# Adaptive Neural Output Feedback Control of Stochastic Nonlinear Systems with Unmodeled Dynamics

Authors: Xiaonan Xia; Tianping Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

An adaptive neural output feedback control scheme is investigated for a class of stochastic nonlinear systems with unmodeled dynamics and unmeasured states. The unmeasured states are estimated by K-filters, and unmodeled dynamics is dealt with by introducing a novel description based on Lyapunov function. The neural networks weight vector used to approximate the black box function is adjusted online. The unknown nonlinear system functions are handled together with some functions resulting from theoretical deduction, and such method effectively reduces the number of adaptive tuning parameters. Using dynamic surface control (DSC) technique, Itô formula, and Chebyshev’s inequality, the designed controller can guarantee that all the signals in the closed-loop system are bounded in probability, and the error signals are semiglobally uniformly ultimately bounded in mean square or the sense of four-moment. Simulation results are provided to verify the effectiveness of the proposed approach.


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## Body

## 1. Introduction

During the past decades, backstepping in [1] and dynamic surface control (DSC) in [2] have become two most popular methods for adaptive controller design. Many adaptive control schemes based on fuzzy/neural networks have been proposed for uncertain nonlinear systems using backstepping or dynamic surface control method in [3–13]. In the existing literature, three types of uncertainties were commonly considered, which included unknown system functions and parameter uncertainties and unmodeled dynamics. Unmodeled dynamics was dealt with by introducing an available dynamic signal in [3]. In addition, it was handled by a description method of Lyapunov function in [4]. In [4, 5], adaptive tracking control schemes were developed by backstepping and DSC for a class of strict-feedback uncertain nonlinear systems, respectively. In [7–10], adaptive control schemes were presented for a class of pure-feedback nonlinear systems. In [11–13], the adaptive tracking approaches for single-input single-output (SISO) nonlinear systems were extended to uncertain large-scale nonlinear systems.When system states are assumed to be unmeasurable, output feedback adaptive control based on filters or observers has attracted much attention. In [14], K-filters were firstly proposed, and adaptive output feedback control was developed using K-filters. Inspired by the work in [14], robust adaptive output feedback control schemes were studied for SISO uncertain nonlinear systems in [15, 16]. In [17], combining backstepping technique with small-gain approach, indirect adaptive output feedback fuzzy control was developed. In [18], decentralized adaptive output-feedback control was designed based on high-gain K-filters and dynamic surface control method for a class of uncertain interconnected nonlinear systems.It is well known that due to the stochastic terms and the extra quadratic variation terms resulting from the Itô differentiation rule, both the structures and the controller design of stochastic systems are commonly more complicated than those of deterministic systems. In the past decade, much effort has focused on the study of adaptive control schemes for uncertain stochastic nonlinear systems and the proof of the control system stability in probability sense. In [19–21], Deng et al. proposed the adaptive control scheme, based on backstepping for stochastic strict feedback or output-feedback nonlinear systems, and introduced a control Lyapunov function formula for stochastic disturbance attenuation earlier. In [22], by employing the stochastic Lyapunov-like theorem, adaptive backstepping state feedback control was developed for a class of stochastic nonlinear systems with unknown backlash-like hysteresis nonlinearities. In [23], the problem of decentralized adaptive output-feedback control was discussed for a class of stochastic nonlinear interconnected systems. In [24, 25], output feedback adaptive fuzzy control approaches were considered using backstepping method for a class of uncertain stochastic nonlinear systems. In [26], by combining stochastic small-gain theorem with backstepping design technique, an adaptive output feedback control scheme was presented for a class of stochastic nonlinear systems with unmodeled dynamics and uncertain nonlinear functions. In [27], a concept of stochastic integral input-to-state stability (SiISS) using Lyapunov function was first introduced, and output feedback control was developed for stochastic nonlinear systems with stochastic inverse dynamics. In [28], two linear output feedback control schemes were studied to make the closed-loop system noise-to-state stable or globally asymptotically stable in probability. In [29], by using the homogeneous domination technique and appropriate Lyapunov functions, an output-feedback stabilizing controller was designed to be globally asymptotically stable in probability. In [30], the small-gain control method was investigated for stochastic nonlinear systems with SiISS inverse dynamics. In [31], based on a reduced-order observer, small-gain type condition on SiISS and stochastic LaSalle theorem, an output feedback controller was developed for stochastic nonlinear systems. In [32], an adaptive output feedback control scheme was investigated by combining K-filters with DSC for a class of stochastic nonlinear systems with dynamic uncertainties and unmeasured states. In [33], adaptive control was developed using the backstepping method for a class of stochastic nonlinear systems with time-varying state delays and unmodeled dynamics.Motivated by the above-mentioned results [4, 14, 32], in this paper, adaptive neural stochastic output feedback control is developed by combining K-filters with dynamic surface control to guarantee the stability of the closed-loop system. The main contributions of the paper lie in the following.(i)
Adaptive neural output feedback control is developed using K-filters and dynamic surface control for a class of stochastic nonlinear systems with unmodeled dynamics and unmeasured states. The advantage of the design is that once the local system constructed by the filter signals is stabilized, all the signals in the closed-loop system are bounded in probability.(ii)
Unmodeled dynamics is dealt with first by introducing a novel description based on Lyapunove function without using the dynamic signal to handle dynamic uncertainty in [32]. The novel description, which provides an effective method for dealing with unmodeled dynamics in output feedback adaptive controller design, is the development of original idea about handling unmodeled dynamics in [4].(iii)
Utilizing the boundedness of continuous function, the unknown nonlinear system functions are handled together with some functions produced in stability analysis, rather than directly approximated before stability analysis in [6, 8, 9, 11, 12]. Therefore the design effectively reduces the order of filters and the number of adjustable parameters of the whole system, without estimating Ξ in [32].(iv)
Using bounded input bounded output (BIBO) stability and the filter special structure, the stability of the closed-loop system is proved. Therefore, the difficulty, that the transfer function cannot be used in a stochastic system while it was widely used to analyze the boundedness of the K-filters signals in the deterministic systems in [4, 14, 16–18], is solved by the proposed stability analysis approach in this paper.The rest of the paper is organized as follows. The problem formulation and preliminaries are given in Section2. The neural filters are designed, and adaptive stochastic output feedback control is developed based on dynamic surface control method. The stability in the closed-loop system in probability sense is analyzed in Section 3. Simulation results are presented to illustrate the effectiveness of the proposed scheme in Section 4. Section 5 contains the conclusions.


## 2. Problem Statement and Preliminaries

Consider the following uncertain stochastic nonlinear systems with unmodeled dynamics:(1)dz=qz,ydtdx1=x2+f1y+Δ1z,y,tdt+g1Tydwhhhhhhhhhhhhhhh⋮dxρ-1=xρ+fρ-1y+Δρ-1z,y,tdthhhhhh+gρ-1Tydwdxρ=xρ+1+fρy+Δρz,y,t+bmσyudthhhhi+gρTydwhhhhhhhhhhhhhhh⋮dxn-1=xn+fn-1y+Δn-1z,y,t+b1σyudthhhhhh+gn-1Tydwdxn=fny+Δnz,y,t+b0σyudt+gnTydwy=x1,
where x=[x1,x2,…,xn]T∈Rn is the state; u∈R is the input, and y∈R is the output; σ(y)≠0 is a known positive continuous function; fi(y) is the unknown smooth function; z∈Rn0 is the unmodeled dynamics, and Δi(z,y,t) is the unknown smooth nonlinear dynamic disturbance; b=[bm,…,b1,b0]T∈Rm+1, B(s)=bmsm+⋯+b1s+b0 is a Hurwitz polynomial; Δi(z,y,t) and q(z,y) are the unknown Lipschitz functions; w is an r-dimensional standard Brownian motion defined on the complete probability space (Ω,F,P) with Ω being a sample space, F being a σ field, and P being a probability measure. In this paper, it is assumed that only output y is available for measurement.The control objective is to design output feedback adaptive controlu for system 1 such that the output y follows the specified desired trajectory yd, and all the signals of the closed-loop system are bounded in probability.Assumption 1 (see [4]).
The unknown nonlinear dynamic disturbancesΔi(z,y,t), i=1,2,…,n, satisfy |Δi(z,y,t)|≤ρi1(|y|)+ρi2(y)z, and ρi1(|y|) and ρi2(y) are the unknown nonnegative smooth functions, and · denotes the Euclidian norm of a vector.Assumption 2.
The systemz˙=q(z,0,t)-q(0,0,t) is globally exponentially stable when z=0; that is, there exists a Lyapunov function W(t,z) satisfying
(2)c1z4≤Wz,t≤c2z4,∂W∂tz,t+∂W∂zz,tqz,0,t-q0,0,t≤-c3z4,∂W∂zz,t≤c4z3,
where c1,c2, c3, c4 are positive constants, and there exists c5≥0 such that q(0,0,t)≤c5, ∀t≥0.Assumption 3.
There exists an unknown functionψ0, and ψ0(0)=0, such that q(z,y,t)-q(z,0,t)≤ψ0y holds.Assumption 4.
The desired trajectoryxd=[yd,y˙d,y¨d]T∈Ωd is known, where Ωd={xd:yd2+y˙d2+y¨d2≤B0}, and B0 is a known constant.Assumption 5.
There exists a known constantbmax⁡ such that the following inequality 0<|bm|≤bmax⁡ holds.Remark 6.
Assumption2 is the extension of the description of unmodeled dynamics in [4], and it can effectively deal with unmodeled dynamics in output feedback adaptive controller design. To the best of authors’ knowledge, this assumption is first addressed.Consider the following stochastic nonlinear system:(3)dx=ft,xdt+hTt,xdw,
where x∈Rn is the system state, w is an r-dimensional standard Brownian motion, f:R+×Rn→Rn, hT:R+×Rn→Rn×r are locally Lipschitz and f(t,0), h(t,0) are uniformly ultimately bounded. For any given V(t,x(t))∈C1,2, associated with the stochastic system 3, the infinitesimal generator l is defined as follows:
(4)lVt,xt=∂Vt,xt∂t+∂Vt,xt∂xTf+12tr⁡h∂2Vt,xt∂xT∂xhT,
where tr⁡(A) is the trace of a matrix A.Definition 7 (see [34]).
The stochastic process{x(t)} is said to be bounded in probability, if lim⁡c→∞⁡sup⁡0≤t<∞⁡P(xt>c)=0.Definition 8.
The solutionx(t) of system 3 is said to be semiglobally uniformly ultimately bounded (SGUUB) in pth moment (p≥1), if for some compact set Ω⊂Rn and any initial state x0=x(t0)∈Ω, there exists a constant ɛ>0 and a time constant T=T(ɛ,x0) such that E[x(t)p]≤ɛ for all t>t0+T, especially, when p=2, it is usually called SGUUB in mean square.Lemma 9 (see [32]).
For any stochastic process{ξ(t)}, if there exists a positive integer p and a positive constant C0 such that E|ξ(t)|p≤C0, ∀t≥0, then {ξ(t)} is bounded in probability.Lemma 10 (see [21]).
Consider system3 and suppose that there exists a C2 function V(t,x(t)): Rn×R→R+, two constants c1>0, c2≥0, class κ∞ functions μ1, μ2 such that
(5)μ1x≤Vt,x≤μ2x,lV≤-c1V+c2
for all x∈Rn and t>t0. Then, (i) for any initial state x0∈Rn, there exists a unique strong solution x(t) for system 3; (ii) the solution x(t) of system 3 is bounded in probability; (iii)  E[V(t0,x)]≤V(t0,x0)e-c1t+c2/c1, ∀t≥t0.In order to design filters and observer,1 can be rewritten as follows:
(6)z˙=qz,y,dx=Ax+fy+FTy,ub+Δdt+gTydw,y=e1Tx,
where
(7)A=0In-100,fy=f1y⋮fny,Δz,y,t=Δ1z,y,t⋮Δnz,y,t,e1=1,0,…,0T,FTy,u=0ρ-1×m+1Im+1σyu.


## 3. Adaptive Robust Controller Design and Stability Analysis

### 3.1. Neural Filters and Controller Design

In order to estimate the statex, we introduce the following filters:
(8)ξ˙=A0ξ+Ly,ξ∈Rn,Ω˙T=A0ΩT+FTy,u,ΩT∈Rn×m+1,
where A0=A-Le1T, L=[l1,…,ln]T, A0 is a Hurwitz matrix; that is
(9)PA0+A0TP=-hI,P=PT>0,
where h>0 is a design constant.Define the state estimate as follows:(10)x^=ξ+ΩTb
The observer error is defined as ɛ=x-x^. Thus
(11)x=ξ+ΩTb+ɛ,(12)dɛ=A0ɛ+fy+Δdt+gTydw.
Denote the columns of ΩT as follows:
(13)ΩT=vm,…,v1,v0,ΩT∈Rn×m+1.
Inspired by the work in [14], the filters are designed as follows:
(14)ξ˙=A0ξ+Ly,ξ∈Rn,λ˙=A0λ+enσyu,λ∈Rn,v˙j=A0vj+en-jσyu,vj∈Rn,j=0,1,…,m.
It is easy to show that
(15)A0jen=en-j,j=0,1,…,m,(16)vj=A0jλ,j=0,1,…,m,
where ei denotes n dimensional vector with the ith element being one and other elements being all zeros, i=1,…,n.Letvi,j be the jth element of the vector vi and λl the lth element of the vector λ, respectively. From [14], we know
(17)vi,j=*⋯*1λ1λ2⋮λi+j,j=1,…,ρ;i=0,1,…,m;λl=0,l>n.
According to 11, we get
(18)x2=ξ2+ωTb+ɛ2=ξ2+vm,2,…,v1,2,v0,2b+ɛ2=bmvm,2+ξ2+ω¯Tb¯+ɛ2,
where ωT denotes the second row of the matrix ΩT, ξ2 denotes the second element of the vector ξ, and ɛ2 is the second element of ɛ.Substituting18 into 1, it yields
(19)dy=bmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1z,y,tdt+g1Tydw,
where
(20)ωT=vm,2,vm-1,2,…,v1,2,v0,2,ω¯T=vm-1,2,…,v1,2,v0,2,b¯T=bm-1,…,b1,b0.
In view of 19 and 14, the system used to design adaptive output feedback DSC in next section is addressed as follows:
(21)dy=bmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1dt+g1Tydw,v˙m,i=vm,i+1-livm,1,i=2,…,ρ-1,v˙m,ρ=σyu+vm,ρ+1-lρvm,1.


### 3.2. Stochastic Adaptive Dynamic Surface Controller Design

In this subsection, according to21 and by using dynamic surface control method, we propose an output feedback stochastic adaptive tracking control scheme. Similar to backstepping, the whole design needs ρ steps.For convenience, some notations are presented below.s¯i=[s1,…,si]T, y¯j=[y2,…,yj]T, where si, yj will be given in the controller design later, i=1,2,…,ρ, j=2,…,ρ. yj=ωj-αj-1, j=2,…,ρ, ωj is the output of a first-order filter with αi-1 as the input, and αi-1 is an intermediate control which will be developed for the corresponding (i-1)th subsystem.Define some Lyapunov functions as follows:(22)Vɛ=ɛTPɛ,VW=1λ0Wz,t,λ0>0,Vsi=14si4,VsWɛ=Vs1+Vɛ+VW,
where W(z,t) is given in Assumption 2.Using Young’s inequality, the infinitesimal generator ofVɛ satisfies
(23)lVɛ=ɛTPA0+A0TPɛ+2ɛTPfy+2ɛTPΔ+tr⁡gyPgTy≤ɛTPA0+A0TPɛ+2ɛTɛ+P2fy2+P2Δ2+tr⁡gyPgTy
According to Assumption 1 and by using Young’s inequality, we obtain
(24)lVɛ≤-h-2ɛTɛ+∑j=1nP2fj2y+∑j=1nP2ρj1y+ρj2yz2+tr⁡gyPgTy≤-h-2ɛTɛ+∑j=1nP2fj2y+∑j=1n2P2ρj12y+ρj22yz2+tr⁡gyPgTy≤-h-2ɛTɛ+∑j=1nP2fj2y+∑j=1n2P2ρj12y+16nλ0c3∑j=1nP4ρj24y+c316λ0z4+tr⁡gyPgTy
According to Assumptions 2 and 3, using Young’s inequality, we get
(25)V˙W=1λ0∂W∂zz,tz˙+∂W∂tz,t≤-5c38λ0z4+16c44λ0c33ψ04y+16c44c54λ0c33.Step  1. Let ω1=yd. Define the first dynamic surface as follows:
(26)s1=x1-ω1.
Using the first equation of 21, we obtain
(27)ds1=bmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1-y˙ddt+g1Tydw.Choose the virtual control lawα1 as follows:
(28)α1=b^mb^m2+β-k1s1-ω¯Tb¯^-ξ2-s13θ^1Tψ1X,
where β>0, k1>0 are design constants, θ^1, b^m, b¯^ are the estimates of θ1, bm, b¯ at time t, respectively, and b~m=bm-b^m, θ~1=θ1-θ^1, b¯~=b¯-b¯^, θ1 and ψ1(X) will be given later. Consider
(29)lα1=∂α1∂ybmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1z,y,t+∂α1∂b^mb^˙m+∂α1∂ξTξ˙+∂α1∂θ^1Tθ^˙1+∂α1∂ydy˙d+∂α1∂y˙dy¨d+12∂2α1∂y2g1Tyg1y,dα1=lα1dt+∂α1∂yg1Tydw.
Therefore, we have
(30)lVs1=s13bmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1-y˙d+32s12g1y2.A first-order filter withα1 as the input is designed as follows:
(31)τ2ω˙2+ω2=α1,ω20=α10.
Let y2=ω2-α1; thus, ω˙2=-y2/τ2. Since vm,2=s2+y2+α1, using Young’s inequality, it yields
(32)lVs1≤-k1-32bmax⁡s14+b~ms13α1+bmax⁡4s24+bmax⁡4y24+s13ω¯Tb¯~-s13θ^1Tψ1X-s13βb^m2+β-k1s1-ω¯Tb¯^-ξ2-s13θ^1Tψ1X+s13ɛ2+s13f1y+s13Δ1-s13y˙d+32s12g1y2.
From Assumption 1, we obtain
(33)s13Δ1≤s13ρ11y+s13ρ12yz≤34s14+14ρ114y+32λ02c33s14ρ124/3y≤+c316λ0z4.
In view of 24, 25, 32, and 33 and by using Young’s inequality, we obtain
(34)lVsWɛ≤-h-2ɛTɛ-k1-32bmax⁡-34s14-c32λ0z4+b~ms13α1+s13ω¯Tb¯~+bmax⁡4s24+bmax⁡4y24+s13S+s13H1X+Qy+14ɛ22-s13θ^1Tψ1X+16c44c54λ0c33+1,
where
(35)Qy=∑j=1nP2fj2y+∑j=1n2P2ρj12y+16nλ0c3∑j=1nP4ρj24y+16c44λ0c33ψ04y+14ρ114y+tr⁡gyPgTy,H1X=32λ02c33s1ρ124/3y-y˙d+916s1g1y4+s13,X=s1,yd,y˙dT∈R3.S(s1,b^m,b¯^,θ^1,ξ,λ¯m+2,yd) is a nonnegative continuous function, and
(36)f1y-βb^m2+β-k1s1-ω¯Tb¯^-ξ2-s13θ^1Tψ1x≤Ss1,b^m,b¯^,θ^1,ξ,λ¯m+2,yd,
where λ¯m+2=[λ1,…,λm+2]T.LetΩX=X∣X≤MX⊂R3 be a given compact set with MX>0 being a design constant, and let θ1Tϕ1(X) be the approximation of the radial basis function neural networks on the compact set ΩX to H1(X). Then, we have H1(X)=θ1Tψ1(X)+B1(X), where B1(X) denotes the approximation error and ψ1(X)=[ψ11(X),…,ψ1M1(X)]T∈RM1 denotes the basis function vector with ψ1j(X) being chosen as the commonly used Gaussian functions, which have the form
(37)ψ1jX=exp⁡-X-μ1j2b1j2j=1,…,M1, and μ1j is the center of the receptive field and b1j is the width of the Gaussian function; θ1 is an adjustable parameter vector.According to34 and by using Young’s inequality, it yields
(38)lVsWɛ≤-h-94ɛTɛ-k1-32bmax⁡-32s14-c32λ0z4+b~ms13α1+s13ω¯Tb¯~+bmax⁡4s24+bmax⁡4y24+14S4+s13θ1Tψ1X+B1X+Qy-s13θ^1Tψ1X+16c44c54λ0c33+1.
There exists a nonnegative continuous function κ(s1,yd,y˙d) satisfying
(39)B1X≤κs1,yd,y˙d.
Using Young’s inequality, we have
(40)lVsWɛ≤-h-94ɛTɛ-k1-32bmax⁡-94s14-c32λ0z4+b~ms13α1+s13ω¯Tb¯~+bmax⁡4s24+bmax⁡4y24+14S4+Qy+s13θ~1Tψ1X+14κ4+C0,
where C0=16c44c54/λ0c33+1.Step  i(2≤i≤ρ-1). Define the ith dynamic surface si=vm,i-ωi, thus
(41)s˙i=vm,i+1-livm,1-ω˙i.Select the virtual control lawαi as follows:
(42)αi=-kisi+livm,1+ω˙i,lαi=-kivm,i+1-livm,1-w˙i+liv˙m,1-lyiτi,dαi=lαidt+∂αi∂yg1Tydw.A first-order filter with the inputαi is designed as follows:
(43)τi+1ω˙i+1+ωi+1=αi,ωi+10=αi0,
where τi+1>0 is a design constant.Letyi+1=ωi+1-αi. Then ω˙i+1=-yi+1/τi+1. Noting vm,i+1=si+1+yi+1+αi, in view of 41 and 42, we obtain
(44)lVsi=si3s˙i=si3vm,i+1-livm,1-w˙i=-kisi4+si3si+1+si3yi+1≤-ki-32si4+14si+14+14yi+14.Step  ρ.The control law will be determined in this step. Define the ρth dynamic surface as sρ=vm,ρ-ωρ. The derivative of sρ is
(45)s˙ρ=σyu+vm,ρ+1-lρvm,1-ω˙ρ.Choose the control law as follows:(46)u=-kρsρ-vm,ρ+1+lρvm,1+ω˙ρσy.
In view of 45 and 46, we have
(47)lVsρ=sρ3s˙ρ=-kρsρ4.
The parameters θ^1, b^m, and b¯^ are updated as follows:
(48)θ^˙1=γ1s13ψ1X-σ1θ^1,(49)b^˙m=γ2s13α1-σ2b^m,b¯^˙=γ3s13ω¯-σ3b¯^,
where γ1, γ2, γ3, σ1, σ2, σ3 are the design constants.


### 3.3. Stability Analysis of Adaptive Control System

In this subsection, we will discuss the stability analysis of the closed-loop system. Firstly we define some Lyapunov functions and compact sets as follows:(50)V1=12s14+2Vɛ+1γ1θ~1Tθ~1+1γ2b~m2+1γ3b¯~Tb¯~+c3λ0z4,Vi=∑j=1i12sj4+2Vɛ+∑j=2i12yj4+1γ1θ~1Tθ~1+1γ2b~m2+1γ3b¯~Tb¯~+c3λ0z4,i=2,…,ρ,Ω1=s1,ɛ,θ^1,b^m,b¯^,z:V1≤p⊂Rp1,Ωi=s¯i,y¯i,ɛ,θ^1,b^m,b¯^,z:Vi≤p⊂Rpi,
where i=2,…,ρ, p>0 is a design constant, pi=2i+M1+n+m+1. It is easy to know that Ω1×Rpρ-p1⊃Ω2×Rpρ-p2⊃⋯⊃Ωρ-1×Rpρ-pρ-1⊃Ωρ.According toy2=ω2-α1, we obtain
(51)ly2=ω˙2-lα1=-y2τ2-lα1,dy2=ly2dt-∂α1∂yg1Tydw.
From 4, we obtain
(52)l14y24=y23ly2+32y22∂α1∂y2g1y2=-y24τ2-y23lα1+32y22∂α1∂y2g1y2.
There exist two nonnegative continuous functions η2(s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2, yd,y˙d,y¨d) and ζ2(s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2, yd,y˙d,y¨d) such that
(53)ly2+y2τ2≤η2s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d,(54)32y22∂α1∂y2g1y2≤ζ2s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d.
From 53, we have
(55)y23ly2≤-y24τ2+y23η2s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d≤-y24τ2+34y24+14η24.
From 52, 54, and 55, we obtain
(56)l14y24≤-y24τ2+34y24+14η24+ζ2.
The infinitesimal generator of yi+1 is
(57)lyi+1=-yi+1τi+1-lαi,yi+13lyi+1=-yi+14τi+1-yi+13lαi,(58)lyi+144=-yi+14τi+1-yi+13lαi+32yi+12∂αi∂y2g1y2.
There exist two nonnegative continuous functions ηi+1(s¯i+1,y¯i+1, θ^1,b^m,b¯^, ξ,λ¯m+2,ε2, yd,y˙d,y¨d) and ζi+1(s¯i+1,y¯i+1, θ^1,b^m,b¯^,ξ, λm+2,ε2,yd, y˙d,y¨d) such that the following inequalities hold:
(59)lyi+1+yi+1τi+1≤ηi+1s¯i+1,y¯i+1,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d,(60)32yi+12αi∂y2g1y2≤ζi+1s¯i+1,y¯i+1,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d.
From 59, we obtain
(61)yi+13lyi+1≤-yi+14τi+1+yi+13ηi+1s¯i+1,y¯i+1,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d≤-yi+14τi+1+3yi+144+ηi+144.
From 58, 60, and 61, we obtain
(62)l14yi+14≤-yi+14τi+1+34yi+14+14ηi+14+ζi+1.
The continuous function S(·) on the compact set Ωd×Ω1 has a maximum M(p), which depends on the constant p, and κ(s1,yd,y˙d) on the compact set Ωd×Ω1 has a maximum N0(p), ηi+1(·) and ζi+1(·) on the compact set Ωd×Ωi+1 have the maximum Ni+1(p) and Ci+1(p) when ξ,λ¯m+2 are bounded.Theorem 11.
Consider the closed-loop system consisting of the plant1 under Assumptions 1–5, the controller 46, and the adaptation laws 48 and 49. For any bounded initial conditions, there exist constants ki, τi, h, γ1, γ2, γ3, σ1, σ2, σ3 satisfying V(0)≤c, such that all of the signals in the closed-loop system are bounded in probability, and s1,…,sρ, y2,…,yρ are SGUUB in four-moment, θ~1, b~m, b¯~ are SGUUB in mean square, and ki, τi, and h satisfy
(63)ki≥32bmax⁡+94+14α0,i=1,2,…,ρ,1τi≥14bmax⁡+1+14α0,i=2,…,ρ,h≥94+α0λmax⁡P,α0=min⁡c32c2,γ1σ1,γ2σ2,γ3σ3,
where c>0 is a positive constant; V will be given later in the proof of Theorem 11.Proof.
Choose the following Lyapunov function candidate:(64)V=VsWɛ+∑i=2ρVsi+14∑i=2ρyi4+12γ1θ~1Tθ~1+12γ2b~m2+12γ3b¯~Tb¯~
For any given positive constant, if EV≤c, according to Lemma 9, we obtain that s1,…,sρ, y2,…,yρ, y, θ^1, b^m, b¯^ are bounded in probability. VW≤VsWɛ=Vs1+VW+Vɛ≤V≤c, and, from Assumption 2, we obtain that (c1/λ0)z4≤1/λ0W≤c; that is, z4≤λ0c/c1, so z is bounded in probability.
Furthermore,64 is rewritten as V=1/λ0W+ 1/2Vρ-c3/2λ0z4; then Vρ=2V-2/λ0W+c3/λ0z4≤(2+c3/c1)c, and choosing p=(2+c3/c1)c, we get Vρ≤p.
From14 and 49, we have that ξ, ω¯, α1 are also bounded in probability. It yields that vm-1,2,…,v0,2 are all bounded in probability. Noting vm,2=s2+y2+α1, we obtain that vm,2 is bounded in probability. From 14, we have that v˙0,1=-l1v0,1+v0,2 and v˙m,1=-l1vm,1+vm,2. Thus we obtain that v0,1, vm,1 are also bounded. Furthermore, from 42, we have that αi(i=2,…,ρ-1) are bounded. According to 16 and 17, we obtain
(65)v0,1v0,2v1,2⋮vm-1,2=100⋯0010⋯0**1⋯0⋮⋱⋮**⋯*1λ1λ2λ3⋮λm+1.
Since v0,1,v0,2,v1,2,…,vm-1,2 are bounded in probability, we have that λ1,…,λm+1 are all bounded in probability. From 17, we get that λm+2 is also bounded. In view of 40, 44, 47–49, and 62, using Young’s inequality, we obtain
(66)lV≤-h-94ɛTɛ-∑i=1ρki-32bmax⁡-94si4-∑i=2ρ1τi-14bmax⁡-1yi4-c32λ0z4-σ1θ~122-σ2b~m22-σ3b¯~22+Qy+14N04p+14M4p+14∑i=2ρNi4p+∑i=2ρCip+σ1θ122+σ2bm22+σ3b¯22+C0.
Substituting 63 into 66, we obtain
(67)lV≤-α0V+μ1+μ0,
where μ1=1/4N04(p)+ 1/4M4(p)+1/4∑i=2ρNi4(p) +∑i=2ρCi(p)+ σ1θ12/2+ σ2bm2/2+ σ3b¯2/2+C0. Since |Q(y)| is a nonnegative continuous function, let |Q(y)|≤μ0, where μ0>0.
Ifα0≥(μ1+μ0)/c, and EV=c, then we have dEV/dt≤0. Thus, if EV(0)≤c, then EV(t)≤c, ∀t>0; that is,
(68)0≤EVt≤μ1+μ0α0+V0-μ1+μ0α0e-α0t≤V0.
Similar to the discussion of Theorem 11 in [32], it is easy to know that the conclusion is true.Remark 12.
This paper differs from [32] in the following several aspects. (1) Unmodeled dynamics is dealt with by introducing a novel description based on Lyapunov function in this paper while the dynamic signal was handled with the help of a dynamic signal in [32]. (2) The unknown nonlinear system functions are handled together with some functions produced in stability analysis, but they were directly approximated before constructing the observer in [32]. Therefore, this brings out a good result that the filter order is reduced. (3) The neural networks weight vector used to approximate the black box function at the first design step is adjusted online in this paper such that much more information of weight vector can be used in adaptive law, whereas only the norm of weight vector acts as adaptive tuning parameter in [32]. (4) Utilizing bounded input bounded output stability and linear equations 65, the stability of the closed-loop system is proved in this paper, which avoids using the transfer function to make stability analysis in [32], which is questionable in probability sense.Remark 13.
The design parameterski, τi and α0 determined by 63 in Theorem 11 are only a sufficient condition. They provide a guideline for the designers. From 63, some suggestions are given for the choice of some key design parameters for any given positive constants B0 and c.(i)
Increasingγ1, γ2, γ3 helps to increase α0, subsequently reduces μ1/α0.(ii)
Decreasingσ1, σ2, σ3 helps to reduce μ1 and reduces μ1/α0.(iii)
Increasingk1,…,kρ helps to increase α0 and reduces μ1/α0.In practical applications, to obtain good tracking performance, some experiments need to be done before the valid parameters are given.


## 3.1. Neural Filters and Controller Design

In order to estimate the statex, we introduce the following filters:
(8)ξ˙=A0ξ+Ly,ξ∈Rn,Ω˙T=A0ΩT+FTy,u,ΩT∈Rn×m+1,
where A0=A-Le1T, L=[l1,…,ln]T, A0 is a Hurwitz matrix; that is
(9)PA0+A0TP=-hI,P=PT>0,
where h>0 is a design constant.Define the state estimate as follows:(10)x^=ξ+ΩTb
The observer error is defined as ɛ=x-x^. Thus
(11)x=ξ+ΩTb+ɛ,(12)dɛ=A0ɛ+fy+Δdt+gTydw.
Denote the columns of ΩT as follows:
(13)ΩT=vm,…,v1,v0,ΩT∈Rn×m+1.
Inspired by the work in [14], the filters are designed as follows:
(14)ξ˙=A0ξ+Ly,ξ∈Rn,λ˙=A0λ+enσyu,λ∈Rn,v˙j=A0vj+en-jσyu,vj∈Rn,j=0,1,…,m.
It is easy to show that
(15)A0jen=en-j,j=0,1,…,m,(16)vj=A0jλ,j=0,1,…,m,
where ei denotes n dimensional vector with the ith element being one and other elements being all zeros, i=1,…,n.Letvi,j be the jth element of the vector vi and λl the lth element of the vector λ, respectively. From [14], we know
(17)vi,j=*⋯*1λ1λ2⋮λi+j,j=1,…,ρ;i=0,1,…,m;λl=0,l>n.
According to 11, we get
(18)x2=ξ2+ωTb+ɛ2=ξ2+vm,2,…,v1,2,v0,2b+ɛ2=bmvm,2+ξ2+ω¯Tb¯+ɛ2,
where ωT denotes the second row of the matrix ΩT, ξ2 denotes the second element of the vector ξ, and ɛ2 is the second element of ɛ.Substituting18 into 1, it yields
(19)dy=bmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1z,y,tdt+g1Tydw,
where
(20)ωT=vm,2,vm-1,2,…,v1,2,v0,2,ω¯T=vm-1,2,…,v1,2,v0,2,b¯T=bm-1,…,b1,b0.
In view of 19 and 14, the system used to design adaptive output feedback DSC in next section is addressed as follows:
(21)dy=bmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1dt+g1Tydw,v˙m,i=vm,i+1-livm,1,i=2,…,ρ-1,v˙m,ρ=σyu+vm,ρ+1-lρvm,1.


## 3.2. Stochastic Adaptive Dynamic Surface Controller Design

In this subsection, according to21 and by using dynamic surface control method, we propose an output feedback stochastic adaptive tracking control scheme. Similar to backstepping, the whole design needs ρ steps.For convenience, some notations are presented below.s¯i=[s1,…,si]T, y¯j=[y2,…,yj]T, where si, yj will be given in the controller design later, i=1,2,…,ρ, j=2,…,ρ. yj=ωj-αj-1, j=2,…,ρ, ωj is the output of a first-order filter with αi-1 as the input, and αi-1 is an intermediate control which will be developed for the corresponding (i-1)th subsystem.Define some Lyapunov functions as follows:(22)Vɛ=ɛTPɛ,VW=1λ0Wz,t,λ0>0,Vsi=14si4,VsWɛ=Vs1+Vɛ+VW,
where W(z,t) is given in Assumption 2.Using Young’s inequality, the infinitesimal generator ofVɛ satisfies
(23)lVɛ=ɛTPA0+A0TPɛ+2ɛTPfy+2ɛTPΔ+tr⁡gyPgTy≤ɛTPA0+A0TPɛ+2ɛTɛ+P2fy2+P2Δ2+tr⁡gyPgTy
According to Assumption 1 and by using Young’s inequality, we obtain
(24)lVɛ≤-h-2ɛTɛ+∑j=1nP2fj2y+∑j=1nP2ρj1y+ρj2yz2+tr⁡gyPgTy≤-h-2ɛTɛ+∑j=1nP2fj2y+∑j=1n2P2ρj12y+ρj22yz2+tr⁡gyPgTy≤-h-2ɛTɛ+∑j=1nP2fj2y+∑j=1n2P2ρj12y+16nλ0c3∑j=1nP4ρj24y+c316λ0z4+tr⁡gyPgTy
According to Assumptions 2 and 3, using Young’s inequality, we get
(25)V˙W=1λ0∂W∂zz,tz˙+∂W∂tz,t≤-5c38λ0z4+16c44λ0c33ψ04y+16c44c54λ0c33.Step  1. Let ω1=yd. Define the first dynamic surface as follows:
(26)s1=x1-ω1.
Using the first equation of 21, we obtain
(27)ds1=bmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1-y˙ddt+g1Tydw.Choose the virtual control lawα1 as follows:
(28)α1=b^mb^m2+β-k1s1-ω¯Tb¯^-ξ2-s13θ^1Tψ1X,
where β>0, k1>0 are design constants, θ^1, b^m, b¯^ are the estimates of θ1, bm, b¯ at time t, respectively, and b~m=bm-b^m, θ~1=θ1-θ^1, b¯~=b¯-b¯^, θ1 and ψ1(X) will be given later. Consider
(29)lα1=∂α1∂ybmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1z,y,t+∂α1∂b^mb^˙m+∂α1∂ξTξ˙+∂α1∂θ^1Tθ^˙1+∂α1∂ydy˙d+∂α1∂y˙dy¨d+12∂2α1∂y2g1Tyg1y,dα1=lα1dt+∂α1∂yg1Tydw.
Therefore, we have
(30)lVs1=s13bmvm,2+ξ2+ω¯Tb¯+ɛ2+f1y+Δ1-y˙d+32s12g1y2.A first-order filter withα1 as the input is designed as follows:
(31)τ2ω˙2+ω2=α1,ω20=α10.
Let y2=ω2-α1; thus, ω˙2=-y2/τ2. Since vm,2=s2+y2+α1, using Young’s inequality, it yields
(32)lVs1≤-k1-32bmax⁡s14+b~ms13α1+bmax⁡4s24+bmax⁡4y24+s13ω¯Tb¯~-s13θ^1Tψ1X-s13βb^m2+β-k1s1-ω¯Tb¯^-ξ2-s13θ^1Tψ1X+s13ɛ2+s13f1y+s13Δ1-s13y˙d+32s12g1y2.
From Assumption 1, we obtain
(33)s13Δ1≤s13ρ11y+s13ρ12yz≤34s14+14ρ114y+32λ02c33s14ρ124/3y≤+c316λ0z4.
In view of 24, 25, 32, and 33 and by using Young’s inequality, we obtain
(34)lVsWɛ≤-h-2ɛTɛ-k1-32bmax⁡-34s14-c32λ0z4+b~ms13α1+s13ω¯Tb¯~+bmax⁡4s24+bmax⁡4y24+s13S+s13H1X+Qy+14ɛ22-s13θ^1Tψ1X+16c44c54λ0c33+1,
where
(35)Qy=∑j=1nP2fj2y+∑j=1n2P2ρj12y+16nλ0c3∑j=1nP4ρj24y+16c44λ0c33ψ04y+14ρ114y+tr⁡gyPgTy,H1X=32λ02c33s1ρ124/3y-y˙d+916s1g1y4+s13,X=s1,yd,y˙dT∈R3.S(s1,b^m,b¯^,θ^1,ξ,λ¯m+2,yd) is a nonnegative continuous function, and
(36)f1y-βb^m2+β-k1s1-ω¯Tb¯^-ξ2-s13θ^1Tψ1x≤Ss1,b^m,b¯^,θ^1,ξ,λ¯m+2,yd,
where λ¯m+2=[λ1,…,λm+2]T.LetΩX=X∣X≤MX⊂R3 be a given compact set with MX>0 being a design constant, and let θ1Tϕ1(X) be the approximation of the radial basis function neural networks on the compact set ΩX to H1(X). Then, we have H1(X)=θ1Tψ1(X)+B1(X), where B1(X) denotes the approximation error and ψ1(X)=[ψ11(X),…,ψ1M1(X)]T∈RM1 denotes the basis function vector with ψ1j(X) being chosen as the commonly used Gaussian functions, which have the form
(37)ψ1jX=exp⁡-X-μ1j2b1j2j=1,…,M1, and μ1j is the center of the receptive field and b1j is the width of the Gaussian function; θ1 is an adjustable parameter vector.According to34 and by using Young’s inequality, it yields
(38)lVsWɛ≤-h-94ɛTɛ-k1-32bmax⁡-32s14-c32λ0z4+b~ms13α1+s13ω¯Tb¯~+bmax⁡4s24+bmax⁡4y24+14S4+s13θ1Tψ1X+B1X+Qy-s13θ^1Tψ1X+16c44c54λ0c33+1.
There exists a nonnegative continuous function κ(s1,yd,y˙d) satisfying
(39)B1X≤κs1,yd,y˙d.
Using Young’s inequality, we have
(40)lVsWɛ≤-h-94ɛTɛ-k1-32bmax⁡-94s14-c32λ0z4+b~ms13α1+s13ω¯Tb¯~+bmax⁡4s24+bmax⁡4y24+14S4+Qy+s13θ~1Tψ1X+14κ4+C0,
where C0=16c44c54/λ0c33+1.Step  i(2≤i≤ρ-1). Define the ith dynamic surface si=vm,i-ωi, thus
(41)s˙i=vm,i+1-livm,1-ω˙i.Select the virtual control lawαi as follows:
(42)αi=-kisi+livm,1+ω˙i,lαi=-kivm,i+1-livm,1-w˙i+liv˙m,1-lyiτi,dαi=lαidt+∂αi∂yg1Tydw.A first-order filter with the inputαi is designed as follows:
(43)τi+1ω˙i+1+ωi+1=αi,ωi+10=αi0,
where τi+1>0 is a design constant.Letyi+1=ωi+1-αi. Then ω˙i+1=-yi+1/τi+1. Noting vm,i+1=si+1+yi+1+αi, in view of 41 and 42, we obtain
(44)lVsi=si3s˙i=si3vm,i+1-livm,1-w˙i=-kisi4+si3si+1+si3yi+1≤-ki-32si4+14si+14+14yi+14.Step  ρ.The control law will be determined in this step. Define the ρth dynamic surface as sρ=vm,ρ-ωρ. The derivative of sρ is
(45)s˙ρ=σyu+vm,ρ+1-lρvm,1-ω˙ρ.Choose the control law as follows:(46)u=-kρsρ-vm,ρ+1+lρvm,1+ω˙ρσy.
In view of 45 and 46, we have
(47)lVsρ=sρ3s˙ρ=-kρsρ4.
The parameters θ^1, b^m, and b¯^ are updated as follows:
(48)θ^˙1=γ1s13ψ1X-σ1θ^1,(49)b^˙m=γ2s13α1-σ2b^m,b¯^˙=γ3s13ω¯-σ3b¯^,
where γ1, γ2, γ3, σ1, σ2, σ3 are the design constants.


## 3.3. Stability Analysis of Adaptive Control System

In this subsection, we will discuss the stability analysis of the closed-loop system. Firstly we define some Lyapunov functions and compact sets as follows:(50)V1=12s14+2Vɛ+1γ1θ~1Tθ~1+1γ2b~m2+1γ3b¯~Tb¯~+c3λ0z4,Vi=∑j=1i12sj4+2Vɛ+∑j=2i12yj4+1γ1θ~1Tθ~1+1γ2b~m2+1γ3b¯~Tb¯~+c3λ0z4,i=2,…,ρ,Ω1=s1,ɛ,θ^1,b^m,b¯^,z:V1≤p⊂Rp1,Ωi=s¯i,y¯i,ɛ,θ^1,b^m,b¯^,z:Vi≤p⊂Rpi,
where i=2,…,ρ, p>0 is a design constant, pi=2i+M1+n+m+1. It is easy to know that Ω1×Rpρ-p1⊃Ω2×Rpρ-p2⊃⋯⊃Ωρ-1×Rpρ-pρ-1⊃Ωρ.According toy2=ω2-α1, we obtain
(51)ly2=ω˙2-lα1=-y2τ2-lα1,dy2=ly2dt-∂α1∂yg1Tydw.
From 4, we obtain
(52)l14y24=y23ly2+32y22∂α1∂y2g1y2=-y24τ2-y23lα1+32y22∂α1∂y2g1y2.
There exist two nonnegative continuous functions η2(s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2, yd,y˙d,y¨d) and ζ2(s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2, yd,y˙d,y¨d) such that
(53)ly2+y2τ2≤η2s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d,(54)32y22∂α1∂y2g1y2≤ζ2s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d.
From 53, we have
(55)y23ly2≤-y24τ2+y23η2s¯2,y2,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d≤-y24τ2+34y24+14η24.
From 52, 54, and 55, we obtain
(56)l14y24≤-y24τ2+34y24+14η24+ζ2.
The infinitesimal generator of yi+1 is
(57)lyi+1=-yi+1τi+1-lαi,yi+13lyi+1=-yi+14τi+1-yi+13lαi,(58)lyi+144=-yi+14τi+1-yi+13lαi+32yi+12∂αi∂y2g1y2.
There exist two nonnegative continuous functions ηi+1(s¯i+1,y¯i+1, θ^1,b^m,b¯^, ξ,λ¯m+2,ε2, yd,y˙d,y¨d) and ζi+1(s¯i+1,y¯i+1, θ^1,b^m,b¯^,ξ, λm+2,ε2,yd, y˙d,y¨d) such that the following inequalities hold:
(59)lyi+1+yi+1τi+1≤ηi+1s¯i+1,y¯i+1,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d,(60)32yi+12αi∂y2g1y2≤ζi+1s¯i+1,y¯i+1,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d.
From 59, we obtain
(61)yi+13lyi+1≤-yi+14τi+1+yi+13ηi+1s¯i+1,y¯i+1,θ^1,b^m,b¯^,ξ,λ¯m+2,ε2,yd,y˙d,y¨d≤-yi+14τi+1+3yi+144+ηi+144.
From 58, 60, and 61, we obtain
(62)l14yi+14≤-yi+14τi+1+34yi+14+14ηi+14+ζi+1.
The continuous function S(·) on the compact set Ωd×Ω1 has a maximum M(p), which depends on the constant p, and κ(s1,yd,y˙d) on the compact set Ωd×Ω1 has a maximum N0(p), ηi+1(·) and ζi+1(·) on the compact set Ωd×Ωi+1 have the maximum Ni+1(p) and Ci+1(p) when ξ,λ¯m+2 are bounded.Theorem 11.
Consider the closed-loop system consisting of the plant1 under Assumptions 1–5, the controller 46, and the adaptation laws 48 and 49. For any bounded initial conditions, there exist constants ki, τi, h, γ1, γ2, γ3, σ1, σ2, σ3 satisfying V(0)≤c, such that all of the signals in the closed-loop system are bounded in probability, and s1,…,sρ, y2,…,yρ are SGUUB in four-moment, θ~1, b~m, b¯~ are SGUUB in mean square, and ki, τi, and h satisfy
(63)ki≥32bmax⁡+94+14α0,i=1,2,…,ρ,1τi≥14bmax⁡+1+14α0,i=2,…,ρ,h≥94+α0λmax⁡P,α0=min⁡c32c2,γ1σ1,γ2σ2,γ3σ3,
where c>0 is a positive constant; V will be given later in the proof of Theorem 11.Proof.
Choose the following Lyapunov function candidate:(64)V=VsWɛ+∑i=2ρVsi+14∑i=2ρyi4+12γ1θ~1Tθ~1+12γ2b~m2+12γ3b¯~Tb¯~
For any given positive constant, if EV≤c, according to Lemma 9, we obtain that s1,…,sρ, y2,…,yρ, y, θ^1, b^m, b¯^ are bounded in probability. VW≤VsWɛ=Vs1+VW+Vɛ≤V≤c, and, from Assumption 2, we obtain that (c1/λ0)z4≤1/λ0W≤c; that is, z4≤λ0c/c1, so z is bounded in probability.
Furthermore,64 is rewritten as V=1/λ0W+ 1/2Vρ-c3/2λ0z4; then Vρ=2V-2/λ0W+c3/λ0z4≤(2+c3/c1)c, and choosing p=(2+c3/c1)c, we get Vρ≤p.
From14 and 49, we have that ξ, ω¯, α1 are also bounded in probability. It yields that vm-1,2,…,v0,2 are all bounded in probability. Noting vm,2=s2+y2+α1, we obtain that vm,2 is bounded in probability. From 14, we have that v˙0,1=-l1v0,1+v0,2 and v˙m,1=-l1vm,1+vm,2. Thus we obtain that v0,1, vm,1 are also bounded. Furthermore, from 42, we have that αi(i=2,…,ρ-1) are bounded. According to 16 and 17, we obtain
(65)v0,1v0,2v1,2⋮vm-1,2=100⋯0010⋯0**1⋯0⋮⋱⋮**⋯*1λ1λ2λ3⋮λm+1.
Since v0,1,v0,2,v1,2,…,vm-1,2 are bounded in probability, we have that λ1,…,λm+1 are all bounded in probability. From 17, we get that λm+2 is also bounded. In view of 40, 44, 47–49, and 62, using Young’s inequality, we obtain
(66)lV≤-h-94ɛTɛ-∑i=1ρki-32bmax⁡-94si4-∑i=2ρ1τi-14bmax⁡-1yi4-c32λ0z4-σ1θ~122-σ2b~m22-σ3b¯~22+Qy+14N04p+14M4p+14∑i=2ρNi4p+∑i=2ρCip+σ1θ122+σ2bm22+σ3b¯22+C0.
Substituting 63 into 66, we obtain
(67)lV≤-α0V+μ1+μ0,
where μ1=1/4N04(p)+ 1/4M4(p)+1/4∑i=2ρNi4(p) +∑i=2ρCi(p)+ σ1θ12/2+ σ2bm2/2+ σ3b¯2/2+C0. Since |Q(y)| is a nonnegative continuous function, let |Q(y)|≤μ0, where μ0>0.
Ifα0≥(μ1+μ0)/c, and EV=c, then we have dEV/dt≤0. Thus, if EV(0)≤c, then EV(t)≤c, ∀t>0; that is,
(68)0≤EVt≤μ1+μ0α0+V0-μ1+μ0α0e-α0t≤V0.
Similar to the discussion of Theorem 11 in [32], it is easy to know that the conclusion is true.Remark 12.
This paper differs from [32] in the following several aspects. (1) Unmodeled dynamics is dealt with by introducing a novel description based on Lyapunov function in this paper while the dynamic signal was handled with the help of a dynamic signal in [32]. (2) The unknown nonlinear system functions are handled together with some functions produced in stability analysis, but they were directly approximated before constructing the observer in [32]. Therefore, this brings out a good result that the filter order is reduced. (3) The neural networks weight vector used to approximate the black box function at the first design step is adjusted online in this paper such that much more information of weight vector can be used in adaptive law, whereas only the norm of weight vector acts as adaptive tuning parameter in [32]. (4) Utilizing bounded input bounded output stability and linear equations 65, the stability of the closed-loop system is proved in this paper, which avoids using the transfer function to make stability analysis in [32], which is questionable in probability sense.Remark 13.
The design parameterski, τi and α0 determined by 63 in Theorem 11 are only a sufficient condition. They provide a guideline for the designers. From 63, some suggestions are given for the choice of some key design parameters for any given positive constants B0 and c.(i)
Increasingγ1, γ2, γ3 helps to increase α0, subsequently reduces μ1/α0.(ii)
Decreasingσ1, σ2, σ3 helps to reduce μ1 and reduces μ1/α0.(iii)
Increasingk1,…,kρ helps to increase α0 and reduces μ1/α0.In practical applications, to obtain good tracking performance, some experiments need to be done before the valid parameters are given.


## 4. Simulation Results

To demonstrate the effectiveness of the proposed approach, two numerical examples are given.Example 1.
Consider the following third-order stochastic nonlinear system with unmodeled dynamics:(69)z˙=qz,y,dx1=x2+y-y31+y2+0.5zdt+ysin⁡y3dw,dx2=x3+y-y31+y2+0.5z+0.235+y2udt+x1sin⁡y3dw,dx3=y2tanh⁡y-y2+2ysin⁡y+0.235+y2u+yzdt+0.5y2dw,y=x1,
where q(z,y)=-2z+ysin⁡t+0.5, m=1, ρ=2. The desired tracking trajectory is taken as yd=0.5sin⁡(0.5t). Select W(z,t)=1/4z4, c1=1/8, c2=1, c3=2, c4=1, c5=0.5; then ∂W/∂t(z,t)+∂W/∂z(z,t)(q(z,0,t)-q(0,0,t))=-2z4, |∂W/∂z(z,t)|=z3, |q(0,0,t)|=0.5; ψ0(|y|)=|y|, |q(z,y,t)-q(z,0,t)|= |ysint|≤ψ0(|y|), and it satisfies the conditions of Assumptions 2 and 3.
The filters are designed as follows:(70)ξ˙1=-l1ξ1+ξ2+l1y,ξ˙2=-l2ξ1+ξ3+l2y,ξ˙3=-l3ξ1+l3y,λ˙1=-l1λ1+λ2,λ˙2=-l2λ1+λ3,λ˙3=-l3λ1+σyu.
The adaptation laws are employed as follows:(71)θ^˙1=γ1s13ψ1X-σ1θ^1,b^˙1=γ2s13α1-σ2b^1,b^˙0=γ3s13v0,2-σ3b^0,
where X=s1,yd,y˙dT.
The virtual control lawα1 is chosen as follows:
(72)α1=b^1b^12+β-k1s1-ξ2-v0,2b^0-s13θ^1Tψ1X.
The control law is employed as follows:(73)u=-k2s2+l2v0,1+ω˙2σy,
where v0,1=λ1, v0,2=λ2, σ(y)=35+y2.
In the simulation,s1=y-yd, s2=v0,2-ω2, l1=6, l2=11, l3=6, k1=40, k2=50, β=0.02, ω2(0)=0.1, τ2=0.01, γ1=γ2=γ3=2, σ1=σ2=σ3=0.05, x(0)= [0.2,0,0]T, ξ(0)= [0,0,0]T, λ(0)= [0,0,0]T, b^0(0)=b^1(0)=1, θ^1(0)=[0.1]1×10T, M1=10. Simulation results are shown in Figures 1, 2, and 3. From Figure 1, it can be seen that fairly good tracking performance is obtained.Figure 1
Outputy (solid line) and desired trajectory yd (dotted line).Figure 2
Tracking errors1.Figure 3
Control signalu.Remark 14.
According to69, we know that b1=0.2 and bmax⁡=1. From the above selected design parameters and 63, it is easy to see that c3/(2c2)=1, α0=0.1. The constant h is only used to analyze the stability in the closed-loop system. Therefore, 63 is true for the above selected design parameters γ1, γ2, γ3, σ1, σ2, σ3, k1, k2, τ2.Example 2.
To compare the simulation results with [32], consider the following same stochastic nonlinear system with unmodeled dynamics in [32]:
(74)z˙=qz,y,dx1=x2+x1-x131+x12+0.5zdt+x1sin⁡x13dw,dx2=x12tanh⁡x1-x12+2x1sinx1hhh+0.20.5+x12u+x1zdt+0.5x12dw,y=x1,
where q(z,y)=-2z+y2, m=0, ρ=2. The desired tracking trajectory is taken as yd=0.5sin⁡(0.5t). The filters are designed as follows:
(75)ξ˙1=-l1ξ1+ξ2+l1y,ξ˙2=-l2ξ1+l2y,λ˙1=-l1λ1+λ2,λ˙2=-l2λ1+σyu.
The adaptation laws are employed as follows:(76)θ^˙1=γ1s13ψ1X-σ1θ^1,b^˙0=γ3s13v0,2-σ3b^0,
where X=s1,yd,y˙dT.
The virtual control lawα1 is chosen as follows:
(77)α1=b^0b^02+β-k1s1-ξ2-λ2b^0-s13θ^1Tψ1X.
The control law is employed as follows:(78)u=-k2s2+l2v0,1+ω˙2σy,
where v0,1=λ1, σ(y)=0.5+y2.
In the simulation,s1=y-yd, s2=v0,2-ω2, l1=5, l2=6, k1=60, k2=60, β=0.02, ω2(0)=0.1, τ2=0.01, γ1=γ3=1.5, σ1=σ3=0.05, x(0)=[0,0]T, z(0)=0, ξ(0)=[0,0]T, λ(0)=[0,0]T, b^0(0)=1, θ^1(0)=[0.1,0.1,0.1,0.1,0.1]T, M1=5. Simulation results are shown in Figures 4–6. If the proposed approach in [32] is utilized, and the design parameters of the adaptive controller are taken, the same values as in [32], the corresponding simulation results are as shown in Figures 7–9.
From Figures4, 5, 7, and 8, it can be seen that better tracking performance can be obtained than [32]. However, 42 equations need to be solved online using the method in [32] while only 14 equations need to be solved online using the approach in this paper. Moreover, we know that increasing k1,k2 helps to improve the tracking precision.Figure 4
Outputy (solid line) and desired trajectory yd (dotted line).Figure 5
Tracking errors1.Figure 6
Control signalu.Figure 7
Outputy (solid line) and desired trajectory yd (dotted line).Figure 8
Tracking errors1.Figure 9
Control signalu.


## 5. Conclusions

Using K-filters and dynamic surface control, an adaptive output feedback neural control scheme has been proposed for a class of stochastic nonlinear systems with unmodeled dynamics. Unmodeled dynamics has been dealt with by introducing the novel description based on Lyapunov function. The unknown nonlinear system functions are handled together with some functions resulting from stability analysis, and the filter order is reduced. The neural network weight vector is adjusted online. Therefore, the more information included in radial basis function can be fully made use of. Using Chebyshev’s inequality and Itô formula, the designed controller can guarantee that all the signals in the closed-loop system are bounded in probability and the error signals are semiglobally uniformly ultimately bounded in the sense of four-moment or mean square. Simulation results illustrate the effectiveness of the proposed approach.



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*Source: 529862-2015-07-22.xml*