Robust Adaptive Neurocontrol of SISO Nonlinear

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J. Humberto Pérez-Cruz 1 and E. Ruiz-Velázquez 1 and José de Jesús Rubio 2 and Carlos A. de Alba-Padilla 1

Recommended by Jung-Fa Tsai

1, Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Boulevord Marcelino García Barragán 1421, 44430 Guadalajara, JAL, Mexico
2, Sección de Estudios de Posgrado e Investigación, ESIME UA-IPN, Avenida de las Granjas 682, Col. Santa Catarina, 02250 México DF, Mexico

Received 21 June 2012; Revised 6 August 2012; Accepted 16 August 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The deadzone is a nonsmooth nonlinearity commonly found in many practical systems such as hydraulic positioning systems [ 1], pneumatic servo systems [ 2], DC servo motors, among others. When the deadzone is not considered explicitly during the design process, the performance of the control system could be degraded by an increase of the steady-state error, the presence of limit cycles, or inclusive instability [ 3- 6]. 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 [ 3], Tao and Kokotovic proposed to employ an adaptive inverse of the deadzone. This scheme was applied to linear systems in transfer function form. Cho and Bai [ 7] extended this work and achieved a perfect asymptotic adaptive cancellation of the deadzone. However, their work assumed that the deadzone output was measurable. In [ 8], the work of Tao and Kokotovic was extended to linear systems in a state space form with nonmeasurable deadzone output. In [ 9], 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 [ 10], the same class of nonlinear systems as in [ 9] 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 [ 10] is 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 is not provided. In order to extend the class of systems previously considered in [ 9, 10], in this paper, we propose the development of two controllers based on universal approximation property of the neural networks. The first scheme utilizes a smooth adaptive inverse of the deadzone as in [ 9]. By means of Lyapunov analyses, the exponential convergence of the tracking error to a bounded zone is proven. The second scheme considers the deadzone as a combination of a linear term and a disturbance-like term as in [ 10]. Thus, the estimation of the deadzone inverse is not required. By using the Lyapunov-like analyses, the asymptotic converge of the tracking error to a bounded zone is demonstrated. Since this control strategy requires the knowledge of a bound for an uncertainty/disturbance term, a procedure to find such bound is provided. In both schemes, the boundedness of all closed-loop signals is guaranteed. A numerical experiment with a second-order nonlinear system shows that a satisfactory performance can be obtained by using any of the two proposed controllers.

2. Preliminaries and Problem Statement

In this study, the system which will be controlled is composed of an unknown nonlinear plant preceded by an actuator with an unknown deadzone in such a way that the deadzone output is the input to the plant. Consider that the n -order dynamics of the nonlinear plant can be represented as follows: [figure omitted; refer to PDF] where the scalar y (t ) is the output of interest, ^{y (i )} (t ) for i =1 , ... ,n -1 represents the i th derivative of y (t ) --each one of these derivatives are assumed measurable, u (t ) ∈ [real] is the deadzone output (and the input to the plant), f :^{[real] n} [arrow right] [real] is an unknown but continuous nonlinear function, b is an unknown constant, and ξ (t ) ∈ [real] is an unknown but bounded disturbance. Defining the state variables as _{x 1} ( t ) : =y ( t ) , _{x 2} (t ) : = y (t ) , ... ,_{x n} (t ) : =^{y (n -1 )} (t ) , ( 2.1) can be expressed as follows: [figure omitted; refer to PDF] where x (t ) ∈^{[real] n} is the measurable state vector for t ∈^{[real] +} : = { t :t ...5;0 } , which is defined as x (t ) : = [_{x 1} (t ) ,_{x 2} (t ) , ... ,_{x n} (t )^{] T} = [_{x 1} (t ) ,_{x 1} (t ) , ... ,^{x 1 (n -1 )} (t )^{] T} . The nonsymmetric deadzone can be represented by [figure omitted; refer to PDF] where _{m r} and _{m l} are the right and left constant slopes for the deadzone characteristic, _{b r} and _{b l} represent the right and left constant breakpoints, u (t ) ∈ [real] and v (t ) ∈ [real] are the output and the input of the deadzone, respectively. Note that v (t ) is the actual control input to the global system formed by the actuator and the plant. In accordance with [ 3, 4], the deadzone model ( 2.3) is a static simplification of diverse physical phenomena with negligible fast dynamics. Hereafter, it is considered that the following assumptions are valid.

Assumption 2.1.

Without loss of generality, the unknown constant b is assumed positive.

Assumption 2.2.

The deadzone output u (t ) is not available for measurement.

Assumption 2.3.

Although the deadzone parameters _{b r} , _{b l} , _{m r} , and _{m l} are unknown constants, we can assure that _{b r} >0 , _{b l} <0 , _{m r} >0 , and _{m l} >0 .

The objective that we are trying to achieve is to determine a control signal v (t ) such that the output of the plant ( 2.2), y (t ) =_{x 1} (t ) , follows a specified reference trajectory _{y r} (t ) , and, at the same time, all closed-loop signals stay bounded.

2.1. Smooth Parameterization of the Deadzone

A direct way of compensating the deleterious effect of the deadzone is by calculating its inverse. From ( 2.3), the deadzone inverse can be obtained as [figure omitted; refer to PDF] Notwithstanding, clearly this is a discontinuous function. A smooth approximation of ( 2.4) was presented in [ 9] as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and _{[straight epsilon] o} is a positive constant chosen by the designer. Since both the parameters and the output of the deadzone are unknown, approximation ( 2.5) cannot be utilized directly. To overcome this problem, a smooth parameterization of the deadzone was proposed in [ 9]. For completeness, this parameterization is explained here. Note that ( 2.3) can be expressed alternatively as [figure omitted; refer to PDF] where θ = [_{m r} ,_{m r}_{b r} ,_{m l} ,_{m l}_{b l}^{] T} , ω (t ) = [ -_{σ r} ( t ) v ( t ) ,_{σ r} ( t ) , -_{σ l} ( t ) v ( t ) ,_{σ l} ( t ) ^{] T} : [figure omitted; refer to PDF] Given that u (t ) , θ , and ω (t ) are unknown, the deadzone output u (t ) is approximated by [figure omitted; refer to PDF] where θ ^ (t ) : = [ _{m r} ^ , _{m r}_{b r} ^ , _{m l} ^ , _{m l}_{b l} ^ ^{] T} is an estimation of θ and ω ^ (t ) : =^{[ -}^{_{[varphi] r}}^{( v (t ) ) v (t ) ,}^{_{[varphi] r}}^{( v (t ) ) , -}^{_{[varphi] l}}^{( v (t ) ) v (t ) ,}^{_{[varphi] l}}^{( v (t ) ) ] T} . From ( 2.7) and ( 2.9), u (t ) can be expressed as [figure omitted; refer to PDF] where _{d N} (t ) : =^{θ T} ( ω ^ (t ) - ω (t ) ) . Although _{d N} (t ) is an unknown term, its boundedness can be guaranteed [ 11]. Consider that the positive constant _{d - N} is an upper bound for _{d N} (t ) , that is, | _{d N} (t ) | ...4;_{d - N} . From ( 2.9), v (t ) can be expressed in terms of _{u d} (t ) as [figure omitted; refer to PDF] This expression can be utilized only if vector θ ^ (t ) is properly determined. To avoid the singularity problem, note that _{m r} ^ and _{m l} ^ must always be different from zero.

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

For the particular case when m : =_{m r} =_{m l} , the deadzone model ( 2.3) can alternatively be described as [ 10, 12] [figure omitted; refer to PDF] where d (t ) is given by [figure omitted; refer to PDF] Note that ( 2.13) is the negative of a saturation function. Thus, although d (t ) could not be exactly known, its boundedness can be assured. Consider that the positive constant d - is an upper bound for d (t ) , that is, | d (t ) | ...4; d - .

3. Neurocontroller Design and Stability Analyses

Based on the universal approximation property of the neural networks, two control schemes are presented in this section to handle the trajectory tracking problem.

Assumption 3.1.

The reference trajectory _{y r} (t ) and their first n -derivatives are continuous and bounded. Besides, all these variables are available for the design.

Given the reference trajectory _{y r} (t ) and their first (n -1 ) -derivatives, the vector _{x r} (t ) can be defined as _{x r} (t ) : = [_{x r ,1} (t ) ,_{x r ,2} (t ) , ... ,_{x r ,n} (t )^{] T} = [_{y r} (t ) ,_{y r} (t ) , ... ,^{y r (n -1 )} (t )^{] T} . Let us define the filtered tracking error r (t ) as [figure omitted; refer to PDF] where _{e 1} (t ) is the first element of the tracking error vector e (t ) which is defined simply as e (t ) : =x (t ) -_{x r} (t ) = [_{e 1} (t ) ,_{e 2} (t ) , ... ,_{e n} (t )^{] T} = [_{e 1} (t ) ,_{e 1} (t ) , ... ,^{e 1 (n -1 )} (t )^{] T} and _{λ r} is a positive constant chosen by the designer. Note that r (t ) can also be expressed as [figure omitted; refer to PDF] where _{Λ r} ∈^{[real] n} is a constant vector given by _{Λ r} : = [^{λ r n -1} , (n -1 )^{λ r n -2} , ... ,1^{] T} .

Remark 3.2.

Note that, from ( 3.1), r (t ) can be considered as the input to a stable linear system whose output is _{e 1} (t ) . Consequently, if r ( t ) ∈_{L ∞} , then _{e 1} (t ) ,e (t ) ∈_{L ∞} . Specifically, e (t ) has the following properties proven in [ 13]: (i) e (t ) converges exponentially to zero, if r (t ) =0 , (ii) if e (0 ) =0 and | r (t ) | ...4; [varsigma] where [varsigma] is a positive constant, then e (t ) belongs to a compact set R given by R = { e (t ) ∈^{[real] n} such that | _{e i} (t ) | ...4;^{2 i -1}^{λ r i -n} [varsigma] , i =1 , ... ,n } for ∀ t ...5;0 , and (iii) if e (0 ) ...0;0 and | r (t ) | ...4; [varsigma] , then e (t ) will converge to R within a time-constant (n -1 ) /_{λ r} .

The first derivative of r (t ) can be calculated as [figure omitted; refer to PDF] where _{Λ - r} : = [0 ,^{λ r n -1} , (n -1 )^{λ r n -2} , ... , (n -1 )_{λ r}^{] T} .

Now then, it is well known that any unknown continuous function can be approximated on a compact set Ω by a neural network as follows [ 14- 16]: [figure omitted; refer to PDF] where σ ( · ) is the activation vector function with sigmoidal components, that is, σ ( · ) : =^[^{_{σ 1}}^{( · ) , ... ,}^{_{σ s}}^{( · ) ] T} : [figure omitted; refer to PDF] where _{a σj} , _{c σj ,i} , and _{d σj} are positive constants which can be specified by the designer, η :^{[real] n} [arrow right] [real] is the approximation error which is bounded by | η (x ) | < η - for all x ∈ Ω , η - is a positive constant, and ^{W *} ∈^{[real] 1 ×s} is an unknown constant vector. Typically, ^{W *} is considered as the optimal weight such that [figure omitted; refer to PDF] By substituting ( 3.4) into ( 2.2), the following alternative representation for the plant dynamics can be obtained: [figure omitted; refer to PDF] By substituting ( 3.7) into ( 3.3), we get [figure omitted; refer to PDF]

3.1. Scheme I

A control scheme which does not require the specific knowledge of the upper bound for the term η (x ) + ξ (t ) +b_{d N} (t ) is developed below.

In order to take into account the effect of the deadzone, the adaptive parameterization model ( 2.10) is substituted into ( 3.8): [figure omitted; refer to PDF] Consider that _{u d} (t ) is chosen as [figure omitted; refer to PDF] where p ^ (t ) is an online estimation of p : =1 /b and _{u - d} (t ) is selected as [figure omitted; refer to PDF] where W (t ) is an online estimation of ^{W *} and k is a positive constant. Note that [figure omitted; refer to PDF] where p ~ (t ) : =p - p ^ (t ) . If ( 3.12) is substituted into ( 3.9), the following is obtained: [figure omitted; refer to PDF] Now, by substituting ( 3.11) into ( 3.13) and reducing like terms, the filtered tracking error dynamics can be expressed as [figure omitted; refer to PDF] where W ~ (t ) : =^{W *} -W (t ) and δ (t ) : = η (x ) + ξ (t ) +b_{d N} (t ) is an unknown term but bounded by the positive constant δ - , that is, | δ (t ) | ...4; δ - .

Once the filtered tracking error dynamics has been determined, the following Lyapunov function candidate is suggested: [figure omitted; refer to PDF] where θ ~ (t ) : = θ - θ ^ (t ) , and _{k 1} , _{k 2} , and _{k 3} are positive constants. The first derivative of V (t ) is [figure omitted; refer to PDF] The first derivative of ( 1 /2_{k 1} ) W ~ (t )^{W ~ T} (t ) can be calculated as follows. [figure omitted; refer to PDF] Since W ~ (t )^{W ~ T} (t ) is a scalar and the transpose of a scalar is the same scalar, then [figure omitted; refer to PDF] Proceeding in a similar way for ( d /dt ) ( ( 1 /2_{k 3} ) ^{θ ~ T} (t ) θ ~ (t ) ) , it can be determined that [figure omitted; refer to PDF] Substituting ( 3.14), ( 3.18), and ( 3.19) into ( 3.16) yields [figure omitted; refer to PDF] Consider that the learning laws for W (t ) , p ^ (t ) , and θ ^ (t ) are chosen as [figure omitted; refer to PDF] where _{[cursive l] 1} , _{[cursive l] 2} , and _{[cursive l] 3} are positive constants, ^{W 0} , ^{p 0} , and ^{θ 0} are ideally (but not necessarily) good estimations of ^{W *} , p , and θ , respectively, and Proj represents a smooth projection operation as in [ 17] or [ 18] in order to avoid that _{m r} ^ and _{m l} ^ can be equal to zero. In accordance with ( 3.21), ( 3.22), and ( 3.23) and taking into account that W ~ (t ) = - W (t ) , p ~ (t ) = - p ^ (t ) , and θ ~ (t ) = - θ ^ (t ) , ( 3.20) can be expressed as [figure omitted; refer to PDF] In [ 11], it is mentioned that the projection operation has the following property: [figure omitted; refer to PDF] On the other hand, the truthfulness of the following inequalities was proven in [ 9] [figure omitted; refer to PDF] Likewise, it can be demonstrated that [figure omitted; refer to PDF] If the inequalities ( 3.25), ( 3.26), ( 3.27) are substituted into ( 3.24), we obtain [figure omitted; refer to PDF] If k >0.5 and defining α : =min { ( 2k -1 ) ,_{[cursive l] 1} ,_{[cursive l] 2} ,_{[cursive l] 3} } and [figure omitted; refer to PDF] the following bound as a function of V (t ) can finally be determined for V (t ) : [figure omitted; refer to PDF] ( 3.30) can be rewritten in the following form: [figure omitted; refer to PDF] Multiplying both sides of the last inequality by exp ( αt ) , it is possible to obtain [figure omitted; refer to PDF] The left-hand side of ( 3.32) can be rewritten as [figure omitted; refer to PDF] or, equivalently, as [figure omitted; refer to PDF] Integrating both sides of the last inequality from 0 to t yields [figure omitted; refer to PDF] Adding V (0 ) to both sides of the last inequality, we obtain [figure omitted; refer to PDF] Multiplying both sides of the inequality ( 3.36) by exp ( - αt ) yields [figure omitted; refer to PDF] and, consequently [figure omitted; refer to PDF] As 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 r ( t ) , W ~ (t ) , p ~ (t ) , and θ ~ (t ) can be guaranteed. Because ^{W *} , p , and θ are bounded, W (t ) =^{W *} - W ~ (t ) , p ^ (t ) =p - p ~ (t ) , and θ ^ (t ) = θ - θ ~ (t ) must be bounded too. If r (t ) ∈_{L ∞} , from Remark 3.2, we can assure that e (t ) ∈_{L ∞} and converges to R . From ( 3.11), it can be seen that _{u - d} (t ) is formed by bounded terms and consequently _{u - d} (t ) ∈_{L ∞} . From ( 3.10), it can be seen that _{u d} (t ) is the product of two bounded variables. Therefore, _{u d} (t ) ∈_{L ∞} . As _{u d} ( t ) , θ ^ ( t ) ∈_{L ∞} , and _{m r} ^ and _{m l} ^ are always different from zero, from ( 2.11), the boundedness of v (t ) can be concluded. Now, note that the following is true: ( 1 /2 ) ^{r 2} (t ) ...4;V (t ) . Taking into account this fact and from ( 3.38), we get [figure omitted; refer to PDF] By taking the limit as t [arrow right] ∞ of the last inequality, we can guarantee that | r (t ) | converges exponentially fast to a zone bounded by the term 2 β / α . Based on this fact together with Remark 3.2, we can conclude that _{e 1} (t ) =y (t ) -_{y r} (t ) converges exponentially fast to a region around zero bounded by the term ( 1 /^{λ r n -1} ) 2 β / α . Thus, the following theorem has been proven.

Theorem 3.3.

If Assumptions 2.1- 3.1are satisfied, k >0.5 , and the control law ( 2.11), ( 3.10), ( 3.11) with the learning laws ( 3.21), ( 3.22), and ( 3.23) are applied to the system formed by ( 2.2)-( 2.3), then the following hold:

(a) the filtered tracking error, tracking error, the weights, the estimations of the control gain reciprocal and the deadzone parameters, and the control signal are bounded: [figure omitted; refer to PDF]

(b) the actual tracking error y (t ) -_{y r} (t ) converges exponentially fast to a region around zero bounded by the term: [figure omitted; refer to PDF]

3.2. Scheme II

A simpler scheme can be obtained by using the model ( 2.12) and the tuning error _{r [straight epsilon]} (t ) instead of the model ( 2.10) and the filtered tracking error r (t ) , respectively. However, the implementation of this scheme requires necessarily the a priori knowledge of a good bound for the term η (x ) + ξ (t ) +bd (t ) . The development of this scheme is explained below.

If the deadzone model ( 2.12) is substituted into ( 3.8), we have [figure omitted; refer to PDF] Note that, by using the model ( 2.12), the actual control input v (t ) appears now directly into the error dynamics ( 3.42) only multiplied by a constant gain bm .

Consider that v (t ) is chosen as [figure omitted; refer to PDF] where q ^ (t ) is an online estimation of q : =1 /bm and v - (t ) is selected as [figure omitted; refer to PDF] where W (t ) is an online estimation of ^{W *} ; k , ^{k *} , and [straight epsilon] are positive constants, and sat ( · ) represents a saturation function given by [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] where q ~ (t ) : =q - q ^ (t ) .

Substituting ( 3.46) into ( 3.42) yields [figure omitted; refer to PDF] If ( 3.44) is substituted into ( 3.47) and reducing like terms, the filtered tracking error dynamics can be expressed as [figure omitted; refer to PDF] where W ~ (t ) : =^{W *} -W (t ) and ζ (t ) : = η (x ) + ξ (t ) +bd (t ) is an unknown term bounded by the positive constant ζ - , that is, | ζ (t ) | ...4; ζ - . Consider the following Lyapunov function candidate: [figure omitted; refer to PDF] where _{k 1} and _{k 2} are positive constants and _{r [straight epsilon]} (t ) is the tuning error defined as [ 10] [figure omitted; refer to PDF] Taking into account ( 3.18), the first derivative of V (t ) can be calculated as [figure omitted; refer to PDF] It can be demonstrated that _{r [straight epsilon]} (t )_{r [straight epsilon]} (t ) =_{r [straight epsilon]} (t ) r (t ) . By substituting ( 3.48) into the last equality and the resulting expression into ( 3.51), we get [figure omitted; refer to PDF] Consider that W ~ (t ) and q ~ (t ) are chosen as [figure omitted; refer to PDF] If ( 3.53) and ( 3.54) are substituted into ( 3.52), we obtain [figure omitted; refer to PDF] Now, from ( 3.50), [figure omitted; refer to PDF] Besides, [figure omitted; refer to PDF] Substituting ( 3.56) and ( 3.57) into ( 3.55) yields [figure omitted; refer to PDF] Considering that _{r [straight epsilon]} (t )sat ( r (t ) / [straight epsilon] ) = | _{r [straight epsilon]} (t ) | , ( 3.58) can be expressed as [figure omitted; refer to PDF] Note that, if ^{k *} and k are selected in such a way that ^{k *} +k [straight epsilon] > ζ - , then [figure omitted; refer to PDF] Because k is a positive constant, the last inequality implies that V (t ) ...4;0 . Based on this fact, it is possible to establish that V (t ) ...4;V (0 ) and, therefore, V (t ) ∈_{L ∞} . Since V (t ) is a nonnegative function, _{r [straight epsilon]} (t ) , W ~ (t ) and q ~ (t ) belongs to _{L ∞} . Because W (t ) =^{W *} - W ~ (t ) , and q ^ (t ) =q - q ~ (t ) , and as ^{W *} and q are constants, the boundedness of W (t ) and q ^ (t ) can be guaranteed. From the definition of tuning error ( 3.50) and as [straight epsilon]sat ( r (t ) / [straight epsilon] ) is a bounded term, if _{r [straight epsilon]} ( t ) ∈_{L ∞} , then r (t ) ∈_{L ∞} . Keeping in view the above fact, and on the basis of Remark 3.2, the boundedness of e (t ) can be assured. Now, it can be seen from ( 3.44) that v - (t ) is formed by bounded terms. Thus, v - (t ) ∈_{L ∞} . Likewise, it can be seen from ( 3.43) that v (t ) is the product of two bounded variables and consequently v (t ) is also bounded. On the other hand, an inspection of ( 3.48) reveals that r (t ) ∈_{L ∞} . From ( 3.50), this means that _{r [straight epsilon]} (t ) is bounded too. Integrating both sides of ( 3.60) from 0 to t yields [figure omitted; refer to PDF] Note that the last inequality can be expressed as [figure omitted; refer to PDF] Since V (t ) is a nonnegative function, the following is true: [figure omitted; refer to PDF] Substituting ( 3.63) into ( 3.62) yields [figure omitted; refer to PDF] taking the limit as t [arrow right] ∞ of both sides of the last inequality, finally, we obtain [figure omitted; refer to PDF] This means that _{r [straight epsilon]} (t ) ∈_{L 2} . As _{r [straight epsilon]} (t ) ∈_{L 2} ∩_{L ∞} and _{r [straight epsilon]} (t ) ∈_{L ∞} , from Barbalat's Lemma, we can conclude that _{r [straight epsilon]} (t ) converges asymptotically to zero. From definition ( 3.50), this implies that r (t ) asymptotically converges to a region around zero bounded by [straight epsilon] . In view of the above and from Remark 3.2, we can conclude that _{e 1} (t ) =y (t ) -_{y r} (t ) converges asymptotically to a region around zero bounded by the term [straight epsilon] /^{λ r n -1} . Thus, the following theorem has been proven.

Theorem 3.4.

If Assumptions 2.1- 3.1are satisfied, ^{k *} +k [straight epsilon] > ζ - , and the control input ( 3.43), ( 3.44) with the learning laws W (t ) =_{k 1}^{σ T} ( x (t ) ) _{r [straight epsilon]} (t ) , q ^ (t ) = -_{k 2} v - (t )_{r [straight epsilon]} (t ) are applied to the system ( 2.2)-( 2.3), then

(a) the tuning error, the filtered tracking error, tracking error, the weights, the bm reciprocal estimation, and the control signal are bounded: [figure omitted; refer to PDF]

(b) the actual tracking error y (t ) -_{y r} (t ) converges asymptotically to a region around zero bounded by the term [figure omitted; refer to PDF] where [straight epsilon] and _{λ r} are positive constants chosen by the designer.

3.2.1. Estimation of a Bound for ζ (t )

Certainly, compared with the scheme I, the structure of the scheme II is simpler. Nevertheless, the implementation of this last scheme requires a good estimation of ζ - in order to guarantee the theoretical performance provided by Theorem 3.4. Here, we propose an offline practical procedure to achieve this goal.

Consider that some experimental data ₍_x_{t i}_{,v (}_{t i}_{) ) i =1 , ... ,N} are available (see Remark 3.5). By substituting ( 2.12) into ( 3.7), we obtain [figure omitted; refer to PDF] Certainly, if _{x n} (t ) , ^{W *} , and bm could be known, ζ (t ) can be completely determined. By hypothesis, _{x n} (t ) is known. However, because of the noise, _{x n} (t ) must not be calculated directly from _{x n} (t ) . Instead, a robust differentiation strategy should be used such as sliding modes [ 19], high-order sliding modes [ 20, 21], or smoothing by least squares, among others. Hereafter, the estimation of _{x n} (t ) is denoted by _{x - n} (t ) . Once _{x - n} (t ) is obtained, let us consider the following regression model: [figure omitted; refer to PDF] where χ (_{t i} ) is simply an error term. By defining _X_{t i} : = [^{σ T} ( x (_{t i} ) ) ,v (_{t i} ) ] , and Φ : = [^{W *} ,bm ] , ( 3.69) can be expressed as [figure omitted; refer to PDF] By using least-squares method, Φ can be estimated as [figure omitted; refer to PDF] where X : = [_X_{t 1} , ... ,_X_{t N}^{] T} and Y =^[^{_{x - n}}⁽^{_{t 1}}^{) , ... ,}^{_{x - n}}⁽^{_{t N}}^{) ] T} . Once Φ - is determined, ζ (t ) can be approximated as [figure omitted; refer to PDF] Next, ζ - can be estimated from ( 3.72).

Remark 3.5.

These experimental data can be generated by trying to use the neurocontroller II with relatively large values for the constant parameters. A first attempt could be to let k =100 and ^{k *} =50 . If the tracking is not satisfactory, then larger values could be tried.

Remark 3.6.

Alternatively, in [ 22], the determination of an error term δ (t ) is achieved by using of sliding modes.

4. Numerical Example

In this section, the proposed neurocontrollers are tested by simulation on the following second order nonlinear system: [figure omitted; refer to PDF] The initial condition for system ( 4.1) is _{x 1} ( 0 ) =1 , _{x 2} ( 0 ) = -1 ; u (t ) is the deadzone output whose parameters are _{m r} =_{m l} =1.5 , _{b r} =2.5 , _{b l} = -2 ; ξ (t ) , the disturbance term is selected as ξ (t ) =sin ( 10t ) . The following reference trajectory is proposed _{y r} (t ) = -cos ( t ) +1.5cos ( 2t ) -0.5 . The first and second derivative of _{y r} (t ) can be calculated analytically. In another case, a robust differentiation method must be used. It is very important to mention that the nonlinear system ( 4.1), the exact values for deadzone parameters, and the disturbance term are assumed completely unknown for the neurocontrollers during the design and simulation process. That is, the system ( 4.1) is only used as a data generator.

With respect to the tuning process, we must recognize that, similarly to many techniques of adaptive control, we do not have a systematic procedure in order to find the proper values for the controller parameters. Instead, an approach by trial and error is used. Thus, through various iterative simulations, the following values for the parameters of the neurocontroller I were found: [figure omitted; refer to PDF] Notice that a great freedom is allowed in order to select ^{W 0} , ^{θ 0} , and ^{p 0} . However, the designer should be aware of that as these parameters take values increasingly different from the optimal ones, the parameter β in Theorem 3.3becomes larger. On the other hand, we have seen that by setting W (0 ) =^{W 0} , θ (0 ) =^{θ 0} , and p (0 ) =^{p 0} , a more regular behavior of the closed-loop system can be obtained [ 23, 24].

The simulation is carried out by means of Simulink with ode45 method, relative tolerance equal to 1e -6 , and absolute tolerance equal to 1e -8 . A Simulink block diagram for the neurocontroller based on scheme I is shown in Figure 1. The results of the tracking process are presented in Figures 2- 4for the first 20 seconds of the simulation. In Figure 2, the output of nonlinear system ( 4.1), y (t ) =_{x 1} (t ) , is represented by dashed line whereas the reference trajectory _{y r} (t ) is represented by solid line. In spite of the difference between y (0 ) and _{y r} (0 ) , the tracking process shows a satisfactory behavior. This can be verified more specifically in Figure 3where the actual tracking error, y (t ) -_{y r} (t ) , is illustrated. In order to appreciate better the quality of the tracking process, a zoom of Figure 3is presented in Figure 4. From Figures 3and 4, we can appreciate that the actual tracking error converges fast to a zone bounded by 0.02. Finally, the control signal v (t ) acting as the input of deadzone is shown in Figure 5.

Figure 1: Simulink block diagram for the neurocontroller I.