# Adaptive Fuzzy Tracking Control for Uncertain Nonlinear Time-Delay Systems with Unknown Dead-Zone Input

Authors: Chiang-Cheng Chiang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The tracking control problem of uncertain nonlinear time-delay systems with unknown dead-zone input is tackled by a robust adaptive fuzzy control scheme. Because the nonlinear gain function and the uncertainties of the controlled system including matched and unmatched uncertainties are supposed to be unknown, fuzzy logic systems are employed to approximate the nonlinear gain function and the upper bounded functions of these uncertainties. Moreover, the upper bound of the uncertainty caused by the fuzzy modeling error is also estimated. According to these learning fuzzy models and some feasible adaptive laws, a robust adaptive fuzzy tracking controller is developed in this paper without constructing the dead-zone inverse. Based on the Lyapunov stability theorem, the proposed controller not only guarantees that the robust stability of the whole closed-loop system in the presence of uncertainties and unknown dead-zone input can be achieved, but it also obtains that the output tracking error can converge to a neighborhood of zero exponentially. Some simulation results are provided to demonstrate the effectiveness and performance of the proposed approach.


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## Body

## 1. Introduction

In general systems, there exist some nonsmooth nonlinearities in the actuators, such as dead-zone, saturation, and backlash [1–7]. The information of the dead-zone is usually poorly known and time variant. Recently, high accuracy position control is required, such as DC servosystems, pressure control systems, power systems, chemical reactor systems, and machine tools [1–3, 8]. However, the dead-zone characteristics in actuators may severely limit the performance of the systems and let the output of the systems not reach our requirements.  The robust adaptive control was proposed to deal with nonlinear systems with unknown dead-zone [2]. In Corradini and Orlando [3], the sliding mode controller was presented to robustly stabilize a nonlinear uncertain input. Robust adaptive dead-zone compensation method was used in a DC servo-motor control system [4]. Variable structure control laws were proposed for uncertain large-scale system with dead-zone input [5]. In [8, 9], adaptive control approach was used to cope with nonlinear systems with nonsymmetric dead-zone input.  The proposed controllers in [10, 11] tackled the plants with unknown dead-zone via dead-zone inverse.  However, the common feature of most previous results [1, 2, 4–6, 8, 9, 12] is the nonlinear gain function which is assumed to be a constant. Although the Previous restrictive assumption can be relaxed in [3, 7, 10, 11], the unmatched uncertainty is not taken into account.  Therefore, the motivation of this paper is to synthesize a controller to handle the tracking control problem for a class of uncertain nonlinear state time-delay systems in the presence of an unknown dead-zone input and unmatched uncertainties without constructing the dead-zone inverse.It is well known that a real system is difficult to be described by the exact mathematical model, owing to the existence of uncertain elements, such as parameter variation, modeling errors, unmodeled dynamics, and external disturbances. These uncertainties may affect the stability of the systems. Robust stabilization of the nonlinear uncertain system has widely been investigated [13–16]. In [13], the purpose of this direct robust adaptive fuzzy controller was to deal with a class of nonlinear systems containing both unconstructed state-dependent unknown nonlinear uncertain and gain functions.  Bartolini et al. [14] suggested the second-order sliding mode controller to cope with the uncertain system nonaffine in the control law and the presence of the unmodeled dynamic actuator. The methods of robust adaptive control [15, 16] were utilized to solve the nonlinear uncertain problem. In [15], the robust adaptive controller for SISO nonlinear uncertain system was presented by the input/output linearization approach.  In the case where the nonlinear uncertain systems include constant linearly parameterized uncertainty and nonlinear state-dependent parametric uncertainty, the direct robust adaptive control framework was developed in [16].In recent years, the design problem of nonlinear time-delay systems has received considerable attention in [17–23] because time-delay characteristic usually confronted in engineering systems may degrade the control performance and make the systems unstable.    By employing the input-output approach and the scaled small gain theorem, the filtering problem for discrete-time T-S fuzzy systems with time-varying delay has been studied [17]. In [18], the stabilization of LTI systems with time delay was considered by using a low-order controller. The stability analysis and robust control for time-delay systems attracted a large number of researchers over the past years [19–21]. Recently, the problem of stability analysis for stochastic neural networks with discrete interval and distributed time-varying was investigated by applying the idea of delay partitioning method [23].On the other hand, the fuzzy control techniques have been widely used in many control problems in recent years [24–26]. The fuzzy logic system is constructed from a collection of fuzzy IF-THEN rules. It becomes a useful way to approximate the unknown nonlinear functions and uncertainties in the nonlinear systems. An adaptive interval type-2 fuzzy sliding mode controller for a class of unknown nonlinear discrete-time systems corrupted by external disturbances was presented [24]. In [25], an adaptive neural-fuzzy control design was examined for tracking of nonlinear affine in the control dynamic systems with unknown nonlinearities.  Based on a novel fuzzy Lyapunov-Krasovskii functional, a delay partitioning method has been developed for the delay-dependent stability analysis of fuzzy time-varying state delay systems [26].In this paper, the problem of output tracking control is investigated for a class of uncertain nonlinear state time-delay systems containing unknown dead-zone input and unmatched uncertainties. The main features of the proposed robust adaptive fuzzy controller are summarized as follows. (i) By utilizing a description of a dead-zone feature, an adaptive law is used to estimate the properties of the dead-zone model intuitively and mathematically, without constructing a dead-zone inverse. (ii) Fuzzy logic systems with some appropriate learning laws are applied to approximate the nonlinear gain function and the upper bounded functions of matched and unmatched uncertainties. (iii) The unknown upper bound of the uncertainties caused by approximation (or fuzzy modeling) error is estimated by a simple adaptive law. (iv) By means of Lyapunov stability theorem, the proposed controller cannot only guarantee the robust stability of the whole closed-loop system but also obtain the good tracking performance.This paper is organized as follows. In Section2, the form of the uncertain nonlinear state time-delay system with unknown dead-zone input is described. The fuzzy logic systems and fuzzy basis functions are also reviewed. Section 3 presents the robust adaptive fuzzy tracking controller to deal with a class of nonlinear uncertain state time-delay systems containing unknown dead-zone input. By Lyapunov stability theorem, the presented controller can ensure the stability of the controlled systems. Simulation results are demonstrated along with the effectiveness and performance of the proposed controller in Section 4. Finally, a conclusion is given in Section 5.


## 2. Problem Statement and Preliminaries

### 2.1. Problem Statement

Consider a class of uncertain nonlinear state time-delay systems containing an unknown dead-zone in the following form:(1)x˙1=x2+Δϕ1(x),x˙2=x3+Δϕ2(x),x˙3=x4+Δϕ3(x),⋮x˙n-1=xn+Δϕn-1(x),x˙n=∑i=1Mθ1if1i(x(t))+Δf1(x(t))+∑j=1Nθ2jf2j(x(t-τ))+Δf2(x(t-τ))+g(x)Z(v(t))+Δϕn(x),y=x1,
or equivalently,
(2)x˙=Ax+B[∑i=1Mθ1if1i(x(t))+Δf1(x(t))+∑j=1Nθ2jf2j(x(t-τ))+Δf2(x(t-τ))+g(x)Z(v(t))∑i=1Mθ1if1i(x(t))+Δf1(x(t))]+Θ(x),y=Cx,
where
(3)A=[010⋯0001⋱⋮⋮⋮⋱⋱000⋯0100⋯00]∈Rn×n,B=[00⋮01]∈Rn×1,CT=[10⋮00]∈Rn×1,
where x(t)=[x1(t),x2(t),…,xn(t)]T∈Rn is the system state vector which is assumed to be available for measurement, and v(t)∈R and y(t)∈R are the input and output of the system, respectively.  τ is the value of time delay.  The unknown nonlinear system functions are assumed to be in the linearly parameterized form and consist of two parts: (i) the sum of θ1if1i(x(t)) for i=1,2,…,M; (ii) the sum of θ2jf2j(x(t-τ)) for j=1,2,…,N.  The parameters θ1i and θ2j are unknown but constant. f1i(x(t)) and f2j(x(t-τ)) are known continuous, linear or nonlinear functions.  Δf1(x(t)) and Δf2(x(t-τ)) are the unknown matched uncertainties. g(x(t)) is the unknown nonlinear gain function, and Θ(x)= [Δϕ1(x),Δϕ2(x),…,Δϕn(x)]T∈Rn×1 is the vector of unknown unmatched uncertainties.  Without loss of generality, it is assumed that the sign of g(x(t)) is positive. Z(v(t)):R→R is the nonlinear input function containing a dead-zone.Now, let the output of the system and its derivatives be expressed as follows:(4)y=x1,y(1)=x˙1=x2+Δϕ1,y(2)=x˙2+(Δϕ1)(1)=x3+Δϕ2+(Δϕ1)(1),y(3)=x˙3+(Δϕ2)(1)+(Δϕ1)(2)=x4+Δϕ3+(Δϕ2)(1)+(Δϕ1)(2),⋮y(n-1)=x˙(n-1)+(Δϕ(n-2))(1)+(Δϕ(n-3))(2)+⋯+(Δϕ2)(n-3)+(Δϕ1)(n-2)=xn+(Δϕ(n-1))+(Δϕ(n-2))(1)+(Δϕ(n-3))(2)+⋯+(Δϕ2)(n-3)+(Δϕ1)(n-2),y(n)=x˙n+(Δϕ(n-1))(1)+(Δϕ(n-2))(2)+(Δϕ(n-3))(3)+⋯+(Δϕ2)(n-2)+(Δϕ1)(n-1)=∑i=1Mθ1if1i(x(t))+Δf1(x(t))+∑j=1Nθ2jf2j(x(t-τ))+Δf2(x(t-τ))+g(x)Z(v(t))+Δϕn+(Δϕ(n-1))(1)+(Δϕ(n-2))(2)+(Δϕ(n-3))(3)+⋯+(Δϕ2)(n-2)+(Δϕ1)(n-1)=∑i=1Mθ1if1i(x(t))+Δf1(x(t))+∑j=1Nθ2jf2j(x(t-τ))+Δf2(x(t-τ))+g(x)Z(v(t))+ΔΦ,
where
(5)ΔΦ=Δϕ1(n-1)+Δϕ2(n-2)+⋯+Δϕn-1(1)+Δϕn.The dead-zone with inputv(t) and output as shown in Figure 1 is described by
(6)Z(v(t))={mr(v(t)-ca)forv(t)≥ca,0forcb<v(t)<ca,ml(v(t)-cb)forv(t)≤cb,
where ca>0,cb<0 and mr>0,ml>0 are parameters and slopes of the dead-zone, respectively.  In order to investigate the key features of the dead-zone in the control problems, the following assumptions should be made.Figure 1
Dead-zone model.Assumption 1.
The dead-zone outputZ(v(t)) is not available to obtain.Assumption 2.
The dead-zone slopes are of the same value; that is,mr=ml=m.Assumption 3.
There exist known constantscamin, camax, cbmin, cbmax, mmin, and mmax such that the unknown dead-zone parameters ca,cb, and m are bounded; that is, ca∈[camin, camax], cb∈[cbmin,  cbmax], and m∈[mmin, mmax].Based on the previous assumptions, the expression (6) can be represented as
(7)Z(v(t))=mv(t)+z(v(t)),
where z(v(t)) can be calculated from (6) and (7) as
(8)z(v(t))={-mcaforv(t)≥ca,-mv(t)forcb<v(t)<ca,-mcbforv(t)≤cb.
From Assumptions 2 and 3, we can conclude that z(v(t)) is bounded and satisfies |z(v(t))|≤ρ, where ρ is the upper bound which can be chosen as
(9)ρ=max{mmaxcamax,-mmaxcbmax},
where cbmin is a negative value.Then, letym be a given bounded reference signal and contain finite derivatives up to the nth order, define the tracking error as
(10)ei=ym(i-1)-y(i-1),fori=1,2,…,n,
and denote e=[e1,e2,…,en]T,  y=[y,y˙,…,y(n-1)]T, and ym=[ym,y˙m,…,ym(n-1)]T.The control objective of this paper is to design a control lawv(t) such that y can follow a given desired reference signal ym and guarantee that all the signals involved in the whole closed-loop system are bounded.


### 2.2. Description of Fuzzy Logic Systems

The basic configuration of the fuzzy logic system consists of four main components: fuzzy rule base, fuzzy inference engine, fuzzifier, and defuzzifier [27].  The fuzzy logic system performs a mapping from U⊂Rn to V⊂R.  Let U=U1×⋯×Un, where Ui⊂R,i=1,2,…,n.  The fuzzy rule base consists of a collection of fuzzy IF-THEN rules as follows:
(11)R(l):IFx1isF1land…andxnisFnl,THENyisGl,
where x=[x1,x2,…,xn]T∈U and y∈V⊂R are the input and output of the fuzzy logic system, and Fil and Gl are fuzzy sets in Ui and V, respectively. The fuzzifier maps a crisp point x=[x1,x2,…,xn]T into a fuzzy set in U. The fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets in V, based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The defuzzifier maps a fuzzy set in V to a crisp point in V.The fuzzy systems with center-average defuzzifier, product inference, and singleton fuzzifier are of the following form:(12)y(x)=∑l=1Mθl(∏i=1nμFil(xi))∑l=1M(∏i=1nμFil(xi)),
where M is the number of rules, θl is the point at which the fuzzy membership function μGl(θl) of fuzzy sets Gl achieves its maximum value, and it is assumed that μGl(θl)=1. Equation (12) can be rewritten as
(13)y(x)=θTξ(x),
where θ=[θ1,θ2,…,θM]T is a parameter vector, and ξ(x)=[ξ1(x),…,ξM(x)]T is a regressive vector with the regressor ξl(x), which is defined as fuzzy basis function
(14)ξl(x)=∏i=1nμFil(xi)∑l=1n(∏i=1nμFil(xi)).


## 2.1. Problem Statement

Consider a class of uncertain nonlinear state time-delay systems containing an unknown dead-zone in the following form:(1)x˙1=x2+Δϕ1(x),x˙2=x3+Δϕ2(x),x˙3=x4+Δϕ3(x),⋮x˙n-1=xn+Δϕn-1(x),x˙n=∑i=1Mθ1if1i(x(t))+Δf1(x(t))+∑j=1Nθ2jf2j(x(t-τ))+Δf2(x(t-τ))+g(x)Z(v(t))+Δϕn(x),y=x1,
or equivalently,
(2)x˙=Ax+B[∑i=1Mθ1if1i(x(t))+Δf1(x(t))+∑j=1Nθ2jf2j(x(t-τ))+Δf2(x(t-τ))+g(x)Z(v(t))∑i=1Mθ1if1i(x(t))+Δf1(x(t))]+Θ(x),y=Cx,
where
(3)A=[010⋯0001⋱⋮⋮⋮⋱⋱000⋯0100⋯00]∈Rn×n,B=[00⋮01]∈Rn×1,CT=[10⋮00]∈Rn×1,
where x(t)=[x1(t),x2(t),…,xn(t)]T∈Rn is the system state vector which is assumed to be available for measurement, and v(t)∈R and y(t)∈R are the input and output of the system, respectively.  τ is the value of time delay.  The unknown nonlinear system functions are assumed to be in the linearly parameterized form and consist of two parts: (i) the sum of θ1if1i(x(t)) for i=1,2,…,M; (ii) the sum of θ2jf2j(x(t-τ)) for j=1,2,…,N.  The parameters θ1i and θ2j are unknown but constant. f1i(x(t)) and f2j(x(t-τ)) are known continuous, linear or nonlinear functions.  Δf1(x(t)) and Δf2(x(t-τ)) are the unknown matched uncertainties. g(x(t)) is the unknown nonlinear gain function, and Θ(x)= [Δϕ1(x),Δϕ2(x),…,Δϕn(x)]T∈Rn×1 is the vector of unknown unmatched uncertainties.  Without loss of generality, it is assumed that the sign of g(x(t)) is positive. Z(v(t)):R→R is the nonlinear input function containing a dead-zone.Now, let the output of the system and its derivatives be expressed as follows:(4)y=x1,y(1)=x˙1=x2+Δϕ1,y(2)=x˙2+(Δϕ1)(1)=x3+Δϕ2+(Δϕ1)(1),y(3)=x˙3+(Δϕ2)(1)+(Δϕ1)(2)=x4+Δϕ3+(Δϕ2)(1)+(Δϕ1)(2),⋮y(n-1)=x˙(n-1)+(Δϕ(n-2))(1)+(Δϕ(n-3))(2)+⋯+(Δϕ2)(n-3)+(Δϕ1)(n-2)=xn+(Δϕ(n-1))+(Δϕ(n-2))(1)+(Δϕ(n-3))(2)+⋯+(Δϕ2)(n-3)+(Δϕ1)(n-2),y(n)=x˙n+(Δϕ(n-1))(1)+(Δϕ(n-2))(2)+(Δϕ(n-3))(3)+⋯+(Δϕ2)(n-2)+(Δϕ1)(n-1)=∑i=1Mθ1if1i(x(t))+Δf1(x(t))+∑j=1Nθ2jf2j(x(t-τ))+Δf2(x(t-τ))+g(x)Z(v(t))+Δϕn+(Δϕ(n-1))(1)+(Δϕ(n-2))(2)+(Δϕ(n-3))(3)+⋯+(Δϕ2)(n-2)+(Δϕ1)(n-1)=∑i=1Mθ1if1i(x(t))+Δf1(x(t))+∑j=1Nθ2jf2j(x(t-τ))+Δf2(x(t-τ))+g(x)Z(v(t))+ΔΦ,
where
(5)ΔΦ=Δϕ1(n-1)+Δϕ2(n-2)+⋯+Δϕn-1(1)+Δϕn.The dead-zone with inputv(t) and output as shown in Figure 1 is described by
(6)Z(v(t))={mr(v(t)-ca)forv(t)≥ca,0forcb<v(t)<ca,ml(v(t)-cb)forv(t)≤cb,
where ca>0,cb<0 and mr>0,ml>0 are parameters and slopes of the dead-zone, respectively.  In order to investigate the key features of the dead-zone in the control problems, the following assumptions should be made.Figure 1
Dead-zone model.Assumption 1.
The dead-zone outputZ(v(t)) is not available to obtain.Assumption 2.
The dead-zone slopes are of the same value; that is,mr=ml=m.Assumption 3.
There exist known constantscamin, camax, cbmin, cbmax, mmin, and mmax such that the unknown dead-zone parameters ca,cb, and m are bounded; that is, ca∈[camin, camax], cb∈[cbmin,  cbmax], and m∈[mmin, mmax].Based on the previous assumptions, the expression (6) can be represented as
(7)Z(v(t))=mv(t)+z(v(t)),
where z(v(t)) can be calculated from (6) and (7) as
(8)z(v(t))={-mcaforv(t)≥ca,-mv(t)forcb<v(t)<ca,-mcbforv(t)≤cb.
From Assumptions 2 and 3, we can conclude that z(v(t)) is bounded and satisfies |z(v(t))|≤ρ, where ρ is the upper bound which can be chosen as
(9)ρ=max{mmaxcamax,-mmaxcbmax},
where cbmin is a negative value.Then, letym be a given bounded reference signal and contain finite derivatives up to the nth order, define the tracking error as
(10)ei=ym(i-1)-y(i-1),fori=1,2,…,n,
and denote e=[e1,e2,…,en]T,  y=[y,y˙,…,y(n-1)]T, and ym=[ym,y˙m,…,ym(n-1)]T.The control objective of this paper is to design a control lawv(t) such that y can follow a given desired reference signal ym and guarantee that all the signals involved in the whole closed-loop system are bounded.


## 2.2. Description of Fuzzy Logic Systems

The basic configuration of the fuzzy logic system consists of four main components: fuzzy rule base, fuzzy inference engine, fuzzifier, and defuzzifier [27].  The fuzzy logic system performs a mapping from U⊂Rn to V⊂R.  Let U=U1×⋯×Un, where Ui⊂R,i=1,2,…,n.  The fuzzy rule base consists of a collection of fuzzy IF-THEN rules as follows:
(11)R(l):IFx1isF1land…andxnisFnl,THENyisGl,
where x=[x1,x2,…,xn]T∈U and y∈V⊂R are the input and output of the fuzzy logic system, and Fil and Gl are fuzzy sets in Ui and V, respectively. The fuzzifier maps a crisp point x=[x1,x2,…,xn]T into a fuzzy set in U. The fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets in V, based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The defuzzifier maps a fuzzy set in V to a crisp point in V.The fuzzy systems with center-average defuzzifier, product inference, and singleton fuzzifier are of the following form:(12)y(x)=∑l=1Mθl(∏i=1nμFil(xi))∑l=1M(∏i=1nμFil(xi)),
where M is the number of rules, θl is the point at which the fuzzy membership function μGl(θl) of fuzzy sets Gl achieves its maximum value, and it is assumed that μGl(θl)=1. Equation (12) can be rewritten as
(13)y(x)=θTξ(x),
where θ=[θ1,θ2,…,θM]T is a parameter vector, and ξ(x)=[ξ1(x),…,ξM(x)]T is a regressive vector with the regressor ξl(x), which is defined as fuzzy basis function
(14)ξl(x)=∏i=1nμFil(xi)∑l=1n(∏i=1nμFil(xi)).


## 3. Adaptive Fuzzy Tracking Controller Design and Stability Analysis

According to (4), (7), and (10), the tracking error dynamic equation can be expressed as
(15)e˙=Ae+B[ym(n)-∑i=1Mθ1if1i(x(t))-Δf1(x(t))-∑j=1Nθ2jf2j(x(t-τ))-Δf2(x(t-τ))-g(x)mv(t)-g(x)z(v(t))-ΔΦ∑i=1Mθ1if1i(x(t))-Δf1(x(t))].Now, let us choose a vectorK=[k1,k2,…,kn]∈R1×n such that  Am=A-BK is Hurwitz; then, the tracking error dynamic equation (15) can be rewritten as
(16)e˙=Ame+B[Ke+ym(n)-∑i=1Mθ1if1i(x(t))-Δf1(x(t))-∑j=1Nθ2jf2j(x(t-τ))-Δf2(x(t-τ))-g(x)mv(t)-g(x)z(v(t))-ΔΦ∑i=1Mθ1if1i(x(t))-Δf1(x(t))].It is worth noting thatΔf1(x(t)), f2j(x(t-τ)), and ΔΦ are unknown uncertainties and satisfy the following assumption.Assumption 4.
|
Δ
Φ
|
≤
h
1
(
x
)
,
|
Δ
f
1
(
x
(
t
)
)
|
≤
h
2
(
x
(
t
)
), and |Δf2(x(t-τ))|≤h3(x(t-τ)), where h1(x),  h2(x(t)), and h3(x(t-τ)) are unknown smooth positive functions and can be estimated by fuzzy logic systems with some adaptive laws which will be determined later.
First, the nonlinear gain functiong(x) and the upper bounded functions h1(x),h2(x(t)), and h3(x(t-τ)) of unmatched and matched uncertainties can be approximated, over a compact set Ωx, by the fuzzy logic systems as follows:
(17)g^(x∣θg)=θgTξ(x),h^1(x∣θh1)=θh1Tξ(x),h^2(x∣θh2)=θh2Tξ(x),h^3(x(t-τ)∣θh3)=θh3Tξ(x(t-τ)),
where ξ(x) and ξ(x(t-τ)) are the fuzzy basis vectors, and θg,θh1,θh2, and θh3 are the corresponding adjustable parameter vectors of each fuzzy logic system.  It is assumed that θg,θh1,θh2, and θh3 belong to compact sets Ωθg,Ωθh1,Ωθh2, and Ωθh3, respectively, which are defined as
(18)Ωθg={θg∈RM:∥θg∥≤N1<∞},Ωθh1={θh1∈RM:∥θh1∥≤N2<∞},Ωθh2={θh2∈RM:∥θh2∥≤N3<∞},Ωθh3={θh3∈RM:∥θh3∥≤N4<∞},
where N1,N2,N3, and N4 are the designed parameters, and M is the number of fuzzy inference rules.  Let us define the optimal parameter vectors θg*,θh1*,θh2*, and θh3* as follows:
(19)θg*=argminθg∈Ωθg{supx∈Ωx|g(x)-g^(x∣θg)|},θh1*=argminθh1∈Ωθh1{supx∈Ωx|h1(x)-h^1(x∣θh1)|},θh2*=argminθh2∈Ωθh2{supx∈Ωx|h2(x)-h^2(x∣θh2)|},θh3*=argminθh3∈Ωθh3{supx∈Ωx|h3(x(t-τ))-h^1(x(t-τ)∣θh1)|supx∈Ωx|h3(x(t-τ))},
where θg*,θh1*,θh2*, and θh3* are bounded in the suitable closed sets Ωθg,Ωθh1,Ωθh2, and Ωθh3, respectively. The parameter estimation errors can be defined as
(20)θ~g=θg-θg*,θ~h1=θh1-θh1*,θ~h2=θh2-θh2*,θ~h3=θh3-θh3*,|ω1|+|ω2|≤ω,
where  ω is an unknown positive constant, and
(21)ω1=(h1(x)-h^1(x∣θh1*))+(h2(x(t))-h^2(x∣θh2*))+(h3x(x(t-τ))-h^3(x∣θh3*)),ω2=(g(x)-g^(x∣θg*))(mv(t)+z(v(t)))
as the minimum approximation errors, which correspond to approximation errors obtained when optimal parameters are used.
Secondly, we define(22)ϕ~=ϕ^-ϕ,θ~1=θ^1-θ1,θ~2=θ^2-θ2,ω~=ω^-ω,
where ϕ^ is an estimate of ϕ, which is defined as ϕ=(m)-1. θ^1 and θ^2 are the estimates of θ1 and θ2, respectively, which are defined as
(23)θ1=[(m)-1θ11,(m)-1θ12,…,(m)-1θ1M]T∈RM,θ2=[(m)-1θ21,(m)-1θ22,…,(m)-1θ2N]T∈RN,
and ω^ is an estimate of ω.Based on the previous discussion and under Assumptions1–4, we are in a position to propose the robust adaptive fuzzy controller in the following form:
(24)v=v1+v2+v3+v4+v5,
where
(25)v1=1g^(x∣θg)ϕ^[Ke+ym(n)+(eTPB)T∥eTPB∥×(h^1(x∣θh1)+h^2(x∣θh2)+h^3(x(t-τ)∣θh3))(eTPB)T∥eTPB∥],v2=-1g^(x∣θg)f1T(x(t))θ^1,v3=-1g^(x∣θg)f2T(x(t-τ))θ^2,v4=1mmin1g^(x∣θg)(eTPB)T∥eTPB∥ω^,(26)v5=ρmmintanh(eTPBε),
where f1(x(t))=[f11,f12,…,f1M]T∈RM and f2(x(t-τ))=[f21,f22,…,f2N]T∈RN, ρ is defined in (9), and P is a symmetric positive definite matrix, which is a solution of the following Lyapunov equation:
(27)AmTP+PAm=-Q,
where Q is a positive definite matrix, and the parameter update laws are as follows:
(28)θ˙g=-γgeTPBξ(x)(v(t)+z1(v(t))),(29)θ˙h1=γh1∥eTPB∥ξ(x(t)),θ˙h2=γh2∥eTPB∥ξ(x(t)),(30)θ˙h3=γh3∥eTPB∥ξ(x(t-τ)),(31)θ^˙1=-γ1eTPBf1(x(t)),(32)θ^˙2=-γ2eTPBf2(x(t-τ)),(33)ω^˙=γω∥eTPB∥,(34)ϕ^˙=η(eTPB){(eTPB)∥eTPB∥h^3[Ke+ym(n)]+(eTPB)∥eTPB∥h^1(x∣θh1)+(eTPB)∥eTPB∥h^2(x(t)∣θh2)+(eTPB)∥eTPB∥h^3(x(t-τ)∣θh3)},
where the scalars γh1,γh2,γh3,γg,γ1,γ2,γω,  and η are positive constants, determining the rates of adaptations, and
(35)z1(v(t))={-caforv(t)≥ca,-v(t)forcb<v(t)<ca,-cbforv(t)≤cb.Remark 1.
Without loss of generality, the adaptive laws used in this paper are assumed that the parameter vectors are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint sets. If the parameter vectors are on the boundaries of the constraint sets but moving toward the outside of the constraint sets, we have to use the projection algorithm [27] to modify the adaptive laws such that the parameter vectors will remain inside of the constraint sets. The proposed adaptive law (28)–(30) can be modified as the following form:
(36)θ˙g={-γg∥eTPB∥ξ(x(t))(v(t)+z1(v(t))),if(∥θg∥<N1)or(∥θg∥=N1and∥eTPB∥θgTξ(x(t))(v(t)+z1(v(t)))≥0),P{-γg∥eTPB∥ξ(x(t))(v(t)+z1(v(t)))},if(∥θg∥=N1and∥eTPB∥θgTξ(x(t))(v(t)+z1(v(t)))<0),
where P{-γg∥eTPB∥ξ(x(t))(v(t)+z1(v(t)))} is defined as
(37)P{-γg∥eTPB∥ξ(x(t))(v(t)+z1(v(t)))}=-γg∥eTPB∥ξ(x(t))(v(t)+z1(v(t)))+γg∥eTPB∥θgθgT∥θg∥2ξ(x(t))×(v(t)+z1(v(t))),θ˙h1={γh1∥eTPB∥ξ(x(t)),if(∥θh1∥<N2)or(∥θh1∥=N2and∥eTPB∥θh1Tξ(x(t))≤0),P{γh1∥eTPB∥ξ(x(t))},if(∥θh1∥=N2and∥eTPB∥θh1Tξ(x(t))>0),
where P{γh1∥eTPB∥ξ(x(t))} is defined as
(38)P{γh1∥eTPB∥ξ(x(t))}=γh1∥eTPB∥ξ(x(t))-γh1∥eTPB∥θh1θh1T∥θh1∥2ξ(x(t)),θ˙h2={γh2∥eTPB∥ξ(x(t)),if(∥θh2∥<N3)or(∥θh2∥=N3and∥eTPB∥θh2Tξ(x(t))≤0),P{γh2∥eTPB∥ξ(x(t))},if(∥θh2∥=N3and∥eTPB∥θh2Tξ(x(t))>0),
where P{γh2∥eTPB∥ξ(x(t))} is defined as
(39)P{γh2∥eTPB∥ξ(x(t))}=γh2∥eTPB∥ξ(x(t))-γh2∥eTPB∥θh2θh2T∥θh2∥2ξ(x(t)),θ˙h3={γh3∥eTPB∥ξ(x(t-τ)),if(∥θh3∥<N4)or(∥θh3∥=N4and∥eTPB∥θh3Tξ(x(t-τ))≤0),P{γh3∥eTPB∥ξ(x(t-τ))},if(∥θh3∥=N4and∥eTPB∥θh3Tξ(x(t-τ))>0),
where P{γh3∥eTPB∥ξ(x(t-τ))} is defined as
(40)P{γh3∥eTPB∥ξ(x(t-τ))}=γh3∥eTPB∥ξ(x(t-τ))-γh3∥eTPB∥θh3θh3T∥θh3∥2ξ(x(t-τ)).The main result of the proposed robust adaptive fuzzy tracking control scheme is summarized in the following theorem.Theorem 2.
Consider the uncertain nonlinear state time-delay system (1) with unknown dead-zone input (7).  If Assumptions 1–4 are satisfied, then the proposed robust adaptive fuzzy tracking controller defined by (24)–(26) with some adaptation laws (28)–(34) ensures that all the signals of the whole closed-loop system are bounded, and the output tracking errors converge to a neighborhood of zero exponentially.Proof.
Consider the Lyapunov function candidate(41)V=12(1meTPe+1γ1θ~1Tθ~1+1γ2θ~2Tθ~2+1m·γh1θ~h1Tθ~h1+1m·γh2θ~h2Tθ~h2+1m·γh3θ~h3Tθ~h3+1γgθ~gTθ~g+1ηϕ~2+1mmin·γωω~2).
Differentiating the Lyapunov function V with respect to time, we can obtain
(42)V˙=12me˙TPe+12meTPe˙+1γ1θ~1Tθ~˙1+1γ2θ~2Tθ~˙2+1mγh1θ~h1Tθ~˙h1+1mγh2θ~h2Tθ~˙h2+1mγh3θ~h3Tθ~˙h3+1γgθ~gTθ~˙g+1ηϕ~ϕ~˙+1mmin·γωω~ω~˙.
From (16) and by the fact that θ~˙1=θ^˙1,θ~˙2=θ^˙2,θ~˙h1=θ˙h1,θ~˙h2=θ˙h2,θ~˙h3=θ˙h3,θ~˙g=θ˙g,ϕ~˙=ϕ^˙, and ω~˙=ω^˙, the previous equation becomes
(43)V˙=12meT[AmTP+PAm]e+1meTPB[Ke+ym(n)-∑i=1Mθ1if1i(x(t))-Δf1(x(t))-∑j=1Nθ2jf2j(x(t-τ))-Δf2(x(t-τ))-g(x)mv(t)-g(x)z(v(t))-ΔΦ∑i=1Mθ1if1i(x(t))]+1γ1θ~1Tθ^˙1+1γ2θ~2Tθ^˙2+1mγh1θ~h1Tθ˙h1+1mγh2θ~h2Tθ˙h2+1mγh3θ~h3Tθ˙h3+1γgθ~gTθ˙g+1ηϕ~ϕ^˙+1mminγωω~ω^˙.
Applying (27) and Assumption 4 to (43) yields
(44)V˙≤-12meTQe+1meTPB×[Ke+ym(n)-∑i=1Mθ1if1i(x(t))-∑j=1Nθ2jf2j(x(t-τ))-g(x)mv(t)-g(x)z(v(t))∑i=1Mθ1if1i(x(t))]+1m∥eTPB∥h1(x)+1m∥eTPB∥h2(x(t))+1m∥eTPB∥h3(x(t-τ))+1γ1θ~1Tθ^˙1+1γ2θ~2Tθ^˙2+1mγh1θ~h1Tθ˙h1+1mγh2θ~h2Tθ˙h2+1mγh3θ~h3Tθ˙h3+1γgθ~gTθ˙g+1ηϕ~ϕ^˙+1mminγωω~ω^˙.
Substituting (17) and (23) into (44), we obtain
(45)V˙≤-12meTQe+eTPB×{1m[Ke+ym(n)]-f1T(x(t))θ1-f2T(x(t-τ))θ2}+1m∥eTPB∥(|ω1|+|ω2|)-1meTPB×[g^(x∣θg)(mv(t)+z(v(t)))]+1meTPB[θ~gTξ(x)(mv(t)+z(v(t)))]+1m∥eTPB∥[h^1(x∣θh1)-θ~h1Tξ(x)]+1m∥eTPB∥[h^2(x(t)∣θh2)-θ~h2Tξ(x(t))]+1m∥eTPB∥[h^3(x(t-τ)∣θh3)-θ~h3Tξ(x(t-τ))]+1γ1θ~1Tθ^˙1+1γ2θ~2Tθ^˙2+1m·γh1θ~h1Tθ˙h1+1m·γh2θ~h2Tθ˙h2+1m·γh3θ~h3Tθ˙h3+1γgθ~gTθ˙g+1ηϕ~ϕ^˙+1mmin·γωω~ω^˙≤-12meTQe+eTPB×{1m[Ke+ym(n)]-f1T(x(t))θ1-f2T(x(t-τ))θ2}+1m∥eTPB∥ω-1meTPB×[g^(x∣θg)(mv(t)+z(v(t)))]+1meTPB[θ~gTξ(x)(mv(t)+m·z1(v(t)))]+1m∥eTPB∥[h^1(x∣θh1)-θ~h1Tξ(x)]+1m∥eTPB∥[h^2(x(t)∣θh2)-θ~h2Tξ(x(t))]+1m∥eTPB∥[h^3(x(t-τ)∣θh3)-θ~h3Tξ(x(t-τ))]+1γ1θ~1Tθ^˙1+1γ2θ~2Tθ^˙2+1m·γh1θ~h1Tθ˙h1+1m·γh2θ~h2Tθ˙h2+1m·γh3θ~h3Tθ˙h3+1γgθ~gTθ˙g+1ηϕ~ϕ^˙+1mmin·γωω~ω^˙.
According to adaptive laws (28)–(30), (45) can be rewritten as
(46)V˙≤-12meTQe+1m∥eTPB∥ω+1meTPB{(eTPB)T∥eTPB∥[Ke+ym(n)]+(eTPB)T∥eTPB∥h^1(x∣θh1)+(eTPB)T∥eTPB∥h^2(x(t)∣θh2)+(eTPB)T∥eTPB∥h^3(x(t-τ)∣θh3)}+eTPB{-g^(x∣θg)v(t)-g^(x∣θg)z(v(t))m}-eTPBf1T(x(t))θ1-eTPBf2T(x(t-τ))θ2+1γ1θ~1Tθ^˙1+1γ2θ~2Tθ^˙2+1ηϕ~ϕ^˙+1mmin·γωω~ω^˙≤-12meTQe+1m∥eTPB∥ω+1meTPB×{[Ke+ym(n)]+(eTPB)T∥eTPB∥h^1(x∣θh1)+(eTPB)T∥eTPB∥h^2(x(t)∣θh2)+(eTPB)T∥eTPB∥h^3(x(t-τ)∣θh3)}-eTPBg^(x∣θg)v(t)+∥eTPB∥g^(x∣θg)×|z(v(t))||m|-eTPBf1T(x(t))θ1-eTPBf2T(x(t-τ))θ2+1γ1θ~1Tθ^˙1+1γ2θ~2Tθ^˙2+1ηϕ~ϕ^˙+1mmin·γωω~ω^˙.
Using the control laws (24)–(26), the previous equation can be rewritten as
(47)V˙≤-12meTQe+1mmin∥eTPB∥·(ω-ω^)+eTPB(ϕ-ϕ^)×{(eTPB)T∥eTPB∥[Ke+ym(n)]+(eTPB)T∥eTPB∥h^1(x∣θh1)+(eTPB)T∥eTPB∥h^2(x(t)∣θh2)+(eTPB)T∥eTPB∥h^3(x(t-τ)∣θh3)}+∥eTPB∥g^(x∣θg)|z(v(t))||m|-eTPB(θ1T-θ^1T)·f1(x(t))-eTPB(θ2T-θ^2T)·f2(x(t-τ))-eTPB·g^(x∣θg)·ρmmin·tanh(eTPBε)+1γ1θ~1Tθ^˙1+1γ2θ~2Tθ^˙2+1ηϕ~ϕ^˙+1mmin·γωω~ω^˙=-12meTQe-1mmin∥eTPB∥·ω~-eTPBϕ~{[Ke+ym(n)]+(eTPB)T∥eTPB∥h^1(x∣θh1)+(eTPB)T∥eTPB∥h^2(x(t)∣θh2)+(eTPB)T∥eTPB∥h^3(x(t-τ)∣θh3)}+∥eTPB∥g^(x∣θg)|z(v(t))||m|+eTPB·θ~1T·f1(x(t))+eTPB·θ~2T·f2(x(t-τ))-eTPB·g^(x∣θg)·ρmmin·tanh(eTPBε)+1γ1θ~1Tθ^˙1+1γ2θ~2Tθ^˙2+1ηϕ~ϕ^˙+1mmin·γωω~ω^˙.
According to adaptive laws (31)–(33), we have
(48)V˙≤-12meTQe+∥eTPB∥g^(x∣θg)|z(v(t))||m|-eTPB·g^(x∣θg)·ρmmin·tanh(eTPBε)≤-12meTQe+∥eTPB∥g^(x∣θg)ρmmin-eTPB·g^(x∣θg)·ρmmin·tanh(eTPBε).
By considering the inequality |ϕ|-ϕtanh(ϕ/ε)≤0.2785ε. We obtain
(49)V˙≤-12meTQe+0.2785g^(x∣θg)·ρmmin·ε=-12meTQe-12γ1θ~1Tθ~1-12γ2θ~2Tθ~2-12mγh1θ~h1Tθ~h1-12mγh2θ~h2Tθ~h2-12mγh3θ~h3Tθ~h3-12γgθ~gTθ~g-12ηϕ~2-12mmin·γωω~2+12γ1θ~1Tθ~1+12γ2θ~2Tθ~2+12mγh1θ~h1Tθ~h1+12mγh2θ~h2Tθ~h2+12mγh3θ~h3Tθ~h3+12γgθ~gTθ~g+12ηϕ~2+12mmin·γωω~2+0.2785g^(x∣θg)·ρmmin·ε.
Let
(50)L=12γ1θ~1Tθ~1+12γ2θ~2Tθ~2+12mγh1θ~h1Tθ~h1+12mγh2θ~h2Tθ~h2+12mγh3θ~h3Tθ~h3+12γgθ~gTθ~g+12ηϕ~2+12mmin·γωω~2+0.2785g^(x∣θg)·ρmmin·ε.
Then,
(51)V˙≤-12meTQe-12γ1θ~1Tθ~1-12γ2θ~2Tθ~2-12mγh1θ~h1Tθ~h1-12mγh2θ~h2Tθ~h2-12mγh3θ~h3Tθ~h3-12γgθ~gTθ~g-12ηϕ~2-12mmin·γωω~2+L≤-12mλmin(Q)eTe-12γ1θ~1Tθ~1-12γ2θ~2Tθ~2-12mγh1θ~h1Tθ~h1-12mγh2θ~h2Tθ~h2-12mγh3θ~h3Tθ~h3-12γgθ~gTθ~g-12ηϕ~2-12mmin·γωω~2+L≤-12mλmin(Q)λmax(P)eTPe-12γ1θ~1Tθ~1-12γ2θ~2Tθ~2-12mγh1θ~h1Tθ~h1-12mγh2θ~h2Tθ~h2-12mγh3θ~h3Tθ~h3-12γgθ~gTθ~g-12ηϕ~2-12mmin·γωω~2+L.
Let λv=λmin(Q)/λmax(P).  We obtain
(52)V˙≤-12mλveTPe-12γ1θ~1Tθ~1-12γ2θ~2Tθ~2-12mγh1θ~h1Tθ~h1-12mγh2θ~h2Tθ~h2-12mγh3θ~h3Tθ~h3-12γgθ~gTθ~g-12ηϕ~2-12mmin·γωω~2+L≤-min{λv,1γ1,1γ2,1m·γh1,1m·γh2,1m·γh3,1γg,1η,1m·γω}×[12meTPe+12γ1θ~1Tθ~1+12γ2θ~2Tθ~2+12mγh1θ~h1Tθ~h1+12mγh2θ~h2Tθ~h2+12mγh3θ~h3Tθ~h3+12γgθ~gTθ~g+12ηϕ~2+12mmin·γωω~2]+L.
Setting c=min{λv,1/γ1,1/γ2,1/(m·γh1),1/(m·γh2),1/(m·γh3),1/γg,1/η,1/(m·γω)}, it yields that
(53)V˙≤-cV+L.
Then, it is easy from (53) to show that
(54)V(t)≤e-ctV(0)+Lc.
Therefore, the output tracking error converges to a neighborhood of zero exponentially.Remark 3.
In the future work, the control problem of uncertain T-S fuzzy time-varying delay systems with unknown dead-zone input is an important topic and is worth to be studied.    Based on a novel fuzzy Lyapunov-Krasovskii functional, a delay partitioning method has been developed for the delay-dependent stability analysis of fuzzy time-varying state delay systems [26].  Obviously, it provides a useful idea to deal with the aforementioned future research.


## 4. An Example and Simulation Results

Consider the second-order uncertain nonlinear time-delay system containing an unknown dead-zone that is modified from the simulation example in [7] as follows:
(55)x˙1=x2+Δϕ1(x),x˙2=x1+f11(x(t))+f22(x(t-τ))+Δf1(x(t))+Δf2(x(t-τ))+g(x)Z(v(t))+Δϕ2(x),y=x1,
where the nonlinear functions f11(x(t))=-0.3sinx1(t),f12(x(t))=0,f21(x(t-τ))=0, and f22(x(t-τ)) = 0.1x12(t-τ) are assumed to be known, and Δf1(x(t))=-0.1x1sin(3x2(t)),Δf2(x(t-τ))=-0.1x1sin(3x2(t-τ)) are unknown system uncertainties with unknown upper bound functions, where τ is the time delay. Δϕ1(x)=0.1x1sin(t) and Δϕ2(x)=0.3x2sin(t) are unknown external disturbances, and g(x(t))=2-sin2(x1(t)). |Δf1(x(t))|≤ h2(x(t)),|Δf2(x(t-τ))|≤h3(x(t-τ)), and Z(v(t)) is an output of a dead-zone.  The goal of control is to maintain the system output y to follow the reference signal ym=0.5[sin(t)+sin(0.5t)].In the simulation, parameters of the dead-zone arem=1, cr=0.5, and cl=-0.5.  And their bounds are chosen as mmax=1.5, mmin=0.6, crmax=0.9, crmin=0.1, clmax=-0.1, and clmin=-0.8.  In the implementation, six fuzzy sets are defined over interval [-3,3] for both x1 and x2, with labels F1, F2, F3, F4, F5, and F6, and their membership functions are
(56)μF1(xi)=11+exp(5(xi+2)),μF2(xi)=exp(-(xi+1.5)2),μF3(xi)=exp(-(xi+0.5)2),μF4(xi)=exp(-(xi-0.5)2),μF5(xi)=exp(-(xi-1.5)2),μF6(xi)=11+exp(-5(xi-2)),i=1,2.In this section, we apply the proposed robust adaptive fuzzy tracking control approach in Section3 to deal with the output tracking control problem of the second-order uncertain nonlinear time-delay system as shown in (55).  Choose K=[10,10] and Q=diag[5,5]; then, we solve the Lyapunov equation (27) to obtain
(57)P=[5.250.250.250.275].
In this example, the sampling time is 0.01 sec.  Initial values are chosen as x(0)=[-2,3]T, θg(0)=1, θh1(0)=0, θh2(0)=0, and θh3(0)=0.  The initial values of the parameters to be estimated are selected as  ϕ^(0)=0.85, θ^1(0)=[00]T, θ^2(0)=[00]T.  γg=2, γh1=1.5, γh2=1.5, γh3=1.5, γ1=1.5, γ2=1.5, γω=1.5, η=1.0, τ=0.5s, and ε=0.06.  The simulation results are shown in Figures 2–5.  Figures 2 and 3 show the trajectories of states x1andx2 and the desired outputs ym1andym2, respectively.  The phase plane of tracking errors of e1 and e2 is shown in Figure 4.  Figure 5 shows the trajectory of the control signal.  Obviously, the proposed robust adaptive fuzzy tracking control scheme can achieve the objective of good tracking performance and robust stability simultaneously in spite of the controlled system containing an unknown dead-zone and uncertainties.Figure 2
The trajectories of statex1 and desired output ym1.Figure 3
The trajectories of statex2 and desired output ym2.Figure 4
The phase plane of  tracking errorse1 and e2.Figure 5
The trajectory of the control inputv(t).


## 5. Conclusion

The dead-zone input characteristics widely exist in the actuators of practical control systems, which are usually poorly known.  The time-delay characteristics are usually confronted in engineering systems.  The two characteristics may severely limit the performance of control.  In this paper, the robust adaptive fuzzy tracking controller is designed to overcome the stabilization problem of a class of uncertain nonlinear state time-delay systems containing unknown dead-zone input and unmatched uncertainties.  By utilizing a description of a dead-zone feature to estimate the properties of the dead-zone model intuitively and mathematically, the adaptive fuzzy tracking controller is proposed without constructing the dead-zone inverse.  The nonlinear uncertainties are approximated by the fuzzy logic system according to the adaptive laws.  Based on the Lyapunov stability theorem, the proposed robust adaptive tracking fuzzy controller can ensure that the output tracking error of the resulting closed-loop system converges to a neighborhood of zero exponentially.  Finally, some simulations results are illustrated to verify the effectiveness and performance of the proposed approach.



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*Source: 363748-2013-02-20.xml*