Adaptive Recurrent Neural Network Enhanced

Asian Journal of Control, Vol. 21, No. 1, pp. 1–15, January 2019 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.1726

ADAPTIVE RECURRENT NEURAL NETWORK ENHANCED VARIABLE STRUCTURE CONTROL FOR NONLINEAR DISCRETE MIMO SYSTEMS Chih-Lyang Hwang

and John Y. Hung

ABSTRACT The stability and performance of robust control for a nonlinear complex dynamic system deteriorates due to large uncertainties. To address this, a discrete variable structure control (DVSC) outside of a convex set is designed to force the operating point into a convergent set, which is verified by Lyapunov stability theory. Due to the dynamic features of the lumped uncertainties, they are learned on-line by a recurrent neural network (RNN) in a specific convex set containing the convergent set of the DVSC. Between this convergent set and the convex set, a switching mechanism determines whether a learning RNN for the lumped uncertainties is executed. An adaptive recurrent-neural-network enhanced discrete variable structure control is then established to improve both transient and steady performances. Simulations, including compared examples and application to the trajectory tracking of a mobile robot, validate the effectiveness and robustness of the proposed control. Key Words:

Recurrent neural network, discrete variable structure control, NARMA, Lyapunov stability theory.

I. INTRODUCTION It is known that a conventionally designed linear controller may not achieve adequate performance over a variety of operating regimes, especially when the system is highly nonlinear [1,2]. Although this difficult condition can be tackled by a linear adaptive control problem with unknown system parameters, its effectiveness is limited. It is also known that a robust controller design based on a nominal system is not enough to stabilize a system with large uncertainty [3]. There are some nonlinear multivariable systems that can be modeled by an interconnected nonlinear matrix gain and a linear dynamic system, for example, the Wiener model, Hammerstein model, hysteresis model [4–6]. Furthermore, a nonlinear autoregressive moving average (NARMA) model is a generalized representation of the input–output behavior of a finite-dimensional nonlinear discrete dynamic system. Compared with a nonlinear state Manuscript received October 18, 2016; revised August 11, 2017; accepted October 7, 2017. C.-L. Hwang (corresponding author, e-mail: [email protected]) is with Department of Electrical Engineering, National Taiwan University of Science and Technology, 43, Section 4, Keelung Road, Taipei, Taiwan, 10607, R.O.C. J. Y. Hung is with Department of Electrical and Computer Engineering, Auburn University, 200 Broun Hall, AL 36849-5201, USA. The paper is supported by the project MOST-106-2918-I-011-003 at Taiwan described as follows: 1. Title: Foreign Short-Term Research Project. 2. Duration: 2017/02/14 to 2017/08/15. 3. Visiting Institute: Department of Electrical and Computer Engineering, Samuel Ginn College of Engineering, Auburn University, USA. 4. Topic: Unmanned Autonomous Guided Vehicle or Manufacturing Process Using RFID or Wireless Sensor Network Systems.

space representation of dynamic systems (e.g., [7,8], the NARMA model does not require a state estimator, which is easier for system identification, and can represent a wider class of nonlinear dynamic systems with time-varying delay for the controller design [9,10]. To deal with the control problems of nonlinear unknown dynamic systems, we categorize them as follows. The first category is that all the system functions are unknown. In this situation, all system functions must be learned on-line to design an indirect [10–13] or direct [14,15] adaptive control. Besides the poor transient response, they need a significant amount of computation to deal with these unknown dynamic control problems [16]. The control cycle time using a low-end processor for these adaptive controls becomes large such that their performances are deteriorated or even unstable [17,3]. Although the adaptive control is suitable for nonlinear dynamics with complex features or difficult derivation or partial learning compensation [18,13], too many system functions are required to learn on-line such that their convergence and then performance become worse [19–21]. Besides the learning of the unknown plant dynamics, filtered signals are employed to circumvent algebraic loop problems encountered in the implementation of the usual controllers [7]. However, this applies only to a single-input–single-output system. Recently, a suboptimal methodology has been developed by fusing neighboring extremals and fuzzy control concepts to deal with large uncertainties and external disturbances [22]. Parts of the suggested control are based on the nominal dynamics of the controlled system; the other parts are then based on various strategies (e.g., fuzzy control

© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd

Asian Journal of Control, Vol. 21, No. 1, pp. 1–15, January 2019

[22], neural network control [23], adaptive compensation [24]. Consequently, the integrated control becomes effective, flexible, robust, and efficient. Hence, the second category to deal with a nonlinear unknown dynamic system is discussed as follows. To begin with, the off-line identification of these unknown system functions through input/output data [5,25,26] is achieved. Although their performances, including identification error and computation time, are excellent, their neural network architectures are not necessarily suitable for the controller design. In contrast, the nonlinear autoregressive moving averaging model is more appropriate for the controller design [10,14]. In this paper, a discrete variable structure control (DVSC) is first designed. It is known to possess excess robustness but possible chattering control input [27,28]. To improve the performance of DVSC, the lumped uncertainties caused by uncertain system functions and external disturbances are learned on-line by a recurrent neural network (RNN) and then compensated such that the switching gain in the DVSC can be smaller to deal with the remaining uncertainties. It is known that the RNN [9,10, 29,30] is more suitable for dynamic mapping than the multilayer neural network or radial basis function neural network. An RNN can cope with time-varying input or output through its own natural temporal operation. Hence, less number of neurons for the RNN can approximate the dynamic mapping to obtain the desired accuracy. Motivated by the above viewpoints, the DVSC is initially designed to accomplish the satisfactory performance for the relatively bounded lumped uncertainties. Then a linearly parameterized approximation of the RNN for the lumped uncertainties is applied to design the learning law for the unknown weight. Moreover, a simple and extra network is established to deal with the residue of linearization parameterization [10]. Finally, the proposed adaptive RNN enhanced discrete variable structure control (ARNNEDVSC) is designed to with two parts: one is the equivalent control including nominal system function and learning lumped uncertainties [23,24], the other is the switching control to improve system robustness [22]. Under mild conditions, the convergent region of tracking error for the proposed ARNNEDSVC can be smaller than that of the DVSC. The semi-global stability of the overall system can be proven by the Lyapunov stability theory. The contributions of this paper are summarized as follows. First, the equivalent control of the ARNNEDSVC based on the real-time nominal system through the original nominal dynamics and the learning lumped uncertainties enhances the system performance of the DVSC. Second, the amplitude of the switching gain in the switching control of the ARNNEDVSC to obtain the specific convergent set of switching surface is verified. Third, based on the convergent set of the DVSC, a switching mechanism determines whether or when the

learning lumped uncertainties should be executed to enhance system performance. Fourth, the compared simulations [15] validate the effectiveness, robustness, and efficiency of the proposed method. Finally, an application to the trajectory tracking control of a differential mobile robot is also presented.

II. MATHEMATICAL PRELIMINARIES AND PROBLEM FORMULATION 2.1 Mathematical preliminaries Throughout this paper, the symbol ‖x‖ represents an Euclidean norm of vector x. Define the trace operator as trfAg ¼ ∑ni¼1 aii ; where A ∈ ℜn × n. The property tr{ABC} =tr{BCA} = tr{CAB} exists, where A, B, and C are three compatible (non-)square matrices. The notation ‖‖F denotes the Frobenius norm, i.e., ‖W‖F = tr{WTW} = tr{WWT}, where W ∈ ℜn × m. The symbol In denotes a unit matrix of dimension n. Furthermore, the following two definitions are given. Definition 1 [31]. It is defined that F(k) : ℜ+ → ℜn is continuous at k, if given ε > 0, there exists kρ > 0 such that |k1 k| < kρ ⇒ ‖F(k1) F(k)‖ < ε. Definition 2 [31]. The solutions of a dynamic system X(k + 1) = F(X, k), X(k) ∈ ℜn are said to be uniformly ultimately bounded (UUB) if there exists positive constants υ and κ, independent of k0, and for every Δ ∈ (0, κ), there is a positive constant K = K(υ, Δ) ≥ 0, such that ‖X(k0)‖ < Δ ⇒ ‖X(k)‖ ≤ υ,∀k ≥ k0 + K. 2.2 Problem formulation Consider a class of nonlinear discrete multivariable dynamic systems by the nonlinear autoregressive moving average (NARMA) model in Fig. 1: Y ðk Þ ¼ F 1 ðX ; k Þ þ F 2 ðX ; k ÞU ðk d Þ þ Ωðk Þ X ðk Þ ¼ Y ðk 1Þ…Y k ny U ðk d 1Þ:…U ðk d nu Þ

F 1 ðX ; k Þ ¼ F 1 ðX Þ þ ΔF 1 ðX ; k Þ;

(1)

(2)

(3)

F 2 ðX ; k Þ ¼ F 2 ðX Þ þ ΔF 2 ðX ; k Þ where d ≥ 1 is a known integer delay; Y(k) and U(k) ∈ ℜn denote the n-dimensional system output and control input, respectively; ny and nu ≥ 1 are the degrees of system output


C.-L. Hwang and J. Y. Hung: ARNNEDVSC

Fig. 1. Control block diagram of the overall system.

and control input, respectively; F 1 ðX Þ∈ℜn and nn F 2 ðX Þ∈ℜ are respectively the nominal vector and matrix functions, which are continuous and known; F 2 ðX Þ is nonsingular for all X(k) [10,14]; ΔF1(X, k) ∈ ℜn and ΔF2(X, k) ∈ ℜn × n are the uncertain vector and matrix functions, respectively; and Ω(k) is the external disturbances, which can be relatively bounded. Remark 1. Without loss of generality, F 2 ðX Þ > 0∀X ðk Þ is assumed. Even with little knowledge of F2(X, k) (e.g., only its sign is known), a constant F 2 ðX Þ is suitable. The delay d is constrained by the stability of the closed-loop system. At the beginning, a discrete variable structure control (DVSC) outside of a convex set for the NARMA model with uncertain system functions and external disturbances is designed to track the desired trajectory Yd(k), which is bounded and continuous. The DVSC contains two parts: one is the equivalent control to deal with the nominal NARMA model, the other is the switching control to cope with ΔF1(X, k), ΔF2(X, k), and Ω(k). The switching control gain to obtain the specific convergent set of switching surface will be verified. To enhance the system performance as compared with the only DVSC, a RNN real-time learns the lumped uncertainties, and then compensates and attenuates their effects. Between the previous convex set and the convergent set by DVSC, a switching mechanism determines whether learning using RNN for the lumped uncertainties is executed or not. It is called the adaptive RNN enhanced discrete variable structure control, which does not require the condition of persistent excitation [23] and the state estimator [7,8]. Finally, the compared simulation examples and an application to the

trajectory tracking control of a mobile robot confirm the effectiveness, robustness, and efficiency of the proposed control.

III. DVSC FOR RELATIVELY BOUNDED UNCERTAINTIES For effectively dealing with the trajectory tracking in the presence of bounded uncertainties caused by uncertain system functions and external disturbance, the following switching surface is designed. S ðk Þ ¼ ∑dj¼1 H j S ðk jÞ þ T 1 E ðk Þ þ T 2 Eðk 1Þ (4) where S(k) ∈ ℜn, Tl = diag(tlii), |t2ii/t1ii| < 1, l = 1, 2, i = 1, 2, …, n, E(k) = Yd(k) Y(k), Hj = diag {hjii}, j = 1, 2, …, d, i = 1, 2, …, n are chosen such that the polynomial equation I n þ ∑dj H j zj ¼ 0 is Hurwitz, and T1 is nonsingular. Remark 2. It is assumed that no pole-zero cancellation from the input E(k) to the output S(k) in (4) occurs. It is better if it contains a pole of 1 to possess the integral feature of the switching surface to eliminate the dc biasing tracking error [2]. On the other hand, if the controlled system already possesses a pole at 1, it is not suitable to assign a pole of 1 (i.e., t2ii/t1ii ≈ 1) for the switching surface. The concept of blocking zero can be included to eliminate the relative uncertainties [32]. Based on the switching surface in (4), the difference of switching surface is derived as follows:



ΔSðk Þ ¼ S ðk Þ S ðk d Þ ¼ ¼

∑dj¼1 H j S ðk ∑dj¼1 H j S ðk

jÞ S ð k d Þ þ T 1 E ð k Þ þ T 2 E ð k 1 Þ jÞ S ð k d Þ þ T 1 ½ Y d ð k Þ Y ð k Þ

þT 2 E ðk 1Þ ¼ ∑dj¼1 H j S ðk jÞ S ðk d Þ þT 1 ½Y d ðk Þ F 1 ðX ; k Þ F 2 ðX ; k ÞU ðk d Þ Ωðk Þ

where 0 < γ = 1 μ(1 + α)2/(1 α)2 < 1,0 < μ < 1, and the switching gain ξ(k) is designed in the following.

þT 2 Eðk 1Þ ¼ Qðk Þ T 1 F 2 ðX ; k ÞU ðk d Þ þ Δðk Þ

(5a) where Qðk Þ ¼ ∑dj¼1 H j S ðk jÞ S ðk d Þ þ T 1 Y d ðk Þ F 1 ðX Þ þ T 2 Eðk 1Þ;

U sw ðk d Þ 8 1 ξðk ÞβðZ Þ T 1 F 2 ðX Þ S ðk d Þ= 1 α2 kS ðk d Þk ; > > < ¼ if kS ðk d Þk > 2ð1 þ αÞβðZ Þ=½ð1 αÞγ; > > : 0; otherwise (9)

(5b)

Theorem 1. Consider the controlled system (1)–(3) and the DVSC expressed by (6) and (9). The switching gain ξ(k) is achieved by the following inequality: 0 ≤ ξ 1 ðk Þ < ξ ðk Þ < ξ 2 ðk Þ

(10)

where Δðk Þ ¼ T 1 ½ΔF 1 ðX ; k Þ þ Ωðk Þ:

(5c)

Based on (5), the discrete variable structure control (DVSC) without the learning compensation is designed as follows: U ðk d Þ ¼ U eq ðk d Þ þ U sw ðk d Þ

(11)

g1 ðk Þ ¼ ð1 αÞ2 kS ðk d Þk=½ð1 þ αÞβðZ Þ ð1 αÞ (12)

(6a) h i g 2 ðk Þ ¼ ð1 αÞ2 β2 ðZ Þ þ μkS ðk d Þk2 =β2 ðZ Þ > 0:

where 1 U eq ðk d Þ ¼ T 1 F 2 ðX Þ Qðk Þ:

(6b)

Since F 2 ðX Þ for all X(k) and T1 are nonsingular, the equivalent control in (6b) exists. Substituting (6) into (5) yields ΔSðk Þ ¼ T 1 F 2 ðX Þ ½I þ Δ1 ðk ÞU sw ðk d Þ þ Δ2 ðZ; k Þ (7a) where 1

Δ1 ðk Þ ¼ F 2 ðX ÞΔF 2 ðX ; k Þ I;

(7b)

Δ2 ðZ; k Þ ¼ Δðk Þ T 1 ΔF 2 ðX ; k ÞU eq ðk d Þ:

(7c)

In addition, these uncertainties are bounded as follows: kΔ1 ðk ÞkF ≤ α < 1; kΔ2 ðZ; k ÞkF ≤ βðZ Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξ 1;2 ðk Þ ¼ g 1 ðk Þ∓ g 21 ðk Þ g2 ðk Þ

(8)

where ZT(k) = [XT(k) UT(k d )], and β(Z ) is a positive function. The first part of (8) indicates that the uncertainty of the input gain (ΔF2(X, k)) can’t be larger than its nominal magnitude. The second part of (8) is caused by ΔF1(X, k), ΔF2(X, k), and Ω(k). It reveals that the lumped uncertainties Δ2(Z, k) are relatively bounded. In sequence, the switching control of the DVSC is as follows:

(13) If the overall system satisfies the conditions (8), then {S(k d)} and {U(k d )} are UUB, and the performance ‖S(k d)‖ ≤ 2(1 + α)β(Z)/[(1 α)γ] as k → ∞ is achieved. Proof. See Appendix A. Lemma 1. For the existence of the condition (10), the following fact (14) is achieved. kS ðk d Þk > 2ð1 þ αÞβðZ Þ=½ð1 αÞγ:

(14)

Proof. See Appendix B. Remark 3. The amplitude of the switching gain (9) and the convergent set of the DVSC (14) are proportional to the lumped uncertainties β(Z). It indicates that they become larger as β(Z) is larger. Remark 4. From (B3), the exponential convergence rate to switching surface 0 < μ < 1 is dependent on μ = (1 γ) (1 α)2/(1 + α)2. The larger γ or α (the parameter α is the 2nd order as compared with the parameter γ) is achieved, the smaller μ is obtained. For fixed β(Z) in (14), the larger γ or the smaller α, the smaller ‖S(k)‖ is obtained as k → ∞ . In summary, for the fixed β(Z) and α (the upper bounds of uncertainties are fixed), the smaller μ, the smaller ‖S(k)‖ as k → ∞ is obtained.



Remark 5. To satisfy the condition (10), the proposed amplitude of the switching gain in (9) can be as follows:

h ^ 0 ðxÞW ^ ðxÞ Φ ^ T2 ðk Þxðk Þ Λ1 ðk Þ ¼ γ1 Φ ^ 0 ðx ÞW ^ ðk Þ S T ðk d Þ=2 ^ T3 ðk Þz1 Φ 2Φ

ξ ðk Þ ¼g 1 ðk Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ c1 g 21 ðk Þ g 2 ðk Þf1 c2 exp½c3 kS ðk Þkg

^ 0 ðxÞ=2 ^ T1 ðk ÞΦ Λ2 ðk Þ ¼ γ2 xðk ÞS T ðk d ÞW

(15) where 0 < < c1 ≤ 2, 0 < c2 ≤ 1, and c3 is the adjustable positive switching gain.

^ 0 ðxÞ ^ ðk Þ S T ðk d ÞW ^ T1 ðk ÞΦ Λ3 ðk Þ ¼ γ3 z1 Φ

IV. ARNNEDVSC FOR HUGE UNCERTAINTIES As the lumped uncertainties Δ2(Z, k) are huge, the DVSC fails to obtain an acceptable result. In this situation, it is approximated by a RNN [10], i.e., T T T Δ2 ðZ; k Þ ¼ W 1 Φ W 2 xðk Þ þ W 3 z1 ðΦðk ÞÞ (16) þ εΔ ðx; k Þ

Pr ðk Þ ¼

Remark 6. The approximation using RNN possesses the input xðk Þ∈ℜð2nþ1Þ1 but not X ðk Þ ¼ X T ðk Þ b T ∈ℜnðn þn Þþ1 for MLNN or RBFNN. It implies that from the computation viewpoint the RNN for the approximation of dynamic mapping is more suitable than that using MLNN or RBFN. The learning laws for the unknown weight matrices are designed as follows: y

u

^ i ðk þ 1Þ ¼ W ^ i ðk Þ; i ¼ 1; 2; 4 ^ i ðk Þ þ Λi ðk Þ ηi W W

(17a)

where

(20a)

8 0; > > pffiffiffiffi > > ^ > if W 3 ðk Þ F < m; or > > > o > pffiffiffiffi n > > ^ 3 ðk Þ ≥ m; tr Λ3 ðk Þ η3 W ^ 3 ðk Þ T W ^ 3 ðk Þ ≤ 0; < if W F > > > > > > > > > > > :

n o h i ^ 3 ðk Þ 2 ; ^ 3 ðk Þ T W ^ 3 ðk Þ = κ þ W tr Λ3 ðk Þ η3 W F o pffiffiffiffi n T ^ ^ ^ if W 3 ðk Þ F ≥ m; tr Λ3 ðk Þ η3 W 3 ðk Þ W 3 ðk Þ > 0

Λ4 ðk Þ ¼ γ4 kS ðk d ÞkΓðk Þ=2

(21)

and 0 ≤ κ, 0 < γi, 0 < ηi < 1, for i = 1, 2, 3, 4. The learning laws (17) possess learning rates γi, i = 1, 2, 3, 4, error function S(k d), and specific basis functions Λi(k), ^ i = 1, 2, 3, 4. The projection termpPffiffiffir ffiðk ÞW 3 ðk Þ in (17b) and ^ (20b) ensures that W 3 ðk Þ F < m as k→∞: Then the main theorem of this paper is as follows. Theorem 2. Consider the controlled system (1)–(3) and the following ARNNEDVSC (22): U ðk d Þ ¼ U eq ðk d Þ þ U sw ðk d Þ

(22a)

where 1

^ 2 ðk Þ þ Ψðk Þ U eq ðk d Þ ¼ T 1 F 2 ðX Þ Qðk Þ þ σðkS kÞ Δ

(22b) ^ T1 ðk Þ Ψðk Þ ¼ ΛT1 ðk Þ=2 η1 W ^ ðk Þ Φ ^ 0 ðk ÞW ^ 0 ðk ÞW ^ ðk Þ ^ T2 ðk Þxðk Þ 2Φ ^ T3 ðk Þz1 Φ Φ ^ 0 ðk Þ ΛT ðk Þ=2 η2 W ^ T1 ðk ÞΦ ^ T2 ðk Þ xðk Þ þW 2 ^ ðk Þ ^ 0 ðk Þ ΛT ðk Þ 2η3 W ^ T ðk ÞΦ ^ T ðk Þ z1 Φ þW 1

^ 3 ðk þ 1Þ ¼ W ^ 3 ðk Þ ^ 3 ðk Þ þ Λ 3 ðk Þ η 3 W W ^ 3 ðk Þ P r ðk ÞW

(19)

(20b)

where xðk Þ ¼ xT ðk Þ b T ¼ Y T ðk 1Þ U T ðk d 1Þ b ∈ℜð2nþ1Þ1 is the input vector with a known constant b; W T1 ðk Þ∈ℜnm is the output-hidden weight matrix; W T2 ðk Þ∈ℜmð2nþ1Þ is the hidden-input weight matrix, W T3 ðk Þ∈ℜmm is the recur rent weight matrix; ΦðxÞ ¼ Φ W T2 ðk Þxðk Þ þ W T3 ðk Þz1 ðΦðk ÞÞ ∈ℜm1 ; z1() is the backward-time shift operator, that is, z1Φ(k) = Φ(k 1), where its ith component ϕi ðxi Þ ¼ ½1 expð2ρxi Þ=½1 þ expð2ρxi Þ; i = 1, 2, .. , m, ρ > 0 T ^ ðk Þ ; ^ 2 ðk Þxðk Þ þ W ^ T3 ðk Þz1 Φ and xi ðk Þ ¼ W and i + kεΔ ðx; k Þk≤ ε; ∀x(k) ∈ Κ, k ∈ Z .

(18)

3

3

þ γ4 S ðk d ÞΓT ðk ÞΓðk Þ=4 ^ T4 ðk ÞΓðk Þ=kS ðk d Þk þ ð1 η4 ÞS ðk d ÞW

(17b)

(22c) and



U sw ðk d Þ 8 1 > ξðk Þe βðk Þ T 1 F 2 ðX Þ S ðk d Þ=½ð1 αÞð1 þ αÞkS ðk d Þk; > > < ¼ if kS ðk d Þk > 2ð1 þ αÞe βðk Þ=½ð1 αÞγ; > > > : 0; otherwise

(22d)

8 if kS ðk d Þk < ns2 ; > < 1; σðkS kÞ ¼ 0; if kS ðk d Þk > ns1 ; > : ðkS ðk d Þk ns2 Þ=ðns1 ns2 Þ; otherwise (22e) ^ 2 ðk Þ ¼ where ns1 > ns2 > S* (see (A5) and (A6)) Δ T T T 1 ^ ^ ^ ^ W 1 Φ W 2 xðk Þ þ W 3 z ðΦðk ÞÞ ;W i ðk Þ; i ¼ 1; 2; 3; 4 are obtained from the learning laws (17)–(21), and 0 < γ = 1 μ(1 + α)2/(1 α)2 < 1, where 0 < μ < 1 denotes the exponential convergence rate to switching surface, and e ^ 2 ðk Þ: In e 2 ðk Þ ¼ Δ 2 ðk Þ Δ βðk Þ; where Δ Δ2 ðk Þ Ψðk Þ ≤ e addition, the switching gains ξ(k) is achieved by the following inequality:

^ 4 ðk Þ in Remark 7. The form of the learning weight W (17a) and (20b) compensates the residue of approximation error for the design of trajectory tracking in the equivalent control (22b). In addition, the signal Ψ(k) in (22c) cancels the unnecessary terms for the satisfaction of the stability of the closed-loop system. It is observed that (i) if ηi, i = 1, 2, 3, 4, are small enough, (ii) if S(k d) → 0, and (iii) e 2 ðk Þ→0; then Ψ(k) is in the neighbor of zero. The asif Δ e e sumption Δ ð k Þ Ψ ð k Þ ≤ βðk Þ is reasonable. In general, 2 e βðk Þ 2ð1 þ αÞe

(27)

Proof. Similar to Appendix B. It is omitted. 0 ≤ ξ 1 ðk Þ < ξ ðk Þ < ξ 2 ðk Þ

(23)

V. SIMULATIONS AND DISCUSSIONS where ξ 1;2 ðk Þ ¼ h1 ðk Þ ∓

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h21 ðk Þ h2 ðk Þ

(24)

h i h1 ðk Þ ¼ ð1 αÞ2 kS ðk d Þk= ð1 þ αÞe βðk Þ ð1 αÞ

Three examples are presented to validate the effectiveness, robustness, and efficiency of the proposed control. The first example is from [15], the second possesses strongly relative bounded system vector functions. The third example is the trajectory tracking of mobile robot.

(25) Example 1. Based on Example 2 in [15], the controlled system is rewritten and modified as the form described in (1):

h 2 i 2 h2 ðk Þ ¼ ð1 αÞ2 e β ðk Þ þ μkS ðk d Þk2 =e β ðk Þ > 0:

"

(26) f 1 ðX Þ ¼

If the overall system satisfies the conditions (8), and

then W^ i ðk Þ; i ¼ 1; 2; 3; 4; U ðk dÞ; S ðk dÞ are UUB, and the performance satisfies kS ðk d Þk < pffiffiffiffi ^ 3 ðk Þ < m; as k → ∞ . 2ð1 þ αÞe βðk Þ=½ð1 αÞγ; W pffiffiffiffi ⌣ W ^ 3 ð0Þ ≤ W 3 < m ; F

" F 2 ðX Þ ¼

See Appendix C.

f 12 ðX Þð1 þ Δf 12 ðX ÞÞ

# ;

f 211 ðX Þð1 þ Δf 211 ðX ÞÞ f 212 ðX Þð1 þ Δf 212 ðX ÞÞ f 221 ðX Þð1 þ Δf 221 ðX ÞÞ f 222 ðX Þð1 þ Δf 222 ðX ÞÞ

# ;

Ωðk Þ ¼ ½ ω1 ðk Þ ω2 ðk Þ T

(28)

F

Proof.

f 11 ðX Þð1 þ Δf 11 ðX ÞÞ

where © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd


f 11 ðX Þ ¼ 0:75y1 ðk 1Þy1 ðk 2Þ=δ11 ðk Þ

and 4 in [15], its resistance to external disturbance (31) and system complexity seems poor (see Fig. 4). In summary, the proposed control is effective and robust as compared with some previous studies.

þ 0:35 sinðy1 ðk 1Þ þ y1 ðk 2ÞÞ; f 12 ðX Þ ¼ 1:5y2 ðk 1Þy2 ðk 2Þ=δ12 ðk Þ; δ11 ðk Þ ¼ 1 þ y21 ðk 1Þ þ y22 ðk 2Þ þ y21 ðk 3Þ; δ12 ðk Þ ¼ 1 þ y22 ðk 1Þ þ y21 ðk 2Þ þ y22 ðk 3Þ; f 211 ðX Þ ¼ 1:4 þ 0:3 cosð0:5u1 ðk 3Þu2 ðk 3Þ y1 ðk 1Þy2 ðk 2ÞÞ 0:3y1 ðk 1Þ;

(29)

f 212 ðX Þ ¼ 0:45 0:3 sinð0:25u1 ðk 4Þu2 ðk 3Þ 0:27y1 ðk 2Þy2 ðk 1ÞÞ 0:3y1 ðk 1Þ; f 221 ðX Þ ¼ 0:56 þ 0:45 sinð0:33u1 ðk 3Þu2 ðk 4Þ 0:22y1 ðk 1Þy2 ðk 2ÞÞ þ 0:35y2 ðk 1Þ;

Example 2. Most of the system vector functions in (28)–(30) are bounded. For verifying the generalization of the proposed control, some system vector functions (e.g., f11(X) and f12(X)) are modified as the strongly relatively bounded ones: f 11 ðX Þ ¼ 0:75y1 ðk 1Þy1 ðk 2Þ=δ11 ðk Þ þ 0:35 sinðy1 ðk 1Þ þ y1 ðk 2ÞÞ þ 0:25y2 ðk 3Þu1 ðk 3Þ; f 12 ðX Þ ¼ 1:5y2 ðk 1Þy2 ðk 2Þ=δ12 ðk Þ þ0:25y1 ðk 2Þu2 ðk 2Þ:

f 222 ðX Þ ¼ 1:6 0:3 cosð0:25u1 ðk 4Þu2 ðk 3Þ þy1 ðk 2Þy2 ðk 1ÞÞ 0:32y1 ðk 2Þ;

Δf 11 ðX Þ ¼ 0:48 cosð0:2y2 ðk 3Þ 0:12u1 ðk 3ÞÞ; Δf 12 ðX Þ ¼ 0:78 sinð0:22y1 ðk 2Þ þ 0:15u2 ðk 4ÞÞ; Δf 211 ðX Þ ¼ 0:77 sinð0:2u2 ðk 3Þ þ 0:12y2 ðk 1ÞÞ; Δf 212 ðX Þ ¼ 0:73 sinð0:17u1 ðk 3Þ 0:18y2 ðk 3ÞÞ; Δf 221 ðX Þ ¼ 0:75 sinð0:22u2 ðk 4Þ 0:2y1 ðk 2ÞÞ; Δf 222 ðX Þ ¼ 0:75 sinð0:22u1 ðk 4Þ 0:2u2 ðk 3ÞÞ; (30) ( Ωðk Þ ¼

½ 0:2 0:25 T as k ≤ 350; ½ 0:25

0:2 T ;

otherwise:

(31)

All initial conditions are zero. The response of the proposed ARNNEDVSC with the switching gain c1 = 0.75, c2 = 0.98, c3 = 400 in (15) with the replacement of gi(k) by hi(k), i = 1, 2, the coefficients of the switching surface (4) H1 = 1.23I2, H2 = 0.235I2, T1 = I2, T2 = 0.825I2, the control parameters α = 0.001,μ = 0.003, e β ¼ 0:005; and the learning parameters κ = 0.5, γ1 = 0.002, γ2 = 0.004, γ3 = 0.01, γ4 = 0.0008, ηi = 0.008, i = 1, 2, 3, 4, and ns1 = 2.0, ns2 = 1.85, is shown in Fig. 2. The green dash-dot lines in Fig. 2(a) and (b) are denoted as the external disturbances, the amplitudes of which are sufficiently large to examine the corresponding response. To demonstrate the tracking ability, the response of Fig. 2 with the initial condition outside of approximated domain (e.g., y1(k0) = 2.5, y2(k0) = 2.5, where k0 = 1, 2, 3) is shown in Fig. 3, which is as our expectation. Although the response using the data-driven adaptive control in the absence of external disturbance, that is, Ω(k) = 0, is satisfactory (see Figs 3

(32)

The response using the adaptive control of [15] for the system (28) – (31) with the modification of (32) is unstable; for brevity, it is omitted. In contrast, the relative response using the proposed ARNNEDVSC is shown in Fig. 5, which is still acceptable. Based on the above investigation, the superiority of the proposed control is confirmed. Example 3. At the beginning, the schematic description of a differential mobile robot (DMR) is shown in Fig. 6. The following states x1(t) = v(t), x2(t) = ω(t), x3(t) = il (t), and x4(t) = ir(t) are defined for the dynamic model including mechanical and electrical subsystems. The matrix form of the controlled system is described as follows [3]: Ẋ ðt Þ ¼ AðX ; t Þ þ BðX ; t ÞU ðt Þ

(33)

Y ðt Þ ¼ C ½X ðt Þ þ ΔX ðt Þ

(34)

where X T ðt Þ ¼ ½ x1 ðt Þ x2 ðt Þ x3 ðt Þ x4 ðt Þ ∈ℜ4 ; Y ðt Þ ¼ ½ y1 ðt Þ y2 ðt Þ T ∈ℜ2 is the system outputs; U ðt Þ ¼ ½ ul ðt Þ ur ðt Þ T ¼ ½ u1 ðt Þ u2 ðt Þ T ∈ℜ2 is the control input; A(X, t) and B(X, t) are the true system vector functions; C2 ∈ ℜ2 × 2 are the output gain matrices; ΔX T ðt Þ ¼ ½ Δx1 ðt Þ Δx2 ðt Þ Δx3 ðt Þ Δx4 ðt Þ is the measurement noises. In addition, the true system vector functions are divided into the nominal and uncertain system vector functions: AðX ; t Þ ¼ AðX Þ þ ΔAðX ; t Þ; BðX ; t Þ ¼ B þ ΔBðX ; t Þ

(35)

where AðX Þ ¼ ½ a1 ðX Þ


a2 ðX Þ a3 ðX Þ

a4 ðX Þ T :

(36)


(a)

(d)

(b)

(e)

(c)

(f)

Fig. 2. Response of the system (28)–(31) by the ARNNEDVSC. [Color figure can be viewed at wileyonlinelibrary.com]

The nominal components in (36) are expressed as follows: a1 ðX Þ ¼ ½N l K tl x3 ðt Þ þ N r K tr x4 ðt Þ=ðMrw Þ f fr ðX Þ=M ; a2 ðX Þ ¼ L½N l K tl x3 ðt Þ N r K tr x4 ðt Þ=ð2Irw Þ τ fr ðX Þ=I; a3 ðX Þ ¼ ½Rl x3 ðtÞ K bl N l ðx1 ðt Þ þ Lx2 ðt ÞÞ=Ll ; a4 ðX Þ ¼ ½Rr x4 ðt Þ K br N r ðx1 ðt Þ Lx2 ðtÞÞ=Lr ; B¯ ¼ diagf1=Ll ; 1=Lr g

(37)

where f fr ðX Þ and τ fr ðX Þ are the nominal friction force and friction torque, respectively. They are modeled as follows: f fr ðX Þ ¼ f slip ðx1 Þρf ðx1 Þ h i þ f stick ðf Þ 1 ρf ðx1 Þ ; τ f ðt Þ ¼ τ slip ðx2 Þρτ ðx2 Þ

(38)

þ τ stick ðτ Þ½1 ρτ ðx2 Þ © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd


(a)

(a)

(b)

(b)

(c)

(c) (d) Fig. 3. Response of the system (28)–(31) for the initial output outside of the approximated domain. [Color figure can be viewed at wileyonlinelibrary.com]

Fig. 4. Response of the system (28)–(31) by [15]. [Color figure can be viewed at wileyonlinelibrary.com]



Fig. 6. Schematic description of a differential mobile robot. [Color figure can be viewed at wileyonlinelibrary.com]

(a)

8 þ þ x1 ðt Þ=xþ f δf 1 exp > s 1 > > > > < þC þ x1 ðt Þ; as x1 ðt Þ > 0; f f slip ðx1 Þ ¼

> f s δf 1 exp x1 ðt Þ=x > 1 > > > : þC f x1 ðt Þ; as x1 ðt Þ ≤ 0; 8 þ

τ s δτ þ 1 exp x2 ðt Þ=xþ > 2 > > > > < þC þ x ðt Þ; as x ðt Þ > 0; 2 τ 2 τ slip ðx2 Þ ¼

> τ s δτ 1 exp x2 ðt Þ=x > 2 > > > : þC τ x2 ðt Þ; as x2 ðt Þ ≤ 0: (40) (b) Fig. 5. Response of the system (28)–(31) with the modification of (32) by the proposed ARNNEDVSC. [Color figure can be viewed at wileyonlinelibrary.com]

where ρf (x1) = 1 as |x1(t)| > δf and ρf (x1) = 0, otherwise; ρτ(x2) = 1 as |x2(t)| > δτ an ρτ(x2) = 0, otherwise. In addition, the stiction force and torque are as follows: 8 þ þ as f ðt Þ > f s > 0; > > > fs ; < þ f stiction ð f Þ ¼ f ðt Þ; as f s ≤ f ðt Þ ≤ f s ; > > > : f ; s as f ðt Þ < f s < 0; (39) 8 þ þ τ ðt Þ > τ s > 0; > τ ; as > s > < þ τ stiction ðτ Þ ¼ τ ðt Þ; as τ s ≤ τ ðt Þ≤ τ s ; > > > : τ ; as s τ ðt Þ < τ s < 0: On the other hand, the slipping force and torque are modeled as follows:

On the other hand, the kinematic relation of mobile robot with non-holonomic constraints (i.e., rolling with no slipping) is described as follows: ẋ ðt Þ ¼ x1 ðt Þ cosðθÞ; ẏ ðt Þ ¼ x1 ðt Þ cosðθÞ; θ̇ ðt Þ ¼ x2 ðt Þ (41) where (x, y, θ) denotes the pose of the DMR. After the achievement of translation velocity and rotation velocity from (33) and (34), the trajectory tracking response can be achieved by the integration with suitable initial condition. Finally, the nominal system parameters including the dimension of the DMR, the parameters of two DC motors, and the coefficients of friction force and torque are estimated in Table I. The corresponding discrete-time model in the form (1)–(3) for the system (33)–(40) is described as follows:



c112 ¼ a4 a1 þ a6 b1 ; c122 ¼ a4 a2 þ a6 b2 ; c113 ¼ a5 a1 þ a7 b1 ;

Table I. System parameters of the DMR. Symbol M I L rw Lr, Ll Rr, Rl K br ; K bl Nr, Nl 0

0

N r; N l K tr ; K tl fþ s ;fs + δf , δf xþ 1 ; x1 Cþ f ; Cf τþ s ; τs + δτ , δτ xþ 2 ; x2 Cþ τ ; Cτ δf, δτ

Description Mass of mobile robot Inertia of moment of mobile robot Distance between two wheels Radius of wheel Inductance of two motors Resistance of two motors Back EMF Total Gear ratios Gear ratios from motor to wheel Torque constants Coulomb friction force

c123 ¼ a5 a2 þ a7 b2 ; d 113 ¼ a5 a3 ; d 123 ¼ a7 b3 ;

Value

c212 ¼ b4 a1 þ b6 b1 ; c222 ¼ b4 a2 þ b6 b2 ; c213 ¼ b5 a1 þ b7 b1 ; c223 ¼ b5 a2 þ b7 b2 ; d 213 ¼ b5 a3 ; d 223 ¼ b7 b3 ;

(44)

0.25m 0.08m 0.082H

and h denotes the sampling time. The response for tracking a circle with 6 m diameter (i.e., r1(t) = 3m, r2(t) = 0.496rad/s) for the discrete-time model of the DMR (i.e., (42)–(44)) with initial condition (x(0), y(0), θ(0)) = (0, 0, π/2) by the DVSC using h = 2.5 ms with suitable control parameters (e.g., c1 = 1.1, c2 = 0.98, c3 = 400, H1 = 0.2I2, H2 = 0.01I2,T1 = I2, T2 = 0.9I2, α = 0.001, μ = 0.003, e β ¼ 0:005 ) is shown in Fig. 7, which is chattering. Then the response of the case shown in Fig. 7 by the proposed ARNNEDVSC, using the learning

30, 30 2, 2

4.53, -4.26N

Stribeck friction force Stribeck velocity

2.07, -1.76N

Coefficients of viscous friction force Coulomb friction torque

0.47Ns/m

Stribeck friction torque Stribeck angular velocity

1.96, -1.78Nm

Coefficients of viscous friction torque Stribeck friction torque

0.5Ns/rad

4.66, -4.38Nm

0.25m/s, 0.25rad/s

f 11 ðX Þ ¼ y1 ðk 1Þ þ c112 y1 ðk 2Þ þ c122 y2 ðk 2Þ þc113 y1 ðk 3Þ þ c123 y2 ðk 3Þ þ d 113 u1 ðk 3Þ þd 123 u2 ðk 3Þ hf fr ðk Þ=M ;

(a)

f 12 ðX Þ ¼ y2 ðk 1Þ þ c212 y1 ðk 2Þ þ c222 y2 ðk 2Þ þc213 y1 ðk 3Þ þ c223 y2 ðk 3Þ þ d 213 u1 ðk 3Þ þd 223 u2 ðk 3Þ hτ fr ðk Þ=I; f 211 ðX Þ ¼ a3 a4 ; f 212 ðX Þ ¼ a6 b3 ; f 221 ðX Þ ¼ a3 b4 ; f 222 ðX Þ ¼ b6 b3 ; (42) where

a0 ¼ 1 hRl =Ll ; a1 ¼ hK bl N l =Ll ; a2 ¼ 0:5La1 ; a3 ¼ h=Ll ; a4 ¼ hK tl N l =Mr; a5 ¼ a0 a4 ; a6 ¼ hK tr N r =Mr; a7 ¼ b0 a6 ;

(b)

b0 ¼ 1 hRr =Lr ; b1 ¼ hK br N r =Lr ; b2 ¼ 0:5Lb1 ; b3 ¼ h=Lr ; b4 ¼ hK tl N l =2Ir; b5 ¼ a0 b4 ; b6 ¼ hK tr N r =2Ir; b7 ¼ b0 b6 ;

(43)

Fig. 7. Tracking response of a circular trajectory by the DVSC. [Color figure can be viewed at wileyonlinelibrary.com]



(a)

(c)

(b)

(d)

Fig. 8. The response of the case in Figure 7 as shown by the ARNNEDVSC. [Color figure can be viewed at wileyonlinelibrary.com]

parameters κ = 0.5 γ1 = 0.001, γ2 = 0.002, γ3 = 0.004, γ4 = 0.0005, ηi = 0.15, i = 1,2,3,4, and ns1 = 1.0, ns2 = 0.9) is presented in Fig. 8, where (xd(t), yd(t)) denotes the desired circular trajectory. The conclusions for Example 3 are as follows: (i) the proposed ARNNEDVSC is better than DVSC; (ii) the smaller learning rate and the larger e-modification rate are set to avoid a larger transient response; (iii) the response in Fig. 8(d) is achieved by the kinematic relation (41); (iv) similarly, the response for the DMR with extra uncertainties (i.e., A(X, t) or ΔB(X, t) or ΔX(t) ≠ 0 for the violation of non-holonomic constraint or different ground conditions) can be achieved; for brevity, it is omitted.

VI. CONCLUSIONS The dynamic features of the lumped uncertainties are real-time learned by a RNN to design an effective learning compensation. To ensure the stability of the closed-loop system, a DVSC outside of a convex set is employed to force the operating point into a verified convergent set

smaller than the previous convex set. The amplitude of the switching gain and the convergent set of the DVSC are proportional to the upper bound of the lumped uncertainties (see Remark 3). In the situation of the fixed uncertainties, the smaller convergent rate to the switching surface is assigned, and the smaller convergent set is achieved (see Remark 4). Between the convex set and the convergent set, a switching function determines whether or when the learning of the lumped uncertainties using an RNN is executed or not. Both transient and steady-state performances are improved by the proposed ARNNEDVSC. In comparison with most adaptive controls, their system vector functions are absolutely bounded [15], which is generally not true for a physical system. The proposed control does not have this constraint. Furthermore, the application to the trajectory tracking of a mobile robot confirms effectiveness, robustness, and generality of the proposed control. One of our future studies is to implement the trajectory tracking control of a mobile robot with the on-line trajectory planning based on the visual or wireless localization system.



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APPENDIX A (THE PROOF OF THEOREM 1) For simplicity, the arguments of variables are omitted if they are not vague. Define the following Lyapunov function: (A1) V ðS Þ ¼ S T S=2: Then the change rate of V is yielded as follows: (A2) ΔV ¼ ΔS ðk ÞΔSðk Þ=2 þ S ðk d ÞΔSðk Þ: To ensure the exponential convergence to the switching surface, ΔV ¼ ΔV þ μV; where 0 < μ < 1is the convergent rate, is considered. T ΔV ¼ U Tsw ðI þ Δ1 ÞT T 1 F 2 T 1 F 2 ðI þ Δ1 ÞU sw =2 ΔT2 T 1 F 2 ðI þ Δ1 ÞU sw þ ΔT2 Δ2 =2 S T T 1 F 2 ðI þ Δ1 ÞU sw þ μS T S=2 T

≤

ξ2 β2

þ 2

D ¼ fS∈ℜm j0≤kS k ≤ S g where S > 2ð1 þ αÞβ=½ð1 αÞγ: Finally, from (4), (6) and (9), {S, U}are UUB. Q.E.D.

(A5) (A6)

APPENDIX B (THE PROOF OF LEMMA 1) Since g1 ðk Þ ¼ ð1 αÞ2 kS ðk d Þk=½ð1 þ αÞβðk Þ ð1 αÞ > 0; the following result (B1) is achieved. (B1) kS ðk d Þk > ð1 þ αÞβðk Þ=ð1 αÞ: 2 Based on the condition g 1 ðk Þ g 2 ðk Þ > 0;the following equation is derived. n o2 g21 ðk Þ g2 ðk Þ ¼ ð1 αÞ2 kS ðk d Þk=½ð1 þ αÞβðk Þ ð1 αÞ h i ð1 αÞ2 β2 ðk Þ þ μkS ðk d Þk2 =β2 ðk Þ

APPENDIXES

T

ΔV ≤ 0 ðor ΔV ≤ μVÞ: (A4) Then, V(k + 1) (1 μ)V(k) ≤ 0. Hence, outside of the domain D making ΔV ≤ μV is described as follows:

ξβ2 ð 1 αÞ

2 ð1 α Þ β ξβkS k μkS k2 þ þ 2 ð1 þ α Þ 2 n o

2 2 ¼ β ξ 2g1 ξ þ g2 = 2ð1 αÞ2 2

(A3) where the inequality in (A3) has used (8), the expressions of g1 and g2 are shown in (12) and (13). As ‖S(k d)‖> 2(1 + α)β(X, Ueq)/[(1 α)γ], g1 > 0 and g21 g 2 > 0 are achieved. Then the result (10) is achieved from the inequality ξ2 2g1ξ + g2 < 0. Hence, the change rate of Lyapunov function becomes

¼ k S ðk d Þk

ð1 αÞ4 kS ðk d Þkγ 2ð1 αÞ3 ð1 þ αÞβðk Þ ð1 þ αÞ2 β 2 ðk Þ

where 0 < γ ¼ 1 μð1 þ αÞ2 =ð1 αÞ2 < 1: From (B2), kS ðk d Þk > 2ð1 þ αÞβðk Þ=½ð1 αÞγ:

>0

(B2) (B3) (B4)

To simultaneously satisfy the conditions (B1) and (B4), the corresponding (14) is achieved. Q.E.D.

APPENDIX C (THE PROOF OF THEOREM 2) If the following mathematical expressions are not vague, their arguments are omitted for simplicity. (i) As ‖S(k d)‖ < ns2, σ(‖S‖) = 1, i.e., x ∈ Κ, is considered as follows. Define the following Lyapunov function: e 1; W e 2; W e 3; W e 4; S V W h T i (C1) ei W e i =γi þ S T S=2 ¼ Z T PZ ¼ ∑4i¼1 tr W where h i e 3 W e 4 kS k e 1 W e 2 W (C2) Z T ¼ W F F F P ¼ diagf 1=γ1

1=γ2

1=γ3

1=γ4

1=2 g:

(C3)

First, the case of Pr = 0is examined. By (17), (22) and e 2; W e 3; W e 4 ; S is as follows: e 1; W (A2), the change rate ofV W



ΔV ¼ ∑4i¼1 ΔV i þ ΔV 5 h i e i Λi þ η i W e Ti W î T W e i Λi þ η i W î W e i =γi ¼ ∑4i¼1 tr W þΔS T ðk ÞΔSðk Þ=2 þ S T ðk d ÞΔSðk Þ

(C4) where ΔSðk Þ ¼ S ðk Þ S ðk d Þ (C5) e2 Ψ ¼ T 1 F 2 ðI þ Δ1 ÞU sw þ Δ e 2 ¼ Δ2 Δ ^ 2 : Based on the where Δ1 is shown in (7b) and Δ similar simplification of [10], the change rate of Lyapunov function becomes ΔV≤ ∑4i¼1 λi V i μV 5 ≤ min ðλi ; μÞV ¼ λ0 V

(C6)

i¼1;2;3;4

where 0 < λ0 < 1. Then, V(k + 1) (1 λ0)V(k) ≤ 0. Hence, outside of the domain D making ΔV ≤ λ0V is described as follows: n o e i ≤ W i ; i ¼ 1; 2; 3; 4; 0 ≤ kS k ≤ S D ¼ Z∈ℜ5 0 ≤ W F (C7) where

h pffiffiffiffiffiffiffiffiffiffiffiffii W i ¼ ð1 ηi Þ þ 1 λi ηi ⌣W i =½ηi ð2 ηi Þ λi (C8)

β=½ð1 αÞγ: (C9) S > 2ð1 þ αÞe Similarly, the case of Pr ≠ 0 can be achieved from Pr in (20b). It makes the original ΔV3 ≤ λ3V3 more negative. (ii) As ‖S(k d)‖ > ns1, σ(‖S‖) = 0. The related stability verification is the same as Theorem 1. (iii) As ns2 ≤ ‖S(k d)‖ ≤ ns1, the result can be achieved from the above parts (i) and (ii) by mean-value theory. From (17)–(22) and the result of Theorem 2, {S, U}

^ i ; i ¼ 1; 2; 3; 4 are UUB. and W Q.E.D. Chih-Lyang Hwang received his B.E. degree in Aeronautical Engineering from Tamkang University, Taipei, Taiwan, in 1981, and his M.E. degree and Ph.D. in Mechanical Engineering from the Tatung Institute of Technology (Tatung University), Taipei, Taiwan, in 1986 and 1990, respectively. From 1990 to 2006, he was with the Department of Mechanical Engineering, Tatung Institute of Technology, where he was involved in teaching

and research in the area of servo, control and control of manufacturing systems and robotic systems, and was also a Professor of Mechanical Engineering from 1996 to 2006. During 1998–1999, he was a Research Scholar at the George W. Woodruf School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA. From 2006 to 2011, he was a Professor in the Department of Electrical Engineering, Tamkang University. Since 2011, he has been a Professor with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei. From August 2016 to July 2017, he has been a Visiting Scholar at Electrical and Computer Engineering of Auburn University, USA. He is the author or co-author of approximately 153 journal and conference papers in the related field. His current research interests include robotics, fuzzy (or neural-network) modeling and control, sliding-mode control, visual or wireless localization, tracking or navigation system, network-based control, and distributed sensor networks. John Y. Hung received his B.S. degree from the University of Tennessee, Knoxville, his M.S.E. degree from Princeton University, Princeton, NJ, and a Ph.D. from the University of Illinois, Urbana-Champaign, in 1979, 1981, and 1989, respectively, all in electrical engineering. From 1981 to 1985, he was with Johnson Controls, Milwaukee, WI, developing microprocessor based controllers for commercial heating, ventilation, and air conditioning systems. From 1985 to 1989 he was a consultant engineer with PolyAnalytics, Inc. In 1989, he joined Auburn University, Auburn, AL, where he is currently Professor and Graduate Program Coordinator in the Department of Electrical and Computer Engineering. His teaching and research interests include nonlinear control systems and signal processing with applications in process control, robotics, electric machinery, and power electronics. Professor Hung has received several awards for his teaching and research, including Best Paper Awards in IEEE Transactions on Industrial Electronics and IEEE Industrial Electronics Magazine, and two US patents in the area of control systems. He served as an Associate Editor of the IEEE Transactions on Control System Technology (1997, 1998), and IEEE Transactions on Industrial Electronics (1996–2005). He was President of the IEEE Industrial Electronics Society (2014, 2015), and currently serves on the IEEE Board of Directors (2017, 2018).
