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Adaptive Control of Uncertain Nonaffine Nonlinear Systems With Input Saturation Using Neural Networks Kasra Esfandiari, Student Member, IEEE, Farzaneh Abdollahi, Senior Member, IEEE, and Heidar Ali Talebi, Senior Member, IEEE Abstract— This paper presents a tracking control methodology for a class of uncertain nonlinear systems subject to input saturation constraint and external disturbances. Unlike most previous approaches on saturated systems, which assumed affine nonlinear systems, in this paper, tracking control problem is solved for uncertain nonaffine nonlinear systems with input saturation. To deal with the saturation constraint, an auxiliary system is constructed and a modified tracking error is defined. Then, by employing implicit function theorem, mean value theorem, and modified tracking error, updating rules are derived based on the well-known back-propagation (BP) algorithm, which has been proven to be the most relevant updating rule to control problems. However, most of the previous approaches on BP algorithm suffer from lack of stability analysis. By injecting a damping term to the standard BP algorithm, uniformly ultimately boundedness of all the signals of the closed-loop system is ensured via Lyapunov’s direct method. Furthermore, the presented approach employs nonlinear in parameter neural networks. Hence, the proposed scheme is applicable to systems with higher degrees of nonlinearity. Using a high-gain observer to reconstruct the states of the system, an output feedback controller is also presented. Finally, the simulation results performed on a Duffing-Holmes chaotic system, a generalized pendulum-type system, and a numerical system are presented to demonstrate the effectiveness of the suggested state and output feedback control schemes. Index Terms— Adaptive control, back-propagation (BP) algorithm, input constraints, neural networks (NNs), nonaffine nonlinear systems.

I. I NTRODUCTION

A

N IMPORTANT problem encountered in practical control systems is input nonlinearities, such as saturation, hysteresis, and dead zone. Input saturation is known to be the most common nonsmooth input nonlinearity, and

Manuscript received April 19, 2014; accepted November 24, 2014. Date of publication December 19, 2014; date of current version September 16, 2015. K. Esfandiari is with the Center of Excellence on Control and Robotics, Department of Electrical Engineering, Amirkabir University of Technology, Tehran 16846, Iran (e-mail: [email protected]; [email protected]). F. Abdollahi is with the Center of Excellence on Control and Robotics, Department of Electrical Engineering, Amirkabir University of Technology, Tehran 16846, Iran, and also with the Department of Electrical and Computer Engineering, Concordia University, Montréal, QC H4B 1R6, Canada (e-mail: [email protected]). H. A. Talebi is with the Center of Excellence on Control and Robotics, Department of Electrical Engineering, Amirkabir University of Technology, Tehran 16846, Iran, and also with the Department of Electrical and Computer Engineering, University of Western Ontario, London, ON N6A 3K7, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2014.2378991

in many applications, actuators usually have limitations on their magnitude, because of the physical characteristics of the actuator or safety considerations. If the saturation is ignored in the designing procedure, the designed controller can severely degrade the performance of the system or even lead it instability, take the system to unsafe mode, and damage the actuators [1]. Consequently, the analysis and the design of control systems subjected to input saturation nonlinearity have attracted an extensive attention during the past decades. To counter the harmful effect of saturation, two major techniques are developed: one technique is based on the adjustment of commanded input signal [2]–[4], and the other is based on constructing an auxiliary system and modifying tracking error based on the auxiliary states. In [5], the idea of using auxiliary system design is employed to compensate saturation constraint in linear time-invariant systems. In addition, different control schemes for nonlinear saturated systems have been developed based on this idea [6]–[9]. In control design with neural networks (NNs), many actuator nonlinearity compensation schemes have been developed. In [10], some valuable adaptive state feedback control schemes for practical systems with input nonlinearities including friction, deadzone, and backlash are presented. Furthermore, some NN-based controllers are presented for nonlinear systems with saturation constraint [4], [11], [12]. In [11], reinforcement learning-based output feedback is developed for multiinput multioutput (MIMO) nonlinear systems and a wavelet adaptive controller is presented for time-delayed nonlinear systems in [12]. It was assumed in [4] that nonlinear dynamics are partially known. More specifically, the control gain function [g(x) in [4]] is completely known. Then, a linear in parameter neural network (LPNN)-based saturation compensation scheme is presented for nonlinear systems in Brunovsky canonical form. Recently, solving the optimal control law for nonlinear systems with input saturation has also attracted attention [13]–[16]. In [14], an offline reinforcement learningbased is presented to solve Hamilton–Jacobi–Bellman equation, which guarantees the sufficient condition for the existence of optimality. Online actor-critic algorithm is presented to find the optimal control law in [17]. An iterative adaptive dynamic algorithm is developed for approximating a nearoptimal controller for a class of discrete-time systems in [15]. However, the presented schemes in [14]–[17] are developed based on the exact knowledge of nonlinear terms. In [13], this assumption is relaxed and only the control gain is assumed

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to be known a priori. The approaches reported in [13]–[16] focused on finding the optimal stabilizing controller for an affine nonlinear system, and they are developed based on the assumptions of a priori knowledge of nonlinear terms, saturation bounds being symmetric, and availability of the states. However, these assumptions are not often satisfied in practical systems. In addition, in the case of adaptive control of uncertain systems without saturation constraint, various adaptive schemes have been presented. In [18], multiple NNs are employed to derive a controller affine nonlinear systems. The system is transformed into a linear form based on the feedback linearization method, and then each nonlinear term is approximated by an LPNN. In [19], an LPNN-based control scheme is developed for uncertain MIMO nonlinear systems in block-triangular form. Although these methodologies showed good performance, the assumption of availability of all the states is not always practical. Moreover, adaptive state feedback control schemes for a class of nonlinear systems in strict-feedback form are presented based on the backstepping method, and the convergence of the error signal to a small neighborhood of the desired trajectory is ensured in [20] and [21]. Some valuable NN control approaches (some of which mentioned above) have been proposed for saturated nonlinear systems. However, most of them are applicable to the relatively simple nonlinear systems in affine form. There exist some practical systems, such as pH neutralization, distillation columns, and chemical reactions [22]; the input control signal of which do not appear in affine form and the prescribed schemes are not applicable. Due to the few choices of mathematical tools in handling nonaffine appearance of the control signal, the control problem of nonaffine systems is a challenging task. In [23]–[26], LPNN controllers are developed to deal with the nonaffine problem under contraction mapping condition. Using mean value and implicit function theorems to handle nonaffine problem, an LPNN controller is proposed for nonaffine nonlinear systems in [27] and [28]. Chen and Ge [29] introduced a disturbance observer to estimate the effects of unknown terms and input saturation, and then derived LPNN learning rules based on the Lyapunov synthesis for nonaffine nonlinear systems. Taylor series expansion is used in [30] to remove nonaffine appearance of the control signal, and then an adaptive fuzzy controller is designed. Most of the nonaffine or saturation compensation schemes mentioned above and references therein focused on LPNNs. The learning rules are derived based on the Lyapunov synthesis. Hence, they are not applicable to the systems with high degrees of nonlinearity and the updating rules are not simple and easy to implement. To tackle these problems in this paper, a nonlinear in parameter neural network (NLPNN) controller is introduced to use global approximation property, and despite of most of the previous approaches, learning rules are rooted in the well-known BP optimization algorithm. BP algorithm is the most relevant learning rule to the control problems, which owes its fame to its promising results and simplicity [31], [32]. The main drawback of the previous

approaches on BP algorithm lies in lack of stability analysis. Most of approaches that studied stability have not investigated control problem [33], [34]. In this paper, by injecting a damping factor to normal BP algorithm and decomposing the system into subsystems, uniformly ultimately boundedness (UUB) of all signals of the closed-loop system is also ensured via Lyapunov’s direct method. Motivated by the aforementioned discussion, we will investigate both state and output feedback designing procedure of an NLPNN-based adaptive controller for a class of uncertain nonaffine systems with nonsymmetric saturation constraint and external disturbance, using BP optimization algorithm. The main contributions of the proposed scheme are as follows. 1) Considering nonaffine appearance of the control signal and nonsymmetric input saturation constraint simultaneously in the plant dynamics, which makes tracking problem tremendously challenging due to the few choices of mathematical tools in handling nonaffine appearance of control signals and nonsymmetric saturation constraint. 2) Since nonaffine appearance of control signal puts high degrees of nonlinearity in the unknown nonlinear functions of system, and therefore it seems crucial to employ NLPNN controller to support global approximation property and to the best of the authors’ knowledge, NLPNN controller is first designed for saturated nonaffine systems in this paper. 3) Adaption laws are derived based on the BP optimization algorithm, which has been proven to be the most relevant algorithm to the control problems, hence adaption rules are simple, easy to implement, and completely rooted in the BP optimization algorithm. 4) UUB of all signals of the closed-loop system using state or output feedback is guaranteed via Lyapunov direct method in the presence of nonsymmetric saturation and unknown external disturbance. It should also be noted that although the presented method is developed for nonaffine saturated systems, it also seems a valuable and novel approach for nonaffine unsaturated nonlinear systems, affine saturated systems, and affine unsaturated nonlinear systems without adding any restrictive assumptions to the system. The rest of this paper is organized in the following manner. In Section II, some preliminaries and problem statement are presented. Adaptive NLPNN-based controller is presented in Section III, and the simulation results are presented in Section IV. Section V contains the conclusion. Notations: Consider A = (tr(A T A))1/2 and x = T (x x)1/2 be Frobenius norm of a matrix and Euclidean norm of a vector, respectively, where tr(.) denotes the trace of (.). The space of L ∞ bounded signals is defined as x(t) ∈ L ∞ if ess sup(|x(t))t < ∞, where ess stands for essential. Then, the L ∞ norm of the signal x(t) is defined as x = ess sup(|x(t)|) < ∞. t

ESFANDIARI et al.: ADAPTIVE CONTROL OF UNCERTAIN NONAFFINE NONLINEAR SYSTEMS

II. P ROBLEM S TATEMENT AND S OME P RELIMINARIES A. System Description Consider a class of single-input single-output uncertain nonaffine nonlinear system given by [29], [35] x˙1 = x 2 x˙2 = x 3 .. . x˙n = f (x, sat(u)) + d(t) y = x1

both of the nonlinear terms ( f (x) and g(x) in [18]) are assumed bounded, and the bounds are supposed to be known a priori. Similarly, in [6], the upper bound of control gain is considered to be known, and it is assumed completely known in [4] and [21]. The control objective is to design an NLPNN-based controller by state and output feedback for (1) such that: 1) the system output tracks the desired trajectory; 2) saturation compensates effectively; and 3) all the signals in the closedloop system remain UUB.

(1)

where x = [x 1 x 2 . . . sx n ]T ∈ R n is the state vector, y ∈ R is the output, f (x, sat(u)) ∈ R is an unknown but smooth nonaffine nonlinear function, d(t) is an unknown external disturbance, and sat(u) ∈ R is the plant input subjected to saturation nonlinearity. Throughout this paper, the following assumptions and lemmas are used. Assumption 1 [1]: Plant input satisfies saturation nonlinearity expressed by ⎧ ⎪ u > u max ⎨u max (2) sat(u) = u u min ≤ u ≤ u max ⎪ ⎩ u min u < u min where u is the designed control law, and u min and u max are the given bounds of saturation nonlinearity. Assumption 2 [1], [30]: The desired trajectory and its nth order derivatives are assumed to be bounded. Assumption 3 [27], [28]: The sign of f u = (∂ f (x, u))/(∂u) is known, and there exist unknown constants l1 and l2 such that 0 < l1 ≤ | f u | ≤ l2

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(3)

without loss of generality, we assume l2 ≥ f u ≥ l1 > 0. Lemma 1 (Implicit Function Theorem [35], [36]): Assume that h(x, u) : R n × R → R is continuously differentiable for all (x, u) ∈ R n × R and there exists a constant l1 such that 0 < l1 ≤ h u |(x0 ,u 0 ) : ∀(x 0 , u 0 ) ∈ R n × R. Then, there exists a continuous (smooth) function u ∗ = g(x) such that h(x, u ∗ ) = 0. For the case 0 > l1 ≥ h u |(x0 ,u 0 ) , the results still hold. Lemma 2 (Mean Value Theorem [37]): Assume that h(x, u) : R n × R → R has a derivative at each point of an open set R n × (a, b) and also assume that it is continuous at both end points u = a and u = b. Then, there is a point u¯ ∈ (a, b) such that ∂h(x, u) ¯ (b − a). h(x, b) − h(x, a) = ∂u Remark 1: Assumption 3 can be seen as a controllability condition, which is satisfied by many practical systems (e.g., robotic systems, chaotic systems, and induction motors). Even, in some literatures, other assumptions such as availability of an approximation of f u or boundedness of the second derivative of f (x, u) respect to u or boundedness of the time derivative of f u are also considered [19], [20], [26], [27]. In the affine case, it is equivalent to control gain that should be positive definite or negative definite [1], [38]. In [18],

B. Neural Networks NN capabilities in approximation made them a valuable tool for control of highly uncertain and nonlinear systems. Based on the universal approximation property, any continuous function f (x) can be arbitrarily estimated well using a two-layer NN. Hence, the nonlinear function f (x) can be expressed as f (x) = W σ (V x) + ε(x) where V are the input to hidden layer interconnection ideal weight vectors, W are the output to hidden layer interconnection ideal weight vectors, x ∈ S ⊂ R n is the input vector with S ⊂ R n being a compact set, ε(x) is a bounded approximation error, and σ (.) is an activation function of hidden layer that is usually considered sigmoidal [39], [40] σi (V x) = −1 +

2 1 + e−2Vi x

(4)

where Vi is the ith row of V and σi (V x) is the ith element of σ (V x). The ideal weights are assumed to be bounded, i.e., W  ≤ W M , V  ≤ VM , and since they are unknown, the implemented NN is an approximation of function f (x), and can be approximated by fˆ(x) = Wˆ σ (Vˆ x) where Wˆ and Vˆ are an approximation of ideal weight vectors. III. A DAPTIVE NLPNN-BASED C ONTROLLER A. Saturation Compensation To remove the effect of the saturation constraint, the following auxiliary system is constructed: ς˙ = K ς + hu, ς (0) = 0

(5)

where ς ∈ R n is the auxiliary state vector, u = u − sat(u) denotes the difference between the desired and actual control signals, which cannot be implemented because of the saturation, and K ∈ R n×n and h ∈ R n are considered as a Hurwitz matrix and a constant vector, respectively. It is clear that ς represents a filtered version of u and remains zero as long as no saturation occurs, and in the presence of saturation, ς becomes nonzero. In other words, ς (t) is a convolution of an exponential term with u(t) and converges to zero if and only if u(t) converges to zero. Hence, by modifying tracking

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error as (7), we are able to remove the effect of the saturation constraint and rewrite (1) in the following form: x˙ = Ax + B( f (x, u) + d(t)) y = Cx where

⎡

0 ⎢ .. ⎢ A=⎢. ⎣0 0 C = [1

1 .. . ··· 0 0

⎤ 0 .. ⎥ .⎥ ⎥, 1⎦ 0

··· .. . ··· ··· ···

r˙ = −kr + f u |u=u¯ × (u − u ∗ ) (6)

⎡ ⎤ 0 ⎢ .. ⎥ ⎢ ⎥ B=⎢.⎥ ⎣0⎦ 1

Note that since we have no a priori knowledge about the derivative of u, we cannot modify tracking error based on u. B. NLPNN-Based State Feedback Controller Based on the availability of all states of the system, the modified tracking error e(t) ˜ is defined as follows:

T y¯ = [y · · · y (n−1) ]T , y¯d = yd · · · yd(n−1) (7)

(12)

where u¯ = θ u + (1 − θ )u ∗ , 0 < θ < 1. To achieve the desired control law, a two-layer NN is employed [39], [40] u ∗ (x NN ) = −W σ (V x NN ) − ε(x NN )

0 ].

e˜ = y¯ − y¯d − ς = e − ς.

Now, using (9), (11), and Lemma 2, mean value theorem, we obtain

(13)

where W and V are the ideal bounded NN weight vectors, ε(x NN ) is a bounded approximation error, σ (.) is the activation function (4), and x NN = [ y¯ y¯d r ς ]T is the input to the NN. The implemented NN is an approximation of (13), and can be expressed as follows: u = −Wˆ σ (Vˆ x NN )

(14)

where Wˆ and Vˆ are an approximation of ideal weight vectors, W˜ = W − Wˆ , and V˜ = V − Vˆ denote NN weight approximation errors, and both of the weight vectors are assumed to be tunable. Substitution of (13) and (14) to (12) yields in r˙ = −kr + f u |u=u¯ × (W˜ σ (Vˆ x NN ) + a(t))

(15)

where a(t) = W (σ (V x NN ) − σ (Vˆ x NN ) + ε(x NN )).

Now, let us define the filtered tracking error as r = [ 1]e˜

(8)

where  ∈ R 1×(n−1) is a constant vector, which is defined such that the roots of the polynomial s n−1 + λn−1 s n−2 + · · · + λ2 s + λ1 = 0 related to the equation of r = 0 are in the open left-half plane, i.e., e˜ → 0 as r → 0. Taking the time derivative of filtered the tracking error (8) yields (9) r˙ = f (x, u) + Yd − kr n−1 (n) where Yd = d(t) − yd − ς˙n + i=1 i (e(i) − ς˙i ) + kr , ς˙i is the ith element of ς˙ , and k is a positive designing parameter. By envisaging (1), (5), and (7), we conclude that ∂Yd = −(λ1 h 1 + · · · + λn−1 h n−1 + h n )u¯ ∂u where

(10)

 0 u min ≤ u ≤ u max u¯ = 1 u < u min or u > u max .

According to Assumption 3 and positive definiteness of λi , i = 1, 2, . . . , n − 1, a sufficient condition for applying the implicit function theorem to nonlinear function F( y¯ , y¯d , ς, u) = Yd ( y¯ , y¯d , ς, u) + f (x, u) is that all elements of h are negative. In other words, we ensure that ∂ F( y¯ , y¯d , ς, u) > 0. ∂u This selection is not a limiting constraint and can be easily satisfied. Hence, one can invoke Lemma 1, implicit function theorem, for the nonlinear function F( y¯ , y¯d , ς, u), and conclude that there exists a desired control signal u ∗ = g( y¯ , y¯d , ς ) such that F( y¯ , y¯d , ς, u ∗ ) = Yd ( y¯ , y¯d , ς, u ∗ ) + f (x, u ∗ ) = 0. (11)

C. Stability Analysis for State Feedback Control The most important issue in designing an NN-based controller is defining an appropriate updating rule, which should be easy to implement and ensure stability of the closed-loop system. However, in most of the previous approaches in the literature, a Lyapunov candidate function is defined, and then updating rules are constructed such that the stability of the closed-loop system is ensured [4], [29]; in this paper, updating rules are directly derived based on the BP optimization algorithm, which has been proven to be easy to implement and the most effective updating rule for control problems. Unlike most of the previous approaches on BP algorithm [31], [32] by injecting a damping term to BP algorithm and decomposing the system in to subsystems, the stability of the overall system is also presented via Lyapunov direct method. Since nonaffine appearance of control signal and saturation constraint puts high degrees of nonlinearity to (1), it is necessary to employ NLPNNs to use full capabilities of NN approximation property. Hence, the proposed scheme is applicable to the systems with a complicated unknown nonlinear function. Throughout the stability analysis, the following equalities, inequalities, and fact are used: sup(σ (.)) = 1, sup(W ) = W M , Wˆ = W − W˜ tr(W˜ T Wˆ ) ≤ W˜ (W M − W˜ ). Fact 1: Since ideal NN weight W , activation function σ (.), and approximation error ε(x NN ) are assumed to be bounded, there exists an unknown constant a¯ such that ¯ a(t) = W (σ (V x NN ) − σ (Vˆ x NN ) + ε(x NN )) ≤ a. Theorem 1: Consider a class of nonlinear nonaffine system described in (1) with nonsymmetric input saturation

ESFANDIARI et al.: ADAPTIVE CONTROL OF UNCERTAIN NONAFFINE NONLINEAR SYSTEMS

constraint (2) satisfying Assumptions 1–3 and auxiliary system (5). Assuming full state measurement, then by employing the NLPNN-based state feedback controller (14) and updating ˜ V, ˜ ς , and e ∈ L ∞ . rules (16) and (17), it is ensured that W, In other words, NN weights errors, auxiliary states, and output tracking error are UUB ∂J ˙ (16) − ρ1 |r |Wˆ Wˆ = −W˙˜ = −η1 ∂ Wˆ ∂J ˙ (17) − ρ2 |r |Vˆ Vˆ = −V˙˜ = −η2 ∂ Vˆ where η1 and η2 are positive learning rates, and ρ1 and ρ2 are small positive constants. Note that the first terms on the righthand side of (16) and (17) are directly derived based on the BP optimization algorithm, and the second terms of (16) and (17) correspond to the e-modification, which are used to ensure UUB of all signals of the closed-loop system [10], [41]. When the tracking error tends to zero, this term also converges to zero. Proof: Let us define the cost function of the NN weight updating rules as J = 1/2r 2 . Based on the chain rule, using (4) and (14) and performing some manipulations, we have ∂ J ∂r ∂u ∂r ∂J (18) = = r (−σ (Vˆ x NN )) ∂r ∂u ∂ Wˆ ∂u ∂ Wˆ ∂σ (Vˆ x NN ) ∂(Vˆ x NN ) ∂u ∂J ∂ J ∂r = ∂r ∂u ∂σ (Vˆ x NN ) ∂(Vˆ x NN ) ∂ Vˆ ∂ Vˆ ∂r T (19) = r (−Wˆ (Im×m − m×m ))T x NN ∂u where m is the number of neurons in the hidden layer and m×m = diag(σi2 ), i = 1, . . . , m. Based on (7) and (8), ∂r /∂u can be computed in two steps. At first, we assume that saturation has been removed using auxiliary system (5) and compute ∂e/∂u, i.e., ς = 0. After that, we will compute ∂ς/∂u, which causes the saturation effect to be seen directly in the updating rules and compensates it effectively. Using (6) and (7), we rewrite ∂r /∂u as T

∂([y y˙ · · · y (n−1) ] ) ∂x ∂e = = . (20) ∂u ∂u ∂u Now, let us invoke Lemma 2 (mean value theorem) and use (11) to rewrite (6) as follows: x˙ = Ax + B(−Yd + f u |u=u¯ × (u − u ∗ ) + d(t)).

(21)

As previously mentioned in this step, we assume ς = 0,  that (i) and therefore, we conclude Yd = d(t)−yd(n) + n−1  e +kr ; i i=1 using (6)–(8), we rewrite (21) as follows: ∗

x˙ = A1 x + B f u |u=u¯ × (u − u ) + B f o where f o = [ 1] (k y¯d + y˙¯d ) and ⎡ 0 1 ··· ⎢ .. .. . . ⎢ . . A1 = ⎢ . ⎣ 0 ··· ··· −kλ1 −kλ2 − λ1 · · ·

0 0 1 −k − λn−1

(22) ⎤ ⎥ ⎥ ⎥. ⎦

Since k and λi , i = 1, . . . , n − 1 are considered as positive designing parameters, A1 is always a nonsingular matrix. Now, let us consider the static approximation of ∂ x/∂u (x˙ = 0)

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and ∂ς/∂u (ς˙ = 0). Using this approximation and considering (5), (20), (22), and Assumption 3, we have ∂ς = −K −1 hu¯ ∂u

∂e = −A−1 1 B, ∂u where

 u¯ =

(23)

1 if u > u max or u < u min 0 if u min ≤ u ≤ u max .

It is worth mentioning that based on Assumption 3 and the fact that replacing f u |u=u¯ by its sign does not change the direction of moving opposite to the gradient of the cost function, in (23), we have replaced f u |u=u¯ by its sign. Therefore, using (7), (8), and (23), we can rewrite the learning rules (16) and (17) in terms of W˜ and V˜ as −1 hu)σ ¯ T (Vˆ x NN ) W˙˜ = η1r [ 1](A−1 1 B−K + ρ1 |r |Wˆ ˙ ¯ V˜ = η r [ 1](A−1 B − K −1 hu) 2

(24)

1

T × (Wˆ (Im×m − m×m ))T x NN + ρ2 |r |Vˆ .

(25)

Now, let us consider the following Lyapunov candidate function: 1 1 (26) L = r 2 + tr(W˜ T ρ1−1 W˜ ). 2 2 Differentiating (26) along the system trajectory, and substituting (15) and (25) to it yields L˙ = r ( f u |u=u¯ × [W˜ σ (Vˆ x NN ) + a(t)] − kr )    −1 hu¯ + tr W˜ T ρ1−1 (η1r [ 1] A−1 1 B−K  × σ T (Vˆ x NN ) + ρ1 |r |Wˆ ) . Using the mentioned equalities, inequalities, Fact 1, Assumption 3, and performing some manipulations, we obtain ¯ L˙ ≤ |r |(−k|r | − W˜ 2 + DW˜  + l2 a)

(27)

−1 hu) ¯ + l2 + W M . where D = ρ1−1 η1 [ 1](A−1 1 B−K By completing squares in (27), we have    2 2 D D + l2 a¯ . (28) L˙ ≤ |r | −k|r | − W˜  − + 2 4

Hence, L˙ is guaranteed to be negative definite as long as |r | ≥

D2 4

or

+ l2 a¯ = rmax k

(29)

 D + W˜  ≥ 2

D2 + l2 a¯ = W˜ max . 4

(30)

From (29) and (30), we can infer that if the modified tracking error or norm of NN weight error W˜ becomes greater than ramx and W˜ max , respectively, then L˙ becomes negative and reduces them, i.e., W˜ , r ∈ L ∞ . However, the proof is not completed by this analysis. Moreover, we have to guarantee that ς, V˜ , (y − yd ) ∈ L ∞ . Since W˜ = W − Wˆ and ideal weights W belong to L ∞ , we conclude that Wˆ and consequently control signal u (14)

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are bounded. Now, we can consider the auxiliary system (5) as a linear system with bounded input u = u − sat(u). It is clear that this auxiliary system is stable since K was chosen Hurwitz, i.e., ς ∈ L ∞ . From the boundedness of the modified tracking error r (29) and by considering (7) and (8), we can conclude that e˜ = e −ς ∈ L ∞ . We can say that e also belongs to L ∞ since ς, e˜ does. To show the boundedness of V˜ , let us rewrite (25) as follows: V˙˜ = f1 − ρ2 |r |V˜

(31)

where −1 hu) ¯ f 1 = η2r [ 1](A−1 1 B−K T × (Wˆ (I − ))T x NN + ρ2 |r |V.

f 1 is bounded, since Wˆ , σ (.), r , and ideal weights V ∈ L ∞ . Therefore, (31) can be regarded as a linear system with bounded input f 1 . It is obvious that (31) is stable since ρ2 |r | is positive and input f1 is bounded, i.e., V˜ ∈ L ∞ . Since we have shown that all the signals of the closed-loop system are UUB, the proof is completed. From (24), (25), and the definition of u, ¯ it is obvious that once saturation occurs, the adaption laws are directly affected and try to compensate saturation constraint and put the system into the safe work zone. Hence, it seems that the presented learning rules will give promising saturation compensation and tracking performance. It also worth pointing out that the bounds on the errors can become arbitrary small by the proper selection of the designing parameters.

tracking error (8), control law (14), and NN input x NN by replacing the immeasurable states by their estimations (n−1) T yˆ¯ = xˆ1 · · · xˆ1 e˜o = yˆ¯ − yd − ς = eˆ − ς (33) rˆ = [ 1]e˜o xˆNN = [ yˆ¯ y¯d rˆ ς ]T u = −Wˆ σ (Vˆ xˆNN ). E. Stability Analysis for Output Feedback Control The structure of the output feedback NLPNN-based controller can be summarized in the following theorem. Theorem 2: Consider a class of nonlinear nonaffine system given in (1) with external disturbance and nonsymmetric input saturation constraint (2) satisfying Assumptions 1–3 and auxiliary system (5). Then, with the application of the high-gain observer (32) to reconstruct states of (1), NLPNN-based output feedback controller (34), and updating rules (35) and (36), it is ensured that NN weights errors, auxiliary states, and output tracking error are UUB   −1 hu¯ σ (Vˆ xˆNN ) W˙˜ = η1rˆ [ 1] A−1 1 B−K (35) + ρ1 |ˆr |Wˆ  −1  ˙ −1 V˜ = η rˆ [ 1] A B − K hu¯ 2

In Section III-B, a state feedback controller has been designed based on the availability of all the states of the system. In practical systems, all the states of the system are not usually measurable or we may not choose to measure all of them. Therefore, it is important to extend the proposed scheme to the output feedback case. In this section, a high-gain observer is employed to reconstruct the state vector using the output of the system y = x 1 . Let us consider the following high-gain observer, which is defined as follows [36], [42]: αi x˙ˆi = xˆ i+1 + i (y − yˆ ), i = 1, . . . , n − 1 ε0 α n x˙ˆn = n (y − yˆ ) ε0 yˆ = xˆ 1

1

T + ρ2 |ˆr|Vˆ (36) ×(Wˆ (Im×m − m×m ))T xˆNN where η1 and η2 are positive learning rates, and ρ1 and ρ2 are small positive constants. Proof: Subtracting (6) from (32) yields

x˙˜ = Ao x˜ + B( f (x, u) + d(t)) where

D. NLPNN-Based Output Feedback Control

(34)

⎡

−α1 /ε0

⎢ ⎢ −α2 /ε2 0 Ao = ⎢ ⎢ .. ⎣ . −αn /ε0n

1 0 .. . 0

··· .. . .. . ···

(37)

⎤ 0 .. ⎥ .⎥ ⎥. ⎥ 1⎦ 0

In the case of no nonlinearity and disturbance, asymptotic error convergence is achieved by designing Ao as a Hurwitz matrix. From (37), it is also obvious that the higher observer gain αi /ε0 is chosen; then, we have smaller influence of the nonlinearity and disturbance on the observer output error and we can obtain excellent accuracy. In other words, we have (for more details, see [36], [42]) lim (s I − Ao )−1 L( f (x, u) + d(t)) = 0

ε0 →0

(32)

where xˆ = [xˆ1 · · · xˆn ]T denotes the estimation of the state vector, αi , i = 1, 2, . . . , n − 1 are positive constants such that the roots of P(s) = s n + α1 s n−1 + · · · + αn have negative real parts, and ε0  1 is a sufficiently small constant, which ensures that all eigenvalues of P(s) tend to −∞ as ε0 → 0. Via the certainty equivalence approach, we modify the filtered

where L(.) denotes the Laplace transform and is the Laplace variable. As clearly expressed in [36], when using highgain observer (32), separation principal is hold for nonlinear systems and the output feedback controller recovers the performance of the state feedback controller when the observer gain is chosen sufficiently high. Hence, separation principal allows us to design a state feedback controller that meets our design specifications, and then, replace the state vector by its estimate provided by high-gain observer (32). Therefore, based on the separation principal [36], the stability

ESFANDIARI et al.: ADAPTIVE CONTROL OF UNCERTAIN NONAFFINE NONLINEAR SYSTEMS

of the state feedback controller (Theorem 1), and the possibility of choosing observer gain sufficiently large enough (Remark 2), the proof is completed. Remark 2: It is important to mention that, even though the high-gain observers give excellent estimation by reducing ε0 , they also suffer from peaking phenomenon, which is an intrinsic feature of any high-gain observer. Hence, when using high-gain observers, the peaking of the estimated states is transmitted to the controller and plant, so this may take the states outside of the region of attraction and cause instability. However, since in our problem the control law (14) is developed such that the control signal cannot become greater than saturation bounds (2), the peaking phenomenon of the estimated states cannot be transmitted to the plant by the controller. Hence, the high-gain observer seems to be the most appropriate observer for (1), and by appropriately choosing the parameters of the auxiliary system (5), ε0 can be chosen arbitrary small to enhance the desired estimation regardless of the peaking phenomenon.

Fig. 1.

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Tracking error by employing the state feedback for example A.

IV. S IMULATION R ESULTS Remark 3: Comparing with [23]–[26], the presented approach utilized NLPNN capabilities, and it does not require contraction mapping condition or availability of an approximation of fu . Moreover, assumptions of boundedness of the time derivative of fu and a priori knowledge about the upper bound of f u are not needed [6], [18]–[20]. To show the effectiveness of the proposed state and output feedback controllers, the simulation results are performed for three nonlinear systems.

Fig. 2. Control signal (input of the actuator) by employing the state feedback for example A.

A. Duffing-Holmes Chaotic Nonlinear System Consider a Duffing-Holmes chaotic nonlinear system described by [29] x˙1 = x 2 x˙2 = − p1 x 1 − px 2 − x 13 + q cos(ωt) + h(x, u) + d(t) (38) where p1 = 0.3 + 0.2 sin(10t), p = 0.2 + 0.2 cos(5t), q = 5 + 0.1 cos(t), ω = 0.5 + 0.1 sin(t), h(x, u) = u + 0.5 cos(u), and the external disturbance d(t) = 0.4 sin(0.2πt) + 0.3 sin(x 1 x 2 ). Since f u = 1 − 0.5 sin(u) > 0, Assumption 3 is satisfied and the proposed controllers are applicable. The proposed controller strategies are applied to (38) to guarantee that the output tracks the desired trajectory yd = sin(t) + cos(0.5t). Bounds on the control signal are considered as u max = 10 and u min = −15. The initial states and NN weights are taken x(0) = [0.2 0]T and small random numbers, respectively. Other designing parameters are chosen as follows:   −5 0 K = , h = [−5 −5]T,  = 500 0 −5 k = 50, ηi = 0.01, ρi = 0.0001, i = 1, 2. In the case of the output feedback, the observer parameters are considered as α1 = 5, α2 = 10, and ε0 = 0.01 and the initial states are considered as x(0) ˆ = [0.8 0]T . Note that the initial states are selected similar to [29], to make a reasonable

Fig. 3.

Tracking error by employing the output feedback for example A.

comparison with the results reported in [29]. The simulation results are shown in Figs. 1–5. Figs. 1 and 3 show that the proposed schemes can track the desired trajectory very well in the presence of nonsymmetric saturation, external disturbance, and nonaffine appearance of the control signal. Moreover, according to Figs. 2 and 4, it is obvious that the proposed schemes have compensated saturation constraint efficiently in both state and output feedback cases and keep the control input in the safety zone; however, in the output feedback case, the state estimation error had an undershoot, as argued in Remark 2.

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Fig. 4. Control signal (input of the actuator) by employing the output feedback for example A.

Fig. 6.

Fig. 5. State estimation error using high-gain observer and its zoomed-in view for example A.

Fig. 7. Control signal (input of the actuator) by employing the state feedback for example B.

To the best of the authors’ knowledge, [29] is the only work that has claimed to design a controller for nonaffine saturated systems, in which an NN and a disturbance observer are employed to approximate the unknown terms and construct the control signal. Hence, the results reported in [29] are chosen for comparison. By comparing the results, one can see that the proposed scheme has a better tracking performance in both cases (state and output feedback), which is the direct result of using NLPNN and BP algorithm. Furthermore, despite the control signal reported in [29], saturation is also compensated effectively in the early moments (transient phase) and exceeding the safe zone is prevented. Moreover, the required assumptions for designing disturbance observer in [29], boundedness of the time derivative of f u , and the designed control signal u are relaxed. In addition, employing NLPNN and extracting learning rules based on the BP optimization algorithm make the proposed learning rules simple, easy to implement, and applicable to the systems with high degrees of nonlinearity.

general pendulum nonlinear system described by [4]

B. General Pendulum To show that the suggested NLPNN controller is also potentially superior to the previous saturation compensation schemes of the nonlinear affine systems, we consider the

Tracking error by employing the state feedback for example B.

x˙1 = x 2 x˙2 = −5x 13 − 2x 2 + sat(u).

(39)

The desired trajectory and the initial condition are considered as yd = sin(t) and x(0) = [0 0]T , respectively. The input saturation limit is four, i.e., |u| ≤ 4, and the high-gain observer parameters are considered as α1 = 5, α2 = 10, and ε = 0.01. The other designing parameters are selected as   −5 0 K = , h = [−5 −5]T,  = 100 0 −5 k = 50, ηi = 0.01, ρi = 0.001, i = 1, 2. The simulation results are shown in Figs. 6–10. Figs. 6 and 8 verify that the proposed NLPNN-based controllers make the states to track the desired trajectory. It is worth noting that by reducing bounds on the control signal, the proposed scheme compensates the saturation very well, while the tracking error slightly increased. In these cases, the existence of such a small tracking error is not a drawback of the proposed scheme, since in the case of perfect tracking, a control signal with an amplitude of greater than four is always required. Figs. 7 and 9 demonstrate the previously mentioned fact that once saturation occurs, the updating rules are directly affected, and

ESFANDIARI et al.: ADAPTIVE CONTROL OF UNCERTAIN NONAFFINE NONLINEAR SYSTEMS

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the other nonlinear function ( f (x) in [4]), and having a priori knowledge about the upper bound of the corresponding error (| f˜(x)| ≤ f M in [4]) are relaxed. It should also be mentioned that even though the given approach in [4] is developed based on the availability of f u and knowing an approximation of the other nonlinear function, the uncertain functions ( f (x) and g(x) in [4]) are assumed to be completely known in their simulations to focus on the saturation compensation. However, their scheme still failed to give a good compensation in the transient phase. C. Numerical Example Fig. 8.

Tracking error by employing the output feedback for example B.

In this section, a simulation is performed on a numerical nonlinear system to focus on the functionality of the proposed scheme, when it is applied to nonlinear systems with high degrees of nonlinearity. Consider the following nonlinear system: x˙1 = x 2 x˙2 =

x1 + x2

+ (1 + x 22 )x 12 + x 1 cos(x 1 ) sin(x 2 ) 1 + (x 1 + x 2 )2  2 + 0.5 sin x 2 e−x1 + Ln(3 + cos(x 1 ) + sin2 (x 1 x 2 ))   − Ln 1 + x 12 x 22 + (2 + cos(x 1 x 2 ))sat(u) + 0.1sat(u 3 ) + sin(0.5sat(u)) + tanh(sat(u)) + d(t)

Fig. 9. Control signal (input of the actuator) under the output feedback for example B.

Fig. 10. State estimation error using high-gain observer and its zoomed-in view for example B.

consequently, the input saturation constraint is compensated effectively (see the initial moments of Figs. 7 and 9). In [4], an LPNN-based controller is developed for affine nonlinear systems. As compared with the results reported there, the proposed approach gives promising saturation compensation (especially, in the transient phase where their scheme fails to compensate the saturation) and better tracking performance. It is worth noting that, in this paper, bounds on the control signal are assumed smaller than the one considered in [4]. Also comparing with [4], a lot of assumptions such as affine appearance in the control signal, availability of f u (g(x) in [4]), knowing an approximation of

(40)

where d(t) = 0.1 sin(t) + 0.5 cos(t +π/6). Suppose that there is no knowledge of the nonlinearities and external disturbance. The objective is to control y to track the desired output yd = 0.5 cos(1.3t + π/3) + 0.3 sin(t), while the control signal is limited to the safe operating zone of |u| ≤ 5. In f u = 2 + cos(x 1 x 2 ) + 0.3u 2 + 0.5 cos(0.5u) + 1 − tanh2 (u); hence, Assumption 3 is satisfied and the proposed control methodology is applicable. In this simulation, the initial conditions of system state vector and NN weight vectors are chosen as x(0) = [ 0.5 0.5 ]T and random numbers, respectively. Other designing parameters are as follows:   −5 0 K = , h = [−5 −5]T,  = 200 0 −5 k = 50, ηi = 0.01, ρi = 0.0001, i = 1, 2. Figs. 11 and 12 show the simulation results in the case of state feedback. Fig. 11 shows the system output y tracks the desired reference signal yd accurately in the presence of unknown external disturbance and high degrees of nonlinearity of the system. Moreover, it can be observed from Fig. 12 that the control signal is bounded and remains in the safe operating zone. In the output feedback case, the initial condition of the observer is chosen to be x(0) ˆ = [1 0]T , and the parameters of the observer are selected as α1 = 5, α2 = 10, and ε = 0.01. Other designing parameters are chosen the same as the state feedback case. The simulation results are shown in Figs. 13–15. From Fig. 13, it is observed that the developed output feedback NLPNN-based controller gives promising tracking performance. It is also observed that the tracking error converges to a small neighborhood of zero, and the unknown

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 10, OCTOBER 2015

Tracking error by employing state feedback for example C.

Fig. 12. Control signal (input of the actuator) by employing the state feedback for example C.

Fig. 14. Control signal (input of the actuator) under the output feedback for example C.

Fig. 15. State estimation error using high-gain observer and its zoomed-in view for example C.

parameters that makes a tradeoff between the error and the transient phase of the control input is , and by increasing/ decreasing it, the tracking error will decrease/increase, while at the early moments, the control signal has more/less oscillations between u max and u min . V. C ONCLUSION

Fig. 13.

Tracking error by employing the output feedback for example C.

nonlinearities and disturbances are effectively compensated. Fig. 14 shows that the proposed control scheme can prevent the control input from reaching the saturation limits. Moreover, the performance of the observer is shown in Fig. 15. Remark 4: From the simulation results, it is obvious that since saturation effects impress the learning rules directly, the suggested controller is able to compensate saturation constraint from the beginning moments. Note that after the time required for NLPNN-based controller to learn nonlinear functions, the control signal is smooth. Moreover, one of the key designing

In this paper, adaptive control issue has been investigated for a challenging class of uncertain nonaffine strict-feedback systems with nonsymmetric input saturation and external disturbance. Using auxiliary system design, high-gain observer, mean value, and implicit function theorem, stable NLPNN controllers are designed for state as well as output feedback control. Nonlinear in parameter characteristics of the proposed controller make the proposed scheme a powerful tool in control of nonlinear systems with arbitrary degrees of nonlinearities, and the updating rules are directly derived based on the BP optimization algorithm. Hence, they are easy to implement. Since bounds are based on the designing parameters, they can become arbitrary small by appropriately choosing the designing parameters. The simulation results confirm these facts and also superiority of this controller in compensating saturation. Further research on the more challenging nonminimum-phase saturated systems is in progress.

ESFANDIARI et al.: ADAPTIVE CONTROL OF UNCERTAIN NONAFFINE NONLINEAR SYSTEMS

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[23] N. Hovakimyan, F.-J. Nardi, and A. J. Calise, “Adaptive output feedback control methodology applicable to non-minimum phase nonlinear systems,” Automatica, vol. 42, no. 1, pp. 513–522, Apr. 2006. [24] A. J. Calise, N. Hovakimyan, and M. Idan, “Adaptive output feedback control of nonlinear systems using neural networks,” Automatica, vol. 37, no. 8, pp. 1201–1211, Aug. 2001. [25] N. Hovakimyan, A. J. Calise, and N. Kim, “Adaptive output feedback control of uncertain multi-input multi-output systems using single hidden layer neural networks,” Int. J. Control, vol. 77, no. 15, pp. 1318–1329, 2004. [26] B.-J. Yang and A. J. Calise, “Adaptive control of a class of nonaffine systems using neural networks,” IEEE Trans. Neural Netw., vol. 18, no. 4, pp. 1149–1159, Jul. 2007. [27] Q. Shen and T. Zhang, “Novel design of adaptive neural network controller for a class of non-affine nonlinear systems,” Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 3, pp. 1107–1116, Mar. 2012. [28] S.-L. Dai, C. Wang, and M. Wang, “Dynamic learning from adaptive neural network control of a class of nonaffine nonlinear systems,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 1, pp. 111–123, Jan. 2014. [29] M. Chen and S. S. Ge, “Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer,” IEEE Trans. Cybern., vol. 43, no. 4, pp. 1213–1225, Aug. 2013. [30] A. Boulkroune, M. M’Saad, and M. Farza, “Adaptive fuzzy tracking control for a class of MIMO nonaffine uncertain systems,” Neurocomputing, vol. 93, pp. 48–55, Sep. 2012. [31] J. Mahdavi, M. R. Nasiri, A. Agah, and A. Emadi, “Application of neural networks and State-space averaging to DC/DC PWM converters in sliding-mode operation,” IEEE/ASME Trans. Mechatronics, vol. 10, no. 1, pp. 60–67, Feb. 2005. [32] H. A. Talebi, R. V. Patel, and H. Asmer, “Neural network based dynamic modeling of flexible-link manipulators with application to the SSRMS,” J. Robot. Syst., vol. 17, no. 7, pp. 385–401, 2000. [33] F. Abdollahi, H. A. Talebi, and R. V. Patel, “A stable neural networkbased observer with application to flexible-joint manipulators,” IEEE Trans. Neural Netw., vol. 17, no. 1, pp. 118–129, Jan. 2006. [34] F. Abdollahi, H. A. Talebi, and R. V. Patel, “Stable identification of nonlinear systems using neural networks: Theory and experiments,” IEEE/ASME Trans. Mechatronics, vol. 11, no. 4, pp. 488–495, Aug. 2006. [35] J.-H. Park, G.-T. Park, S.-H. Kim, and C.-J. Moon, “Direct adaptive self-structuring fuzzy controller for nonaffine nonlinear system,” Fuzzy Sets Syst., vol. 153, no. 3, pp. 429–445, Aug. 2005. [36] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ, USA: Prentice-Hall, 2001. [37] S. Lang, Real Analysis. Reading, MA, USA: Addison-Wesley, 1983. [38] J. Li, T. Li, and Y. Li, “NN-based adaptive dynamic surface control for a class of nonlinear systems with input saturation,” in Proc. 7th IEEE Conf. Ind. Electron. Appl., Jul. 2012, pp. 570–575. [39] G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control, Signals Syst., vol. 2, no. 4, pp. 303–314, 1989. [40] H. A. Talebi, F. Abdollahi, R. V. Patel, and K. Khorasani, Neural Network-Based State Estimation of Nonlinear Systems (Lecture Notes in Control and Information Sciences), vol. 395. New York, NY, USA: Springer-Verlag, 2010. [41] K. S. Narendra and A. M. Annaswamy, “A new adaptive law for robust adaptation without persistent excitation,” IEEE Trans. Autom. Control, vol. 32, no. 2, pp. 134–145, Feb. 1987. [42] A. Tornambé, “High-gain observers for non-linear systems,” Int. J. Syst. Sci., vol. 23, no. 9, pp. 1475–1489, 1992.

Kasra Esfandiari (S’13) received the B.Sc. degree in electrical engineering/control systems from Shiraz University, Shiraz, Iran, in 2012, and the M.Sc. (Hons.) degree in electrical engineering/ control systems from the Amirkabir University of Technology, Tehran, Iran, in 2014. He is currently a Research Assistant with the Stability of Dynamical Systems Laboratory, Department of Electrical Engineering, Amirkabir University of Technology. His current research interests include adaptive control, intelligent control, nonlinear control, robust control, robotics, and multiagent systems.

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Farzaneh Abdollahi (S’98–M’09–SM’14) received the B.Sc. degree from the Isfahan University of Technology, Isfahan, Iran, in 1999, the M.Sc. degree from the Amirkabir University of Technology, Tehran, Iran, in 2003, and the Ph.D. degree from Concordia University, Montreal, QC, Canada, in 2008, all in electrical engineering. She is currently an Assistant Professor with the Amirkabir University of Technology and a Research Assistant Professor with Concordia University. Her current research interests include neural networks, robotics, control of nonlinear systems, control of multiagent networks, and robust and switching control.

Heidar Ali Talebi (M’02–SM’08) received the B.Sc. degree from Ferdowsi University, Mashhad, Iran, in 1988, the M.Sc. degree from Tarbiat Modarres University, Tehran, Iran, in 1991, and the Ph.D. degree from Concordia University, Montreal, QC, Canada, in 1997, all in electrical engineering. He was a Post-Doctoral Fellow and Research Fellow with Concordia University and the University of Western Ontario, London, ON, Canada. In 1999, he joined the Amirkabir University of Technology, Tehran. He is currently a Professor with the Department of Electrical Engineering, and his research interests include control, robotics, medical robotics, fault diagnosis and recovery, intelligent systems, adaptive control, nonlinear control, and real-time systems.