# Robust Adaptive Control for a Class of Uncertain Nonlinear Systems with Time-Varying Delay

Authors: Ruliang Wang; Jie Li; Shanshan Zhang; Dongmei Gao; Huanlong Sun
Journal: The Scientific World Journal
 (2013)

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## Abstract

We present adaptive neural control design for a class of perturbed nonlinear MIMO time-varying delay systems in a block-triangular form. Based on a neural controller, it is obtained by constructing a quadratic-type Lyapunov-Krasovskii functional, which efficiently avoids the controller singularity. The proposed control guarantees that all closed-loop signals remain bounded, while the output tracking error dynamics converge to a neighborhood of the desired trajectories. The simulation results demonstrate the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

In the practical control process, control system is usually required to meet the stability and the corresponding performance index, which affects the system stability factors mainly including the uncertainties and time delays. On the study of the uncertain time-delay many scholars have achieved valuable fruits [1, 2]. Paper [3] has analyzed and designed the optimal H∞ feedback controller by the LMI method. In recent decades, the delay nonlinear systems with neural network research have received extensive attention [4–20]. Paper [4] has solved the problem of chaotic synchronization phenomenon by the neural network method. In [5–11], the study of the nonlinear continuous system and discrete nonlinear system is based on adaptive neural network control. The tracking and stabilization problem of nonlinear systems has been studied by neural network backstepping method [12, 13]. In [14], neural network control has been applied to a piece of triangle structure of multiple-input multiple-output nonlinear time-delay system, in which a dynamic system neural network is used mainly for unknown function approximation and separation. In multiple input and multiple output nonlinear system, [15] presents a new adaptive neural network controller design method but does not consider with external disturbance and time-varying delay. In [18], the problem of the adaptive neural networks control for a class of nonlinear state-delay systems with unknown virtual control coefficients is considered. In [19], a control scheme combined with backstepping, radial basis function (RBF) neural networks and adaptive control are proposed for the stabilization of nonlinear system with input and state delay.This paper mainly aims at studying the simultaneous presence of uncertainties and time-varying delay MIMO nonlinear system. By defining the new quadratic Lyapunov-Krasovskii functionals, it has analyzed and designed the adaptive neural network controller by neural network approximation method in [15, 16].


## 2. Description of the Problem

Let us consider the following block-triangular structure with the disturbance of nonlinear MIMO systems with time-varying delays:(1)x˙j,ij=fj,ij(x-j,ij)+gj,ij(x-j,ij)xj,ij+1+hj,ij(x-τj,ij)+ωj,ij(t),x˙j,mj=fj,mj(X,u-j-1)+gj,mj(X,u-j-1)uj+hj,mj(Xτ)+ωj,mj(t),yj=xj,1,j=1,…,n,ij=1,…,mj-1,
where x-j,ij=[xj,1,…,xj,ij]T∈Rij are the state variable for the ij differential equations of the jth subsystem; X=[x1T,…,xnT]T, where xj=[xj,1,…,xj,mj]T∈Rmj are the state vector of the jth subsystem; xτj,ij=xj,ij(t-τj,ij(t)), where τj,ij(t) are unknown time-varying delay of the states, and |τj,ij(t)|≤τj,ij,|τ˙j,ij(t)|≤τ1<1,τ0=max{τj,ij∣1≤j≤n,1≤ij≤mj}, x-τj,ij=[xτj,1,…,xτj,ij]T, Xτ=[xτ1,1,…,xτ1,n1,…,xτn,1,…,xτn,mn]T, the output y=[y1,…,yn]T∈Rn; u-j=[u1,…,uj]T are input vector of the jth subsystem; fj,ij(·), gj,ij(·), and hj,ij(·) are unknown smooth nonlinear function. ωj,ij(t) is the disturbance input and |ωj,ij(t)|≤dj,ij<1. Let xj,ij(t)=βj,ij(t), with t∈[-τ0,0]; assume βj,ij(t) is smooth and bounded.We make the following assumptions for the system (1).Assumption 1.
The desired trajectoriesydj,j=1,2,…,n, have the nth derivation and the derivation is continuous and bounded.Assumption 2.
We usegj,ij(·) to represent some given function. There exist constant gj0 and unknown smooth functions g-j,ij(·), such that 0<gj0≤|gj,ij(·)|≤g-j,ij(·)<∞. Without loss of generality, we further assume that gj,ij(·)>gj0>0.Lemma 3 (see [16]).
There exists smooth positive functionψj(ηj):Rmj→R(j=1,2,…,n) with ψj(0)=0 for all continuous functions h(η1,…,ηn):Rm1×⋯×Rmn→R with h(0,…,0)=0, where ηj∈Rmj(j-1,2,…,n,mj>0), such that |h(η1,…,ηn)|≤∑j=1n‍ψj(ηj).Lemma 4 (see [14]).
On any normal numberξ>0 and random variable l∈R have liml→0tanh2(l/ξ)/l=0.In this paper, the following radial basis function neural network is used to approximate unknown continuous function (in [13] once had been put forward):
(2)f(Z)=WTS(Z)+θ(Z),|θ(Z)|≤ε,(ε>0),
where the input vector Z∈ΩZ⊂Rn; W=[w1,w2,…,wl]T is the weight vector; the number of neural network node l>1 and S(Z)=[s1(Z),s2(Z),…,sl(Z)]T, where si=exp[-(Z-μi)T(Z-μi)/ϕi2],i=1,2,…,l,μi=[μi1,μi2,…,μin]T is the center of the receptive field, and ϕi is the width of the Gaussian function.


## 3. Adaptive Neural Network Controller Design

In this section, we will introduce a novel adaptive NN control design procedure. There aremj design steps in the design procedure for the jth subsystem. In each step, the unknown nonlinear function f-j,ij(Zj,ij) will be approximated by a radial neural network approximation function. Define an unknown constant as
(3)αj=1gj0max{∥Wj,ij∥2:1≤ij≤mj},
where the constant gj0 is defined as in Assumption 2; function f-j,ij and vector Zj,ij will be specified in each step. Furthermore, for j=1,2,…,n and ij=1,2,…,mj-1, choose the virtual control laws as follows:
(4)λj,ij=-(kj,ij+1)zj,ij-12aj,ij2α^jzj,ijST(Zj,ij)S(Zj,ij),
where kj,ij>0 and aj,ij>0 are design parameters, α^j represent the estimation of the unknown constant αj, and S(·) is the basis function vector, and define the variables zj,ij as follows:
(5)zj,ij=xj,ij-λj,ij-1,zj,1=xj,1-ydj,
for j=1,…,n,ij=2,…,mj. Choose the adaptive laws α^˙j as follows:
(6)α^˙j=∑ij=1mj‍rj2aj,ij2zj,ij2ST(Zj,ij)S(Zj,ij)-bjα^j,
where rj>0 and bj>0 are design parameters.Step  j·1(1≤j≤n). For the first differential equation of the jth subsystem, choose the Lyapunov function candidate
(7)Vzj,1=12zj,12+gj02rjα~2,
where zj,1=xj,1-ydj,α~=αj-α^j. Taking the time derivative of Vzj,1, we obtain
(8)V˙zj,1=zj,1(fj,1+gj,1λj,1-y˙dj+hj,1(x-τj,1)+ωj,1(t))+zj,1gj,1zj,2-gj0rjα~jα^˙j.
With Lemma 3, existence of positive function Qj,lj,ij(xτj,l)l=1,2,…,ij, such that
(9)|hj,ij(x-τj,ij)|≤∑l=1ij‍Qj,lj,ij(xτj,l).
Then, we have
(10)zj,1hj,1(x-τj,1)≤|zj,1|Qj,1j,1≤12zj,12+12[Qj,1j,1(xτj,1)]2.
Substituting (10) into (8) yields
(11)V˙zj,1≤zj,1(fj,1+gj,1λj,1-y˙dj+12zj,1+ωj,1(t))+zj,1gj,1zj,2+12[Qj,1j,1(xτj,1)]2-gj0rjα~jα^˙j.
To overcome the time-varying delay terms of (11), consider the following Lyapunov-Krasovskii functional:
(12)Vj,1=Vzj,1+Vuj,1,
where
(13)Vuj,1=∫t-τj,1(t)t‍12(1-τ1)[Qj,1j,1(xj,1(s))]2ds.
Take the time derivative of Vuj,1:
(14)V˙uj,1≤12(1-τ1)[Qj,1j,1(xj,1(t))]2-12[Qj,1j,1(xj,1(t-τj,1(t)))]2;
from (11) and (14), one has
(15)V˙j,1≤zj,1(f-j,1(Zj,1)+gj,1λj,1+ωj,1(t))-gj0rjα~jα^˙j+zj,1gj,1zj,2+[1-2tanh2(zj,1ηj,1)]Uj,1,
where
(16)Zj,1=[xj,1,ydj,y˙dj,α^j]T,Uj,1=12(1-τ1)[Qj,1j,1(xj,1)]2,f-j,1(Zj,1)=fj,1-y˙dj+12zj,1+2zj,1tanh2(zj,1ηj,1)Uj,1,
and ηj,1 is a positive constant.From Lemma4, the function (1/z)tanh2(z/η) is defined at z=0 and can be approximated by a neural network. Therefor the function f-j,1 will be approximated by the NN Wj,1TS(Zj,1), such that, for given εj,1>0,
(17)f-j,1(Zj,1)=Wj,1TS(Zj,1)+θj,1(Zj,1),|θj,1(Zj,1)|≤εj,1,
where θj,1(Zj,1) is the approximation error. Furthermore, a straightforward calculation shows that
(18)zj,1f-j,1(Zj,1)≤12aj,12gj0zj,12αjST(Zj,1)S(Zj,1)+12aj,12+12gj0zj,12+12εj,12gj0-1.
In additions, from (6), we obtain that for any initial conditions α^j(t0)≥0, α^j(t)>0 for all t>t0. Therefor
(19)zj,1gj,1λj,1≤-gj02aj,12α^jzj,12ST(Zj,1)S(Zj,1)-(kj,1+1)gj0zj,12,(20)zj,1ωj,1(t)≤12gj0zj,12+12dj,12gj0-1.
Substituting (18)–(20) into (15) yields that
(21)V˙j,1≤kj,1gj0zj,12+12(aj,12+εj,12gj0-1+dj,12gj0-1)+gj0rjα~j(rj2aj,12zj,12ST(Zj,1)S(Zj,1)-α^˙j)+zj,1gj,1zj,2+[1-2tanh2(zj,1ηj,1)]Uj,1.Step  j·ij(ij=2,…,mj-1). Define the Lyapunov-Krasovskii functional as
(22)Vzj,ij=12zj,ij2;
differentiating Vzj,ij yields
(23)V˙zj,ij=zj,ij((x-τj,ij)fj,ij+gj,ijxj,ij+1-λ˙j,ij-1+hj,ij(x-τj,ij)+ωj,ij(t)).
From (10), we have
(24)zj,ijhj,ij≤∑k=1ij‍(12zj,ij2+12[Qj,kj,ij(xτj,k)]2);λ˙j,ij-1(Zj,ij-1) can be expressed as
(25)λ˙j,ij-1=∑k=1ij-1‍∂λj,ij-1∂xj,k(fj,k+gj,kxj,k+1+ωj,k)+∑k=0ij-1‍∂λj,ij-1∂ydj(k)ydj(k+1)+∂λj,ij-1∂α^jα^˙j+∑k=1ij-1‍∂λj,ij-1∂xj,khj,k(x-τj,k).
Similar to (24), we can get
(26)-zj,ij∑k=1ij-1‍∂λj,ij-1∂xj,khj,k(x-τj,k)≤∑k=1ij-1‍∑l=1k‍12zj,ij2[∂λj,ij-1∂xj,k]2+∑k=1ij-1‍∑l=1k‍12[Qj,lj,k(xτj,l)]2.
Substituting (24)–(26) into (23) yields that
(27)V˙zj,ij≤zj,ij(∑k=1ij-1‍fj,ij+gj,ijxj,ij+1+ωj,ij-∑k=1ij-1‍∂λj,ij-1∂xj,k(fj,k+gj,kxj,k+1+ωj,k)+∑k=1ij‍12zj,ij-∑k=0ij-1‍∂λj,ij-1∂ydj(k)ydj(k+1)+∑k=1ij-1‍∑l=1k‍12zj,ij[∂λj,ij-1∂xj,k]2)-∂λj,ij-1∂α^jα^˙j+∑k=1ij-1‍∑l=1k‍12[Qj,lj,k(xτj,l)]2+∑k=1ij‍12[Qj,kj,ij(xτj,k)]2.
To overcome the delay terms in (27), let us consider the following Lyapunov-Krasovskii functional:
(28)Vj,ij=Vzj,ij+Vuj,ij,
where
(29)Vuj,ij=∑k=1ij‍∫t-τj,kt‍12(1-τ1)[Qj,kj,ij(xj,k(s))]2ds+∑k=1ij‍∑l=1k‍∫t-τj,lt‍12(1-τ1)[Qj,lj,k(xj,l(s))]2ds.
Differentiating Vuj,ij yields
(30)V˙uj,ij=∑k=1ij‍12(1-τ1)[Qj,kj,ij(xj,k(t))]2+∑k=1ij-1‍∑l=1k‍12(1-τ1)[Qj,lj,k(xj,l(t))]2-∑k=1ij‍12(1-τ1)[Qj,kj,ij(xτj,ij)]2(1-τ˙j,k)-∑k=1ij-1‍∑l=1k‍12(1-τ1)[Qj,lj,k(xτj,l)]2(1-τ˙j,l)≤Uj,ij-∑k=1ij‍12[Qj,kj,ij(xτj,ij)]2-∑k=1ij-1‍∑l=1k‍12[Qj,lj,k(xτj,l)]2≤zj,ij2zj,ijtanh2(zj,ijηj,ij)Uj,ij+[1-2tanh2(zj,ijηj,ij)]Uj,ij-∑k=1ij‍12[Qj,kj,ij(xτj,ij)]2-∑k=1ij-1‍∑l=1k‍12[Qj,lj,k(xτj,l)]2,
where
(31)Uj,ij=∑k=1ij‍12(1-τ1)[Qj,kj,ij(xj,k(t))]2+∑k=1ij-1‍∑l=1k‍12(1-τ1)[Qj,lj,k(xj,l(t))]2.
Then, combining (27) and (30) results in
(32)V˙j,ij≤zj,ij(φj,ij-∂λj,ij-1∂α^jα^˙j)+gj,ijzj,ijzj,ij+1+[1-2tanh2(zj,ijηj,ij)]Uj,ij+zj,ij(f-j,ij+gj,ijλj,ij+1+ωj,ij),
where
(33)f-j,ij=fj,ij-∑k=1ij-1‍∂λj,ij-1∂xj,k(fj,k+gj,kxj,k+1+ωj,k)-∑k=0ij-1‍∂λj,ij-1∂ydj(k)ydj(k+1)+∑k=1ij‍12zj,ij+∑k=1ij-1‍∑l=1k‍12zj,ij[∂λj,ij-1∂xj,k]2+2zj,ijtanh2(zj,ijηj,ij)Uj,ij-φj,ij.
The NN Wj,ijTS(Zj,ij) is used to approximate f-j,ij such that for given εj,ij>0 we have
(34)f-j,ij=Wj,ijTS(Zj,ij)+θj,ij(Zj,ij),|θj,ij(Zj,ij)|≤εj,ij,
where θj,ij(Zj,ij) represent the approximation error. Similar to (18) and (20), we have
(35)V˙j,ij≤-kj,ijgj0zj,ij2+12(aj,ij2+εj,ij2gj0-1+dj,ij2gj0-1)+gj0rjα~jrj2aj,ij2zj,ij2ST(Zj,ij)S(Zj,ij)+zj,ij(φj,ij-∂λj,ij-1∂α^jα^˙j)+gj,ijzj,ijzj,ij+1+[1-2tanh2(zj,ijηj,ij)]Uj,ij.Step  j·mj(1≤j≤n). In the final step of the jth subsystem to construct the actual control law uj, let us consider the following Lyapunov-Krasovskii function:
(36)Vj,ij=12zj,mj2+Vuj,mj,
where
(37)Vuj,mj=∑j=1n‍∑k=1mj‍∫t-τj,kt‍12(1-τ1)[Qj,kj,mj(xj,k(s))]2ds+∑k=1mj-1‍∑l=1k‍∫t-τj,lt‍12(1-τ1)[Qj,lj,k(xj,l(s))]2ds
and zj,mj=xj,mj-λj,mj-1. Similar to (32) we get(38)V˙j,mj≤zj,mj(φj,mj-∂λj,mj-1∂α^jα^˙j)+[1-2tanh2(zj,mjηj,mj)]Uj,mj+zj,mj(f-j,mj+gj,mjuj+ωj,mj),
where f-j,mj(zj,mj) can be defined by (33) with ij=mj.We use the NNWj,mjTS(Zj,mj) to approximate f-j,mj such that, for given εj,mj>0, we have
(39)f-j,mj=Wj,mjTS(Zj,mj)+θj,mj(Zj,mj),(Zj,mj)|θj,mj(Zj,mj)|≤εj,mj,
where θj,mj(Zj,mj) express the approximation error.Choose the control lawuj as
(40)uj=-(kj,mj+1)zj,mj-12aj,mj2α^jzj,mjST(Zj,mj)S(Zj,mj).
Similar to (21) we have
(41)V˙j,mj≤12(aj,mj2+εj,mj2gj0-1+dj,mj2gj0-1)+gj0rjα~jrj2aj,mj2zj,mj2ST(Zj,mj)S(Zj,mj)-kj,mjgj0zj,mj2+zj,mj(φj,mj-∂λj,mj-1∂α^jα^˙j)+[1-2tanh2(zj,mjηj,mj)]Uj,mj.
Let Vn,mn=∑j=1n‍∑k=1mj‍Vj,k. Combining (21), (35), and (41) gives that
(42)V˙n,mn≤-∑j=1n‍∑k=1mj‍kj,kgj0zj,k2+∑j=1n‍∑k=1mj‍12(aj,k2+εj,k2gj0-1+dj,k2gj0-1)+∑j=1n‍gj0rjα~j(∑k=1mj‍rj2aj,k2zj,k2ST(Zj,k)S(Zj,k)-α^˙j)+∑j=1n‍∑k=1mj‍[1-2tanh2(zj,kηj,k)]Uj,k+∑j=1n‍∑k=2mj‍zj,k(φj,k-∂λj,k-1∂α^jα^˙j).
The control law design is thus completed.


## 4. Stability Analysis

Now, the main result in this paper can be presented as follows.Theorem 5.
Consider the nonlinear time-delay system (1) with the NN adaption law (6) and the control law (40) satisfying Assumptions 1–2. All the closed-loop trajectories can guarantee boundedness if the unknown function f-j,ij can be approximated by neural network and the approximating error θj,ijis boundedness.Proof.
Define functionsφj,k, such that
(43)-∑j=1n‍∑k=2mj‍zj,k(φj,k-∂λj,k-1∂α^jα^˙j)≤0.
Let 0<ST(·)S(·)<L, where L is the number of neural network weights.
From (6), we can get
(44)-∑k=2mj‍zj,k∂λj,k-1∂α^jα^˙j≤∑k=2mj‍zj,k(∑l=1k-1‍bjα^j∂λj,k-1∂α^j-∑l=1k-1‍∂λj,k-1∂α^jrj2aj,l2zj,l2ST(Zj,l)S(Zj,l))+∑k=2mj‍zj,k(rjL2aj,k2zj,k2∑l=2k‍|zj,l∂λj,l-1∂α^j|).
By choosing function φj,k as
(45)φj,k=-bjα^j∂λj,k-1∂α^j-rjL2aj,k2zj,k2∑l=2k‍|zj,l∂λj,l-1∂α^j|+∑l=1k-1‍∂λj,k-1∂α^jrj2aj,l2zj,l2ST(Zj,l)S(Zj,l),
(43) holds.
In a similar way, we can get(46)∑j=1n‍gj0rjα~j(∑k=1mj‍rj2aj,k2zj,k2ST(Zj,k)S(Zj,k)-α^˙j)≤∑j=1n‍gj0rj(-α~j2+αj2).
Now, choose the Lyapunov function as V=Vn,mn.
Combining (42)–(46) gives that
(47)V˙n,mn≤-∑j=1n‍∑k=1mj‍kj,kgj0zj,k2-∑j=1n‍gj0bjrjα~j2+∑j=1n‍∑k=1mj‍[1-2tanh2(zj,kηj,k)]Uj,k+D,
where
(48)D=∑j=1n‍∑k=1mj‍12(aj,k2+εj,k2gj0-1+dj,k2gj0-1)+∑j=1n‍gj0rjαj2
is a constant. Thus, by (47) the boundedness follows immediately from the same line used in the proof in [9–11].


## 5. Simulation Examples

In this section, we will give one example to demonstrate the effectiveness of the proposed method in this paper. Let us consider the following example:(49)x˙1,1=-x1,1+(1+cos2(x1,1))x1,2+xτ1,12+ω1,1(t),x˙1,2=x1,1x1,2+x2,1+x2,2+(1+0.5cos2(x2,2))u1+xτ1,2+ω1,2(t),x˙2,1=-x2,1+x2,2+xτ2,1+ω2,1(t),x˙2,2=(x1,2+x2,1)x2,2-x1,1u1+(2+sin2(u1))u2+xτ1,1xτ2,2+ω2,2(t),
where xτj,ij=xj,ij(t-τj,ij),j=1,2,ij=1,2.And the time delays are chosen as(50)τ1,1=0.9+0.1sin(t),τ1,2=1+0.5sin(t),τ2,1=0.4+0.1cos(t),τ2,2=2+0.1cos(t),
given the reference output signals as yd1=0.5(sin(t)+sin(0.5t)),yd2=0.5sin(t)+sin(0.5t). The control law is given by (40). The NN adaptation law is given by (6). Choose the design parameters
(51)k1,1=k1,2=k2,1=k2,2=20,a1,1=a1,2=2,a2,1=a2,2=1,r1=r2=400,b1=b2=0.025.
Take the external disturbance as
(52)ω1,1(t)=ω1,2(t)=0.04sin(2πt),ω2,1(t)=ω2,2(t)=0.04cos(2πt).The simulation is run under the initial conditionsxj,ij(ϑ)=0,-τ0≤ϑ≤0,j=1,2,ij=1,2, and [α^1(0),α^2(0)]T=[0,0]T. The result of control scheme is displayed in Figures 1–5. Figures 1 and 2 demonstrate the outputs of system and the reference signals. The responses of state variables x1,2 and x2,2 are shown in Figure 3. The control input signals u1 and u2 are illustrated in Figures 4 and 5which depict the boundedness of adaptive parameters α^1 and α^2.Figure 1
System outputy1(t)(“−”) and the reference yd1(t)(“−·−”).Figure 2
System outputy2(t)(“−”) and the reference yd2(t) (“−·−”).Figure 3
Responses of state variablesx1,2 and x2,2.Figure 4
Control input signalsu1 and u2.Figure 5
Adaptive parametersα^1 and α^2.


## 6. Conclusion

For a class of perturbed nonlinear MIMO time-varying delay systems in a block-triangular form, an adaptive neural control design is presented. Although there are some fluctuations of the systems and control output under the influence of interference, the required performance can be achieved in a short period of time by using the controller designed in this paper and guarantees the boundedness of all the signals in the closed-loop system. It is further extended on the bases in [14, 15], which makes it suitable for wider range of applications. The effectiveness of the proposed approach is provided by a simulation example.



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