nσ2aj2zj6.
Then, substituting (49) into (48) and taking (50) into account, (48) can be rewritten as
(51)LVi≤∑j=1i-1(-kjzj4+ρj)+zi3(fi(x¯i,xi+1)+wi)-(ki+32)zi4+bmθ~σ(∑j=1i-1σ2aj2zj6-θ^˙)-∑m=1i-1∂αm∂θ^zm+13∑j=i+1nσ2aj2zj6+34li2,
where
(52)wi=bM44zi-∑j=1i-1∂αi-1∂xjfj(x¯j,xj+1)-∑j=0i-1∂αi-1∂yd(j)yd(j+1)-12∑p,q=1i-1∂2αi-1∂xp∂xqhpThq+34li-2zi∥hi-∑j=1i-1∂αi-1∂xjhj∥4-∂αi-1∂θ^(∑j=1iσ2aj2zj6-σθ^)-σ2ai2zi3∑m=1i-2∂αm∂θ^zm+13+(ki+32)zi.
Noting ∂wi/∂xi+1=0, it follows that
(53)∂[fi(x¯i,xi+1)+wi]∂xi+1≥bm>0.
Based on Lemma 1 in [37], there exists a smooth ideal control input xi+1=α¯i(x¯i,wi) which makes
(54)fi(x¯i,α¯i)+wi=0.
In addition, applying mean value theorem [48], there exists μi (0<μi<1) such that
(55)fi(x¯i,xi+1)=fi(x¯i,α¯i)+gμi(xi+1-α¯i),
where gμi=(∂fi(x¯i,xi+1)/∂xi+1)|xi+1=xμi,xμi=μixi+1+(1-μi)α¯i. Assumption 5 on gi(x¯i,xi+1) is still valid for gμi. Combining (51)–(55) yields
(56)LVi≤∑j=1i-1(-kjzj4+ρj)+zi3gμizi+1+zi3gμi(αi-α¯i)-(ki+32)zi4+bmθ~σ(∑j=1i-1σ2aj2zj6-θ^˙)-∑m=1i-1∂αm∂θ^zm+13∑j=i+1nσ2aj2zj6+34li2,
where zi+1=xi+1-αi. Subsequently, using fuzzy logic system WiTS(Zi) to model α¯i over a compact set ΩZi⊂R2i+2 such that for any given ɛi>0, α¯i can be expressed as
(57)α¯i=WiTSi(Zi)+δi(Zi),|δi(Zi)|≤ɛi,
with δi(Zi) being the approximation error. Furthermore, constructing virtual control signal αi in (13) and following the same line as the procedures used from (26) to (27), one has
(58)-zi3gμiα¯i≤bm2ai2zi6θ+12ai2+34zi4+14bM4ɛi4,zi3gμiαi≤-bm2ai2zi6θ^.
By substituting (58) into (56) and using Young’s inequality, we have
(59)LVi≤∑j=1i(-kjzj4+ρj)+bmθ~σ(∑j=1iσ2aj2zj6-θ^˙)-∑m=1i-1∂αm∂θ^zm+13∑j=i+1nσ2aj2zj6+14bM4zi+14,
where ρj=(1/2)aj2+(1/4)bM4ɛj4+(3/4)lj2.
Step n. In this step, the actual control input signal u will be obtained. By zn=xn-αn-1 and Itô formula, we have
(60)dzn=(fn(x¯n,u)-ℓαn-1)dt+(hn-∑j=1n-1∂αn-1∂xjhj)Tdw,
where ℓαn-1 is defined in (46) with i=n. Choosing the Lyapunov function
(61)Vn=Vn-1+14zn4,
then, the following inequality can be easily verified by using (2), the completion of squares, and taking (59) with i=n-1 into account:
(62)LVn≤∑j=1n-1(-kjzj4+ρj)+bmθ~σ(∑j=1n-1σ2aj2zj6-θ^˙)+zn3(fn(x¯n,u)+wn)-(kn+34)zn4+34ln2,
where
(63)wn=ℓαn-1-∑m=1n-2∂αm∂θ^zm+13σ2an2zn3+14bM4zn+34ln-2zn∥hn-∑j=1n-1∂αn-1∂xjhj∥4+(kn+34)zn.
From the definition of wn, Assumption 5 and Lemma 1 in [37], for every value x¯n and wn, there exists a smooth ideal control input u=α¯n(x¯n,wn) such that fn(x¯n,α¯n)+wn=0. By using mean value theorem [48], there exists μn (0<μn<1) such that
(64)fn(x¯n,u)=fi(x¯n,α¯n)+gμn(u-α¯n),
where gμn=(∂fn(x¯n,u)/∂u)|u=xμn,xμn=μnu+(1-μn)α¯n.
Furthermore, (62) can be rewritten as
(65)LVn≤∑j=1n-1(-kjzj4+ρj)+bmθ~σ(∑j=1n-1σ2aj2zj6-θ^˙)+zn3gμn(u-α¯n)-(kn+34)zn4+34ln2.
Next, using fuzzy logic system WnTS(Zn) to approximate the unknown function α¯n on the compact set ΩZn⊂R2n+2, constructing the actual control u in (13) with i=n, and repeating the same procedures as (58)-(59), one has
(66)LVn≤∑j=1n(-kjzj4+ρj)+bmθ~σ(∑j=1nσ2aj2zj6-θ^˙).
Further, choosing the adaptive law θ^˙ in (14) and using the inequality (γbm/σ)θ~θ^≤-(γbm/2σ)θ~2+(γbm/2σ)θ2 result in
(67)LVn≤-∑j=1nkjzj4-γbm2σθ~2+∑j=1nρj+γbm2σθ2,
where ρj=(1/2)aj2+(1/4)bM4ɛj4+(3/4)lj2,j=1,2,…,n.
Now, the actual control signal u is constructed. The main result will be summarized by the following theorem.
Theorem 10.
Consider the pure-feedback stochastic nonlinear system (5), the controller (13), and adaptive law (14) under Assumptions 5 and 6. Assume that there exists sufficiently large compacts ΩZi,i=1,2,…,n, such that Zi∈ΩZi, for all t≥0; then, for bounded initial conditions [z¯n(0)T,θ^(0)]T∈Ω0 (where Ω0 is an appropriately chosen compact set),
all the signals in the closed-loop system are bounded in probability;
there exists a finite time T1 such that the quartic mean square tracking error enters inside the area for all t>T1,
(68)Ω1={y(t)∈R∣E[|y-yd|4]≤8ρ,∀t>T1},
where the time T1 will be given later.
Proof.
(i) For the stability analysis of the closed-loop system, choose the Lyapunov function as V=Vn. From (67), it follows that
(69)LV≤-a0V+b0,
where a0=min{4kj,γ,j=1,2,…,n} and b0=∑j=1nρj+(γbm/2σ)θ2. Therefore, based on Lemma 1 in [37], zj and θ~ remain bounded in probability. Because θ is a constant, θ^ is bounded in probability. It can be further seen that αj is a function of zj and θ^. So, αj is also bounded in probability. Hence, we conclude that all the signals xj in the closed-loop system (5) remain bounded in the sense of probability.
(ii) From (69), the following inequality can be obtained directly by [24, Theorem 4.1]
(70)dE[V(t)]dt≤-a0E[V(t)]+b0.
Let ρ:=b0/a0>0, then (70) satisfies
(71)0≤E[V(t)]≤V(0)e-a0t+ρ.
Then, it can be easily verified that there exists a time T1=max{0,(1/a0)ln(V(0)/ρ)} such that
(72)E[|y-yd|4]≤4E[V(t)]≤8ρ,∀t>T1.
4. Simulation Results
In this section, to illustrate the effectiveness of the proposed control scheme, we consider the following second-order pure-feedback stochastic nonlinear system:
(73)dx1=(x2+0.05sin(x2))dt+0.1cos(x1)dw,dx2=((2+x121+x12+x22)u+0.1u3)dt+0.2sin(x1x2)dw,y=x1.
The control objective is to design an adaptive controller for the system such that all the signals in the closed-loop system remain bounded and the system output y tracks the given reference signal yd=0.5(sin(t)+sin(0.5t)). According to Theorem 10, choose the virtual control law α1 in (13), actual control input u in (13) with i=2, and adaptive law in (14). The design parameters are taken as follows: a1=a2=0.06, γ=0.0015, and σ=25. The initial conditions are given by [x1(0),x2(0)]T=[0.2,0.5]T and θ^(0)=0.
The simulation results are shown in Figures 1–4. Figure 1 shows the system output y and the reference signal yd. Figure 2 and Figure 3 show that the state variable x2 and adaptive parameter θ^ are bounded. Figure 4 displays the control input signal u.
System output y and reference signal yd.
The state variable x2.
The adaptive parameter θ^.
The true control input u.
5. Conclusions
In this paper, a novel fuzzy-based adaptive control scheme has been presented for pure-feedback stochastic nonlinear systems. The proposed controller guarantees that all the signals in the closed-loop systems are bounded in probability and tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. The main advantages of this control scheme are that the controller is simpler than the existing ones and only one adaptive parameter needs to be estimated online for an n-order system. Numerical results have been provided to show the effectiveness of the suggested approach. Our future research will mainly focus on the multi-input and multioutput (MIMO) pure-feedback stochastic nonlinear systems based on the result in this paper.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by the Natural Science Foundation of China (61304002, 61304003, and 11371071), the Program for New Century Excellent Tallents in University (NECT-13-0696), the Program for Liaoning Innovative Research Team in University under Grant (LT2013023), the Program for Liaoning Excellent Talents in University under Grant (LR2013053), and the Education Department of Liaoning Province under the general project research under Grant (no. L2013424).
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