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

and Applied Analysis 3 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: Plant: ẋ t f x t g x t u t ξ t , 2.1 Deadzone: ui t DZi vi t ⎧ ⎪ ⎨ ⎪ ⎩ mi vi t − bi,r vi t ≥ bi,r , 0 bi,l < vi t < bi,r , mi vi t − bi,l vi t ≤ bi,l, 2.2 where x t ∈ n is the measurable state vector for t ∈ : {t : t ≥ 0}, f : n → n is an unknown but continuous nonlinear vector function, g : n → n×q is an unknown but continuous nonlinear matrix function, ξ t ∈ n represents an unknown but bounded deterministic disturbance, the ith element of the vector u t ∈ , 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, andmi 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 ∈ q is the actual 4 Abstract and Applied Analysis 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. The ith deadzone output, that is, ui t is not available for measurement. Assumption 2.3. Although the ith 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 signal v 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 that xr t is generated by the following exosystem:


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 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.

Preliminaries
In this study, the system to be controlled consists of an unknown multi-input nonlinear plant with unknown deadzones in the following form: where x t ∈ n is the measurable state vector for t ∈ : {t : t ≥ 0}, f : n → n is an unknown but continuous nonlinear vector function, g : n → n×q is an unknown but continuous nonlinear matrix function, ξ t ∈ n represents an unknown but bounded deterministic disturbance, the ith element of the vector u t ∈ q , that is, u i t , represents the output of the ith deadzone, v i t is the input to the ith deadzone, b i,r and b i,l represent the right and left constant breakpoints of the ith deadzone, and m i 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 ∈ q 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 2.2.The ith deadzone output, that is, u i t is not available for measurement.Assumption 2.3.Although the ith deadzone parameters b i,r , b i,l , and m i are unknown constants, we can assure that b i,r > 0, b i,l < 0, and m i > 0 for all i ∈ {1, 2, . . .q}.

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

Deadzone Representation as a Linear Term and a Disturbance-Like Term
The deadzone model 2.2 can alternatively be described as 23, 29 : where d i t is given by

2.5
Note that 2.5 is the negative of a saturation function.

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.
where a φij , c φij,l , and d φij are positive constants which can be specified by the designer, and ω : n × q → n is the unmodeled dynamics which can be defined simply as ω x t , u t : Specifically, the following learning laws are here used: where k 1 , 1 , k 2 , and 2 are positive constants selectable by the designer.
Based on the learning laws 3.7 and 3.8 , the following result is here established.
The first derivative of V t is 3.14 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 ∈ n×n is the identity matrix, yields

3.16
We can see that 3.17 Substituting 3.17 into 3.16 and reducing the like terms yields Now, it can be proven that 10 tr S * T S * .

3.19
Likewise, it is easy to show that

3.20
If 3.19 and the inequality 3.20 are substituted into 3.18 , we obtain 2k 2 tr S * T S * .

3.22
In view of α : 2 /2k 2 tr{S * T S * }, the following bound as a function of V t can finally be determined for V t , V t ≤ −αV t β.

3.23
Abstract and Applied Analysis 9 Equation 3.23 can be rewritten in the following form V t αV t ≤ β.

3.24
Multiplying both sides of the last inequality by exp αt , it is possible to obtain exp αt V t α exp αt V t ≤ β exp αt .

3.25
The left-hand side of 3.25 can be rewritten as or equivalently as Integrating both sides of the last inequality from 0 to t yields β exp ατ dτ.

3.28
Adding V 0 to both sides of the last inequality, we obtain exp αt V t ≤ V 0 t 0 β exp ατ dτ.

3.29
Multiplying both sides of the inequality 3.29 by exp −αt yields and, consequently

3.31
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.

Controller Design
In this section, a proper control law v t in order to solve the tracking problem is determined.Note that the dynamics of the exosystem 2.3 can be alternatively represented as ẋr t Ax r t W * r σ r x r t ω r x r t , 4.1 where A ∈ n×n is the same Hurwitz matrix as in 3.6 , W * r ∈ n×m r is an unknown constant weight matrix, σ r • is an activation vector function with sigmoidal components, that is, σ r • : σ r1

4.16
Note that 4.17

Abstract and Applied Analysis 13
On the other hand, by construction, σ x t and σ r x r t are bounded.Consider that s 1 and s r are the corresponding upper bounds, that is, |σ x t | ≤ s 1 and |σ r x r t | ≤ s r both s 1 and s r 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 where γ 1 > 0.5 and γ 2 are two positive constants, we can obtain

4.20
Now, in accordance with Theorem 3.3, W 1 t ∈ L ∞ .Based on this fact together with the Assumption 4.1, the boundedness of the term W 1 t s 1 |ω r x r t | can be concluded.Consider that the unknown positive constant ε is an upper bound for that term, that is, On the other hand, if the constants r and λ r are selected in such a way that then the following can be established

4.23
Based on 4.23 , it can be proven that This means that

4.27
As by definition α r and β r are positive constants, the right-hand side of the last inequality is bounded by V 0 β r /α r .Next, V 2 t ∈ L ∞ and consequently e t , W r t , and W r t ∈ L ∞ .As by hypothesis x r 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: Taking into account 4.28 and from 4.27 , we get Now, the ultimate objective is to achieve that the state x t of the unknown system 2.1 -2.2 follows the reference trajectory x r t .Thus, we need to know if the actual tracking error x t − x r t converges or not to a some value.Note that

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 ẋ t −x t sin x t 0.2 co s 2 x t u t ξ t .

5.1
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, b r 2.5, and b l −2; ξ t , the disturbance term is selected as ξ t 0.5 sin 13t .The following reference trajectory is employed y r t sin t − 1.5 sin 2t .The parameters for the neural identifier and the control law are selected by trial and error as x 0 0, a 2000, k 1 500000, l 1 1, W 1 0 0, k 2 200, l 2 50, S 0 0.5, σ x t φ x t 2/ 1 exp −x t −1, γ 300, l r 31, W r 0 −1, σ r x r t 2/ 1 exp −x r t −1, γ 1 1, γ 2 1499, s r 1, μ 8, k 249.375, and λ r 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 x r 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.

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.
, and d σj are positive constants which can be specified by the designer, φ • : n → r×q is a sigmoidal function, that is, 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 W * 2 φ x t Mv t can be alternatively expressed as S * φ x t v t , where S * ∈ n×r is an unknown weight matrix.In view of the above, 3.4 can be rewritten as ∈ n is the state of the neural network, v t ∈ q is the control input, and W 1 t ∈ n×m and S t ∈ n×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 W 1 t and S t by proper learning laws such that the identification error Δ t : x t − x t can be reduced.
u t .Assumption 3.1.On a compact set Ω ⊂ n , 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 ẋ t Ax t W * 1 σ x t W * 2 φ x t Mv t W * 2 φ x t d t ω x t , u t ξ t .3.4 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 W * 2 φ x t d t is also bounded.Let us define the following expression: ζ t : W * 2 φ x t d t ω x t , u t ξ t .* φ x t v t ζ t .
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 Because W * 1 and S * are bounded, W 1 t W 1 t W * 1 , and S t S t S * must be bounded too and the first part of * , and ζ.
• , . . ., σ rm r • Assumption 4.1.On a compact set Ω ⊂ n , the error term ω r x r t is bounded by the positive constant not necessarily a priori known ω r , that is, |ω r x r t | ≤ ω r .
r j for j 1, ..., m r , 4.2 where a σ r j , c σ r j,i , and d σ r j are positive constants which can be specified by the designer, and ω r : n → n is an error term which can be defined simply asω r x t : B x r t − Ax r t − W * r σ r x r t .4.3where γ is a positive constant.The first derivative ofV 2 t is • V 2 t γe T t ė t tr ˙ W T t S t φ x t φ T x t S T t W r t σ r xr t 1 S t 2 φ T x t 2 − γke T t S t φ x t e t − γe T t W r t σ r x r t − γe T t ω r x r t T z ≤ |y| |z| for y ∈ n , z ∈ n and Y 2 tr{Y T r t | |Δ t | |e t |. 4.30 Clearly, |x t − x r t | ∈ L ∞ .Finally, by substituting 3.32 and 4.29 into 4.30 , we have Given the Assumptions 2.1-4.1, 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 matrix W r t , the virtual tracking error, the actual tracking error, the state of the neural network, the system state, and the control input are bounded: W r t , e t , x t − x r t , x t , x t , v t ∈ L ∞ , 4.32 b the actual tracking error |x t − x r t | converges exponentially to a zone bounded by the term