# Tracking Control Based on Recurrent Neural Networks for Nonlinear Systems with Multiple Inputs and Unknown Deadzone

Authors: J. Humberto Pérez-Cruz; José de Jesús Rubio; E. Ruiz-Velázquez; G. Solís-Perales
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper deals with the problem of trajectory tracking for a broad class of uncertain nonlinear systems with multiple inputs each one subject to an unknown symmetric deadzone. On the basis of a model of the deadzone as a combination of a linear term and a disturbance-like term, a continuous-time recurrent neural network is directly employed in order to identify the uncertain dynamics. By using a Lyapunov analysis, the exponential convergence of the identification error to a bounded zone is demonstrated. Subsequently, by a proper control law, the state of the neural network is compelled to follow a bounded reference trajectory. This control law is designed in such a way that the singularity problem is conveniently avoided and the exponential convergence to a bounded zone of the difference between the state of the neural identifier and the reference trajectory can be proven. Thus, the exponential convergence of the tracking error to a bounded zone and the boundedness of all closed-loop signals can be guaranteed. One of the main advantages of the proposed strategy is that the controller can work satisfactorily without any specific knowledge of an upper bound for the unmodeled dynamics and/or the disturbance term.


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## Body

## 1. Introduction

After more than a half century of ongoing research, the adaptive control of linear and nonlinear systems with linearly parameterized unknown constants is currently a solid area within an automatic control theory. In order to extend these results to more general classes of systems, during the last twenty years, intense research has been carried out relying on the universal approximation capability of the artificial neural networks [1–7].An artificial neural network can be simply considered as a nonlinear generic mathematical formula whose parameters are adjusted in order to represent the behavior of a static or dynamic system [5]. These parameters are called weights. Generally speaking, ANN can be classified as feedforward (static) ones, based on the back propagation technique [2], or as recurrent (dynamic) ones [4]. In the first network type, system dynamics is approximated by a static mapping. These networks have two major disadvantages: a slow learning rate and a high sensitivity to training data. The second approach (recurrent ANN) incorporates feedback into its structure. Due to this feature, recurrent neural networks can overcome many problems associated with static ANN, such as global extrema search, and consequently have better approximation properties [8]. Depending on their structure, recurrent neural networks can be classified as discrete-time ones or continuous-time ones.Much of the first effort of research about the theory and application of the control based on continuous-time recurrent neural networks was synthesized in [4, 6, 9, 10]. In [9], a strategy of indirect adaptive control based on a parallel recurrent neural network was presented. In that study, the asymptotic convergence of the average integral identification error to a bounded zone was guaranteed. In order to prove this result, a Riccati matrix equation was employed. Based on the neural model of the uncertain system, a local optimal-type controller was developed. In spite of the significant contributions presented in that study, the usage of the Riccati matrix equation can be some restrictive and certain important questions such as the possible singularity of the control law were not considered. On the basis of this work, the exponential convergence of the identification error to a bounded zone could be guaranteed in [11–13]. However, the need of a Riccati matrix equation could not be avoided. In [10], a tracking controller based on a series-parallel neural network model was proposed. In that study, the assumptions about the uncertain system were less restrictive than in [9], Riccati matrix equation was not necessary, and the possibility of the singularity problem for the control law was conveniently avoided. In contrast, the control law proposed by [10] is some complex. In spite of the importance of the aforementioned works, the case when the presence of a deadzone degrades the performance of an automatic control system was not taken into account.The deadzone is a nonsmooth nonlinearity commonly found in many practical systems such as hydraulic positioning systems [14], pneumatic servo systems [15], and DC servo motors and so on. When the deadzone is not considered explicitly during the design process, the performance of the control system could be degraded due to an increase of the steady-state error, the presence of limit cycles, or inclusive instability [16–19]. A direct way of compensating the deleterious effect of the deadzone is by calculating its inverse. However, this is not an easy question because in many practical situations, both the parameters and the output of the deadzone are unknown. To overcome this problem, in a pioneer work [16], Tao and Kokotović proposed to employ an adaptive inverse of the deadzone. This scheme was applied to linear systems in a transfer function form. Cho and Bai [20] extended this work and achieved a perfect asymptotic adaptive cancellation of the deadzone. However, their work assumed that the deadzone output was measurable. In [21], the work of Tao and Kokotović was extended to linear systems in a state space form with nonmeasurable deadzone output. In [22], a new smooth parameterization of the deadzone was proposed and a class of SISO systems with completely known nonlinear functions and with linearly parameterized unknown constants was controlled by using backstepping technique. In order to avoid the construction of the adaptive inverse, in [23], the same class of nonlinear systems as in [22] was controlled by means of a robust adaptive approach and by modeling the deadzone as a combination of a linear term and a disturbance-like term. The controller design in [23] was based on the assumption that maximum and minimum values for the deadzone parameters are a priori known. However, a specific procedure to find such bounds was not provided. Based on the universal approximation property of the neural networks, a wider class of SISO systems in Brunovsky canonical form with completely unknown nonlinear functions and unknown constant control gain was considered in [24–26]. Apparently, the generalization of these results to the case when the control gain is varying, state dependent is trivial. Nevertheless, the solution to this problem is not so simple due to the singularity possibility for the control law. In [27, 28], this problem was overcome satisfactorily.All the aforementioned works about deadzone studied a very particular class of systems, that is, systems in strict Brunovsky canonical form with a unique input. In this paper, by combining, in an original way, the design strategies from [9, 10, 23], we can handle a broad class of uncertain nonlinear systems with multiple inputs each one subject to an unknown symmetric deadzone. On the basis of a model of the deadzone as a combination of a linear term and a disturbance-like term, a continuous-time recurrent neural network is directly employed in order to identify the uncertain dynamics. By using a Lyapunov analysis, the exponential convergence of the identification error to a bounded zone is demonstrated. Subsequently, by a proper control law, the state of the neural network is compelled to follow a bounded reference trajectory. This control law is designed in such a way that the singularity problem is conveniently avoided as in [10] and the exponential convergence to a bounded zone of the difference between the state of the neural identifier and the reference trajectory can be proven. Thus, the exponential convergence of the tracking error to a bounded zone and the boundedness of all closed-loop signals can be guaranteed. This is the first time, up to the best of our knowledge, that recurrent neural networks are utilized in the context of uncertain system control with deadzone.


## 2. Preliminaries

In this study, the system to be controlled consists of an unknown multi-input nonlinear plant with unknown deadzones in the following form:(2.1)Plant:x˙(t)=f(x(t))+g(x(t))u(t)+ξ(t),(2.2)Deadzone:ui(t)=DZi(vi(t))={mi(vi(t)-bi,r)vi(t)≥bi,r,0bi,l<vi(t)<bi,r,mi(vi(t)-bi,l)vi(t)≤bi,l,
where x(t)∈ℜn is the measurable state vector for t∈ℜ+:={t:t≥0}, f:ℜn→ℜn is an unknown but continuous nonlinear vector function, g:ℜn→ℜn×q is an unknown but continuous nonlinear matrix function, ξ(t)∈ℜn represents an unknown but bounded deterministic disturbance, the ith element of the vector u(t)∈ℜq, that is, ui(t), represents the output of the ith deadzone, vi(t) is the input to the ith deadzone, bi,r and bi,l represent the right and left constant breakpoints of the ith deadzone, and mi is the constant slope of the ith deadzone. In accordance with [16, 17], the deadzone model (2.2) is a static simplification of diverse physical phenomena with negligible fast dynamics. Note that v(t)∈ℜq is the actual control input to the global system described by (2.1) and (2.2). Hereafter it is considered that the following assumptions are valid.Assumption 2.1.
The plant described by (2.1) is controllable.Assumption 2.2.
Theith deadzone output, that is, ui(t) is not available for measurement.Assumption 2.3.
Although theith deadzone parameters bi,r, bi,l, and mi are unknown constants, we can assure that bi,r>0, bi,l<0, and mi>0 for all i∈{1,2,…q}.


### 2.1. Statement of the Problem

The objective that we are trying to achieve is to determine a control signalv(t) such that the state x(t) follows a given bounded reference trajectory xr(t), and, at the same time, all closed-loop signals stay bounded.Assumption 2.4.
Without the loss of generality, we consider thatxr(t) is generated by the following exosystem:
(2.3)x˙r(t)=B(xr(t)),
where B:ℜn→ℜn is an unknown but continuous nonlinear vector function.


### 2.2. Deadzone Representation as a Linear Term and a Disturbance-Like Term

The deadzone model (2.2) can alternatively be described as [23, 29]:
(2.4)ui(t)=mivi(t)+di(t),
where di(t) is given by
(2.5)di(t)={-mibi,r,vi(t)≥bi,r,-mivi(t),bi,l<vi(t)<bi,r,-mibi,l,vi(t)≤bi,l.
Note that (2.5) is the negative of a saturation function. Thus, although di(t) could not be exactly known, its boundedness can be assured. Consider that the positive constant d-i is an upper bound for di(t), that is, |di(t)|≤d-i.Based on (2.4), the relationship between u(t) and v(t) can be expressed as
(2.6)u(t)=Mv(t)+d(t),
where M:=diag(m1,m2,…mq) and d(t)∈Rq is given by d(t):=[d1(t),d2(t),…,dq(t)]T. Clearly, d(t)∈L∞. Consider that the positive constant d- is an upper bound for d(t).


## 2.1. Statement of the Problem

The objective that we are trying to achieve is to determine a control signalv(t) such that the state x(t) follows a given bounded reference trajectory xr(t), and, at the same time, all closed-loop signals stay bounded.Assumption 2.4.
Without the loss of generality, we consider thatxr(t) is generated by the following exosystem:
(2.3)x˙r(t)=B(xr(t)),
where B:ℜn→ℜn is an unknown but continuous nonlinear vector function.


## 2.2. Deadzone Representation as a Linear Term and a Disturbance-Like Term

The deadzone model (2.2) can alternatively be described as [23, 29]:
(2.4)ui(t)=mivi(t)+di(t),
where di(t) is given by
(2.5)di(t)={-mibi,r,vi(t)≥bi,r,-mivi(t),bi,l<vi(t)<bi,r,-mibi,l,vi(t)≤bi,l.
Note that (2.5) is the negative of a saturation function. Thus, although di(t) could not be exactly known, its boundedness can be assured. Consider that the positive constant d-i is an upper bound for di(t), that is, |di(t)|≤d-i.Based on (2.4), the relationship between u(t) and v(t) can be expressed as
(2.6)u(t)=Mv(t)+d(t),
where M:=diag(m1,m2,…mq) and d(t)∈Rq is given by d(t):=[d1(t),d2(t),…,dq(t)]T. Clearly, d(t)∈L∞. Consider that the positive constant d- is an upper bound for d(t).


## 3. Neural Identifier

In this section, the identification problem of the unknown global dynamics described by (2.1) and (2.2) using a recurrent neural network is considered.Note that an alternative representation for (2.1) is given by
(3.1)x˙(t)=Ax(t)+W1*σ(x(t))+W2*ϕ(x(t))u(t)+ω(x(t),u(t))+ξ(t),
where A∈ℜn×n is a Hurwitz matrix, W1*∈ℜn×m and W2*∈ℜn×r are unknown constant weight matrices, and σ(·) is the activation vector function with sigmoidal components, that is, σ(·):=[σ1(·),…,σm(·)]⊤(3.2)σj(x(t)):=aσj1+exp(-∑i=1ncσj,ixi(t))-dσjforj=1,…,m,
where aσj, cσj,i, and dσj are positive constants which can be specified by the designer, ϕ(·):ℜn→ℜr×q is a sigmoidal function, that is,
(3.3)ϕij(x(t)):=aϕij1+exp(-∑l=1ncϕij,lxl(t))-dϕijfori=1,…,r,j=1,…,q,
where aϕij, cϕij,l, and dϕij are positive constants which can be specified by the designer, and ω:ℜn×ℜq→ℜn is the unmodeled dynamics which can be defined simply as ω(x(t),u(t)):=f(x(t))+g(x(t))u(t)-Ax(t)-W1*σ(x(t))-W2*ϕ(x(t))u(t).Assumption 3.1.
On a compact setΩ⊂ℜn, unmodeled dynamics ω(x(t),u(t)) is bounded by ω-, that is, |ω(x(t),u(t))|≤ω-. The disturbance ξ(t) is also bounded, that is, |ξ(t)|≤Υ. Both ω- and Υ are positive constants not necessarily a priori known.By substituting (2.6) into (3.1), we get
(3.4)x˙(t)=Ax(t)+W1*σ(x(t))+W2*ϕ(x(t))Mv(t)+W2*ϕ(x(t))d(t)+ω(x(t),u(t))+ξ(t).Remark 3.2.
It can be observed that by using the model (2.6), the actual control input v(t) appears now directly into the dynamics.Since, by construction,ϕ(x(t)) is bounded, the term W2*ϕ(x(t))d(t) is also bounded. Let us define the following expression: ζ(t):=W2*ϕ(x(t))d(t)+ω(x(t),u(t))+ξ(t). Clearly, this expression is bounded. Let us denote an upper bound for ζ(t) as ζ-. This bound is a positive constant not necessarily a priori known. Now, note that the term W2*ϕ(x(t))Mv(t) can be alternatively expressed as S*ϕ(x(t))v(t), where S*∈ℜn×r is an unknown weight matrix. In view of the above, (3.4) can be rewritten as
(3.5)x˙(t)=Ax(t)+W1*σ(x(t))+S*ϕ(x(t))v(t)+ζ(t).
Now, consider the following series-parallel structure for a continuous-time recurrent neural network
(3.6)x^˙(t)=Ax^(t)+W1(t)σ(x(t))+S(t)ϕ(x(t))v(t),
where x^t∈ℜn is the state of the neural network, v(t)∈ℜq is the control input, and W1(t)∈ℜn×m and S(t)∈ℜn×r are the time-varying weight matrices. The problem of identifying system (2.1)-(2.2) based on the recurrent neural network (3.6) consists of, given the measurable state x(t) and the input v(t), adjusting online the weights W1(t) and S(t) by proper learning laws such that the identification error Δ(t):=x^(t)-x(t) can be reduced. Specifically, the following learning laws are here used:
(3.7)W·1(t)=-k1Δ(t)σT(x(t))-ℓ1W1(t),(3.8)S·(t)=-k2Δ(t)vT(t)ϕT(x(t))-ℓ2S(t),
where k1, ℓ1, k2, and ℓ2 are positive constants selectable by the designer.Based on the learning laws (3.7) and (3.8), the following result is here established.Theorem 3.3.
If the Assumptions2.2, 2.3, and 3.1 are satisfied, the constant a is selected greater than 0.5, and the weight matrices W1(t), S(t) of the neural network (3.6) are adjusted by the learning laws (3.7) and (3.8), respectively, then (a)
the identification error and the weights of the neural network (3.6) are bounded:
(3.9)Δ(t),W1(t),S(t)∈L∞,(b)
the norm of the identification error, that is,|x^(t)-x(t)| converges exponentially fast to a zone bounded by the term
(3.10)2βα,
where α:=min{(2a-1),ℓ1,ℓ2}, β:=(1/2)ζ-2+(ℓ1/2k1)tr{W1*TW1*}+(ℓ2/2k2)tr{S*TS*}.Proof of Theorem3.3.
First, let us determine the dynamics of the identification error. The first derivative ofΔ(t) is simply
(3.11)Δ˙(t)=x^˙(t)-x˙(t).
Substituting (3.6) and (3.5) into (3.11) yields
(3.12)Δ˙t=Ax^(t)+W1(t)σ(x(t))+S(t)ϕ(x(t))v(t)-Ax(t)-W1*σ(x(t))-S*ϕ(x(t))v(t)-ζ(t)=AΔ(t)+W~1(t)σ(x(t))+S~(t)ϕ(x(t))v(t)-ζ(t),
where W~1(t):=W1(t)-W1* and S~(t):=S(t)-S*.
Consider the following Lyapunov function candidate(3.13)V(t)=12ΔT(t)Δ(t)+12k1tr{W~1T(t)W~1(t)}+12k2tr{S~T(t)S~(t)}.
The first derivative of V(t) is
(3.14)V˙(t)=ΔT(t)Δ˙(t)+1k1tr{W~˙1T(t)W~1(t)}+1k2tr{S~˙T(t)S~(t)}.
Substituting (3.12) into (3.14) and taking into account that, for simplicity, A can be selected as A=-aI, where a is a positive constant greater than 0.5 and I∈ℜn×n is the identity matrix, yields
(3.15)V˙(t)=-a|Δ(t)|2+ΔT(t)W~1(t)σ(x(t))+ΔT(t)S~(t)ϕ(x(t))v(t)-ΔT(t)ζ(t)+1k1tr{W~˙1T(t)W~1(t)}+1k2tr{S~˙T(t)S~(t)}.
Since W~1(t):=W1(t)-W1* and S~(t):=S(t)-S*, the first derivatives for W~˙1(t) and S~˙(t) are clearly W~˙1(t)=W˙1(t) and S~˙(t)=S˙(t), respectively. However, W˙1(t) and S˙(t) are given by the learning laws (3.7) and (3.8). Therefore, by substituting (3.7) into W~˙1(t)=W˙1(t) and (3.8) into S~˙(t)=S˙(t) and the corresponding expressions into the right-hand side of (3.15), it is possible to obtain
(3.16)V˙(t)=-a|Δ(t)|2+ΔT(t)W~1(t)σ(x(t))+ΔT(t)S~(t)ϕ(x(t))v(t)-ΔT(t)ζ(t)+tr{-σ(x(t))ΔT(t)W~1(t)}-ℓ1k1tr{W1T(t)W~1(t)}+tr{-ϕ(x(t))v(t)ΔT(t)S~(t)}-ℓ2k2tr{ST(t)S~(t)}.
We can see that
(3.17)tr{-σ(x(t))ΔT(t)W~1(t)}=-tr{σ(x(t))ΔT(t)W~1(t)}=-tr{ΔT(t)W~1(t)σ(x(t))}=-ΔT(t)W~1(t)σ(x(t)),tr{-ϕ(x(t))v(t)ΔT(t)S~(t)}=-tr{ϕ(x(t))v(t)ΔT(t)S~(t)}=-tr{ΔT(t)S~(t)ϕ(x(t))v(t)}=-ΔT(t)S~(t)ϕ(x(t))v(t).
Substituting (3.17) into (3.16) and reducing the like terms yields
(3.18)V˙(t)=-a|Δ(t)|2-ΔT(t)ζ(t)-ℓ1k1tr{W1T(t)W~1(t)}-ℓ2k2tr{ST(t)S~(t)}.
Now, it can be proven that [10]
(3.19)tr{W1T(t)W~1(t)}=12tr{W1T(t)W1(t)}+12tr{W~1T(t)W~1(t)}-12tr{W1*TW1*},tr{ST(t)S~(t)}=12tr{ST(t)S(t)}+12tr{S~T(t)S~(t)}-12tr{S*TS*}.
Likewise, it is easy to show that
(3.20)-ΔT(t)ζ(t)≤12|Δ(t)|2+12|ζ(t)|2≤12|Δ(t)|2+12ζ-2.
If (3.19) and the inequality (3.20) are substituted into (3.18), we obtain
(3.21)V·(t)≤-a|Δ(t)|2+12|Δ(t)|2+12ζ-2-ℓ12k1tr{W1T(t)W1(t)}-ℓ12k1tr{W~1T(t)W~1(t)}+ℓ12k1tr{W1*TW1*}-ℓ22k2tr{ST(t)S(t)}-ℓ22k2tr{S~T(t)S~(t)}+ℓ22k2tr{S*TS*}
or
(3.22)V˙(t)≤-(2a-1){12|Δ(t)|2}-ℓ1(12k1tr{W~1T(t)W~1(t)})-ℓ2(12k2tr{S~T(t)S~(t)})+12ζ-2+ℓ12k1tr{W1*TW1*}+ℓ22k2tr{S*TS*}.
In view of α:=min{(2a-1),ℓ1,ℓ2}, β:=(1/2)ζ-2+(ℓ1/2k1)tr{W1*TW1*}+(ℓ2/2k2)tr{S*TS*}, the following bound as a function of V(t) can finally be determined for V˙(t),
(3.23)V˙(t)≤-αV(t)+β.
Equation (3.23) can be rewritten in the following form
(3.24)V˙(t)+αV(t)≤β.
Multiplying both sides of the last inequality by exp(αt), it is possible to obtain
(3.25)exp(αt)V˙(t)+αexp(αt)V(t)≤βexp(αt).
The left-hand side of (3.25) can be rewritten as
(3.26)ddt(exp(αt)V(t))≤βexp(αt)
or equivalently as
(3.27)d(exp(αt)V(t))≤βexp(αt)dt.
Integrating both sides of the last inequality from 0 to t yields
(3.28)exp(αt)V(t)-V(0)≤∫0tβexp(ατ)dτ.
Adding V(0) to both sides of the last inequality, we obtain
(3.29)exp(αt)V(t)≤V(0)+∫0tβexp(ατ)dτ.
Multiplying both sides of the inequality (3.29) by exp(-αt) yields
(3.30)V(t)≤exp(-αt)V(0)+exp(-αt)∫0tβexp(ατ)dτ
and, consequently
(3.31)V(t)≤V(0)exp(-αt)+βα(1-exp(-αt)).
As by definition α and β are positive constants, the right-hand side of the last inequality can be bounded by V(0)+(β/α). Thus, V(t)∈L∞ and since by construction V(t) is a nonnegative function, the boundedness of Δ(t), W~1(t), and S~(t) can be guaranteed. Because W1* and S* are bounded, W1(t)=W~1(t)+W1*, and S(t)=S~(t)+S* must be bounded too and the first part of Theorem 3.3 has been proven. With respect to the second part of this theorem, from (3.13), it is evident that (1/2)|Δ(t)|2≤V(t). Taking into account this fact and from (3.31), we get
(3.32)|Δ(t)|≤2V(0)exp(-αt)+2βα(1-exp(-αt)).
By taking the limit as t→∞ of the inequality (3.32), we can guarantee that |Δ(t)| converges exponentially fast to a zone bounded by the term 2β/α and the last part of Theorem 3.3 has been proven.Remark 3.4.
It is very important to mention that the identification process based on Theorem3.3 can be accomplished without the a priori knowledge about W1*, S*, and ζ-.


## 4. Controller Design

In this section, a proper control lawv(t) in order to solve the tracking problem is determined.Note that the dynamics of the exosystem (2.3) can be alternatively represented as
(4.1)x˙r(t)=Axr(t)+Wr*σr(xr(t))+ωr(xr(t)),
where A∈ℜn×n is the same Hurwitz matrix as in (3.6), Wr*∈ℜn×mr is an unknown constant weight matrix, σr(·) is an activation vector function with sigmoidal components, that is, σr(·):=[σr1(·),…,σrmr(·)]⊤(4.2)σrj(x(t)):=aσrj1+exp(-∑i=1ncσrj,ixi(t))-dσrjforj=1,…,mr,
where aσrj, cσrj,i, and dσrj are positive constants which can be specified by the designer, and ωr:ℜn→ℜn is an error term which can be defined simply as
(4.3)ωr(x(t)):=B(xr(t))-Axr(t)-Wr*σr(xr(t)).Assumption 4.1.
On a compact setΩ⊂ℜn, the error term ωr(xr(t)) is bounded by the positive constant not necessarily a priori known ω-r, that is, |ωr(xr(t))|≤ω-r.Let us define the virtual tracking errore(t) as
(4.4)e(t):=x^(t)-xr(t).
The first derivative of (4.4) is simply
(4.5)e˙(t)=x^˙(t)-x˙r(t).
Substituting (3.6) and (4.1) into (4.5) yields
(4.6)e˙(t)=Ax^(t)+W1(t)σ(x(t))+S(t)ϕ(x(t))v(t)-Axr(t)-Wr*σr(xr(t))-ωr(xr(t)).
By adding and subtracting the term Wr(t)σr(xr(t)) into (4.6), we obtain
(4.7)e˙(t)=Ae(t)+W1(t)σ(x(t))+S(t)ϕ(x(t))v(t)+W~r(t)σr(xr(t))-Wr(t)σr(xr(t))-ωr(xr(t)),
where W~r(t):=Wr(t)-Wr*.Consider the following Lyapunov function candidate:(4.8)V2(t)=12γeT(t)e(t)+12tr{W~rT(t)W~r(t)},
where γ is a positive constant. The first derivative of V2(t) is
(4.9)V·2(t)=γeT(t)e˙(t)+tr{W~˙rT(t)W~r(t)}.
Substituting (4.7) into (4.9) and taking into account that A was selected in Section 3 as A=-aI yields
(4.10)V˙2(t)=-γa|e(t)|2+γeT(t)W1(t)σ(x(t))+γeT(t)S(t)ϕ(x(t))v(t)+γeT(t)W~r(t)σr(xr(t))-γeT(t)Wr(t)σr(xr(t))-γeT(t)ωr(xr(t))+tr{W~˙rT(t)W~r(t)}.
If the learning law for Wr(t) is selected as
(4.11)W˙r(t)=-γe(t)σrT(xr(t))-ℓrWr(t),
where ℓr is a positive constant and the control law v(t) is chosen as
(4.12)v(t)=1λrϕT(x(t))ST(t)Wr(t)σr(xr(t))1+∥S(t)∥2∥ϕ(x(t))∥2-ke(t),
where λr and k are positive constants and taking into account that W~˙r(t)=W˙r(t) then, (4.10) becomes
(4.13)V˙2(t)=-γa|e(t)|2+γeT(t)W1(t)σ(x(t))+γλreT(t)S(t)ϕ(x(t))ϕT(x(t))ST(t)Wr(t)σr(xr(t))1+∥S(t)∥2∥ϕT(x(t))∥2-γkeT(t)S(t)ϕ(x(t))e(t)+γeT(t)W~r(t)σr(xr(t))-γeT(t)Wr(t)σr(xr(t))-γeT(t)ωr(xr(t))-γtr{σr(xr(t))eT(t)W~r(t)}-ℓrtr{WrT(t)W~r(t)}.
It can be proven that
(4.14)tr{σr(xr(t))eT(t)W~r(t)}=tr{eT(t)W~r(t)σr(xr(t))}=eT(t)W~r(t)σr(xr(t)),tr{WrT(t)W~r(t)}=12tr{WrT(t)Wr(t)}+12tr{W~rT(t)W~r(t)}-12tr{Wr*TWr*}.
By substituting (4.14) into (4.13) and reducing the like terms, we obtain
(4.15)V˙2(t)=-γa|e(t)|2+γeT(t)W1(t)σ(x(t))+γλreT(t)S(t)ϕ(x(t))ϕT(x(t))ST(t)Wr(t)σr(xr(t))1+∥S(t)∥2∥ϕT(x(t))∥2-γkeT(t)S(t)ϕ(x(t))e(t)-γeT(t)Wr(t)σr(xr(t))-γeT(t)ωr(xr(t))-ℓr2tr{WrT(t)Wr(t)}-ℓr2tr{W~rT(t)W~r(t)}+ℓr2tr{Wr*TWr*}.
Taking into account that ±yTz≤|y||z| for y∈ℜn, z∈ℜn and ∥Y∥2=tr{YTY} for Y∈ℜL1×L2, (4.15) becomes
(4.16)V˙2(t)≤-γa|e(t)|2+γ|e(t)|∥W1(t)∥|σ(x(t))|+γλr|e(t)|∥S(t)∥2∥ϕ(x(t))∥2∥Wr(t)∥|σr(xr(t))|1+∥S(t)∥2∥ϕT(x(t))∥2+γk|e(t)|2∥S(t)∥∥ϕ(x(t))∥+γ|e(t)|∥Wr(t)∥|σr(xr(t))|+γ|e(t)||ωr(xr(t))|-ℓr2∥Wr(t)∥2-ℓr2∥W~r(t)∥2+ℓr2∥Wr*∥2.
Note that
(4.17)∥S(t)∥2∥ϕ(x(t))∥21+∥S(t)∥2∥ϕT(x(t))∥2≤1.
On the other hand, by construction, σ(x(t)) and σr(xr(t)) are bounded. Consider that s1 and sr are the corresponding upper bounds, that is, |σ(x(t))|≤s1 and |σr(xr(t))|≤sr (both s1 and sr can be calculated). Likewise, by construction, ϕ(x(t)) is bounded and S(t) is bounded from Theorem 3.3. Consider that μ is an upper bound for ∥S(t)∥∥ϕ(x(t))∥, that is, ∥S(t)∥∥ϕ(x(t))∥≤μ. In view of the above and selecting a>μk and
(4.18)γ(a-μk)=γ1+γ2,
where γ1>0.5 and γ2 are two positive constants, we can obtain
(4.19)V˙2(t)≤-γ1|e(t)|2-γ2|e(t)|2+γ|e(t)|∥W1(t)∥s1+γλr|e(t)|∥Wr(t)∥sr+γ|e(t)|∥Wr(t)∥sr+γ|e(t)||ωr(xr(t))|-ℓr2∥Wr(t)∥2-ℓr2∥W~r(t)∥2+ℓr2∥Wr*∥2
or
(4.20)V˙2(t)≤-γ1|e(t)|2-γ2|e(t)|2+γsr(1+1λr)|e(t)|∥Wr(t)∥-ℓr2∥Wr(t)∥2+γ|e(t)|{∥W1(t)∥s1+|ωr(xr(t))|}-ℓr2∥W~r(t)∥2+ℓr2∥Wr*∥2.
Now, in accordance with Theorem 3.3, W1(t)∈L∞. Based on this fact together with the Assumption 4.1, the boundedness of the term ∥W1(t)∥s1+|ωr(xr(t))| can be concluded. Consider that the unknown positive constant ε is an upper bound for that term, that is, ∥W1(t)∥s1+|ωr(xr(t))|≤ε. Thus, it is easy to show that
(4.21)γ|e(t)|{∥W1(t)∥s1+|ωr(xr(t))|}≤γ|e(t)|ε≤12|e(t)|2+12γ2ε2.
On the other hand, if the constants ℓr and λr are selected in such a way that
(4.22)ℓr>γ2sr22γ2,λr≥γsr2γ2ℓr-γsr
then the following can be established
(4.23)γsr(1+1λr)≤2γ2ℓr.
Based on (4.23), it can be proven that
(4.24)-γ2|e(t)|2+γsr(1+1λr)|e(t)|∥Wr(t)∥-ℓr2∥Wr(t)∥2≤-(γ2|e(t)|-ℓr2∥Wr(t)∥)2≤0.
Substituting (4.21) and (4.24) into (4.20) yields
(4.25)V˙2(t)≤-γ1|e(t)|2+12|e(t)|2-ℓr2∥W~r(t)∥2+12γ2ε2+ℓr2∥Wr*∥2.
Defining αr:=min{(2γ1-1)/γ,ℓr},  βr:=(1/2)γ2ε2+(ℓr/2)∥Wr*∥2, (4.25) becomes
(4.26)V˙2(t)≤-αrV2(t)+βr.
This means that
(4.27)V2(t)≤V2(0)exp(-αrt)+βrαr(1-exp(-αrt)).
As by definition αr and βr are positive constants, the right-hand side of the last inequality is bounded by V(0)+(βr/αr). Next, V2(t)∈L∞ and consequently e(t),W~r(t), and Wr(t)∈L∞.As by hypothesisxr(t)∈L∞, the boundedness of e(t) guarantees the boundedness of x^(t). Remember that Theorem 3.3 guarantees that Δ(t)∈L∞. By the definition of Δ(t), that is, Δ(t)=x^(t)-x(t) and considering that x^(t)∈L∞, the boundedness of x(t) can be concluded. From (4.12), we can see that the control law v(t) is selected in such a way that the denominator is never equal to zero although ∥S(t)∥=0 and/or ∥ϕ(x(t))∥=0. Besides, we can verify that v(t) is formed by bounded elements. Next, the control input v(t) must be bounded too. On the other hand, note that the following is true:
(4.28)12γ|e(t)|2≤V2(t).
Taking into account (4.28) and from (4.27), we get
(4.29)|e(t)|≤2γV2(0)exp(-αrt)+2βrγαr(1-exp(-αrt)).
Now, the ultimate objective is to achieve that the state x(t) of the unknown system (2.1)-(2.2) follows the reference trajectory xr(t). Thus, we need to know if the actual tracking error x(t)-xr(t) converges or not to a some value. Note that
(4.30)|x(t)-xr(t)|=|x(t)-x^(t)+x^(t)-xr(t)|≤|x^(t)-x(t)|+|x^(t)-xr(t)|=|Δ(t)|+|e(t)|.
Clearly, |x(t)-xr(t)|∈L∞. Finally, by substituting (3.32) and (4.29) into (4.30), we have
(4.31)|x(t)-xr(t)|≤2V(0)exp(-αt)+2βα(1-exp(-αt))+2γV2(0)exp(-αrt)+2βrγαr(1-exp(-αrt)).
By taking the limit as t→∞ of the last inequality, we can guarantee that |x(t)-xr(t)| converges exponentially fast to a zone bounded by the term 2β/α+2βr/γαr. Thus, the following theorem has been provenTheorem 4.2.
Given the Assumptions2.1–4.1, if the control law (4.12) is used together with the learning laws (3.8) and (4.11) then it can be guaranteed that (a)
the weight matrixWr(t), the virtual tracking error, the actual tracking error, the state of the neural network, the system state, and the control input are bounded:
(4.32)Wr(t),e(t),x(t)-xr(t),x^(t),x(t),v(t)∈L∞,(b)
the actual tracking error|x(t)-xr(t)| converges exponentially to a zone bounded by the term
(4.33)2βα+2βrγαr,
where α and β are defined as in Theorem 3.3 and αr:=min{(2γ1-1)/γ,ℓr}, βr:=(1/2)γ2ε2+(ℓr/2)∥Wr*∥2.


## 5. Numerical Example

In this section, a simple but illustrative simulation example is presented in order to show the feasibility of the suggested approach. Consider the first order nonlinear system given by(5.1)x˙(t)=-x(t)sin(x(t))+(0.2+cos2(x(t)))u(t)+ξ(t).
The initial condition for system (5.1) is x(0)=1; u(t) is the deadzone output; the parameters of the deadzone are m=1.6, br=2.5, and bl=-2; ξ(t), the disturbance term is selected as ξ(t)=0.5sin(13t). The following reference trajectory is employed yr(t)=sin(t)-1.5sin(2t). The parameters for the neural identifier and the control law are selected by trial and error as  x^(0)=0, a=2000, k1=500000, l1=1, W1(0)=0, k2=200, l2=50, S(0)=0.5, σ(x(t))=ϕ(x(t))=2/(1+exp(-x(t)))-1, γ=300,  lr=31, Wr(0)=-1, σr(xr(t))=2/(1+exp(-xr(t)))-1, γ1=1, γ2=1499, sr=1, μ=8, k=249.375, and λr=62. The simulation is carried out by means of Simulink with ode45 method, relative tolerance equal to 1e-7, and absolute tolerance equal to 1e-9. The results of the tracking process are presented in Figures 1–3 for the first 20 seconds. In Figure 1, the output of the nonlinear system (5.1), x(t), is represented by a dashed line whereas the reference trajectory xr(t) is represented by a solid line. In Figure 2, the control signal v(t) acting as the input of the deadzone is shown. In Figure 3, a zoom of Figure 2 is presented. From Figure 3, we can appreciate that the control law v(t) avoids properly the deadzone.Figure 1
Tracking process: reference trajectory: solid line; system output: dashed line.Figure 2
Control signalv(t).Figure 3
Zoom of Figure2.


## 6. Conclusions

In this paper, an adaptive scheme based on a continuous-time recurrent neural network is proposed in order to handle the tracking problem for a broad class of nonlinear systems with multiple inputs each one subject to an unknown symmetric deadzone. The need of an inverse adaptive commonly required in many previous works is conveniently avoided by considering the deadzone as a combination of a linear term and a disturbance-like term. Thus, the identification of the unknown dynamics together with the deadzone can be carried out directly by using a recurrent neural network. The exponential convergence of the identification error norm to a bounded zone is thoroughly proven by a Lyapunov analysis. Subsequently, the state of the neural network is compelled to follow a reference trajectory by using a control law designed in such a way that the singularity problem is conveniently avoided without the need of any projection strategy. By another Lyapunov analysis, the exponential convergence of the difference between the neural network state and the reference trajectory is demonstrated. As the tracking error is bounded by the identification error and the difference between the neural network state and the reference trajectory, the exponential convergence of the tracking error to a bounded zone is also proven. Besides, the boundedness of the system state, the neural network state, the weights, and the control signal can be guaranteed. The proposed control scheme presents two important advantages:(i)
the specific knowledge of a bound for the unmodeled dynamics and/or the disturbance term is not necessary,(ii)
the determination of the first derivative for the reference trajectory is not required.



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*Source: 471281-2012-12-23.xml*