# Robust Decentralized Adaptive Neural Control for a Class of Nonaffine Nonlinear Large-Scale Systems with Unknown Dead Zones

Authors: Huanqing Wang; Qi Zhou; Xuebo Yang; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The problem of robust decentralized adaptive neural stabilization control is investigated for a class of nonaffine nonlinear interconnected large-scale systems with unknown dead zones. In the controller design procedure, radical basis function (RBF) neural networks are applied to approximate packaged unknown nonlinearities and then an adaptive neural decentralized controller is systematically derived without requiring any information on the boundedness of dead zone parameters (slopes and break points). It is proven that the developed control scheme can ensure that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded in the sense of mean square. Simulation study is provided to further demonstrate the effectiveness of the developed control scheme.


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## Body

## 1. Introduction

During the past several decades, a large number of research results have been obtained on the problems of stability analysis and control design for nonlinear systems because of the extensive existence of nonlinearity in the practical systems. So far, there are many control methods proposed to control design of nonlinear systems, such as adaptive backstepping control [1–4], fault-tolerant control [5–9], H∞ control [10–16], and fuzzy control [17–21]. In the control of uncertain complex nonlinear systems, backstepping-based neural networks or fuzzy adaptive control technique is an efficient and practical strategy, and many interesting results have been obtained for a class of uncertain nonlinear strict-feedback systems; for example, see [22–37]. In addition, adaptive neural or fuzzy backstepping control approach can also be extended to control a class of nonaffine pure-feedback nonlinear systems representing a class of more general lower triangular systems, in which no affine appearance of the state variables can be used as virtual control signals. This makes it quite difficult and more meaningful to control the pure-feedback systems. By using small-gain theorem, in [38], an adaptive neural control scheme is presented for a class of completely nonaffine pure-feedback systems. Afterwards, some researchers further consider other types of pure-feedback nonlinear systems, such as pure-feedback systems with time-delay [39] and with dead-zone [40].Large-scale system is considered as a dynamical system which is composed of some lower-order subsystems with interconnections and often exists in many practical applications such as electric power systems, computer network systems, and aerospace systems. Decentralized adaptive control of unknown nonlinear interconnected systems has attracted much research attention because this problem is important both theoretically and practically. Up to now, many interesting adaptive decentralized control approaches for large-scale nonlinear systems have been proposed in [41, 42] and the references therein. By combining backstepping technique together with adaptive neural or fuzzy control, many research papers have been published in [28–32] for affine nonlinear large-scale interconnected systems and in [43, 44] for nonaffine nonlinear large-scale systems. However, these researches have not considered the effect of the system input signal with dead zone. The existence of dead zone input nonlinearity severely degrades system performance. So, dead zone nonlinearity has to be considered when controller is designed in many industrial processes, such as valves, DC servo motors, and other devices. In [45], an adaptive robust control scheme is proposed for a class of nonlinear systems, in which the dead zone nonlinearity is expressed by a combination of a line and a disturbance-like term and its parameters are tuned by using adaptive technique. Then, some backstepping-based adaptive control schemes are developed for nonlinear systems with unknown dead zone input [46–50].Based on the above observations, this paper focuses on the problem of adaptive neural decentralized control for a class of nonaffine large-scale nonlinear systems with unknown dead zones. The proposed adaptive neural controller guarantees that all the signals in the closed-loop systems remain semiglobally uniformly ultimately bounded in the sense of mean square, and the error signals eventually converge to small neighborhood around the origin. The main advantages of this research lie in that (i) the dead-zone inverse as well as the prior knowledge of bounds of dead-zone parameters (slopes and break-points) is not required; (ii) only one adaptive parameter is involved in the developed controller for each subsystem. As a result, the computational burden is significantly alleviated. In this way, the proposed control law could be easily implemented in practical applications. Simulation results are provided to further illustrate the effectiveness of the proposed control approach.The paper is organized as follows. Section2 provides the problem formulation and preliminaries. The control design and analysis of state feedback controller are given in Section 3, following a simulation example in Section 4. Section 5 concludes this paper.


## 2. Problem Formulation and Preliminaries

Consider a class of pure-feedback nonlinear interconnected large-scale systems with dead zones andN subsystems, and the ith (i=1,2,…,N) subsystem is in the following form:
(1)x˙i,j=fi,j(x-i,j,xi,j+1)+φi,j(y-),1≤j≤ni-1,x˙i,ni=fi,ni(x-i,ni,ui)+φi,ni(y-),yi=xi,1,
where x-i,j=[xi,1,xi,2,…,xi,j]T and y-=[y1,y2,…,yN]T. xi=[xi,1,xi,2,…,xi,ni]T∈Rni and yi∈R are the state variables and the output of the ith subsystem, respectively. fi,j(·):Rj+1→R,(j=1,2,…,ni) are unknown smooth nonlinear functions; φi,j(·):RN→R(j=1,2,…,ni) are unknown interconnections between the ith subsystem and other subsystems, with fi,j(0)=φi,j(0)=0; ui∈R is the output of an unknown dead zone and defined as
(2)ui=D(vi)={gir(vi),vi≥bir,0,bil<vi<bir,gil(vi),vi≤bil,
where bil<0 and bir>0 are the unknown parameters and vi(t)∈R is the input of the dead zone.For the unknown dead zone input, the following assumption is required.Assumption 1 (see [40]).
The functions,gil(vi) and gir(vi), are smooth and there exist unknown positive constants, kil0,kil1,kir0, and kir1, such that
(3)0<kil0≤gil′(vi)≤kil1,∀vi∈(-∞,bil],0<kir0≤gir′(vi)≤kir1,∀vi∈[bir,+∞),
and βi0≤min⁡{kil0,kir0} is an unknown positive constant, where gil′(vi)=(dgil(z)/dz)|z=vi and gir′(vi)=(dgir(z)/dz)|z=vi.Remark 2.
Assumption1 is similar to the assumption in [40] where βi0 is a known constant. However, Assumption 1 does not require it to be known. So, Assumption 1 relaxes the limitation in [40].Based on Assumption1, the dead zone (2) can be rewritten as [40]
(4)ui=D(vi)=KiT(t)Φi(t)vi+di(vi),
where
(5)Φi(t)=[φir(t),φil(t)]T,φir(t)={1,vi(t)>bil,0,vi(t)≤bil,φil(t)={1,vi(t)<bir,0,vi(t)≥bir,Ki(t)=[Kir(vi(t)),Kil(vi(t))]T,Kir(vi(t))={0,vi(t)≤bil,gir′(ξir(vi(t))),bil<vi(t)<+∞,Kil(vi(t))={gil′(ξil(vi(t))),-∞<vi(t)<bir,0,vi(t)≥bir,di(vi)={-gir′(ξir(vi))bir,vi≥bir,-[gil′(ξil(vi))+gir′(ξir(vi))]vi,bil<vi<bir,-gil′(ξil(vi))bil,vi≤bil,
where ξil(vi)∈(vi,bil), if vi<bil;  ξil(vi)∈(bil,vi), if bil≤vi<bir;  ξir(vi)∈(bir,vi), if bir<vi; ξir(vi)∈(vi,bir), and if bil<vi≤bir, and |di(vi)|≤pi*, pi* is an unknown positive constant with p*=(kir1+kil1)max⁡{bir,-bil}.Remark 3.
As shown in [40], there exist some other expressions for the case of linear dead zone outside the deadband, but (4) includes the most actual cases and KiT(t)Φi(t)∈[βi0,kil1+kir1]⊂(0,+∞).Based on mean value theorem [51], fi,j(·) in (1) can be described as
(6)fi,j(x-i,j,xi,j+1)=fi,j(x-i,j,xi,j+10)+hμi,j(xi,j+1-xi,j+10),fi,ni(x-i,ni,ui)=fi,ni(x-i,ni,ui0)+hμi,ni(ui-ui0),
where smooth function fi,j(·) is explicitly analyzed between fi,j(x-i,j,xi,j+1) and fi,j(x-i,j,xi,j+10), hμi,j:=hi,j(x-i,j,xμi,j)=(∂fi,j(x-i,j,xi,j+1)/∂xi,j+1)|xi,j+1=xμi,j,xi,ni+1=ui,  xμi,j=μi,jxi,j+1+(1-μi,j)xi,j+10,  0<μi,j<1,i=1,2,…,N,j=1,2,…,ni.Next, by substituting (6) and (4) into (1), and choosing xi,j+10=0,  ui0=0, we obtain
(7)x˙i,j=hμi,jxi,j+1+fi,j(x-i,j,0)+φi,j(y-),1≤j≤ni-1,x˙i,ni=hμi,niKiT(t)Φi(t)vi+hμi,nidi(vi)+fi,ni(x-i,ni,0)+φi,ni(y-),yi=xi,1.Assumption 4 (see [40]).
The signs ofhμi,j, 1≤i≤N,1≤j≤ni, do not change, and there exist unknown constants bm and c such that
(8)0<bm≤|hμi,j|≤c<∞.Remark 5.
Assumption4 means that hμi,j are strictly either positive or negative. Without loss of generality, it is assumed that 0<bm≤hμi,j. In addition, by means of Assumptions 1 and 4, it can be further supposed that
(9)0<b≤hμi,j,1≤i≤N,1≤j≤ni-1,0<b≤hμi,niKiT(t)Φi(t),
where b=min⁡{bm,bmβi0} is an unknown constant.Assumption 6 (see [31]).
For uncertain nonlinear functionsφi,j(y-) in (1), there exist unknown smooth functions φi,j,l(yl) such that for 1≤i≤N,  1≤j≤ni,
(10)|φi,j(y-)|2≤∑l=1N‍φi,j,l2(yl),
where φi,j,l(0)=0,l=1,2,…,N.Noting thatφi,j,l(yl) in (10) is smooth function with φi,j,l(0)=0, so there exist unknown smooth functions φ-i,j,l(yl) such that
(11)|φi,j(y-)|2≤∑l=1N‍yl2φ-i,j,l2(yl).
In what follows, RBF neural networks are applied to approximate any continuous function f(X):Rn→R,
(12)fnn(X)=WTϕ(X),
where X∈ΩZ⊂Rq is the input vector with q being the neural networks input dimension, weight vector W=[w1,w2,…,wl]T∈Rl, l>1 is the neural networks node number, and ϕ(X)=[ϕ1(X),ϕ2(X),…,ϕl(X)]T means the basis function vector with ϕi(X) being chosen as the commonly used Gaussian function of the form
(13)ϕi(X)=exp⁡[-(X-μi)T(X-μi)ηi2],i=1,2,…,l,
where μi=[μi1,μi2,...,μiq]T is the center of the receptive field and ηi is the width of the Gaussian function. In [52], it has been indicated that with sufficiently large node number l, the RBF neural networks (12) can approximate any continuous function f(X) over a compact set ΩZ⊂Rq to arbitrary any accuracy ε>0 as
(14)f(X)=W*Tϕ(X)+δ(X),∀z∈Ωz∈Rq,
where W* is the ideal constant weight vector and defined as
(15)W*:=argmin⁡W∈R-l{sup⁡Z∈ΩZ|f(X)-WTϕ(X)|},
and δ(X) denotes the approximation error and satisfies |δ(X)|≤ε.Lemma 7 (see [53]).
Consider the Gaussian RBF networks (12) and (13). Let ρ:=(1/2)min⁡i≠j∥μi-μj∥; then an upper bound of ∥ϕ(X)∥ is taken as
(16)∥ϕ(X)∥≤∑k=0∞‍3q(k+2)q-1e-2ρ2k2/η2:=s.It has been shown in [38] that the constant s in Lemma 7 has a limited value and is independent of the variable X and the dimension of neural weights l.


## 3. Adaptive Neural Control Design

In this section, an adaptive neural backstepping control scheme will be developed. During the controller design, for theith subsystem, RBF neural network Wi,jTϕ(Xi,j) will be utilized to approximate the packaged unknown function Fi,j(Xi,j) at step j. Both the virtual control signals αi,j(Xi,j) and adaption laws θ^˙i will be constructed in the following forms:
(17)αi,j(Xi,j)=-ki,jzi,j-12ai,j2zi,jθ^iϕi,jT(Xi,j)ϕi,j(Xi,j),(18)θ^˙i=∑j=1ni‍λi2ai,j2zi,j2ϕi,jT(Xi,j)ϕi,j(Xi,j)-γiθ^i,
where i=1,2,…,N,  j=1,2,…,ni, ki,j, ai,j, λi, and γi are positive design parameters, Xi,1=xi,1,Xi,j=[x-i,jT,θ^i]T,  (j=2,…,ni) with x-i,j=[xi,1,xi,2,…,xi,j]T, and zi,j satisfy the following coordinate transformation:
(19)zi,j=xi,j-αi,j-1
with αi,0=0. θ^i is used to estimate an unknown constant θi which will be defined as
(20)θi=max⁡{1b∥Wi,j∥2;j=1,2,…,ni},
where b is specified in Remark 5, and ∥Wi,j∥ will be defined at the jth step. Specially, αi,ni is the actual control input signal ui.For simplicity, in the following, the time variablet and the state vector x-i,j will be omitted from the corresponding functions and let ϕi,j(Xi,j)=ϕi,j.Step
1. Based on zi,1=xi,1,  zi,2=xi,2-αi,1, the error dynamic zi,1 satisfies
(21)z˙i,1=hμi,1zi,2+hμi,1αi,1+fi,j(x-i,j,0)+φi,1(y-).Consider a Lyapunov function candidate asVi,1=(1/2)zi,12=(1/2)yi2. Then, the time derivative of V1 along (21) satisfies
(22)V˙i,1=yi(hμi,1zi,2+hμi,1αi,1+fi,1(x-i,1,0)+φi,1(y-)).
By using (11) and the completion of squares, we get
(23)yiφi,1(y-)≤12yi2+12∑l=1N‍yl2φ-i,1,l2(yl).
Further, (22) can be rewritten as
(24)V˙i,1=yi(hμi,1zi,2+hμi,1αi,1+fi,1(x-i,1,0)+12yi)+12∑l=1N‍yl2φ-i,1,l2(yl).Step
j
(
2
≤
j
≤
n
i
-
1
). According to (19), one has
(25)z˙i,j=hμi,jzi,j+1+hμi,jαi,j+fi,j(x-i,j,0)+φi,j(y-)-α˙i,j-1,
where
(26)α˙i,j-1=∑k=1j-1‍∂αi,j-1∂xi,k(fi,k(x-i,k+1)+φi,k(y-))+∂αi,j-1∂θ^iθ^˙i.
Take Lyapunov function Vi,j=(1/2)zi,j2; then we have
(27)V˙i,j=zi,j(∑k=1j-1‍∂αi,j-1∂xi,k(fi,k(x-i,k+1)+φi,k(y-))-∂αi,j-1∂θ^iθ^˙igμi,jzi,j+1+gμi,jαi,j+fi,j(x-i,j,0)+φi,j(y-)hh-∑k=1j-1‍∂αi,j-1∂xi,k(fi,k(x-i,k+1)+φi,k(y-))-∂αi,j-1∂θ^iθ^˙i).
Following the same line as the procedures used in (23), one has
(28)-zi,j∑k=1j-1‍∂αi,j-1∂xi,kφi,k(y-)≤12zi,j2∑k=1j-1‍(∂αi,j-1∂xi,k)2+12∑k=1j-1‍∑l=1N‍yl2φ-i,k,l2(yl),(29)zi,jφi,j(y-)≤12zi,j2+12∑l=1N‍yl2φ-i,j,l2(yl).
It can be easily verified from (27) to (29) that
(30)V˙i,j≤zi,j(∑k=1j-1‍(∂αi,j-1∂xi,k)2-∂αi,j-1∂θ^ihμi,jzi,j+1+hμi,jαi,j+fi,j(x-i,j,0)hh-∑k=1j-1‍∂αi,j-1∂xi,kfi,k(x-i,k+1)+12zi,jhh+12zi,j∑k=1j-1‍(∂αi,j-1∂xi,k)2-∂αi,j-1∂θ^iθ^˙i)+12∑k=1j‍∑l=1N‍yl2φ-i,k,l2(yl).Step
n
i. Similar to (25), the following equation can be obtained:
(31)z˙i,ni=hμi,niKiT(t)Φi(t)vi+hμi,nidi(vi)+fi,ni(x-i,ni,0)+φi,ni(y-)-α˙i,ni-1,
where α˙i,ni-1 is shown in (26) with j=ni. Choose a Lyapunov function as
(32)Vi,ni=12zi,ni2+b2λiθ~i2,
where θ~i=θi-θ^i is the parameter error and λi>0 is a design parameter.Then, the following result holds.(33)V˙i,ni=zi,ni(∑k=1ni-1‍∂αi,ni-1∂xi,khμi,niKiT(t)Φi(t)vi+hμi,nidi(vi)+fi,ni(x-i,ni,0)+φi,ni(y-)-∂αi,ni-1∂θ^iθ^˙i-∑k=1ni-1‍∂αi,ni-1∂xi,k(fi,k(x-i,k+1)+φi,k(y-)))-bλiθ~iθ^˙i.
Repeating the same derivations as (28)–(30) produces
(34)V˙i,ni≤zi,ni(∑k=1ni-1‍(∂αi,ni-1∂xi,k)2hμi,niKiT(t)Φi(t)vi+hμi,nidi(vi)hh+fi,ni(x-i,ni,0)-∑k=1ni-1‍∂αi,ni-1∂xi,kfi,k(x-i,k+1)hh+12zi,ni∑k=1ni-1‍(∂αi,ni-1∂xi,k)2+12zi,ni-∂αi,ni-1∂θ^iθ^˙i)+12∑k=1ni‍∑l=1N‍yl2φ-i,k,l2(yl)-bλiθ~iθ^˙i.
Now, consider a Lyapunov function for the whole system as
(35)V=∑i=1N‍∑j=1ni‍Vi,j=∑i=1N‍(12yi2+∑j=2ni‍12zi,j2+b2λiθ~i2).
Then, combining (24) together with (30) and (34), one has
(36)V˙≤∑i=1N‍yi{∑l=1N‍∑s=1nl‍∑k=1s‍hμi,1αi,1+fi,1(x-i,1,0)+12yih+12yi∑l=1N‍∑s=1nl‍∑k=1s‍φ-l,k,i2(yi)}+∑i=1N‍∑j=2ni-1‍zi,j{∑k=1j-1‍(∂αi,j-1∂xi,k)2hμi,jαi,j+hμi,jzi,j-1hh+fi,j(x-i,j,0)-∑k=1j-1‍∂αi,j-1∂xi,kfi,k(x-i,k+1)hh+12zi,j+12zi,j∑k=1j-1‍(∂αi,j-1∂xi,k)2}+∑i=1N‍zi,ni{∑k=1j-1‍(∂αi,j-1∂xi,k)2hμi,niKiT(t)Φi(t)vi+hμi,nidi(vi)hh+fi,ni(x-i,ni,0)-∑k=1ni-1‍∂αi,ni-1∂xi,kfi,k(x-i,k+1)hh+12zi,ni∑k=1ni-1‍(∂αi,ni-1∂xi,k)2+12zi,ni}-∑i=1N‍∑j=2ni‍zi,j∂αi,j-1∂θ^iθ^˙i-∑i=1N‍bλiθ~iθ^˙i,
where the fact of
(37)12∑i=1N‍∑s=1ni‍∑k=1s‍∑l=1N‍yl2φ-i,k,l2(yl)=12∑i=1N‍∑l=1N‍∑s=1nl‍∑k=1s‍yi2φ-l,k,i2(yi)
has been used in above inequality.By using adaptive laws defined in (18) and rearranging sequence, we can obtain
(38)-∑i=1N‍∑j=2ni‍zi,j∂αi,j-1∂θ^iθ^˙i=-∑i=1N‍∑j=2ni‍zi,j∂αi,j-1∂θ^i(∑k=1ni‍λi2ai,k2zi,k2ϕi,kTϕi,k-γiθ^i)=∑i=1N‍∑j=2ni‍zi,j∂αi,j-1∂θ^iγiθ^i-∑i=1N‍∑j=2ni‍zi,j∂αi,j-1∂θ^i∑k=1j-1‍λi2ai,k2zi,k2ϕi,kTϕi,k-∑i=1N‍∑j=2ni‍zi,j∂αi,j-1∂θ^i∑k=jni‍λi2ai,k2zi,k2ϕi,kTϕi,k≤∑i=1N‍∑j=2ni‍zi,j∂αi,j-1∂θ^iγiθ^i-∑i=1N‍∑j=2ni‍zi,j∂αi,j-1∂θ^i∑k=1j-1‍λi2ai,k2zi,k2ϕi,kTϕi,k+∑i=1N‍∑j=2ni‍λi2ai,j2zi,j2(∑k=2j‍|zi,k∂αi,k-1∂θ^i|).
Taking |di(vi)|≤pi* into account, one has
(39)zi,nihμi,nidi(vi)≤12zi,ni2hμi,ni2+12pi*2.By substituting (38) and (39) into (36), it follows
(40)V˙≤∑i=1N‍yi(hμi,1αi,1+Fi,1(Xi,1))+∑i=1N‍∑j=2ni-1‍zi,j(hμi,jαi,j+Fi,j(Xi,j))+∑i=1N‍zi,ni(hμi,niKiT(t)Φi(t)vi+Fi,ni(Xi,ni))+12∑i=1N‍pi*2-12∑i=1N‍∑j=1ni‍zi,j2-∑i=1N‍bλiθ~iθ^˙i,
where the packaged functions Fi,j(Xi,j),  i=1,2,…,N, are specified as
(41)Fi,1(Xi,1)=fi,1(x-i,1,0)+yi+12yi∑l=1N‍∑s=1nl‍∑k=1s‍φ-l,k,i2(yi),Fi,j(Xi,j)=hμi,jzi,j-1+fi,j(x-i,j,0)-∑k=1j-1‍∂αi,j-1∂xi,kfi,k(x-i,k+1)+zi,j+12zi,j∑k=1j-1‍(∂αi,j-1∂xi,k)2+∂αi,j-1∂θ^iγiθ^i-∂αi,j-1∂θ^i∑k=1j-1‍λi2ai,k2zi,k2ϕi,kTϕi,k+λi2ai,j2zi,j(∑k=2j‍|zi,k∂αi,k-1∂θ^i|),j=2,…,ni-1,Fi,ni(Xi,ni)=fi,ni(x-i,ni,0)-∑k=1ni-1‍∂αi,ni-1∂xi,kfi,k(x-i,k+1)+12zi,ni∑k=1ni-1‍(∂αi,ni-1∂xi,k)2+12zi,nihμi,ni2+zi,ni+∂αi,ni-1∂θ^iγiθ^i-∂αi,ni-1∂θ^i∑k=1ni-1‍λi2ai,k2zi,k2ϕi,kTϕi,k+λi2ai,ni2zi,ni(∑k=2ni‍|zi,k∂αi,k-1∂θ^i|).
Because fi,j, hμi,j, and φ-l,k,i are unknown smooth functions, Fi,j(Xi,j),i=1,2,…,N,j=1,2,…,ni cannot be used to define the virtual control input αi,j and the real controller ui. Then, RBF neural network Wi,jTϕi,j(Xi,j) is utilized to model Fi,j(Xi,j), such that for a given positive constant εi,j,
(42)Fi,j(Xi,j)=Wi,jTϕi,j(Xi,j)+δi,j(Xi,j),
where δi,j(Xi,j) is approximation error and satisfies |δi,j(Xi,j)|<εi,j. Subsequently, by Young's inequality, one has
(43)zi,jFi,j(Xi,j)=zi,jWi,jT∥Wi,j∥ϕi,j∥Wi,j∥+zi,jδi,j(Xi,j)≤b2ai,j2zi,j2θiϕi,jTϕi,j+12ai,j2+12zi,j2+12εi,j2,
where i=1,2,…,N,  j=1,2,…,ni and the unknown constant θi=max⁡⁡{(1/b)∥Wi,j∥2;j=1,2,…,ni}.Substituting (42) into (40) and using (43) produces
(44)V˙≤∑i=1N‍zi,1(hμi,1αi,1+b2ai,12zi,1θiϕi,1Tϕi,1)+∑i=1N‍∑j=2ni-1‍zi,j(hμi,jαi,j+b2ai,j2zi,jθiϕi,jTϕi,j)+∑i=1N‍zi,ni(hμi,niKiT(t)Φi(t)vi+b2ai,ni2zi,niθiϕi,niTϕi,ni)+12∑i=1N‍pi*2+∑i=1N‍∑j=1ni‍(12ai,j2+12εi,j2)-∑i=1N‍bλiθ~iθ^˙i.
Now, construct the virtual control signals αi,j=-ki,jzi,j-(1/2ai,j2)zi,jθ^iϕi,jTϕi,j, where i=1,2,…,N,j=1,2,…,ni, and αi,ni=ui. Then, by using (9), we can obtain
(45)zi,jhμi,jαi,j≤-ki,jbzi,j2-b2ai,j2zi,j2θ^iϕi,jTϕi,j,hhhhhhhhhhhhhhhhhhhhj=1,…,ni-1,zi,nihμi,niKiT(t)Φi(t)vi≤-ki,nibzi,ni2-b2ai,ni2zi,ni2θ^iϕi,niTϕi,ni,
where i=1,2,…,N.Substituting (45) into (44) and using adaptive laws θ˙i in (18) result in
(46)V˙≤∑i=1N‍∑j=1ni‍zi,j(-ki,jbzi,j+b2ai,j2zi,jθ~iϕi,jTϕi,j)+∑i=1N‍∑j=1ni‍(12ai,j2+12εi,j2)+12∑i=1N‍pi*2-∑i=1N‍bλiθ~i(∑j=1ni‍λi2ai,j2zi,j2ϕi,jTϕi,j-γiθ^i)≤-∑i=1N‍∑j=1ni‍ki,jbzi,j2+∑i=1N‍γibλiθ~iθ^i+∑i=1N‍∑j=1ni‍(12ai,j2+12εi,j2)+12∑i=1N‍pi*2≤-∑i=1N‍(∑j=1ni‍ki,jbzi,j2+γib2λiθ~i2)+∑i=1N‍∑j=1ni‍(12ai,j2+12εi,j2+γib2λiθi2)+12∑i=1N‍pi*2,
where θ~iθ^i≤-(1/2)θ~i2+(1/2)θi2 has been used in the above inequality.Define(47)a0=min⁡{2ki,jb,γi,i=1,2,…,N,j=1,2,…,ni};b0=∑i=1N‍∑j=1ni‍(12ai,j2+12εi,j2+γib2λiθi2)+12∑i=1N‍pi*2,
we have
(48)V˙≤-a0V+b0,t≥0.Furthermore, multiplying (48) by ea0t yields
(49)ddt(Vea0t)≤b0ea0t.
Integrating (49) over [0,t] gives
(50)0≤V(t)≤b0a0+(V(0)-b0a0)e-a0t,
which implies that
(51)V(t)≤V(0)+b0a0,∀t>0,
which means that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded.Furthermore, it is easily obtained that(52)V(t)≤b0a0,t⟶+∞.
Therefore, based on the definition of V in (50), the error signals zi,j and θ~i eventually converge to the compact set Ωs, which is specified as
(53)Ωs={zi,j,θ~i∣|zi,j|≤2b0a0,|θ~i|≤2λibb0a0,hh1≤i≤N,1≤j≤ni2λibb0a0}.At the present stage, adaptive decentralized control procedure has been completed via backstepping technique. The above design procedures and stable analysis are summarized in the following theorem.Theorem 8.
Under Assumptions1–6, consider the closed-loop nonlinear system consisted of (1), unknown dead zone nonlinearities (2), controller (17), and adaptive law (18). Under the action of controller (17), for any initial conditions [ziT(0),θ^i(0)]T∈Ω0 (where Ω0 is an appropriately chosen compact set), all the signals in the closed-loop system are semiglobally uniformly ultimately bounded in the sense of mean square, and the error signals zi,j and θ~i eventually converge to the compact set Ωs in (53).


## 4. Simulation Example

In this section, in order to illustrate the effectiveness of the proposed control scheme, consider the nonaffine interconnected nonlinear system with dead zones as(54)x˙1,1=(1+x1,12)x1,2+x1,23+y2ln⁡(1+y12),x˙1,2=(5+sin(x1,1x1,2))u1+0.2u15+sin(y12)y2,y1=x1,1,x˙2,1=(3+sin(x2,1))x2,2+0.3x2,25+y1cos⁡(y22),x˙2,2=(2+x2,12)u2+15sin(u2)+y23y1,y2=x2,1,
where ui=D(vi) is defined as
(55)u1=D(v1)={1.5(v1-2.5),v1≥2.5,0,-1<v1<2.5,0.5(v1+1),v1≤-1,u2=D(v2)={0.8(v2-2),v2≥2,0,-1.8<v2<2,1.2(v2+1.8),v2≤-1.8,
where x1,1, x1,2, x2,1, and x2,2 are the state variables and yi is the system output. ui and vi are the output and input of the dead zone nonlinearity, respectively. It is obvious that the system is of nonaffine structure and satisfies Assumptions 1, 4, and 6. By using Theorem 8, the virtual control law, the actual controller, and the adaptive laws are constructed as
(56)αi,1=-ki,1zi,1-12ai,12zi,1θ^iϕi,1Tϕi,1,ui=-ki,2zi,2-12ai,22zi,2θ^iϕi,2Tϕi,2,θ^˙i=∑j=12‍λi2ai,j2zi,j2ϕi,jTϕi,j-γiθ^i,
where zi,1=xi,1,  zi,2=xi,2-αi,1, and  i=1,2. The simulation is run under the initial conditions [x1,1(0),x1,2(0),x2,1(0),x2,2(0)]T=[0.4,-0.1,0.2,0.3]T, and [θ^1(0),θ^2(0)]T=[0.1,0.2]T. In the simulation, design parameters are taken as follows: k1,1=k1,2=k2,1=k2,2=11,a1,1=a1,2=a2,1=a2,2=3,  γ1=γ2=1, and λ1=λ2=2.The simulation results are indicated by Figures1–4. Figure 1 shows the state variables x1,1 and x1,2 of the first subsystems. Figure 2 shows the second subsystems state variables x2,1 and x2,2. Figure 3 displays the control signals v1 and v2, and Figure 4 shows the response curve of the adaptive parameters θ^1 and θ^2. Apparently, simulation results show that good convergence performances are achieved and all the signals in the closed-loop system are bounded.Figure 1
State variablesx1,1 and x1,2.Figure 2
State variablesx2,1 and x2,2.Figure 3
The control signalsv1 and v2.Figure 4
Adaptive lawsθ^1 and θ^2.


## 5. Conclusion

In this paper, a robust decentralized adaptive neural control approach has been developed for a class of large-scale nonaffine nonlinear systems with dead zones. The proposed decentralized controller can guarantee that all the signals in the closed-loop systems are semiglobally uniformly ultimately bounded in the sense of mean square, and the error signals eventually converge to small neighborhood around the origin. The main advantage of the proposed controller is that the prior knowledge of bounds of dead zone slopes is not required. In addition, only one adaptive parameter needs to be updated online for eachn-order subsystem. In this way, the computational burden is significantly alleviated. Simulation results have been provided to further illustrate the effectiveness of our results.There are some problems remaining to be considered, for example, how to generalize the result in this paper to stochastic pure-feedback nonlinear systems with dead zones and how to design an output-feedback adaptive neural controller for original system (1).



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*Source: 841306-2014-03-19.xml*