Nonlinear Dyn (2013) 73:427–437 DOI 10.1007/s11071-013-0798-7

O R I G I N A L PA P E R

Robust tracking and model following based on discrete-time neuro-sliding mode control for uncertain time-delay systems Ming-Chang Pai

Received: 1 August 2012 / Accepted: 22 January 2013 / Published online: 6 February 2013 © Springer Science+Business Media Dordrecht 2013

Abstract This paper proposes a discrete-time neurosliding mode control (NSMC) scheme to realize the problem of robust tracking and model following for a class of uncertain time-delay systems. It is shown that the proposed scheme guarantees the stability of closed-loop system and achieves zerotracking error in the presence of state delays, input delays, parameter uncertainties, and external disturbances. The selection of sliding surface and the existence of sliding mode are two important issues, which have been addressed. This scheme not only assures robustness against time-delays, system uncertainties and disturbances, but also avoids chattering phenomenon and reaching phase. Moreover, the knowledge of upper bound of uncertainties is not required. Both the theoretical analysis and illustrative example demonstrate the validity of the proposed scheme. Keywords Discrete-time · Neuro-sliding mode control · Time-delay · Zero-tracking error · Chattering phenomenon

M.-C. Pai () Department of Automation Engineering, Nan Kai University of Technology, Tsao-Tun, Nantou 54210, Taiwan, R.O.C. e-mail: [email protected]

1 Introduction Over the past decades, the robust tracking and model following problem for dynamical systems with significant uncertainties have been widely investigated [1–8]. In [1], a class of linear state feedback controllers is designed to achieve robust tracking of dynamic signals for uncertain linear dynamic systems. In [2], a nonlinear switching controller is developed to deal with the problem of robust tracking and model following for uncertain time-delay systems. In [3], a linear robust tracking controller is developed to make the ultimate bound of the tracking error arbitrarily small. In [4], a nonlinear state feedback controller is proposed for robust tracking of dynamical signals. In [5], an adaptive robust tracking controller is developed to guarantee that tracking error decreases asymptotically to zero. In [6], a controller with an adaptive compensation term is developed for robust tracking of uncertain dynamic delay systems. In [7], an adaptive sliding mode controller is proposed for robust tracking and model following of linear systems with time-varying parameter uncertainties, multiple delayed state perturbations, and external disturbance. In [8], a composite nonlinear feedback control method is proposed for robust tracking and model following of linear systems with timevarying uncertain parameters and disturbances. However, these methods are developed in continuous-time system. To the best of the author’s knowledge, very little attention has been paid to the problem of robust tracking and model following for a class of uncertain

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time-delay systems in discrete-time domain, which is still open in the literature. This has motivated my research. On the other hand, using computers or DSP chips to implement the controller has become more and more important nowadays. Therefore, research in discretetime control has been intensified in recent years, and it is quite natural to extend the technique of continuous control to discrete-time systems. Sliding mode control (SMC) has attractive features such as fast response, good transient performance, insensitiveness to the matching parameter uncertainties and external disturbances [9–11] so that SMC is an effective robust control approach for uncertain systems. Several design methods of discrete-time SMC have been proposed in the literature [12–18]. The objective of this paper is to develop a discretetime NSMC scheme for uncertain time-delay systems to track dynamic inputs of a nondelay reference model. The algorithm is base on discrete-time SMC and neural network control [19–21]. It is shown that the proposed scheme not only guarantees the stability of the closed-loop system, but also asymptotically achieves zero-tracking error in the presence of time-delays, parameter uncertainties, and external disturbances. The proposed scheme has the following attractive features: (1) the control design is rather straightforward and the zero-tracking error is achieved. (2) The order of the motion equation in the quasi-sliding mode is equal to the order of the original system, rather than reduced by the number of dimension of the control input. The robustness of the system can be guaranteed throughout the entire response of the system starting from the initial time instance. (3) The discretetime NSMC needs not a switching type of control law. Chattering phenomenon is eliminated. Moreover, the knowledge of upper bound of uncertainties is not required. This paper is organized as follows. Section 2 briefly states the problem of tracking and model following. Section 3 provides the proposed discrete-time NSMC scheme. The selection of sliding surface, the design of neuro-sliding mode controller, the stability of the overall closed-loop system, and zero-tracking error will be addressed. Section 4 provides results from numerical simulations, and verifies the efficacy of the proposed scheme. Finally, a conclusion is provided in Sect. 5.

M.-C. Pai

2 Problem formulation and assumptions Consider a class of uncertain time-delay dynamic systems discredited with a small sampling time T > 0, and the resultant discrete-time systems can be described by:   x(k + 1) = A + A(k) x(k) + Ad x(k − d)   + B + B(k) u(k) (1) + Bh u(k − h) + f (k) y(k) = Cx(k) where x(k) ∈ R n is the state, u(k) ∈ R 1 is the control input, y(k) ∈ R P is the output, f (k) ∈ R n×1 is the external disturbance, A, Ad , B, Bh , and C are known constant matrices of appropriate dimensions, A(k) and B(k) represent the time-varying parameter uncertainties in the system model, d > 0 and h > 0 represent known constant time-delay in state and input, respectively. The reference input ym (k) is the output from the discrete-time reference model described by xm (k + 1) = Am xm (k) ym (k) = Cm xm (k)

(2)

where xm (k) ∈ R nm is states of reference model, and ym (k) has the same dimension as y(k). Furthermore, it is assumed that the model states are bounded, i.e. xm (k) ≤ ρ for all k. In this paper, similar to [3–5], the requirement for the proposed controller to track the reference output (2) is the existence of G ∈ R n×nm and H ∈ R 1×nm such that      A B G GAm = (3) C 0 H Cm If a solution cannot be found to satisfy (3), then a different reference model or output matrix C must be chosen. The solution of G and H in (3) is discussed in [1]. To facilitate further development, the following assumptions for system (1) are assumed to be valid: Assumption 1 The pair (A, B) is controllable. Assumption 2 There exist matrix functions of appropriate dimension Dd , Dh , DA (k), DB (k), and Df (k)

Robust tracking and model following based on discrete-time neuro-sliding mode control for uncertain

such that Ad = BDd , Bh = BDh , A = BDA (k), B = BDB (k), and f (k) = BDf (k). With Assumption 2, the uncertain time-delay system (1) can be rewritten as x(k + 1) = Ax(k) + Bu(k) + Bd(k) y(k) = Cx(k)

(4)

where the generalized disturbance d(k) is constructed 

as d(k) = Dd x(k − d) + Dh u(k − h) + DA (k)x(k) + DB (k)u(k) + Df (k). The objective of this paper is, based on discretetime NSMC scheme, to find a robust tracking controller such that the output of uncertain discrete timedelay system (4) follows the output of model (2) in the presence of state delays, input delays, parameter uncertainties, and external disturbances simultaneously.

state vector z(k) to zero, then the tracking error will approach to zero, i.e., z(k) → 0(C < ∞) ⇒ e(k) → 0 or y(k) → ym (k) from (7). Next, the discrete-time NSMC method is used to design the auxiliary control function v(k) so that the auxiliary system (8) is stable. Remark 1 The sliding mode characteristics of discrete-time SMC systems are different from those of continuous-time SMC systems. It is noted that the motion of a discrete-time SMC system can approach the switching surface but cannot stay on it in practice. Thus, only the quasi-sliding mode is ensured [12, 13]. 3.1 Design of sliding surface In this paper, the sliding function is defined as σ (k) = Sz(k) − ε(k) ε(k) = ε(k − 1) + S(A + BK)z(k − 1),

3 Main results In this section, a robust tracking controller is proposed to guarantee that the output y(k) of uncertain discrete time-delay system (4) follows the output ym (k) of reference model (2). For this purpose, the structure of the tracking controller is considered as u(k) = H xm (k) + v(k)

(5)

where H satisfies the condition (3), and v(k) is an auxiliary control function, which will be given later as a discrete-time neuro-sliding mode controller. To facilitate further development, a new auxiliary state vector z(k) is defined as z(k) = x(k) − Gxm (k)

(6)

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(9a) ε(0) = 0 (9b)

where S ∈ R 1×n is chosen such that SB is nonsingular, K ∈ R 1×n is designed later such that the auxiliary system (8) in the quasi-sliding mode is asymptotically stable. Consider a forward expression of (9a), (9b) σ (k + 1) = Sz(k + 1) − ε(k + 1)

(10a)

ε(k + 1) = ε(k) + S(A + BK)z(k)

(10b)

Substituting (10b) and (8) into (10a), and using the concept of equivalent control, the equivalent control veq (k) can be found by solving for σ (k + 1) = 0 veq (k) = Kz(k) + (SB)−1 ε(k) − d(k)

(11)

where SB is assumed to be nonsingular. From (9a) and σ (k) = 0, it leads to

where G satisfies the condition (3). Since CG = Cm in (3), the tracking error vector e(k) and auxiliary state vector z(k) can be expressed as

ε(k) = Sz(k)

e(k) = y(k) − ym (k) = Cz(k)

Substituting (12) into (11), the equivalent control (11) can be expressed as

(7)

From (6) and (2)–(5), the auxiliary system can be expressed as z(k + 1) = Az(k) + Bv(k) + Bd(k)

(8)

If it is possible to design a control law v(k) that makes the asymptotic convergence of auxiliary

veq (k) = Kz(k) + (SB)−1 Sz(k) − d(k)

(12)

(13)

Substituting (13) into (8), the dynamic equation of system (8) in the quasi-sliding mode can be obtained as   z(k + 1) = A + B(SB)−1 S + BK z(k) (14)

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an output layer which has a linear activation function. The control input can be defined as v=

n

zj wj + b

(18)

j =1

In order to fulfill the requirement σ (k + 1) − φσ (k) = 0, the neural network control is applied to minimize the error function E= Fig. 1 Structure of the neural network controller

Equation (14) can be considered as a linear state feedback problem. The matrices K and S can be designed by using the pole placement method.

bnew = bold − η

After designing the sliding surface, the next phase is to design the control law v(k) such that quasi-sliding mode is reached and stayed thereafter. A sufficient condition for a discrete-time SMC to assure both sliding motion and convergence onto the hyperplane given in [12] is     σ (k + 1) < σ (k) (15) In order to achieve the condition (15), the reaching law is defined as (16)

where 0 < φ < 1 is a design parameter to adjust the rate of asymptotic convergence to the hyperplane. Or σ (k + 1) − φσ (k) = 0

By designing the weight wj and bias b such that E → 0 and E = 0 is a stable solution, the condition σ (k + 1) − φσ (k) = 0 will be satisfied and the quasisliding mode will be achieved. The weight update and bias update can be respectively expressed as wjnew = wjold − η

3.2 Design of discrete-time neuro-sliding mode controller

σ (k + 1) = φσ (k)

T   1 σ (k + 1) − φσ (k) σ (k + 1) − φσ (k) (19) 2

(17)

In this paper, a least square minimization using neural network control is applied to fulfill the above requirement (17). The structure of the neural network controller is presented in Fig. 1, where zj is the j th row of the auxiliary state z, j = 1, . . . , n, wj refers to the weight of the signal that comes from the j th row of the auxiliary state z, and b refers to the bias term. This structure is a type of the neural network structures called ADALINE. There are an input layer and

∂E ∂wj

∂E ∂b

(20) (21)

where η > 0 is a learning constant. ∂E Using the chain rule, the term ∂w in (20) can be j written as ∂E ∂E ∂v = ∂wj ∂v ∂wj

(22)

Substituting (18) and (19) into (22), it yields  T ∂E = σ (k + 1) − φσ (k) ∂wj ×

∂(σ (k + 1) − φσ (k)) zj (k) ∂v

(23)

Using (8), (9a), (9b), and (23), it yields  T ∂E = σ (k + 1) − φσ (k) SBzj (k) ∂wj

(24)

Similarly, the bias update can be computed using the same procedure as T ∂E  = σ (k + 1) − φσ (k) SB ∂b

(25)

Since the back propagation weight update algorithm is used, system may not reach global minimum and may stay in some local minimum. Next, by investigating the shape of the error function (19), it will

Robust tracking and model following based on discrete-time neuro-sliding mode control for uncertain

be shown that the proposed algorithm does not have this problem. It is known that if a function’s second derivative does not change sign with respect to a function variable, then the function does not have a change in the curvature sign through that variable, which means that the function does not have a local minimum through that variable. Taking the second derivative of the error function (19) with respective to the weight wj , it yields ∂ 2 E ∂(σ (k + 1) − φσ (k))T = SBzj (k) ∂wj ∂wj2

(26)

as the control law (18) and the weight update (20) with (24) and bias update (21) with (25) are applied. Next, it shows that the sliding motion z(k) of the auxiliary system (8) is stable. Since the matrices K and S can be designed such that the matrix A + B(SB)−1 S + BK is stable, it is obvious that the quasi-sliding mode of the auxiliary system (8) is stable. That implies z(k) → 0 as k → ∞. From (7), it yields e(k) ≤ Cz(k). Since C < ∞, then it follows that z(k) → 0 ⇒ e(k) → 0 or y(k) → ym (k) as k → ∞. The proof is completed. 

(27)

The procedures of the proposed robust tracking controller to realize the tracking and following model problem for uncertain time-delay systems are summarized as follows:

Using the chain rule in (26) ∂ 2 E ∂(σ (k + 1) − φσ (k))T ∂v = SBzj (k) ∂v ∂wj ∂wj2

Substituting (8), (9a), (9b), and (18) into (27), it yields ∂ 2E = B T S T SBzj2 = SB22 zj2 ∂wj2

(28)

Similarly, the second derivative of the error function (19) with respective to the bias b is computed as ∂ 2E = B T S T SB = SB22 ∂b2

431

(29)

From (28) and (29), it is shown that the second derivatives of the error function (19) with respective to weight and bias variables are always positive and do not change sign. The error function (19) does not have a local minimum. Hence, by selecting a proper learning constant η, the proposed network is capable of minimizing the error function (19) up to the global minimum, i.e., E = 0. Theorem 1 Consider the model following problem of system (1) with Assumptions 1 and 2. If the sliding surface (9a), (9b), the control law (5) with (18) and the weight update (20) with (24) and bias update (21) with (25) are used, and there exist matrices K and S such that the eigenvalues of the matrix A + B(SB)−1 S + BK are within the unit circle in the Z-plane, then the zero-tracking error is guaranteed, i.e., e(k) → 0 as k → ∞. Proof From (18)–(29), it can be seen that the auxiliary system (8) in the quasi-sliding mode can be achieved

Step 1: Find the solutions G and H of algebraic matrix equation (3). If no solution exists, then a different choice of the reference model or the output matrix C must be made. Step 2: Choose the sliding surface matrix S in (9a), (9b) such that SB is nonsingular. Step 3: Use the pole placement method to calculate the gain matrix K such that the matrix A + B(SB)−1 S + BK is stable. Step 4: Choose the design parameter φ in the reaching law (16). Step 5: Select a proper learning constant η and update the weight and bias by using (20), (21), (24), and (25). Step 6: Calculate the tracking controller u(k) from (5), (18), (20), and (21). 4 Illustrative example Example 1 Consider an uncertain time-delay system described by the following equations:

    0 1 0 0 x(t) ˙ = + x(t) 1 −2 0.5 cos(t) 0.5 cos(t)   0 0 + x(t − 0.1) 0.5 0.5

    0 0 + + u(t) (30) 1 0.5 sin(t)     0 0 + u(t − 0.2) + 0.1 sin(t) y(t) = [ 1

0 ]x(t)

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The reference model is given by   0 1 x (t) x˙m (t) = −1 0 m ym (t) = [ 1

(31)

0.5 ]xm (t)

The initial condition for (30) and (31) are given as follows: x(0) = [ 8

5 ]T ,

xm (0) = [ 1

1 ]T

(32)

To illustrate the utilization of the proposed approach, the Matlab program function c2d with Ts = 0.01 second is used in (30) and (31), and the discretetime system and reference model can be obtained in (33) and (34), respectively,     1 0.01 0 x(k + 1) = x(k) + u(k) 0.01 0.9802 0.01   0 + d(k) (33) 0.01 y(k) = [ 1

0 ]x(k)

where the generalized disturbance d(k) is expressed as d(k) = [0.5 0.5]x(k − 10) + 0.1u(k − 20) + [0.5 cos(k) 0.5 cos(k)]x(k) + 0.5 sin(k)u(k) + sin(k).   1 0.01 xm (k + 1) = xm (k) −0.01 1 (34) ym (k) = [ 1

0.5 ]xm (k)

Following the design procedures in the above section, the proposed robust tracking controller is given by the following steps. Step 1: From (3), the solution of the matrices G and H can be obtained respectively as follows:   1 0.5 G= , H = [ −3 1 ] −0.5 1 Steps 2–3: The constant matrices S and K in the sliding function (9a), (9b) are designed respectively as S = [1

1 ],

K = [ −104.5

−102.5 ]

such that SB is nonsingular and eigenvalues of A + B(SB)−1 S + BK are located at [0.99 0.97].

Step 4: The design parameter φ in the reaching law (16) is chosen as 0.7. Step 5: Select a proper learning constant as η = 80 and update the weights and bias by using (20), (21), (24), and (25) with w1 (0) = 1, w2 (0) = 1 and b(0) = 0. Step 6: Calculate the tracking controller u(k) from (5), (18), (20), and (21). With the designed parameter setting and initial conditions (32), the results of simulation are shown in Figs. 2 and 3. Figure 2(a) shows the tracking trajectory of the system. Figure 2(b) shows the error between the desired and actual outputs. Those results not only confirm the stability of the closedloop system, but also clearly show that the proposed technique achieves zero-tracking error in the presence of time-delays, parameter uncertainties, and external disturbances. Figure 2(c) shows the sliding surface variable. It clearly shows that the trajectory of the auxiliary system (8) converges to the quasi-sliding mode. Figure 2(d) shows the control input with reasonable magnitudes. Since the proposed method needs not a switching type of control law, it can be seen that no chattering phenomenon would occur. Figure 3(a)–3(c) indicate the time evolution of the weights w1 , w2 , and the time evolution of bias b, respectively. From the above results, it clearly shows that the proposed technique has excellent tracking performance and robustness properties in the presence of time-delays, parameter uncertainties, and external disturbances simultaneously. Next, we will make a comparison between the proposed method and the method developed in [8], which was developed in the continuous time domain. For this purpose, a composite nonlinear feedback controller designed by [8] is applied to systems (30)– (32) without time-delays. Following the design procedure in [8], the gain matrix K can be obtained by choosing η =  1.6,  γ = 1, and the positive-definite 0 matrix Q = 10 . The eigenvalues of A + BK are 0 1 located at −2.3539 ± 0.6049i. The trajectory of the tracking error obtained by CNF-Form 4 controller is shown in Fig. 4. Next, the proposed approach is applied to the same system. The design procedure is analogous to that in the system with state and input delays. The gain matrix K is designed such that the eigenvalues of A + B(SB)−1 S + BK in the discrete time domain are equivalent to those of

Robust tracking and model following based on discrete-time neuro-sliding mode control for uncertain

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Fig. 2 (a) System output y(k) and model output ym (k), (b)tracking error e(k), (c) sliding surface variable σ (k), (d) control input u(k)

Fig. 3 Time evolution of (a) weight w1 , (b) weight w2 , (c) bias b

A + BK in the continuous time domain. The trajectory of the tracking error obtained by the proposed controller is shown in Fig. 4. From Fig. 4, it is obvious that the proposed method provides better tracking performance and faster response over the composite nonlinear feedback controller of [8]. Furthermore, the composite nonlinear feedback technique of [8] was developed in continuous time domain and cannot deal with time-delays. Therefore, the proposed method outperforms the method developed in [8]. Example 2 Consider an uncertain time-delay system described by the following equations:

⎛⎡

⎤ 0 1 0 x(t) ˙ = ⎝⎣ 0 1 2⎦ −1 −2 0 ⎡ 0 0 + ⎣ 0.015 sin(t) 0 0.15 sin(t) 0 ⎡

⎤⎞ 0 0.02 sin(t) ⎦⎠ x(t) 0.2 sin(t)

⎤ 0 0 0 + ⎣ 0 0.01 0.015 ⎦ x(t − 0.1) 0 0.1 0.15 ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 + ⎝⎣ 0.1 ⎦ + ⎣ 0.03 sin(2t) ⎦⎠ u(t) 1 0.3 sin(2t)

(35)

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Fig. 4 The trajectories of the tracking errors from Mobayen and Majds’ method [8] and the proposed method

⎡

⎤ ⎡ ⎤ 0 0 + ⎣ 0.01 ⎦ u(t − 0.2) + ⎣ 0.01 sin(3t) ⎦ 0.1 0.1 sin(3t) y(t) = [ 1

(38) and (39), respectively x(k + 1) ⎡

0 ]x(t)

0

Note that the nominal system in (35) is unstable since the eigenvalues of the nominal system given in (35) are λ1,2,3 = −0.4329, 0.7164 ± 2.0266i. The reference model is given by  x˙m (t) =



0 1 x (t) −1 0 m

ym (t) = [ 1

(36)

0.5 ]xm (t)

The initial condition for (35) and (36) are given as follows:

⎤ 1 0.01 0.0001 = ⎣ −0.0001 1.0098 0.0201 ⎦ x(k) −0.01 −0.0201 0.9998 ⎡ ⎤ ⎡ ⎤ 0 0 + ⎣ 0.0011 ⎦ u(k) + ⎣ 0.0011 ⎦ d(k) 0.01 0.01   y(k) = 1 0 0 x(k)

(38)

where the generalized disturbance d(k) is expressed as d(k) = [0 0.1 0.15]x(k − 10) + 0.1u(k − 20) + [0.15 sin(k) 0 0.2 sin(k)]x(k) + 0.3 sin(2k)u(k) + 0.2 sin(3k).   1 0.01 xm (k + 1) = xm (k) −0.01 1 (39) ym (k) = [ 1

0.5 ]xm (k)

(37)

Following the design procedures in the above section, the proposed robust tracking controller is given by the following steps.

To illustrate the utilization of the proposed approach, the Matlab program function c2d with Ts = 0.01 second is used in (35) and (36), and the discretetime system and reference model can be obtained in

Step 1: From (3), the solution of the matrices G and H can be obtained respectively as follows: ⎡ ⎤ 1 0 ⎦, G=⎣ 0 1 −0.5786 −0.5711

x(0) = [ 0

0

0 ]T ,

xm (0) = [ 1

0 ]T

Robust tracking and model following based on discrete-time neuro-sliding mode control for uncertain

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Fig. 5 (a) System output y(k) and model output ym (k), (b) tracking error e(k), (c) sliding surface variable σ (k), (d) control input u(k)

Fig. 6 Time evolution of (a) weight w1 , (b) weight w2 (c) weight w3 , (d) bias b

H = [ 1.5711 1.4214 ]. Steps 2–3: The constant matrices S and K in the sliding function (9a), (9b) are designed respectively as S = [1

1

1 ],

K = [ −95.5537 −98.6184 −96.5259 ] such that SB is nonsingular and eigenvalues of A + B(SB)−1 S + BK are located at [ 0.9676 0.9841 + 0.0114i

0.9841 − 0.0114i ].

Step 4: The design parameter φ in the reaching law (16) is chosen as 0.2. Step 5: Select a proper learning constant as η = 200 and update the weights and bias by using (20), (21), (24), and (25) with w1 (0) = 10, w2 (0) = 10, w3 (0) = 10 , and b(0) = 10. Step 6: Calculate the tracking controller u(k) from (5), (18), (20), and (21). With the designed parameter setting and initial conditions (37), the results of simulation are shown in Figs. 5 and 6. Figure 5(a) shows the tracking trajectory of the system. Figure 5(b) shows the error between the desired and actual outputs. Those results not only

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Fig. 7 The trajectories of the tracking errors from Mobayen and Majds’ method [8] and the proposed method

confirm the stability of the closed-loop system, but also clearly show that the proposed technique achieves zero-tracking error in the presence of time-delays, parameter uncertainties and external disturbances. Figure 5(c) shows the sliding surface variable. It clearly shows that the trajectory of the auxiliary system (8) converges to the quasi-sliding mode. Figure 5(d) shows the control input with reasonable magnitudes. Since the proposed method needs not a switching type of control law, it can be seen that no chattering phenomenon would occur. Figures 6(a)–6(d) indicate the time evolution of the weights w1 , w2 , w3 , and the time evolution of bias b, respectively. From the above results, it clearly shows that the proposed technique has excellent tracking performance and robustness properties in the presence of time-delays, parameter uncertainties, and external disturbances simultaneously. Next, we will make a comparison between the proposed method and the method developed in [8], which was developed in the continuous time domain. For this purpose, a composite nonlinear feedback controller designed by [8] is applied to systems (35)–(37) without time-delays. Following the design procedure in [8], the gain matrix K can be obtained by choosing η = 1.5, γ = 1, and the positive-definite matrix Q = diag{20, 6, 1}. The eigenvalues of A + BK are located at [−3.2898 − 1.5924 ± 1.1575i]. The trajectory of the tracking error obtained by CNF-Form 4 controller is shown in Fig. 7. Next, the proposed approach is applied to the same system. The design procedure is analogous to that in the system with state and

input delays. The gain matrix K is designed such that the eigenvalues of A + B(SB)−1 S + BK in the discrete time domain are equivalent to those of A + BK in the continuous time domain. The trajectory of the tracking error obtained by the proposed controller is shown in Fig. 7. From Fig. 7, it is obvious that the proposed method provides better tracking performance and faster response over the composite nonlinear feedback controller of [8]. Furthermore, the composite nonlinear feedback technique of [8] was developed in continuous time domain and can not deal with timedelays. Therefore, the proposed method outperforms the method developed in [8].

5 Conclusions In this paper, the robust tracking and model following problem for uncertain time-delay systems via a discrete-time NSMC scheme has been considered. It has been shown that the proposed control scheme provides good tracking performance and robustness even in the presence of time-delays, parameter uncertainties and external disturbances. It has been shown that the stability of the closed-loop system is guaranteed and the zero-tracking error is achieved. Moreover, the knowledge of upper bound of uncertainties is not required and chattering phenomenon is eliminated. Simulation studies verified the theoretical analysis and demonstrated the effectiveness of the proposed control methodology.

Robust tracking and model following based on discrete-time neuro-sliding mode control for uncertain Acknowledgements The author would like to thank the National Science Council of the Republic of China for financial support through Grant No. NSC 100-2221-E-252-006.

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