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Adaptive NN Tracking Control of Uncertain Nonlinear Discrete-Time Systems With Nonaffine Dead-Zone Input Yan-Jun Liu and Shaocheng Tong
Abstract—In the paper, an adaptive tracking control design is studied for a class of nonlinear discrete-time systems with dead-zone input. The considered systems are of the nonaffine pure-feedback form and the dead-zone input appears nonlinearly in the systems. The contributions of the paper are that: 1) it is for the first time to investigate the control problem for this class of discrete-time systems with dead-zone; 2) there are major difficulties for stabilizing such systems and in order to overcome the difficulties, the systems are transformed into an n-step-ahead predictor but nonaffine function is still existent; and 3) an adaptive compensative term is constructed to compensate for the parameters of the dead-zone. The neural networks are used to approximate the unknown functions in the transformed systems. Based on the Lyapunov theory, it is proven that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded and the tracking error converges to a small neighborhood of zero. Two simulation examples are provided to verify the effectiveness of the control approach in the paper. Index Terms—Adaptive NN control, dead-zone input, nonlinear control theory, nonlinear discrete-time systems
I. I NTRODUCTION HE input nonlinearities are commonly encountered in various systems, such as electric servomotors and mechanical actuators etc. Such nonlinearity is poorly unknown and often limits system performance. In the control fields, its study has been drawing much interest. For example, the adaptive controller was proposed in [1] for multiinput and multioutput (MIMO) nonlinear systems with nonsymmetric saturation and known control coefficient matrix. An auxiliary system was constructed to avoid the effect of saturation input. The dead-zone is one of the input nonlinearities. The control problems of the systems with dead-zone have received a great deal of attention. A pioneer adaptive design was studied in [2] for the plants with dead-zone. Subsequently, a number
T
Manuscript received March 24, 2014; revised May 6, 2014 and May 25, 2014; accepted June 2, 2014. The work was supported in part by the National Natural Science Foundation of China under Grant 61104017 and Grant 61374113 and in part by the Program for Liaoning Innovative Research Team in University under Grant LT2012013. This paper was recommended by Associate Editor M. J. Er. The authors are with the College of Science, Liaoning University of Technology, Jinzhou, Liaoning 121001, China (e-mail: [email protected]; [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/TCYB.2014.2329495
of important results were reported for the systems with deadzone. An adaptive tracking of nonlinear SISO systems with nonsymmetric dead-zone was discussed in [3]. A decentralized adaptive control was proposed to solve the tracking problem of interconnected MIMO systems with time-delay and the input of each loop preceded by an unknown dead-zone [4]. However, the linearly parameterized condition is required in these results. Recently, the adaptive controller designs for the systems with unknown functions had been studied by using the neural network (NN) and the fuzzy systems. Many works were made in [5]–[10] for uncertain nonlinear SISO systems with unknown functions which are approximated by using the fuzzy systems and the NN. The approaches in [11]–[14] are to reduce the adjustable parameters. In [15], the adaptive NNbased dynamic surface control was handled for uncertainty permanent magnet synchronous motors with the disturbance in load torque. Subsequently, the adaptive control approaches using the approximators were studied in [16]–[18] for nonlinear stochastic SISO systems. In order to stabilize nonlinear MIMO systems, some significant adaptive intelligent control algorithms were designed in [19]–[23]. The adaptive NN controllers were developed in [24] and [25] for wheeled inverted pendulum models with unknown dynamics. The above elegant schemes were developed for nonlinear continuous-time systems and they cannot be applied to those nonlinear discretetime systems. Several important full state and output feedback adaptive neural algorithms were presented in [26]–[36] for nonlinear discrete-time systems. The optimal stabilization control for discrete-time and continuous-time systems was proposed in [37] based on adaptive dynamic programming. The representative works using the FLSs and the NNs were studied in [38] and [39] for some real systems. In these papers, some promising techniques were employed to improve the performance of mobile manipulators and nonlinear teleoperators, which were important in the practical applications. The authors in the above results do not consider the deadzone input. A dead-zone fuzzy compensator was developed for industrial positioning systems by using the fuzzy systems [40]. In [41], an adaptive controller was proposed for nonlinear systems with a nonlinear dead-zone and multiple time-delays by using the high-order NN. The adaptive output feedback tracking problems were addressed for nonlinear systems with unknown dead-zones based on the fuzzy or neural control [42]–[44] and adaptive control methods were
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proposed for nonlinear systems with other types of input nonlinearities [45]–[47]. However, these elegant schemes were developed for nonlinear continuous-time systems and they cannot be applied to those nonlinear discrete-time systems with dead-zone. In [48], a new controller structure using an adaptive deadzone inverse was proposed for discrete-time plants with unknown dead-zones. A fuzzy logic compensator was presented for the control of nonlinear MIMO dynamics systems in the discrete-time form with dead-zone [49]. Zhang et al. [50] studied an adaptive NN critic control using reinforcement learning for nonlinear discrete-time systems with nonsymmetric dead-zone input. The backstepping control approach for nonlinear discrete-time triangular structure systems in the presence of input nonlinearities, such as saturation and deadzone were studied in [51]. Adaptive NN tracking design was got in [52] for discrete-time systems with dead-zone and unknown gain function. In [53], a predictive control was yielded to control continuous stirred tank reactor (CSTR) system in discrete-time with dead-zone. Unfortunately, the systems considered in these results are required to satisfy affine structure, i.e., the dead-zone input appears linearly in nonlinear discrete-time function. Therefore, this assumption definitely limits the potential applications of those results in practice. This paper will try to study an adaptive NN control approach for a class of uncertain nonlinear discrete-time systems with dead-zone input. The unknown functions are approximated by using the NN. The Lyapunov functions in the difference form are used for analyzing the stability of systems. Two simulation examples are given to verify effectiveness of the approach. The main contributions of the paper are summarized as follows. 1) The dead-zone input appears nonlinearly in the unknown functions of the systems and the variables which can be viewed as virtual controllers appear also nonlinearly in the systems. Thus, the nonlinear pure-feedback discrete-time systems have more general form than those in the previous works. To the best of our knowledge, to date, few effective techniques are provided to control such a class of systems with dead-zone input. 2) In contrast to a large amount of works on continuoustime, there are little results are useful for controlling discrete-time systems. The controller design and the concept definition have essential differences between the discrete-time and the continuous-time. Specifically, because this paper considers the systems including the discrete-time dead-zone input, the control design becomes more difficult and complex than the continuous-time forms. 3) In order to overcome the obstacles in the control design, the systems are transformed into an n-step-ahead predictor. But the transformed systems are still of the nonaffine structure. The unknown functions in the predictor description can be approximated by using the NN. The discrete-time dead-zone input can be compensated by designing an adaptive compensatory term.
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II. P ROBLEM AND F ORMULATION A. System Description Consider the discrete-time system, which is described as [29] ⎧ ⎨ xi (k + 1) = ϕi (¯xi (k) , xi+1 (k)) , i = 1, . . . , n − 1, xn (k + 1) = ϕn (¯xn (k) , D (v (k)) , d (k)) (1) ⎩ y (k) = x1 (k) where x¯ i (k) = [x1 (k) , . . . , xi (k)]T , i = 1, . . . , n are the states of the systems; ϕi (·, ·) , i = 1, . . . , n − 1, and ϕn (·, ·, ·) are the unknown functions to be continuous with respect to all the arguments and continuously differentiable with respect to the second argument; y (k) ∈ R is the system output; d (k) represents the external disturbance with the upper bound dM , i.e., |d (k)| ≤ dM ; u (k) = D (v (k)) stands for the system input and the output of the dead-zone where v (k) is the input of dead-zone. The dead-zone u (k) can be defined as [48] ⎧ ⎨ mr (v (k) − br ) , v (k) ≥ br u (k) = D (v (k)) = 0, −bl < v (k) < br (2) ⎩ ml (v (k) + bl ) , v (k) ≤ −bl where mr and ml are the right and left slopes of the dead-zone, respectively; br and bl stand for the breakpoints; mr and ml , br and bl are the positive constants. The dead-zone can be rewritten as u (k) = m (k) v (k) − b (k) where
m (k) =
and
(3)
mr , v (k) > 0 ml , v (k) ≤ 0
⎧ ⎨ mr br , v (k) ≥ br b (k) = m (k) v (k) , −bl < v (k) < br ⎩ −ml bl , v (k) ≤ −bl .
According to the result in [3], it is easy to know ¯ = max (mr , ml ) min (mr , ml ) = m ≤ |m (k)| ≤ m |b (k)| ≤ b¯ = max (mr br , ml bl ) . Remark 1: In the previous results of the discrete-time systems, the dead-zone input D (v (k)) appears linearly in the systems such as [48]–[53]. When the dead-zone input does not satisfy this restriction, the above results are very difficult to be applied. In this paper, D (v (k)) is nonlinearly described in the systems. Thus, the proposed approach can be used to control a general class of nonlinear discrete-time systems with the dead-zone input. The control objective is to design a neural controller such that: 1) all the signals in the closed-loop systems are bounded and 2) the output y (k) can track a reference trajectory yd (k) to a bounded compact set where yd (k) is bounded and known smooth function. To achieve this control objective, we make the following assumptions. Assumption 1 [29]: The functions ϕi (·, 0), i = 1, . . . , n − 1, and ϕn (·, 0, ·) are Lipschitz functions.
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Assumption 2 [28], [29]: There are the constants h¯ i > hi > 0 such that h¯ i ≤ |hi (.)| ≤ hi where ∂ϕi (¯xi (k) , xi+1 (k)) ∂xi+1 (k) ∂ϕn (¯xn (k) , u (k) , d (k)) . hn (¯xn (k) , u (k) , d (k)) = ∂u (k) hi (¯xi (k) , xi+1 (k)) =
Assumption 2 implies that hi (.) , i = 1, . . . , n are strictly either positive or negative. Without losing generality, it is assumed that hi ≤ hi (·) ≤ h¯ i . Let h = n n ¯ ¯ i=1 hi and h = i=1 hi . We introduce the following lemma. Lemma 1 [29]: Let V (k) = ni=1 Vi (k) with Vi (k) ≥ 0, ∀k ∈ Z0+ . If the following inequality holds: V (k + 1) ≤
n
pi (k1 ) Vi (k1 ) + q (k1 )
(4)
i=1
where k1 = k − n + 1, k ≥ n − 1, n ≥ 1, |pi (k1 )| ≤ p¯ < 1, and |q (k1 )| ≤ q¯ . Then, it has V (k) ≤ V¯ (0) + q¯ / (1 − p¯ ) and
C. System Transformation Based on the result in [29], the prediction system can be obtained as ⎧
x1 (k + n) = τ1,1 n−1,1 (¯xn (k)) , x2 (k + n − 1) ⎪ ⎪
⎪ ⎪ ⎪ ⎨ x2 (k + n − 1) = τ1,2 n−2,2 (¯xn (k)) , x3 (k + n − 2) .. (5) . ⎪ ⎪ ⎪ + 1) = τ x , u , d x (k (¯ (k) (k) (k)) ⎪ 1,n n ⎪ ⎩ n y (k + n) = x1 (k + n) where τ1,i (¯xi+1 (k)) = ϕi (¯xi (k) , xi+1 (k)) τ1,n (¯xn (k) , u (k) , d (k)) = ϕn (¯xn (k) , u (k) , d (k)) n−i,i (¯xn (k)) = x¯ i (k + n − i) , i = 1, . . . , n − 1. Replacing x2 (k + n − 1) in the first equation of (5) with the right-hand side of the second equation, it has
x1 (k + n) = τ1,1 n−1,1 (¯xn (k)) , τ1,2 n−2,2 (¯xn (k)) , x3 (k + n − 2) . Continuing to iteratively replace xj (k + n − j + 1) in the above equation with the right-hand side of the jth equation in (5), j = 3, . . . , n − 1, until u (k) appears at the last step, we can obtain y (k + n) = x1 (k + n) = τ (¯xn (k) , u (k) , d (k))
lim sup {V (k)} ≤ q¯ / (1 − p¯ )
k→∞
3
(6)
where
where V¯ (0) = max {V (j)}. 0≤j≤n−1
Proof: The proof is placed in Appendix A.
τ (.) = τ1,1 n−1,1 (¯xn (k)) , τ1,2 n−1,1 (¯xn (k))
τ1,3 . . . , τ1,n (¯xn (k) , u (k) , d (k)) . . . .
Equation (6) can be defined as B. Radial Basis Function Neural Networks It has been shown that for any given continuous function g (X) : Rn → R on a compact set U ∈ Rn and an arbitrary ε > 0, there exist the RBFNN σ T (X) θ so that sup g (X) − σ T (X) θ < ε X∈U
where θ ∈ Rl is the adjustable weight vector, l > 1 is the NN node number, X ∈ Rn is the input vector, and σ (X) = [σ1 (X) , · · · , σl (X)]T is Gaussian basis function vector. Gaussian basis function is expressed as
X − πi 2 σi (X) = exp − , i = 1, · · · , l μ2i where πi = [πi1 , · · · , πin ]T and μi are the center and the width of Gaussian function, respectively. Based on the approximation property of the RBFNN, the continuous time function g (X) can be represented as ∗
∗
g (X) = σ (X) θ + ε (X) T
where θ ∗ denotes the optimal weight vector and ε∗ (X) is the minimal approximation error. In general, ε∗ (X) is assumed to bounded by a constant ε¯ , this is, |ε∗ (X)| ≤ ε¯ .
y (k + n) = τs (¯xn (k) , u (k)) + ds (k)
(7)
where ds (k) = τ (¯xn (k) , u (k) , d (k))−τ (¯xn (k) , u (k) , 0) and τs (¯xn (k) , u (k)) = τ (¯xn (k) , u (k) , 0). From definition of τ (.), we know that τ (.) is still satisfying Assumption 1, i.e., τ (.) satisfies Lipschitz condition. Then, there must be a constant Ld such that |τ (¯xn (k) , u (k) , d (k)) − τ (¯xn (k) , u (k) , 0)| ≤ Ld |d (k)| . Accordingly, we have |ds (k)| ≤ Ld |d (k)| ≤ Ld dM . Assumption 3 [29]: The functions ϕi (·, 0), i = 1, . . . , n − 1 and ϕn (·, 0, ·) are Lipschitz functions. According to Assumption 2, it can be obtained that ∂τ1,i (¯xi+1 (k)) /∂xi+1 (k) = hi (·) > 0.
(8)
Based on (5) and (8), it can be easily to get ¯ 0 < h < ∂τs (¯xn (k) , u (k)) /∂u (k) < h.
(9)
III. C ONTROLLER D ESIGN AND S TABILITY A NALYSIS The controller will be designed for (7). Define the tracking error z (k) = y (k) − yd (k). Then, we get z (k + n) = y (k + n) − yd (k + n) = τs (¯xn (k) , D (v (k))) + ds (k) − yd (k + n) . (10)
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From (10), it can be obtained that ∂ τs (¯xn (k) , D (v (k))) − yd (k + n) = h (·) > 0. ∂D (v (k)) Then, there exists a ideal signal u∗ (k) = D (v∗ (k)) so that
τs x¯ n (k) , D v∗ (k) − yd (k + n) = 0. (11) The ideal signal D (v∗ (k)) can be approximated by the NN as
(12) D v∗ (k) = σ T (X (k)) θ ∗ + ε (X (k)) T T where X (k) = x¯ n (k) , yd (k + n) ∈ X ∈ Rn+1 is the input vector of the NN, σ (X (k)) ∈ Rl is the basis function vector, ε (X (k)) is the approximation error, and θ ∗ ∈ Rl is the ideal NN weight. Let θˆ (k) estimate θ ∗ and θ˜ (k) = θˆ (k) − θ ∗ is the estimation error. Adding and subtracting τs (¯xn (k) , D (v∗ (k))) on the right side of (10), and using (11), it gets z (k + n) = τs (¯xn (k) , D (v (k)))
−τs x¯ n (k) , D v∗ (k) + ds (k) . By using the Mean Value Theorem, it has
z (k + n) = h x¯ n (k) , D vc (k)
× D (v (k)) − D v∗ (k) + ds (k) where
(13)
D vc (k) ∈ min D v∗ (k) , D (v (k)) ∗ max D v (k) , D (v (k)) .
Remark 2: It can be seen from (13) that D (vc (k)) is nonlinearly included in h (¯xn (k) , D (vc (k))). Thus, h (¯xn (k) , D (vc (k))) is still nonaffine function in essence. In view of this property, it is difficult to design a stable adaptive neural controller. The main difficulties are that h (¯xn (k) , D (vc (k))) cannot appear in packaged unknown functions which are approximated by the NN. Note that h (¯xn (k) , D (vc (k))) satisfies still Assumption 2. For convenience, h (k) is used to denote h (¯xn (k) , D (vc (k))) in the following. Using (3) and (12), (13) can be rewritten as z (k + n) = hm (k) v (k) − h (k) b (k) + ds (k) −hm (k) σ T (X (k)) θ ∗ + ε (X (k))
(14)
¯ where hm (k) = h (k) m (k). Let hm = mh and h¯ m = m ¯ h. The controller v (k) is defined as v (k) = β (k) z (k) + ηˆ (k) + v¯ (k)
Define the adaptation laws as follows: θˆ (k + 1) = θˆ (k1 ) − γθ θˆ (k1 ) + αθ σ (X (k1 )) z (k + 1)] ηˆ (k + 1) = ηˆ (k1 ) − αη z (k + 1) + γη ηˆ (k1 )
where αθ , αη , γθ , γη > 0 are design parameters to be specified later on. Remark 3: In the previous results without considering the dead-zone, only adaptation law for the optimal neural weight is designed. Because the unknown dead-zone appears in the input of the systems, the adaptation law for ηˆ (k + 1) in (18) is designed to compensate for unknown parameter of dead-zone. It will result in a complicated design procedure owing to the existence of the dead-zone. For nonlinear discrete-time systems with dead-zone, some approaches are made in [48]–[53]. These approaches are obtained by using the dead-zone inversion or the dead-zone to be viewed as a disturbance. The ways are not good to cancel the effect of dead-zone. In this paper, an adaptive compensative algorithm is effectively to overcome of the effect of dead-zone. The following theorem is explained to show the stability of the closed-loop system. Theorem 1: Consider the systems, which are described by (1). On the compact sets X and under Assumptions 1 and 2, by constructing the controller (15), the adaptation laws (17) and (18), the overall closed-loop system is stability in the sense that all the signals are semi-globally uniformly ultimately bounded (SGUUB) and the system output tracks the reference signal to a small neighborhood of zero by choosing the design parameters appropriately. Proof: Please see the Appendix. IV. S IMULATION E XAMPLE In this section, the examples are provided to demonstrate the feasibility of the proposed method. Example 1: Consider nonlinear systems ⎧ ⎨ x1 (k+1) = ϕ1 (x1 (k) , x2 (k)) x2 (k+1) = ϕ2 (¯x2 (k) , D (v (k)) , d (k)) (19) ⎩ y (k) = x1 (k) where ϕ1 (.) = 1.4
(15)
where |β (k)| ≤ β¯ < 1 is a scaling factor and v¯ (k) = ¯ m, σ T (X (k)) θˆ (k) and ηˆ (k) is the estimation of η = b/ ¯ and let η˜ (k) = ηˆ (k) − η. Using (15), (14) can be rewritten as z (k + 1) = H (k1 ) + hm (k1 ) η˜ (k1 ) + σ T (X (k1 )) θ˜ (k1 ) + β (k1 ) z (k1 ) (16) where k1 = k − n + 1 and H (k1 ) = hm (k1 ) (η − ε (X (k1 ))) + ds (k1 ) − h (k) b (k). According to Assumption 2 and the boundedness of approximation error, it has
¯ |H (k1 )| ≤ h¯ 2b¯ + m¯ ¯ ε + 1 + Ld dM = H.
(17) (18)
x12 (k)
+ 0.1x22 (k) + 0.5x2 (k) 1 + x12 (k) x1 (k) + 0.5 sin (D (v (k))) ϕ2 (.) = 2 1 + x1 (k) + x22 (k) + ⎧ D (v (k)) + 0.5 + 0.1 cos (0.05k) cos (x1 (k)) ⎨ 0.5 (v (k) − 0.3) , if v (k) ≥ 0.3 D (v (k)) = 0, if − 0.4 < v (k) < 0.3 (20) ⎩ 0.3 (v (k) + 0.4) , if v (k) ≤ −0.4. The initial values for the states are chosen as x1 (0) = −0.2 and x2 (0) = 0.1. The control objective is to design an adaptive controller for the system (1) such that: 1) the output y (k) tracks the reference signal yd (k) = 0.5 sin ((π/5) kT) + 0.5 cos ((π/10) kT) + 0.9 with T = 0.05 as closely as possible and 2) the boundedness of all the signals in the closed-loop system is guaranteed.
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Fig. 1.
y(k) (solid line) and yd (k) (dashed line) in Example 1.
Fig. 2.
(a) x2 (k) and (b) v (k) in Example 1.
Fig. 3.
(a) θˆ (k) and (b) ηˆ (k) in Example 1.
5
From the above design, the controller is defined as v (k) = β (k) z (k) + ηˆ (k) + v¯ (k) σT
where v¯ (k) = given as follows:
(21)
(X (k)) θˆ (k) and the adaptation laws are
θˆ (k + 1) = θˆ (k − 1) − γθ θˆ (k − 1) + αθ σ (X (k − 1)) z (k + 1)] (22) ηˆ (k + 1) = ηˆ (k − 1) − αη z (k + 1) + γη ηˆ (k − 1) (23)
with z (k) = y (k) − yd (k). The initial value of the adaptation laws are shown as θˆ (0) = 0 and ηˆ (0) = 0.01. The design parameters are selected as αθ = 0.1I, αη = 0.1, γθ = 0.01, γη = 0.01, and β (k) = 0.1. In the simulation studies, the centers and widths are selected on a regular lattice in the respective compact sets. The NN contain 25 nodes with the centers πl evenly spaced in [−1.5, 1.5] × [−2, 2] × [−2.5, 2.5], and widths μi = 1 where the input variable of the NN is given as X (k) = T x1 (k) , x2 (k) , yd (k + 2) . Figs. 1–3 show the simulation results which are obtained by applying the control controller v (k) to the system (19). Fig. 1 illustrates the system tracking trajectories. It can be seen that a good tracking performance is obtained. The system state x2 (k) and the control signal v (k) are described in Fig. 2. The trajectories of the adaptation laws θˆ (k) and ηˆ (k) are shown in Fig. 3. From Figs. 2 and 3, we can see that the variables x2 (k), v (k), θˆ (k), and ηˆ (k) are bounded. Example 2: This simulation will be to consider CSTR in discrete-time [29], [53] ⎧ x1 (k + 1) = x1 (k) + [−x1 (k) + ⎪ ⎪ Ca (1 − x1 (k)) ⎪ γ x2 (k)/(γ +x2 (k)) × 0.05 ⎪ × e ⎪ ⎪ ⎨ x2 (k + 1) = x2 (k) + [−x2 (k) (24) + BCa (1 − x1 (k)) eγ x2 (k)/(γ +x2 (k)) ⎪ ⎪ ⎪ ⎪ − Cr (x2 (k) − D (v (k)))] × 0.05 + d (k) ⎪ ⎪ ⎩ y (k) = x1 (k) where x1 (k), x2 (k), and y (k) are the states and the output of the systems, respectively; d (k) = 0.01 cos (0.05k) cos (x1 (k));
D (v (k)) is given as (20). The reference signal yd (k) is described by yd (k) = 0.1sin π5k T + 0.3 with T = 0.1. The NN are chosen to be the same as Example 1. The initial value of the adaptation laws are shown as θˆ (0) = 0 and ηˆ (0) = 0.1. The design parameters are selected as αθ = 0.2I, αη = 0.3, γθ = 0.02, γη = 0.02, and β (k) = 0.1. The initial condition for the states is chosen as x1 (0) = 0.1, x2 (0) = −0.5. The simulation Figs. 4–6 are obtained by taking the approach to (24). The tracking performance, trajecto the ries of x2 (k),v (k), and the boundedness of θˆ (k), ηˆ (k) are illustrated in Figs. 4–6, respectively. From these figures, we conclude that a good tracking performance is got and all the signals in the closed-loop system are bounded. V. C ONCLUSION This paper proposed an adaptive controller for a class of nonlinear discrete-time systems by using the NN. The
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Fig. 4.
y(k) (solid line) and yd (k) (dashed line) in Example 2.
Fig. 6.
(a) θˆ (k) and (b) ηˆ (k) in Example 2.
Based on (4), we have V (sn + n + j) ≤ =
m i=1 m
p1 (sn + j) Vi (sn + j) + q (sn + j) j
j
pi (s) Vi (s) + qj (s)
(26)
i=1 j
where pi (s) = pi (sn + j) and qj (s) = q (sn + j). From (25) and (26), we obtain V j (s + 1) ≤
m
j
j
pi (s) Vi (s) + qj (s) .
i=1
j Noting that pi (s) ≤ p¯ and qj (s) ≤ q¯ , based on [29, Lemma 4] to the above equation and it leads to Fig. 5.
_ q¯ q¯ ≤ V (0) + 1 − p¯ 1 − p¯ j q¯ . lim sup V (s) ≤ l→∞ 1 − p¯
(a) x2 (k) and (b) v (k) in Example 2.
nonlinear system under study contains the dead-zone input, unknown functions, and the external disturbance. Therefore, the nonlinear systems considered in this paper are more general than those in the previous literature. It has been proven that the all variables of the control system are uniformly bounded and the tracking error converges to a small neighborhood of zero. The effectiveness of the proposed controller has been confirmed by using two simulation examples. A PPENDIX A T HE P ROOF OF L EMMA 1 This proof has been given in [29]. For the readers’ convenience, we append the proof in this paper. j j Proof: Let V j (s) = m i=1 Vi (s) where Vi (s) = Vi (sn + j) and s ∈ Z0+ . One has V j (0) ≤ V¯ (0). Then, it obtains V j (s + 1) =
m i=1
Vi (s + 1) n + j = V (sn + n + j) . (25)
V j (s) ≤ V j (0) +
Then, ∀k ≥ n − 1, there exist j = k(modn), j ∈ {0, 1, . . . , n − 1}, and s = (k − j) /n so that V (k) =
m
Vi (sn + j) =
i=1
m
_
j
Vi (s) = V j (s) ≤ V (0) +
i=1
q¯ 1 − p¯
and lim sup {V (k)} ≤
k→∞
q¯ . 1 − p¯
A PPENDIX B T HE P ROOF OF T HEOREM 1 Consider the following Lyapunov function: V (k) = Vz (k) + Vθ (k) + Vη (k)
(27)
where Vz (k) = z2 (k) /hm , Vθ (k) = θ˜ T (k) θ˜ (k) /αθ , and Vη (k) = η˜ 2 (k) /αη .
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It can be derived from (17) and (18) that θ˜ (k + 1) = θ˜ (k1 ) − αθ σ (X (k1 )) z (k + 1) + γθ θˆ (k1 ) (28) η˜ (k + 1) = η˜ (k1 ) − αη z (k + 1) + γη ηˆ (k1 ) . (29) Using (28), we have
7
αθ σ (X (k1 )) 2 z2 (k + 1) ≤ αθ lz2 (k + 1) 2γθ σ T (X (k1 )) θˆ (k1 ) z (k + 1) ≤ αθ lz2 (k + 1) γ 2 θˆ 2 (k1 ) + θ αθ γη2 ηˆ 2 (k1 ) 2γη z (k + 1) ηˆ (k1 ) ≤ αη z2 (k + 1) + αη 2 ¯ (k1 ) βz ¯ m z2 (k + 1) − 2β (k1 ) z (k1 ) z (k + 1) ≤ + βh hm 2 2 2 2θ˜ T (k1 ) θˆ (k1 ) = θ˜ (k1 ) + θˆ (k1 ) − θ ∗
θ˜ T (k + 1) θ˜ (k + 1) αθ 2 ˜ θ (k1 ) = + 2γθ σ T (X (k1 )) θˆ (k1 ) z (k + 1) αθ 2η˜ (k1 ) ηˆ (k1 ) = η˜ 2 (k1 ) + ηˆ 2 (k1 ) − η2 . 2γθ θ˜ T (k1 ) θˆ (k1 ) 2 2 − + αθ σ (X (k1 )) z (k + 1) αθ γ 2 θˆ 2 (k1 ) By using Vz (k + 1) = z2 (k + 1) /hm and combining with the . (30) −2σ T (X (k1 )) θ˜ (k1 ) z (k + 1) + θ above facts, we obtain αθ 2 Using (29), we have ˜ θ (k1 ) ¯ 2 (k1 ) βz V (k + 1) = (1 − γθ ) + η˜ 2 (k + 1) αθ hm Vη (k + 1) = αη η˜ (k1 ) 2
+ 1 − γη +γθ (γθ − 1) η˜ 2 (k1 ) αη = − 2η˜ (k1 ) z (k + 1) 2 ˆ αη θ (k1 )
ηˆ 2 (k1 ) 2γη η˜ (k1 ) ηˆ (k1 ) × γ + γ − 1 2 η η − + αη z (k + 1) αθ αη αη
+ 3αθ l + 2αη + hm β¯ − 1/hm z2 (k + 1) + q¯ + 2γη z (k + 1) ηˆ (k1 ) + γη2 ηˆ 2 (k1 ) /αη . (31) (33) From (16), it has 2 2 ∗ ¯ γη η H γθ θ where q¯ = + + . − η˜ (k1 ) + σ T (X (k1 )) θ˜ (k1 ) z (k + 1) 2 αθ αη αθ lh¯ m If the design parameters satisfy that 0 < γθ < 1, 0 < γη < 1 , z2 (k + 1) H (k1 ) z (k + 1) and 3hm αθ l + 2hm αη + h2m β¯ < 1, we have =− + hm (k1 ) hm (k1 ) ¯ z (k) + (1 − γθ ) Vθ (k) V (k + 1) = βV + β (k1 ) z (k1 ) z (k + 1) . (32)
+ 1 − γη Vη (k) + q¯ . (34) By employing (30)-(32), we have Because β¯ < 1, 1 − γθ < 1, and 1 − γη < 1, (34) satisfies ¯ Lemma 1. (k) ≤ V (0) + q¯ / (1 − p¯ ) where Thus, we have V Vθ (k + 1) + Vη (k + 1) ¯ 1 − γθ , 1 − γη . Further, one has p¯ = max β, θ˜ T (k1 ) θ˜ (k1 ) 2γθ θ˜ T (k1 ) θˆ (k1 ) ⎧ ⎫ 2 − = ⎪ ⎪ ⎨ z2 (k) αθ αθ 2 θ˜ (k) η˜ (k) ⎬ q¯ . (35) lim sup + + ≤ 2z2 (k + 1) 2 2 ⎪ hm ⎪ 1 − p¯ k→∞ αθ αη ⎭ ⎩ + αθ σ (X (k1 )) z (k + 1) − hm (k1 ) γ 2 θˆ 2 (k1 ) The above inequality implies that the signals are z (k), θ˜ (k), + 2γθ σ T (X (k1 )) θˆ (k1 ) z (k + 1) + θ αθ and η˜ (k) bounded. Accordingly, it can be obtained that 2γη η˜ (k1 ) ηˆ (k1 ) θˆ (k) , ηˆ (k)are also bounded. According to (35), it has that − + 2β (k1 ) z (k1 ) z (k + 1) z (k) ≤ q¯ hm / (1 − p¯ ) = . It can be seen that depends αη on the design parameters. Then, if the design parameters are 2H (k1 ) z (k + 1) + 2γη z (k + 1) ηˆ (k1 ) + appropriately chosen, can converge to a small bounded set. hm (k1 ) η˜ 2 (k1 ) γη2 ηˆ 2 (k1 ) R EFERENCES + + + αη z2 (k + 1) . αη αη Vθ (k + 1) =
The following inequalities can be obtained: 2z2 (k + 1) 2z2 (k + 1) ≤− hm (k1 ) hm ¯ 2H (k1 ) z (k + 1) H ≤ αθ lz2 (k + 1) + hm (k1 ) αθ lh¯ 2m
−
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[26] S. S. Ge, J. Zhang, and T. H. Lee, “Adaptive neural network control for a class of MIMO nonlinear systems with disturbances in discretetime,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 4, pp. 1630–1645, Aug. 2004. [27] S. S. Ge, G. Y. Li, J. Zhang, and T. H. Lee, “Direct adaptive control for a class of MIMO nonlinear systems using neural networks,” IEEE Trans. Autom. Control, vol. 49, no. 11, pp. 2001–2006, Nov. 2004. [28] C. G. Yang et al., “Output feedback NN control for two classes of discrete-time systems with unknown control directions in a unified approach,” IEEE Trans. Neural Netw., vol. 19, no. 11, pp. 1873–1886, Aug. 2008. [29] S. S. Ge, C. G. Yang, and T. H. Lee, “Adaptive predictive control using neural network for a class of pure-feedback systems in discrete time,” IEEE Trans. Neural Netw., vol. 19, no. 9, pp. 1599–1614, Sep. 2008. [30] H. G. Zhang, Y. H. Luo, and D. R. Liu, “Neural network-based nearoptimal control for a class of discrete-time affine nonlinear systems with control constraint,” IEEE Trans. Neural Netw., vol. 20, no. 9, pp. 1490–1503, Sep. 2009. [31] Y. N. Li, C. G. Yang, S. S. Ge, and T. H. Lee, “Adaptive output feedback NN control of a class of discrete-time MIMO nonlinear systems with unknown control directions,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 2, pp. 507–517, Apr. 2011. [32] H. X. Li and H. Deng, “An approximate internal model-based neural control for unknown nonlinear discrete processes,” IEEE Trans. Neural Netw., vol. 17, no. 3, pp. 659–670, May 2006. [33] D. R. Liu, H. Javaherian, O. Kovalenko, and T. Huang, “Adaptive critic learning techniques for engine torque and air–fuel ratio control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 4, pp. 988–993, Aug. 2008. [34] D. Liu, D. Wang, D. Zhao, Q. Wei, and N. Jin, “Neural-network-based optimal control for a class of unknown discrete-time nonlinear systems using globalized dual heuristic programming,” IEEE Trans. Autom. Sci. Eng., vol. 9, no. 3, pp. 628–634, Jul. 2012. [35] D. Liu and Q. Wei, “Finite-approximation-error-based optimal control approach for discrete-time nonlinear systems,” IEEE Trans. Cybern., vol. 43, no. 2, pp. 779–789, Apr. 2013. [36] D. Liu and Q. Wei, “Policy iterative adaptive dynamic programming algorithm for discrete-time nonlinear systems,” IEEE Trans. Neural Netw. Learn. Syst., vol. 25, no. 3, pp. 621–634, Mar. 2014. [37] H. G. Zhang, D. R. Liu, Y. H. Luo, and D. Wang, Adaptive Dynamic Programming for Control-Algorithms and Stability. London, U.K.: Springer-Verlag, 2012. [38] Z. J. Li and C. Y. Su, “Neural-adaptive control of single-master multiple slaves teleoperation for coordinated multiple mobile manipulators with time-varying communication delays and input uncertainty,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 9, pp. 1400–1413, Sep. 2013. [39] Z. J. Li, L. Ding, H. Gao, G. R. Duan, and C. Y. Su, “Trilateral tele-operation of adaptive fuzzy force/motion control for nonlinear teleoperators with communication random delays,” IEEE Trans. Fuzzy Syst., vol. 21, no. 4, pp. 610–623, Aug. 2013. [40] F. L. Lewis, W. K. Tim, L. Z. Wang, and Z. X. Li, “Deadzone compensation in motion control systems using adaptive fuzzy logic control,” IEEE Trans. Control Syst. Technol., vol. 7, no. 6, pp. 731–742, Nov. 1999. [41] J. Na, X. M. Ren, G. Herrmann, and Z. Qiao, “Adaptive neural dynamic surface control for servo systems with unknown dead-zone,” Control Eng. Pract., vol. 19, no. 11, pp. 1328–1343, 2011. [42] D. J. Li, “Neural network control for a class of continuous stirred tank reactor process with dead-zone input,” Neurocomputing, vol. 131, pp. 453–459, May 2014. [43] S. C. Tong and Y. M. Li, “Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones,” IEEE Trans. Fuzzy Syst., vol. 20, no. 1, pp. 168–180, Feb. 2012. [44] S. C. Tong and Y. M. Li, “Adaptive fuzzy output feedback control of MIMO nonlinear systems with unknown dead-zone inputs,” IEEE Trans. Fuzzy Syst., vol. 21, no. 3, pp. 134–146, Feb. 2013. [45] H. Q. Wang, B. Chen, X. P. Liu, and K. F. Liu, “Robust adaptive fuzzy tracking control for pure-feedback stochastic nonlinear systems with input constraints,” IEEE Trans. Cybern., vol. 43, no. 6, pp. 2093–2104, Dec. 2013. [46] B. B. Ren, S. S. Ge, T. H. Lee, and C. Y. Su, “Adaptive neural control for a class of nonlinear systems with uncertain hysteresis inputs and time-varying state delays,” IEEE Trans. Neural Netw., vol. 20, no. 7, pp. 1148–1164, Jul. 2009. [47] M. Chen, S. S. Ge, and B. V. E. How, “Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities,” IEEE Trans. Neural Netw., vol. 21, no. 5, pp. 796–812, May 2010.
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[48] G. Tao and P. V. Kokotovic, “Discrete-time adaptive control of plants with unknown output dead-zones,” Automatica, vol. 31, no. 2, pp. 287–291, Feb. 1995. [49] J. Campos and F. L. Lewis, “Deadzone compensation in discrete-time using adaptive fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 7, no. 6, pp. 697–707, Dec. 1999. [50] X. Zhang, H. G. Zhang, D. R. Liu, and Y. S. Kim, “Neural-networkbased reinforcement learning controller for nonlinear systems with non-symmetric dead-zone inputs,” in Proc. IEEE Symp. Adaptive Dyn. Program. Reinforcement Learn., Nashville, TN, USA, 2009, pp. 124–129. [51] V. K. Deolia, S. Purwar, and T. N. Sharma, “Control of discrete-time nonlinear systems using backstepping technique in the presence of saturation and dead-zone constraints,” in Proc. 4th Int. Conf. Comput. Intell. Commun. Netw., Mathura, India, 2012, pp. 594–599. [52] Y. J. Liu, L. Liu, and S. C. Tong, “Adaptive neural network tracking design for a class of uncertain nonlinear discrete-time systems with deadzone,” Sci. China Inf. Sci., vol. 57, no. 3, pp. 032206:1–032206:12, Mar. 2014. [53] D. J. Li and L. Tang, “Adaptive control for a class of chemical reactor systems in discrete-time form,” Neural Comput. Appl., vol. 24, no. 6, pp. 1807–1814, 2014.
Yan-Jun Liu received the B.S. degree in applied mathematics and the M.S. degree in control theory and control engineering from the Shenyang University of Technology, Shenyang, China, in 2001 and 2004, respectively. He received the Ph.D. degree in control theory and control engineering from the Dalian University of Technology, Dalian, China, in 2007. He is currently an Associate Professor with the College of Science, Liaoning University of Technology, Liaoning, China. His current research interests include adaptive fuzzy control, nonlinear control, neural network control, and reinforcement learning.
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Shaocheng Tong received the B.S. degree with the Department of Mathematics from Jinzhou Normal College, Jinzhou, China, in 1982, the M.S. degree with the Department of Mathematics from the Dalian Marine University, Dalian, China, in 1988, and the Ph.D. degree in control theory and control engineering from the Northeastern University, Shenyang, China, in 1997. He is currently a Professor with the College of Science, Liaoning University of Technology, Liaoning, China. His current research interests include fuzzy and neural networks control theory and nonlinear control, adaptive control, and system identification.
1
Adaptive NN Tracking Control of Uncertain Nonlinear Discrete-Time Systems With Nonaffine Dead-Zone Input Yan-Jun Liu and Shaocheng Tong
Abstract—In the paper, an adaptive tracking control design is studied for a class of nonlinear discrete-time systems with dead-zone input. The considered systems are of the nonaffine pure-feedback form and the dead-zone input appears nonlinearly in the systems. The contributions of the paper are that: 1) it is for the first time to investigate the control problem for this class of discrete-time systems with dead-zone; 2) there are major difficulties for stabilizing such systems and in order to overcome the difficulties, the systems are transformed into an n-step-ahead predictor but nonaffine function is still existent; and 3) an adaptive compensative term is constructed to compensate for the parameters of the dead-zone. The neural networks are used to approximate the unknown functions in the transformed systems. Based on the Lyapunov theory, it is proven that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded and the tracking error converges to a small neighborhood of zero. Two simulation examples are provided to verify the effectiveness of the control approach in the paper. Index Terms—Adaptive NN control, dead-zone input, nonlinear control theory, nonlinear discrete-time systems
I. I NTRODUCTION HE input nonlinearities are commonly encountered in various systems, such as electric servomotors and mechanical actuators etc. Such nonlinearity is poorly unknown and often limits system performance. In the control fields, its study has been drawing much interest. For example, the adaptive controller was proposed in [1] for multiinput and multioutput (MIMO) nonlinear systems with nonsymmetric saturation and known control coefficient matrix. An auxiliary system was constructed to avoid the effect of saturation input. The dead-zone is one of the input nonlinearities. The control problems of the systems with dead-zone have received a great deal of attention. A pioneer adaptive design was studied in [2] for the plants with dead-zone. Subsequently, a number
T
Manuscript received March 24, 2014; revised May 6, 2014 and May 25, 2014; accepted June 2, 2014. The work was supported in part by the National Natural Science Foundation of China under Grant 61104017 and Grant 61374113 and in part by the Program for Liaoning Innovative Research Team in University under Grant LT2012013. This paper was recommended by Associate Editor M. J. Er. The authors are with the College of Science, Liaoning University of Technology, Jinzhou, Liaoning 121001, China (e-mail: [email protected]; [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/TCYB.2014.2329495
of important results were reported for the systems with deadzone. An adaptive tracking of nonlinear SISO systems with nonsymmetric dead-zone was discussed in [3]. A decentralized adaptive control was proposed to solve the tracking problem of interconnected MIMO systems with time-delay and the input of each loop preceded by an unknown dead-zone [4]. However, the linearly parameterized condition is required in these results. Recently, the adaptive controller designs for the systems with unknown functions had been studied by using the neural network (NN) and the fuzzy systems. Many works were made in [5]–[10] for uncertain nonlinear SISO systems with unknown functions which are approximated by using the fuzzy systems and the NN. The approaches in [11]–[14] are to reduce the adjustable parameters. In [15], the adaptive NNbased dynamic surface control was handled for uncertainty permanent magnet synchronous motors with the disturbance in load torque. Subsequently, the adaptive control approaches using the approximators were studied in [16]–[18] for nonlinear stochastic SISO systems. In order to stabilize nonlinear MIMO systems, some significant adaptive intelligent control algorithms were designed in [19]–[23]. The adaptive NN controllers were developed in [24] and [25] for wheeled inverted pendulum models with unknown dynamics. The above elegant schemes were developed for nonlinear continuous-time systems and they cannot be applied to those nonlinear discretetime systems. Several important full state and output feedback adaptive neural algorithms were presented in [26]–[36] for nonlinear discrete-time systems. The optimal stabilization control for discrete-time and continuous-time systems was proposed in [37] based on adaptive dynamic programming. The representative works using the FLSs and the NNs were studied in [38] and [39] for some real systems. In these papers, some promising techniques were employed to improve the performance of mobile manipulators and nonlinear teleoperators, which were important in the practical applications. The authors in the above results do not consider the deadzone input. A dead-zone fuzzy compensator was developed for industrial positioning systems by using the fuzzy systems [40]. In [41], an adaptive controller was proposed for nonlinear systems with a nonlinear dead-zone and multiple time-delays by using the high-order NN. The adaptive output feedback tracking problems were addressed for nonlinear systems with unknown dead-zones based on the fuzzy or neural control [42]–[44] and adaptive control methods were
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proposed for nonlinear systems with other types of input nonlinearities [45]–[47]. However, these elegant schemes were developed for nonlinear continuous-time systems and they cannot be applied to those nonlinear discrete-time systems with dead-zone. In [48], a new controller structure using an adaptive deadzone inverse was proposed for discrete-time plants with unknown dead-zones. A fuzzy logic compensator was presented for the control of nonlinear MIMO dynamics systems in the discrete-time form with dead-zone [49]. Zhang et al. [50] studied an adaptive NN critic control using reinforcement learning for nonlinear discrete-time systems with nonsymmetric dead-zone input. The backstepping control approach for nonlinear discrete-time triangular structure systems in the presence of input nonlinearities, such as saturation and deadzone were studied in [51]. Adaptive NN tracking design was got in [52] for discrete-time systems with dead-zone and unknown gain function. In [53], a predictive control was yielded to control continuous stirred tank reactor (CSTR) system in discrete-time with dead-zone. Unfortunately, the systems considered in these results are required to satisfy affine structure, i.e., the dead-zone input appears linearly in nonlinear discrete-time function. Therefore, this assumption definitely limits the potential applications of those results in practice. This paper will try to study an adaptive NN control approach for a class of uncertain nonlinear discrete-time systems with dead-zone input. The unknown functions are approximated by using the NN. The Lyapunov functions in the difference form are used for analyzing the stability of systems. Two simulation examples are given to verify effectiveness of the approach. The main contributions of the paper are summarized as follows. 1) The dead-zone input appears nonlinearly in the unknown functions of the systems and the variables which can be viewed as virtual controllers appear also nonlinearly in the systems. Thus, the nonlinear pure-feedback discrete-time systems have more general form than those in the previous works. To the best of our knowledge, to date, few effective techniques are provided to control such a class of systems with dead-zone input. 2) In contrast to a large amount of works on continuoustime, there are little results are useful for controlling discrete-time systems. The controller design and the concept definition have essential differences between the discrete-time and the continuous-time. Specifically, because this paper considers the systems including the discrete-time dead-zone input, the control design becomes more difficult and complex than the continuous-time forms. 3) In order to overcome the obstacles in the control design, the systems are transformed into an n-step-ahead predictor. But the transformed systems are still of the nonaffine structure. The unknown functions in the predictor description can be approximated by using the NN. The discrete-time dead-zone input can be compensated by designing an adaptive compensatory term.
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II. P ROBLEM AND F ORMULATION A. System Description Consider the discrete-time system, which is described as [29] ⎧ ⎨ xi (k + 1) = ϕi (¯xi (k) , xi+1 (k)) , i = 1, . . . , n − 1, xn (k + 1) = ϕn (¯xn (k) , D (v (k)) , d (k)) (1) ⎩ y (k) = x1 (k) where x¯ i (k) = [x1 (k) , . . . , xi (k)]T , i = 1, . . . , n are the states of the systems; ϕi (·, ·) , i = 1, . . . , n − 1, and ϕn (·, ·, ·) are the unknown functions to be continuous with respect to all the arguments and continuously differentiable with respect to the second argument; y (k) ∈ R is the system output; d (k) represents the external disturbance with the upper bound dM , i.e., |d (k)| ≤ dM ; u (k) = D (v (k)) stands for the system input and the output of the dead-zone where v (k) is the input of dead-zone. The dead-zone u (k) can be defined as [48] ⎧ ⎨ mr (v (k) − br ) , v (k) ≥ br u (k) = D (v (k)) = 0, −bl < v (k) < br (2) ⎩ ml (v (k) + bl ) , v (k) ≤ −bl where mr and ml are the right and left slopes of the dead-zone, respectively; br and bl stand for the breakpoints; mr and ml , br and bl are the positive constants. The dead-zone can be rewritten as u (k) = m (k) v (k) − b (k) where
m (k) =
and
(3)
mr , v (k) > 0 ml , v (k) ≤ 0
⎧ ⎨ mr br , v (k) ≥ br b (k) = m (k) v (k) , −bl < v (k) < br ⎩ −ml bl , v (k) ≤ −bl .
According to the result in [3], it is easy to know ¯ = max (mr , ml ) min (mr , ml ) = m ≤ |m (k)| ≤ m |b (k)| ≤ b¯ = max (mr br , ml bl ) . Remark 1: In the previous results of the discrete-time systems, the dead-zone input D (v (k)) appears linearly in the systems such as [48]–[53]. When the dead-zone input does not satisfy this restriction, the above results are very difficult to be applied. In this paper, D (v (k)) is nonlinearly described in the systems. Thus, the proposed approach can be used to control a general class of nonlinear discrete-time systems with the dead-zone input. The control objective is to design a neural controller such that: 1) all the signals in the closed-loop systems are bounded and 2) the output y (k) can track a reference trajectory yd (k) to a bounded compact set where yd (k) is bounded and known smooth function. To achieve this control objective, we make the following assumptions. Assumption 1 [29]: The functions ϕi (·, 0), i = 1, . . . , n − 1, and ϕn (·, 0, ·) are Lipschitz functions.
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Assumption 2 [28], [29]: There are the constants h¯ i > hi > 0 such that h¯ i ≤ |hi (.)| ≤ hi where ∂ϕi (¯xi (k) , xi+1 (k)) ∂xi+1 (k) ∂ϕn (¯xn (k) , u (k) , d (k)) . hn (¯xn (k) , u (k) , d (k)) = ∂u (k) hi (¯xi (k) , xi+1 (k)) =
Assumption 2 implies that hi (.) , i = 1, . . . , n are strictly either positive or negative. Without losing generality, it is assumed that hi ≤ hi (·) ≤ h¯ i . Let h = n n ¯ ¯ i=1 hi and h = i=1 hi . We introduce the following lemma. Lemma 1 [29]: Let V (k) = ni=1 Vi (k) with Vi (k) ≥ 0, ∀k ∈ Z0+ . If the following inequality holds: V (k + 1) ≤
n
pi (k1 ) Vi (k1 ) + q (k1 )
(4)
i=1
where k1 = k − n + 1, k ≥ n − 1, n ≥ 1, |pi (k1 )| ≤ p¯ < 1, and |q (k1 )| ≤ q¯ . Then, it has V (k) ≤ V¯ (0) + q¯ / (1 − p¯ ) and
C. System Transformation Based on the result in [29], the prediction system can be obtained as ⎧
x1 (k + n) = τ1,1 n−1,1 (¯xn (k)) , x2 (k + n − 1) ⎪ ⎪
⎪ ⎪ ⎪ ⎨ x2 (k + n − 1) = τ1,2 n−2,2 (¯xn (k)) , x3 (k + n − 2) .. (5) . ⎪ ⎪ ⎪ + 1) = τ x , u , d x (k (¯ (k) (k) (k)) ⎪ 1,n n ⎪ ⎩ n y (k + n) = x1 (k + n) where τ1,i (¯xi+1 (k)) = ϕi (¯xi (k) , xi+1 (k)) τ1,n (¯xn (k) , u (k) , d (k)) = ϕn (¯xn (k) , u (k) , d (k)) n−i,i (¯xn (k)) = x¯ i (k + n − i) , i = 1, . . . , n − 1. Replacing x2 (k + n − 1) in the first equation of (5) with the right-hand side of the second equation, it has
x1 (k + n) = τ1,1 n−1,1 (¯xn (k)) , τ1,2 n−2,2 (¯xn (k)) , x3 (k + n − 2) . Continuing to iteratively replace xj (k + n − j + 1) in the above equation with the right-hand side of the jth equation in (5), j = 3, . . . , n − 1, until u (k) appears at the last step, we can obtain y (k + n) = x1 (k + n) = τ (¯xn (k) , u (k) , d (k))
lim sup {V (k)} ≤ q¯ / (1 − p¯ )
k→∞
3
(6)
where
where V¯ (0) = max {V (j)}. 0≤j≤n−1
Proof: The proof is placed in Appendix A.
τ (.) = τ1,1 n−1,1 (¯xn (k)) , τ1,2 n−1,1 (¯xn (k))
τ1,3 . . . , τ1,n (¯xn (k) , u (k) , d (k)) . . . .
Equation (6) can be defined as B. Radial Basis Function Neural Networks It has been shown that for any given continuous function g (X) : Rn → R on a compact set U ∈ Rn and an arbitrary ε > 0, there exist the RBFNN σ T (X) θ so that sup g (X) − σ T (X) θ < ε X∈U
where θ ∈ Rl is the adjustable weight vector, l > 1 is the NN node number, X ∈ Rn is the input vector, and σ (X) = [σ1 (X) , · · · , σl (X)]T is Gaussian basis function vector. Gaussian basis function is expressed as
X − πi 2 σi (X) = exp − , i = 1, · · · , l μ2i where πi = [πi1 , · · · , πin ]T and μi are the center and the width of Gaussian function, respectively. Based on the approximation property of the RBFNN, the continuous time function g (X) can be represented as ∗
∗
g (X) = σ (X) θ + ε (X) T
where θ ∗ denotes the optimal weight vector and ε∗ (X) is the minimal approximation error. In general, ε∗ (X) is assumed to bounded by a constant ε¯ , this is, |ε∗ (X)| ≤ ε¯ .
y (k + n) = τs (¯xn (k) , u (k)) + ds (k)
(7)
where ds (k) = τ (¯xn (k) , u (k) , d (k))−τ (¯xn (k) , u (k) , 0) and τs (¯xn (k) , u (k)) = τ (¯xn (k) , u (k) , 0). From definition of τ (.), we know that τ (.) is still satisfying Assumption 1, i.e., τ (.) satisfies Lipschitz condition. Then, there must be a constant Ld such that |τ (¯xn (k) , u (k) , d (k)) − τ (¯xn (k) , u (k) , 0)| ≤ Ld |d (k)| . Accordingly, we have |ds (k)| ≤ Ld |d (k)| ≤ Ld dM . Assumption 3 [29]: The functions ϕi (·, 0), i = 1, . . . , n − 1 and ϕn (·, 0, ·) are Lipschitz functions. According to Assumption 2, it can be obtained that ∂τ1,i (¯xi+1 (k)) /∂xi+1 (k) = hi (·) > 0.
(8)
Based on (5) and (8), it can be easily to get ¯ 0 < h < ∂τs (¯xn (k) , u (k)) /∂u (k) < h.
(9)
III. C ONTROLLER D ESIGN AND S TABILITY A NALYSIS The controller will be designed for (7). Define the tracking error z (k) = y (k) − yd (k). Then, we get z (k + n) = y (k + n) − yd (k + n) = τs (¯xn (k) , D (v (k))) + ds (k) − yd (k + n) . (10)
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From (10), it can be obtained that ∂ τs (¯xn (k) , D (v (k))) − yd (k + n) = h (·) > 0. ∂D (v (k)) Then, there exists a ideal signal u∗ (k) = D (v∗ (k)) so that
τs x¯ n (k) , D v∗ (k) − yd (k + n) = 0. (11) The ideal signal D (v∗ (k)) can be approximated by the NN as
(12) D v∗ (k) = σ T (X (k)) θ ∗ + ε (X (k)) T T where X (k) = x¯ n (k) , yd (k + n) ∈ X ∈ Rn+1 is the input vector of the NN, σ (X (k)) ∈ Rl is the basis function vector, ε (X (k)) is the approximation error, and θ ∗ ∈ Rl is the ideal NN weight. Let θˆ (k) estimate θ ∗ and θ˜ (k) = θˆ (k) − θ ∗ is the estimation error. Adding and subtracting τs (¯xn (k) , D (v∗ (k))) on the right side of (10), and using (11), it gets z (k + n) = τs (¯xn (k) , D (v (k)))
−τs x¯ n (k) , D v∗ (k) + ds (k) . By using the Mean Value Theorem, it has
z (k + n) = h x¯ n (k) , D vc (k)
× D (v (k)) − D v∗ (k) + ds (k) where
(13)
D vc (k) ∈ min D v∗ (k) , D (v (k)) ∗ max D v (k) , D (v (k)) .
Remark 2: It can be seen from (13) that D (vc (k)) is nonlinearly included in h (¯xn (k) , D (vc (k))). Thus, h (¯xn (k) , D (vc (k))) is still nonaffine function in essence. In view of this property, it is difficult to design a stable adaptive neural controller. The main difficulties are that h (¯xn (k) , D (vc (k))) cannot appear in packaged unknown functions which are approximated by the NN. Note that h (¯xn (k) , D (vc (k))) satisfies still Assumption 2. For convenience, h (k) is used to denote h (¯xn (k) , D (vc (k))) in the following. Using (3) and (12), (13) can be rewritten as z (k + n) = hm (k) v (k) − h (k) b (k) + ds (k) −hm (k) σ T (X (k)) θ ∗ + ε (X (k))
(14)
¯ where hm (k) = h (k) m (k). Let hm = mh and h¯ m = m ¯ h. The controller v (k) is defined as v (k) = β (k) z (k) + ηˆ (k) + v¯ (k)
Define the adaptation laws as follows: θˆ (k + 1) = θˆ (k1 ) − γθ θˆ (k1 ) + αθ σ (X (k1 )) z (k + 1)] ηˆ (k + 1) = ηˆ (k1 ) − αη z (k + 1) + γη ηˆ (k1 )
where αθ , αη , γθ , γη > 0 are design parameters to be specified later on. Remark 3: In the previous results without considering the dead-zone, only adaptation law for the optimal neural weight is designed. Because the unknown dead-zone appears in the input of the systems, the adaptation law for ηˆ (k + 1) in (18) is designed to compensate for unknown parameter of dead-zone. It will result in a complicated design procedure owing to the existence of the dead-zone. For nonlinear discrete-time systems with dead-zone, some approaches are made in [48]–[53]. These approaches are obtained by using the dead-zone inversion or the dead-zone to be viewed as a disturbance. The ways are not good to cancel the effect of dead-zone. In this paper, an adaptive compensative algorithm is effectively to overcome of the effect of dead-zone. The following theorem is explained to show the stability of the closed-loop system. Theorem 1: Consider the systems, which are described by (1). On the compact sets X and under Assumptions 1 and 2, by constructing the controller (15), the adaptation laws (17) and (18), the overall closed-loop system is stability in the sense that all the signals are semi-globally uniformly ultimately bounded (SGUUB) and the system output tracks the reference signal to a small neighborhood of zero by choosing the design parameters appropriately. Proof: Please see the Appendix. IV. S IMULATION E XAMPLE In this section, the examples are provided to demonstrate the feasibility of the proposed method. Example 1: Consider nonlinear systems ⎧ ⎨ x1 (k+1) = ϕ1 (x1 (k) , x2 (k)) x2 (k+1) = ϕ2 (¯x2 (k) , D (v (k)) , d (k)) (19) ⎩ y (k) = x1 (k) where ϕ1 (.) = 1.4
(15)
where |β (k)| ≤ β¯ < 1 is a scaling factor and v¯ (k) = ¯ m, σ T (X (k)) θˆ (k) and ηˆ (k) is the estimation of η = b/ ¯ and let η˜ (k) = ηˆ (k) − η. Using (15), (14) can be rewritten as z (k + 1) = H (k1 ) + hm (k1 ) η˜ (k1 ) + σ T (X (k1 )) θ˜ (k1 ) + β (k1 ) z (k1 ) (16) where k1 = k − n + 1 and H (k1 ) = hm (k1 ) (η − ε (X (k1 ))) + ds (k1 ) − h (k) b (k). According to Assumption 2 and the boundedness of approximation error, it has
¯ |H (k1 )| ≤ h¯ 2b¯ + m¯ ¯ ε + 1 + Ld dM = H.
(17) (18)
x12 (k)
+ 0.1x22 (k) + 0.5x2 (k) 1 + x12 (k) x1 (k) + 0.5 sin (D (v (k))) ϕ2 (.) = 2 1 + x1 (k) + x22 (k) + ⎧ D (v (k)) + 0.5 + 0.1 cos (0.05k) cos (x1 (k)) ⎨ 0.5 (v (k) − 0.3) , if v (k) ≥ 0.3 D (v (k)) = 0, if − 0.4 < v (k) < 0.3 (20) ⎩ 0.3 (v (k) + 0.4) , if v (k) ≤ −0.4. The initial values for the states are chosen as x1 (0) = −0.2 and x2 (0) = 0.1. The control objective is to design an adaptive controller for the system (1) such that: 1) the output y (k) tracks the reference signal yd (k) = 0.5 sin ((π/5) kT) + 0.5 cos ((π/10) kT) + 0.9 with T = 0.05 as closely as possible and 2) the boundedness of all the signals in the closed-loop system is guaranteed.
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Fig. 1.
y(k) (solid line) and yd (k) (dashed line) in Example 1.
Fig. 2.
(a) x2 (k) and (b) v (k) in Example 1.
Fig. 3.
(a) θˆ (k) and (b) ηˆ (k) in Example 1.
5
From the above design, the controller is defined as v (k) = β (k) z (k) + ηˆ (k) + v¯ (k) σT
where v¯ (k) = given as follows:
(21)
(X (k)) θˆ (k) and the adaptation laws are
θˆ (k + 1) = θˆ (k − 1) − γθ θˆ (k − 1) + αθ σ (X (k − 1)) z (k + 1)] (22) ηˆ (k + 1) = ηˆ (k − 1) − αη z (k + 1) + γη ηˆ (k − 1) (23)
with z (k) = y (k) − yd (k). The initial value of the adaptation laws are shown as θˆ (0) = 0 and ηˆ (0) = 0.01. The design parameters are selected as αθ = 0.1I, αη = 0.1, γθ = 0.01, γη = 0.01, and β (k) = 0.1. In the simulation studies, the centers and widths are selected on a regular lattice in the respective compact sets. The NN contain 25 nodes with the centers πl evenly spaced in [−1.5, 1.5] × [−2, 2] × [−2.5, 2.5], and widths μi = 1 where the input variable of the NN is given as X (k) = T x1 (k) , x2 (k) , yd (k + 2) . Figs. 1–3 show the simulation results which are obtained by applying the control controller v (k) to the system (19). Fig. 1 illustrates the system tracking trajectories. It can be seen that a good tracking performance is obtained. The system state x2 (k) and the control signal v (k) are described in Fig. 2. The trajectories of the adaptation laws θˆ (k) and ηˆ (k) are shown in Fig. 3. From Figs. 2 and 3, we can see that the variables x2 (k), v (k), θˆ (k), and ηˆ (k) are bounded. Example 2: This simulation will be to consider CSTR in discrete-time [29], [53] ⎧ x1 (k + 1) = x1 (k) + [−x1 (k) + ⎪ ⎪ Ca (1 − x1 (k)) ⎪ γ x2 (k)/(γ +x2 (k)) × 0.05 ⎪ × e ⎪ ⎪ ⎨ x2 (k + 1) = x2 (k) + [−x2 (k) (24) + BCa (1 − x1 (k)) eγ x2 (k)/(γ +x2 (k)) ⎪ ⎪ ⎪ ⎪ − Cr (x2 (k) − D (v (k)))] × 0.05 + d (k) ⎪ ⎪ ⎩ y (k) = x1 (k) where x1 (k), x2 (k), and y (k) are the states and the output of the systems, respectively; d (k) = 0.01 cos (0.05k) cos (x1 (k));
D (v (k)) is given as (20). The reference signal yd (k) is described by yd (k) = 0.1sin π5k T + 0.3 with T = 0.1. The NN are chosen to be the same as Example 1. The initial value of the adaptation laws are shown as θˆ (0) = 0 and ηˆ (0) = 0.1. The design parameters are selected as αθ = 0.2I, αη = 0.3, γθ = 0.02, γη = 0.02, and β (k) = 0.1. The initial condition for the states is chosen as x1 (0) = 0.1, x2 (0) = −0.5. The simulation Figs. 4–6 are obtained by taking the approach to (24). The tracking performance, trajecto the ries of x2 (k),v (k), and the boundedness of θˆ (k), ηˆ (k) are illustrated in Figs. 4–6, respectively. From these figures, we conclude that a good tracking performance is got and all the signals in the closed-loop system are bounded. V. C ONCLUSION This paper proposed an adaptive controller for a class of nonlinear discrete-time systems by using the NN. The
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Fig. 4.
y(k) (solid line) and yd (k) (dashed line) in Example 2.
Fig. 6.
(a) θˆ (k) and (b) ηˆ (k) in Example 2.
Based on (4), we have V (sn + n + j) ≤ =
m i=1 m
p1 (sn + j) Vi (sn + j) + q (sn + j) j
j
pi (s) Vi (s) + qj (s)
(26)
i=1 j
where pi (s) = pi (sn + j) and qj (s) = q (sn + j). From (25) and (26), we obtain V j (s + 1) ≤
m
j
j
pi (s) Vi (s) + qj (s) .
i=1
j Noting that pi (s) ≤ p¯ and qj (s) ≤ q¯ , based on [29, Lemma 4] to the above equation and it leads to Fig. 5.
_ q¯ q¯ ≤ V (0) + 1 − p¯ 1 − p¯ j q¯ . lim sup V (s) ≤ l→∞ 1 − p¯
(a) x2 (k) and (b) v (k) in Example 2.
nonlinear system under study contains the dead-zone input, unknown functions, and the external disturbance. Therefore, the nonlinear systems considered in this paper are more general than those in the previous literature. It has been proven that the all variables of the control system are uniformly bounded and the tracking error converges to a small neighborhood of zero. The effectiveness of the proposed controller has been confirmed by using two simulation examples. A PPENDIX A T HE P ROOF OF L EMMA 1 This proof has been given in [29]. For the readers’ convenience, we append the proof in this paper. j j Proof: Let V j (s) = m i=1 Vi (s) where Vi (s) = Vi (sn + j) and s ∈ Z0+ . One has V j (0) ≤ V¯ (0). Then, it obtains V j (s + 1) =
m i=1
Vi (s + 1) n + j = V (sn + n + j) . (25)
V j (s) ≤ V j (0) +
Then, ∀k ≥ n − 1, there exist j = k(modn), j ∈ {0, 1, . . . , n − 1}, and s = (k − j) /n so that V (k) =
m
Vi (sn + j) =
i=1
m
_
j
Vi (s) = V j (s) ≤ V (0) +
i=1
q¯ 1 − p¯
and lim sup {V (k)} ≤
k→∞
q¯ . 1 − p¯
A PPENDIX B T HE P ROOF OF T HEOREM 1 Consider the following Lyapunov function: V (k) = Vz (k) + Vθ (k) + Vη (k)
(27)
where Vz (k) = z2 (k) /hm , Vθ (k) = θ˜ T (k) θ˜ (k) /αθ , and Vη (k) = η˜ 2 (k) /αη .
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It can be derived from (17) and (18) that θ˜ (k + 1) = θ˜ (k1 ) − αθ σ (X (k1 )) z (k + 1) + γθ θˆ (k1 ) (28) η˜ (k + 1) = η˜ (k1 ) − αη z (k + 1) + γη ηˆ (k1 ) . (29) Using (28), we have
7
αθ σ (X (k1 )) 2 z2 (k + 1) ≤ αθ lz2 (k + 1) 2γθ σ T (X (k1 )) θˆ (k1 ) z (k + 1) ≤ αθ lz2 (k + 1) γ 2 θˆ 2 (k1 ) + θ αθ γη2 ηˆ 2 (k1 ) 2γη z (k + 1) ηˆ (k1 ) ≤ αη z2 (k + 1) + αη 2 ¯ (k1 ) βz ¯ m z2 (k + 1) − 2β (k1 ) z (k1 ) z (k + 1) ≤ + βh hm 2 2 2 2θ˜ T (k1 ) θˆ (k1 ) = θ˜ (k1 ) + θˆ (k1 ) − θ ∗
θ˜ T (k + 1) θ˜ (k + 1) αθ 2 ˜ θ (k1 ) = + 2γθ σ T (X (k1 )) θˆ (k1 ) z (k + 1) αθ 2η˜ (k1 ) ηˆ (k1 ) = η˜ 2 (k1 ) + ηˆ 2 (k1 ) − η2 . 2γθ θ˜ T (k1 ) θˆ (k1 ) 2 2 − + αθ σ (X (k1 )) z (k + 1) αθ γ 2 θˆ 2 (k1 ) By using Vz (k + 1) = z2 (k + 1) /hm and combining with the . (30) −2σ T (X (k1 )) θ˜ (k1 ) z (k + 1) + θ above facts, we obtain αθ 2 Using (29), we have ˜ θ (k1 ) ¯ 2 (k1 ) βz V (k + 1) = (1 − γθ ) + η˜ 2 (k + 1) αθ hm Vη (k + 1) = αη η˜ (k1 ) 2
+ 1 − γη +γθ (γθ − 1) η˜ 2 (k1 ) αη = − 2η˜ (k1 ) z (k + 1) 2 ˆ αη θ (k1 )
ηˆ 2 (k1 ) 2γη η˜ (k1 ) ηˆ (k1 ) × γ + γ − 1 2 η η − + αη z (k + 1) αθ αη αη
+ 3αθ l + 2αη + hm β¯ − 1/hm z2 (k + 1) + q¯ + 2γη z (k + 1) ηˆ (k1 ) + γη2 ηˆ 2 (k1 ) /αη . (31) (33) From (16), it has 2 2 ∗ ¯ γη η H γθ θ where q¯ = + + . − η˜ (k1 ) + σ T (X (k1 )) θ˜ (k1 ) z (k + 1) 2 αθ αη αθ lh¯ m If the design parameters satisfy that 0 < γθ < 1, 0 < γη < 1 , z2 (k + 1) H (k1 ) z (k + 1) and 3hm αθ l + 2hm αη + h2m β¯ < 1, we have =− + hm (k1 ) hm (k1 ) ¯ z (k) + (1 − γθ ) Vθ (k) V (k + 1) = βV + β (k1 ) z (k1 ) z (k + 1) . (32)
+ 1 − γη Vη (k) + q¯ . (34) By employing (30)-(32), we have Because β¯ < 1, 1 − γθ < 1, and 1 − γη < 1, (34) satisfies ¯ Lemma 1. (k) ≤ V (0) + q¯ / (1 − p¯ ) where Thus, we have V Vθ (k + 1) + Vη (k + 1) ¯ 1 − γθ , 1 − γη . Further, one has p¯ = max β, θ˜ T (k1 ) θ˜ (k1 ) 2γθ θ˜ T (k1 ) θˆ (k1 ) ⎧ ⎫ 2 − = ⎪ ⎪ ⎨ z2 (k) αθ αθ 2 θ˜ (k) η˜ (k) ⎬ q¯ . (35) lim sup + + ≤ 2z2 (k + 1) 2 2 ⎪ hm ⎪ 1 − p¯ k→∞ αθ αη ⎭ ⎩ + αθ σ (X (k1 )) z (k + 1) − hm (k1 ) γ 2 θˆ 2 (k1 ) The above inequality implies that the signals are z (k), θ˜ (k), + 2γθ σ T (X (k1 )) θˆ (k1 ) z (k + 1) + θ αθ and η˜ (k) bounded. Accordingly, it can be obtained that 2γη η˜ (k1 ) ηˆ (k1 ) θˆ (k) , ηˆ (k)are also bounded. According to (35), it has that − + 2β (k1 ) z (k1 ) z (k + 1) z (k) ≤ q¯ hm / (1 − p¯ ) = . It can be seen that depends αη on the design parameters. Then, if the design parameters are 2H (k1 ) z (k + 1) + 2γη z (k + 1) ηˆ (k1 ) + appropriately chosen, can converge to a small bounded set. hm (k1 ) η˜ 2 (k1 ) γη2 ηˆ 2 (k1 ) R EFERENCES + + + αη z2 (k + 1) . αη αη Vθ (k + 1) =
The following inequalities can be obtained: 2z2 (k + 1) 2z2 (k + 1) ≤− hm (k1 ) hm ¯ 2H (k1 ) z (k + 1) H ≤ αθ lz2 (k + 1) + hm (k1 ) αθ lh¯ 2m
−
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Yan-Jun Liu received the B.S. degree in applied mathematics and the M.S. degree in control theory and control engineering from the Shenyang University of Technology, Shenyang, China, in 2001 and 2004, respectively. He received the Ph.D. degree in control theory and control engineering from the Dalian University of Technology, Dalian, China, in 2007. He is currently an Associate Professor with the College of Science, Liaoning University of Technology, Liaoning, China. His current research interests include adaptive fuzzy control, nonlinear control, neural network control, and reinforcement learning.
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Shaocheng Tong received the B.S. degree with the Department of Mathematics from Jinzhou Normal College, Jinzhou, China, in 1982, the M.S. degree with the Department of Mathematics from the Dalian Marine University, Dalian, China, in 1988, and the Ph.D. degree in control theory and control engineering from the Northeastern University, Shenyang, China, in 1997. He is currently a Professor with the College of Science, Liaoning University of Technology, Liaoning, China. His current research interests include fuzzy and neural networks control theory and nonlinear control, adaptive control, and system identification.
