# Adaptive Neural Control for a Class of Outputs Time-Delay Nonlinear Systems

Authors: Ruliang Wang; Jie Li
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper considers an adaptive neural control for a class of outputs time-delay nonlinear systems with perturbed or no. Based on RBF neural networks, the radius basis function (RBF) neural networks is employed to estimate the unknown continuous functions. The proposed control guarantees that all closed-loop signals remain bounded. The simulation results demonstrate the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

The study of the time-delay systems has been one of the most active research topics in recent years [1–15]. The time-delay systems can be divided into four types: systems with input delay [1–5], systems with state delay [6–9, 16–18], systems with both input and state delays, and systems with both input and output delays [19]. The effect of time delay on stability and asymptotic performance has been investigated in [20]. In [21], Lyapunov-Krasovskii functionals were used with backstepping to obtain a robust controller for a class of single-input single-output (SISO) nonlinear time-delay systems with known bounds on the functions of delayed states, but it was commented that results could not be constructively obtained in [22]. In [23], the problem of the adaptive neural-networks control for a class of nonlinear state-delay systems with unknown virtual control coefficients is considered. In [24], An adaptive control scheme combined with radius basis function (RBF) neural networks, backstepping, and adaptive control is proposed for the output tracking control problem of a class of MIMO nonlinear system with input delay and disturbances. Neural networks are employed to estimate the unknown continuous functions; the control scheme ensures that the closed-loop system is semiglobally uniformly ultimately bounded (SGUUB). In [11] A control scheme combined with backstepping, radius basis function (RBF) neural networks, and adaptive control is proposed for the stabilization of nonlinear system with input and state delay.In this paper, we present an adaptive neural controller design procedure for a class of output time-delay nonlinear systems with perturbed, based on backstepping, adaptive control, and neural networks. RBF neural network is employed to the unknown continuous function. A numerical example is provided to show the effectiveness of the control scheme.


## 2. Problem Formulation and Preliminaries

Consider the nonlinear time-delay system is described as follows:(2.1)x˙i=xi+1+gi(y)+fi(y(t-τ))+νi(t),1≤i≤n-1,x˙n=u(t)+gn(y)+fn(y(t-τ))+νn(t),y=x1,
where x=[x1,x2,…,xn]T∈Rn is state, u∈R is control and y∈R is output vectors, respectively. νi(t)(i=1,2,…,n) is a time-varying disturbance. fi(y(t-τ)), (i=1,2,…,n), gi(y), (i=1,2,…,n) are unknown continuous functions.Assumption 2.1.
The unknown functionfi(y) satisfies fi2(y)≤ki, where ki(i=1,2,…,n) is a known constant.Assumption 2.2.
The time-varying disturbanceνi(t) satisfies |νi(t)|≤di<1, 1≤i≤n, where di(i=1,2,…,n) is a known constant.Lemma 2.3.
x
(
t
)
∈
Ω
M
,
t
∈
[
-
τ
max
,
τ
], where ΩM={x∣∥x∥≤M}, M is an unknown constant.


## 3. RBF NN Approximation

In this paper, for a givenδ>0 and any continuous function Hi(ηi) defined on Ωi, there is a perfect RBF neural network, which satisfies
(3.1)Fi(ζi)=WiTSi(ζi)+δi(ζi),
where |δi(ζi)|≤δWi∈Rmi is the weight vector of the neural networks, mi is the number of the NN nodes, ζi∈Ωi is the input vector, Si(ζi)=[si1,…,simi]T is defined by
(3.2)sij(ζi)=exp[-(ζi-μij)T(ζi-μij)ξ2].
According to the discussion in [21, 22], denote the best weight vector as follows:
(3.3)Wi*=argminwi∈Rmisupζi∈Ωi|WiTSi(ζi)-Hi(ζi)|
which is unknown and needs to be estimated in control design. Let W^i be the estimate of Wi*, and define W~i=W^i-Wi*.


## 4. Main Result

In this section, we will consider system (2.1).(I) whenνi(t)=0,i=1,2,…,n,Let us define error variableszi assistant functions and the virtual control αi, respectively, as follows:
(4.1)z1=x1,zi=xi-αi-1,2≤i≤n.
Define the following sets:
(4.2)Ωzi={zi∈R∣|zi|<λi},Ωzio={zi∈R∣|zi|≥λi},
where λi is a small constant. Define assistant functions as
(4.3)F1=g1(x)+12λ12U1(x1z1),Fi=gi(x)+12λi2zi∑j=1i-1Uj(x1)-∑j=1i-1∂αi-1∂W^jW^˙j-∑j=1i-1∂αi-1∂xj(xj+1+gj(x1)-12∂αi-1∂xjzj),2≤i≤n.
Define the virtual control as
(4.4)αi=-ki0zi-W^iTSi(ξi),1≤i≤n,
where
(4.5)k10=c1+b12+32+12λ12(λ22+τmaxk1),ki0=ci+bi2+2+12λi2(λi+12+τmax∑j=1i-1‍kj),2≤i≤n-1,kn0=cn+bn2+1+12λn2(λi+12+τmax∑j=1n-1‍kj),ξ1=[x1,x^1]T,ξi=[x-iT,x^-iT,αi-1,(∂i-1∂x-i-1)T(∂i-1∂x^-i-1)T,ψi-1]T,2≤i≤n,ψi-1=∑j=1i-1∂αi-1∂W^j∂W^˙j,2≤i≤n.Theorem 4.1.
System (2.1) with both input delay and state delay satisfies Assumptions 2.1 and 2.2. The virtual control can be selected as (4.4). If the control law and the adaptive law are selected as follows:
(4.6)u(t)=αn,(4.7)W^˙i=Γi(ziSi(ξi)-σiW^i),1≤i≤n,
then the closed-loop system is semi-globally uniformly ultimately bounded.Proof.
Define the Lyapunov-Kresovskii functionalV(t) as
(4.8)Vzi(t)=12zi2(t),VWi(t)=12W~iTΓi-1W~i,(4.9)VUi(t)=12∑j=1i∫t-τhtUj(y(σ))dσ,(4.10)Vi(t)=Vzi(t)+VUi(t)+VWi(t),1≤i≤n,(4.11)V(t)=∑i=1nVi(t).Step 1. For the first differential equation of the the first subsystem, by (4.1), (4.3), we can get
(4.12)z˙1=x˙1=x2+g1(x1)+f1(x1(t-τh))=z2+α1+F1(ζ1)-12λ12U1(x1)z1+g1(x1(t-τh)).
By
(4.13)z1f1(x1(t-τh))≤12z12+12f12(x1(t-τh))
differentiating (4.10) and using (4.12), the inequality below can be obtained easily. 							(4.14)V˙1=z1z˙1+W~1TΓ1-1W^˙1+12U1(x1)-12U1(x1(t-τh))=z1(z2+α1+F1(ζ1)-12λ12U1(x1)z1+f1(x1(t-τh)))+W~1TΓ1-1W^˙1+12U1(x1)-12U1(x1(t-τh))≤z1z2+z1α1+z1F1(ζ1)+12z12+12U1(1-z12λ12)+W~1TΓ1-1W^˙1.
(1) If z1∈Ωz1o, then |z1|≥λ1. Thus, substituting (4.4) and (4.7) into (4.14) results in
(4.15)V˙1≤z1z2+z1(-k10z1-W^1TS1(ζ1))+z1F1(ζ1)+12z12+12U1(1-z12λ12)+W~1TΓ1-1W^˙1≤12∥z1∥2+12∥z2∥2-(c1+b12+32)z12+z1F1(ζ1)-z122λ1(τmaxk1+λ22)-z1W^1TS1(ζ1)+12z12+12∥z1∥2+W~(z1S1(ζ1)-σ1W^1)≤12z22-(c1+b12)z12-z122λ12(τmaxk1+λ22)+z1ε1(ξ1)-σ1W~1TW^1≤12z22-(c1+b12)z12+b12z12+12b1(ε1*)2-12(τmaxk1+λ22)-σ1W~1TW^1≤-c1z12-VU1(T)-σ12∥W~1∥2+σ12∥W1*∥2+12b1(ε1*)2+12(z22-λ22)≤-k1V1+bv1+Θ1,
where k1=min{2c1,σ1/λmax(Γ1-1),1}, bv1=(σ1/2)∥W1*∥2+(1/2b1)(ε1*)2. Θ1=(1/2)(z22-λ22).
If there is no itemΘ1 in (4.15), then
(4.16)0≤V1(t)≤(V1(0)-δv)e-k1t+δv,
where δv=bv1/k1. Thus V1 is bounded.
(2) Ifz1∈Ωz1, then |z1|<λ1, z1 is bounded. By the integral median theorem, we can obtain
(4.17)12∫t-τht‍U1(y(σ))dσ=12τhU1(y(θ)),θ∈(t-τh,t).
By Assumption 2.1 and (4.9), it can be concluded that VU1 is bounded.
DifferentiatingVw1,
(4.18)Vw1=W~1T(z1S1(ξ1)-σ1W^1)≤-12(σ1-kw1)∥W~1∥2+12(σ1∥W1*∥2+1kw1z12∥S1(ζ1)∥2)≤-k1Vw1+b1,
where
(4.19)k1=σ1-kw1λmax(Γ1-1),b1=12(σ1∥W1*∥2+m1kw1λ12).m1 is the number of neurons of the neural networks. Choose the parameter so that kw1<σ1,k1>0. Therefore
(4.20)0≤Vw1(t)≤(Vw1(0)-δw1)e-k1t+δw1,
where δw1=b1/k1. Thus Vw1(t) is bounded. Because Vz1,VU1,Vw1 are all bounded, V1 is bounded when z1∈Ωz1.
Step
i
. For the ith (2≤i≤n-1) subsystem, by utilizing (4.1)(4.3), we have
(4.21)z˙i=x˙i-α˙i-1=xi+1+gi(x1)+fi(x1((t-τh)))-α˙i-1=zi+1+αi+gi(x1)+fi(x1(t-τh))-∑j=1i-1‍∂αi-1∂xj(xj+1+gi(x1)+fi(x1(t-τh)))-∑j=1i-1‍∂αi-1∂w^jW^˙j.
Differentiating (4.10) along track (4.21), we have
(4.22)V˙i≤zi(zi+1+αi+Fi(ζi))+12zi2+12∑j=1i‍fj2(x1(t-τh))-zi22λi2∑j=1i-1‍Uj(x1)+12∑j=1i-1‍(Uj(x1)-Uj(x1(t-τh)))≤z1(z2+α1+F1(ζ1)-12λ12U1(x1)z1)+f1(x1(t-τh))+W~1TΓ1-1W^˙1+12U1(x1)-12U1(x1(t-τh))+W~iTΓi-1W^˙i+zivi(t)≤z1z2+z1α1+z1F1(ζ1)+12z12+12U1(1-z12λ12)+W~iTΓi-1W^˙i+zivi(t)≤z1z2+z1α1+z1F1(ζ1)+12z12+12zi2+∑j=1i-1‍Uj(x1)(1-zi2λi2)+W~iTΓi-1W^˙i.
(1) Ifzi∈Ωzio, then |zi|≥λi. Thus, substituting (4.4) and (4.7) into (4.22) results in
(4.23)V˙i≤zizi+1+ziFi(ζi)12zi2-(ci+bi2+2)zi2-zi22λi2[τmax∑j=1i-1‍kj+λi+12]-ziW^iTSi(ζi)+W~iT(ziSi(ζi)-σiW^i)≤zizi+1-(ci+bi2+1)zi2-12[τmax∑j=1i-1‍kj+λi+12]+ziεi(ξi)-σiW~iTW^i+12di2≤-cizi2-τmax2∑j=1i-1‍kj-σ2∥W~i∥2+σ2∥Wi*∥2+12bi(εi*)2+12(-zi2+zi+12-λi+12)≤-kiVi+bvi+Θi,
where ki=min{2ci,σi/(λmax(Γi-1)),1}, bvi=(σi/2)∥Wi*∥2+(1/2bi)(εi*)2. Θi=(1/2)(-zi2+zi+12-λi+12).
If there is no itemΘi in (4.23), then
(4.24)0≤Vi(t)≤(Vi(0)-δv)e-kit+δv,
where δv=bvi/ki. Thus Vi is bounded.
(2) Ifzi∈Ωzi, similar to step 1, we have Vi is bounded.
Step
n. This is the last step for the nth subsystem, similarly to the ith subsystem, if zn∈Ωzno, then |zn|≥λn. Thus we have
(4.25)V˙n≤-cnzn2-τmax2∑j=1n-1‍kj-σ2∥W~n∥2+σ2∥Wn*∥2+12bn(εn*)2-12zn2≤-knVn+bvn+Θn,
where kn=min{2cn,σn/λmax(Γn-1),1}, bvn=(σn/2)∥Wn*∥2+(1/2bn)(εn*)2. Θn=-(1/2)zn2.
By (4.25), it is easy to have
(4.26)0≤Vn(t)≤(Vn(0)-δv)e-knt+δv,
where δv=bvn/kn. Thus Vn is bounded.
(1) Ifzn∈Ωzn, similar to step 1, we have Vn is bounded.
TheVi,(1≤i≤n) is bounded when zi∈Ωzi,(1≤i≤n). In zi∈Ωzio,(1≤i≤n):
(4.27)V˙(t)≤∑i=1n‍V˙i≤-∑i=1n‍kiVi+∑i=1n‍bvi+∑i=1n‍Θi≤-kvV(t)+bv,
where kv=min{k1,k2,…,kn}, bv=∑i=1n‍bvi.
Then(4.28)0≤V(t)≤(V(0)-δv)e-kvt+δv,
where δv=bv/kv. Thus V(t) is bounded.(II) Whenνi(t)≠0,i=1,2,…,n.Let us define error variableszi assistant functions and the virtual control αi, respectively, as follows:
(4.29)z1=x1,zi=xi-αi-1,2≤i≤n.
Define the following sets:
(4.30)Ωzi={zi∈R∣|zi|<λi},Ωzio={zi∈R∣|zi|≥λi},
where λi is a small constant. Define assistant functions as
(4.31)F1=g1(x)+12λ12U1(x1z1),Fi=gi(x)+12λi2zi∑j=1i-1‍Uj(x1)-∑j=1i-1‍∂αi-1∂W^jW^˙j-∑j=1i-1‍∂αi-1∂xj(xj+1+gj(x1)-12∂αi-1∂xjzj),2≤i≤n.
Define the virtual control as
(4.32)αi=-ki0zi-W^iTSi(ξi),1≤i≤n,
where
(4.33)k10=c1+b12+32+12λ12(λ22+τmaxk1+d12),ki0=ci+bi2+2+12λi2(λi+12+τmax∑j=1i-1‍kj+di2),2≤i≤n-1kn0=cn+bn2+1+12λn2(λi+12+τmax∑j=1n-1‍kj+dn2),(4.34)ξ1=[x1,x^1]Tξi=[x-iT,x^-iT,αi-1,(∂i-1∂x-i-1)T(∂i-1∂x^-i-1)T,ψi-1]T,2≤i≤n,ψi-1=∑j=1i-1‍∂αi-1∂W^j∂W^˙j,2≤i≤n.Theorem 4.2.
System (2.1) with both input delay and state delay satisfies Assumptions 2.1 and 2.2. The virtual control can be selected as (4.32). If the control law and the adaptive law are selected as follow:
(4.35)u(t)=αn,(4.36)W^˙i=Γi(ziSi(ξi)-σiW^i),1≤i≤n,
then the closed-loop system is semi-globally uniformly ultimately bounded.Proof.
Define the Lyapunov-Kresovskii functionalV(t) as
(4.37)Vzi(t)=12zi2(t),VWi(t)=12W~iTΓi-1W~i,(4.38)VUi(t)=12∑j=1i‍∫t-τht‍Uj(y(σ))dσ,(4.39)Vi(t)=Vzi(t)+VUi(t)+VWi(t),1≤i≤n,(4.40)V(t)=∑i=1n‍Vi(t).Step 1. For the first differential equation of the first subsystem, by (4.29), (4.31), We can get
(4.41)z˙1=x˙1=x2+g1(x1)+f1(x1(t-τh))+ω1(t)=z2+α1+F1(ζ1)-12λ12U1(x1)z1+h1(x1(t-τh))+ω1(t).
By
(4.42)z1f1(x1(t-τh))≤12z12+12f12(x1(t-τh))
differentiating (4.39) and using (4.41), the inequality below can be obtained easily. 							(4.43)V˙1=z1z˙1+W~1TΓ1-1W^˙1+12U1(x1)-12U1(x1(t-τh))=z1(z2+α1+F1(ζ1)-12λ12U1(x1)z1+ω1(t)+f1(x1(t-τh)))+W~1TΓ1-1W^˙1+12U1(x1)-12U1(x1(t-τh))≤z1z2+z1α1+z1F1(ζ1)+12z12+12U1(1-z12λ12)+W~1TΓ1-1W^˙1+z1ω1(t).
(1) Ifz1∈Ωz1o, then |z1|≥λ1. Thus, substituting (4.32) and (4.36) into (4.43) results in
(4.44)V˙1≤z1z2+z1(-k10z1-W^1TS1(ζ1))+z1F1(ζ1)+12z12+12U1(1-z12λ12)+W~1TΓ1-1W^˙1+z1ν1(t)≤12∥z1∥2+12∥z2∥2-(c1+b12+32)z12+z1F1(ζ1)-z122λ1(τmaxk1+λ22+d12)-z1W^1TS1(ζ1)+12z12+12∥z1∥2+W~(z1S1(ζ1)-σ1W^1)+12d12≤12z22-(c1+b12)z12-z122λ12(τmaxk1+λ22)+z1ε1(ξ1)-σ1W~1TW^1≤12z22-(c1+b12)z12+b12z12+12b1(ε1*)2+12d12-12(τmaxk1+λ22+12d12)-σ1W~1TW^1≤-c1z12-VU1(T)-σ12∥W~1∥2+σ12∥W1*∥2+12b1(ε1*)2+12(z22-λ22)≤-k1V1+bv1+Θ1,
where k1=min{2c1,σ1/(λmax(Γ1-1)),1}, bv1=(σ1/2)∥W1*∥2+(1/2b1)(ε1*)2. Θ1=(1/2)(z22-λ22).
If there is no itemΘ1 in (4.44), then
(4.45)0≤V1(t)≤(V1(0)-δv)e-k1t+δv,
where δv=bv1/k1. Thus V1 is bounded.
(2) Ifz1∈Ωz1, then |z1|<λ1, z1 is bounded. By the integral median theorem, we can obtain
(4.46)12∫t-τhtU1(y(σ))dσ=12τhU1(y(θ)),θ∈(t-τh,t).
By Assumption 2.1 and (4.38), it can be concluded that VU1 is bounded.
DifferentiatingVw1,
(4.47)Vw1=W~1T(z1S1(ξ1)-σ1W^1)≤-12(σ1-kw1)∥W~1∥2+12(σ1∥W1*∥2+1kw1z12∥S1(ζ1)∥2)≤-k1Vw1+b1,
where
(4.48)k1=σ1-kw1λmax(Γ1-1),b1=12(σ1∥W1*∥2+m1kw1λ12).m1 is the number of neurons of the neural networks. Choose the parameter so that kw1<σ1,k1>0. Therefore
(4.49)0≤Vw1(t)≤(Vw1(0)-δw1)e-k1t+δw1,
where δw1=b1/k1. Thus Vw1(t) is bounded. Because Vz1,VU1,Vw1are all bounded, V1 is bounded when z1∈Ωz1.
Step
i. For the ith (2≤i≤n-1) subsystem, by utilizing (4.29), (4.31), we have
(4.50)z˙i=x˙i-α˙i-1=xi+1+gi(x1)+fi(x1((t-τh)))-α˙i-1+νi(t)=zi+1+αi+gi(x1)+fi(x1(t-τh))+νi(t)-∑j=1i-1∂αi-1∂xj(xj+1+gi(x1)+fi(x1(t-τh)))-∑j=1i-1∂αi-1∂w^jw^˙j.
Differentiating (4.39) along track (4.50),we have
(4.51)V˙i≤zi(zi+1+αi+Fi(ζi))+12zi2+12∑j=1i‍fj2(x1(t-τh))-zi22λi2∑j=1i-1‍Uj(x1)+12∑j=1i-1‍(Uj(x1)-Uj(x1(t-τh)))≤z1(z2+α1+F1(ζ1)-12λ12U1(x1)z1+νi(t))+f1(x1(t-τh))+W~1TΓ1-1W^˙1+12U1(x1)-12U1(x1(t-τh))+W~iTΓi-1W^˙i+ziνi(t)≤z1z2+z1α1+z1F1(ζ1)+12z12+12U1(1-z12λ12)+W~iTΓi-1W^˙i+ziνi(t)≤z1z2+z1α1+z1F1(ζ1)+12z12+12zi2+12di2+∑j=1i-1‍Uj(x1)(1-zi2λi2)+W~iTΓi-1W^˙i.
(1) Ifzi∈Ωzio, then |zi|≥λi. Thus, substituting (4.32) and (4.36) into (4.51) results in
(4.52)V˙i≤zizi+1+ziFi(ζi)12zi2-(ci+bi2+2)zi2-zi22λi2[di2+τmax∑j=1i-1‍kj+λi+12]-ziW^iTSi(ζi)+W~iT(ziSi(ζi)-σiW^i)≤zizi+1-(ci+bi2+1)zi2-12[τmax∑j=1i-1‍kj+di2+λi+12]+12di2+ziεi(ξi)-σiW~iTW^i+12di2≤-cizi2-τmax2∑j=1i-1‍kj-σ2∥W~i∥2+σ2∥Wi*∥2+12bi(εi*)2+12(-zi2+zi+12-λi+12)≤-kiVi+bvi+Θi,
where ki=min{2ci,σi/λmax(Γi-1),1}, bvi=(σi/2)∥Wi*∥2+(1/2bi)(εi*)2. Θi=(1/2)(-zi2+zi+12-λi+12).
If there is no itemΘi in (4.52), then
(4.53)0≤Vi(t)≤(Vi(0)-δv)e-kit+δv,
where δv=bvi/ki. Thus Vi is bounded.
(2) Ifzi∈Ωzi, similar to step 1, we have Vi is bounded.
Step
n. This is the last step for the nth subsystem, similarly to the ith subsystem, If zn∈Ωzno, then |zn|≥λn. Thus we have
(4.54)V˙n≤-cnzn2-τmax2∑j=1n-1‍kj-σ2∥W~n∥2+σ2∥Wn*∥2+12bn(εn*)2-12zn2≤-knVn+bvn+Θn,
where kn=min{2cn,σn/λmax(Γn-1),1},bvn=(σn/2)∥Wn*∥2+(1/2bn)(εn*)2. Θn=-(1/2)zn2.
By (4.54), it is easy to have
(4.55)0≤Vn(t)≤(Vn(0)-δv)e-knt+δv,
where δv=bvn/kn. Thus Vn is bounded.(II) Ifzn∈Ωzn, similar to step 1, we have Vn is boundedTheVi,(1≤i≤n) is bounded. When zi∈Ωzi,(1≤i≤n). In zi∈Ωzio,(1≤i≤n):
(4.56)V˙(t)≤∑i=1n‍V˙i≤-∑i=1n‍kiVi+∑i=1n‍bvi+∑i=1n‍Θi≤-kvV(t)+bv,
where kv=min{k1,k2,…,kn},bv=∑i=1n‍bvi.Then(4.57)0≤V(t)≤(V(0)-δv)e-kvt+δv,
where δv=bv/kv. Thus V(t) is bounded.Simulation Example
Consider the nonlinear system with input and state delays as follows:(4.58)x˙1=x2+0.3y2+sin(2y(t-0.2))+ω1,x˙2=u(t-0.20)+0.2y2-sin(2y(t-0.2))+ω2,y=x1.
Define virtual control as
(4.59)α1=-[c1+b12+32+12λ12(τmaxk1+λ22+d12)]z1-W^1TS1(ξ1),α2=-[c2+b22+1+12λ22(τmaxk1+d22)]z2-W^2TS2(ξ2),
where c1=c2=30, b1=b2=2, λ1=λ2=2, τmax=0.6, k1=2.d1=d2=0.6, Γ1=Γ2=600, σ1=σ2=0.006, ω1=0.05sin(2πt), ω2=0.05cos(2πt). 							(4.60)W^˙1=Γ1(z1S1(ξ1)-σ1W^1),W^˙2=Γ2(z2S2(ξ2)-σ2W^2),z1=x1,z2=x2-α1.
The result of control scheme is in Figures 1 and 2.Figure 1
The control inputu(t).Figure 2
System statex1(t)(“-”) and x2(t)(“-.-”).


## 5. Conclusion

For a class of outputs time-delay nonlinear systems with perturbed or not, a control scheme combined with adaptive control, backstepping, and neural network is proposed. The radius basis function (RBF) neural networks is employed to estimate the unknown continuous functions. It is shown that the proposed method guarantees the semi-globally uniformly ultimately boundedness of all signals in the adaptive closed-loop systems. Simulation results are provided to illustrate the performance of the proposed approach.



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*Source: 852161-2012-09-29.xml*