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 trajectories ydj, j=1,2,…,n, have the nth derivation and the derivation is continuous and bounded.
Assumption 2.
We use gj,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 function ψj(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 number ξ>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 are mj 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 Lemma 4, 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 NN Wj,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 law uj 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.