Command Filter-Based Adaptive Neural Tracking Controller Design

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Command Filter-Based Adaptive Neural Tracking Controller Design for Uncertain Switched Nonlinear Output-Constrained Systems Ben Niu, Yanjun Liu, Guangdeng Zong, Zhaoyu Han, and Jun Fu, Senior Member, IEEE

Abstract—In this paper, a new adaptive approximation-based tracking controller design approach is developed for a class of uncertain nonlinear switched lower-triangular systems with an output constraint using neural networks (NNs). By introducing a novel barrier Lyapunov function (BLF), the constrained switched system is first transformed into a new system without any constraint, which means the control objectives of the both systems are equivalent. Then command filter technique is applied to solve the so-called “explosion of complexity” problem in traditional backstepping procedure, and radial basis function NNs are directly employed to model the unknown nonlinear functions. The designed controller ensures that all the closed-loop variables are ultimately boundedness, while the output limit is not transgressed and the output tracking error can be reduced arbitrarily small. Furthermore, the use of an asymmetric BLF is also explored to handle the case of asymmetric output constraint as a generalization result. Finally, the control performance of the presented control schemes is illustrated via two examples. Index Terms—Command filter, neural network (NN), output constraints, switched nonlinear systems.

I. I NTRODUCTION N THE real world, all practical control systems are affected by constraints in one way or the other. How to handle such constraints has drawn much attention from the automatic control field [1]–[5]. In reality, control systems are

I

Manuscript received April 14, 2016; revised August 25, 2016 and October 28, 2016; accepted December 30, 2016. This work was supported in part by the National Natural Science Foundation of China under Grant 61673073, Grant 61473139, Grant 61622303, Grant 61273123, and Grant 61673072, in part by the Liaoning Provincial Natural Science Foundation, China, under Grant 201602009, in part by the Program for New Century Excellent Talents in University under Grant NCET-13-0878, and in part by the Taishan Scholar Project of Shandong Province of China under Grant tsqn20161033. This paper was recommended by Associate Editor P. Shi. (Corresponding author: Jun Fu.) B. Niu and Z. Han are with the College of Mathematics and Physics, Bohai University, Jinzhou 121013, China (e-mail: [email protected]; [email protected]). Y. Liu is with the College of Science, Liaoning University of Technology, Jinzhou 121001, China (e-mail: [email protected]). G. Zong is with the School of Engineering, Qufu Normal University, Rizhao 276826, China (e-mail: [email protected]). J. Fu is with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2016.2647626

frequently required that the output, or the output tracking error, should be limited to some compact sets during operation. Such requirements are not only coming from physical limitations, but also from safety considerations [6]. Since the theoretical and actual significance, the handling of output constraints has been paid more and more attention. Therefore, various numerical algorithms have been proposed to control of output-constrained systems, such as set invariance theory, reference governors, model predictive control, and so on. Even more remarkable, barrier Lyapunov function (BLF) is regarded as a constraint-handling tool has been employed on output-constrained nonlinear systems [7]. Nevertheless, one shortcoming of the proposed BLFs in [7] lies in that when the asymmetric output constraint is considered, a subsection continuous switched asymmetric BLF is constructed to design controller. Thus, several special techniques should be utilized to assure the continuity and differentiability of the developed controller. During the last several decades, the tracking controller design problem for general nonlinear lower-triangular systems without switching has been well studied under various conditions in the area of adaptive control [8]–[17]. It is generally known that the backstepping-based adaptive control method is an effective tool for controller design (see [18]–[27] and the references therein). However, the traditional backstepping design procedure requires the repeated differentiations of virtual controls, which causes a severe drawback called “explosion of complexity.” To efficiently eliminate the explosion of complexity problem, Swaroop et al. [28] originally proposed the dynamic surface control (DSC) method for nonswitched nonlinear lower-triangular systems. By introducing the DSC skill into adaptive backsteppingbased controller design framework, a number of adaptive approximation-based control methods for nonlinear lowertriangular systems have been obtained by using neural networks (NNs) or fuzzy logic system [29]–[34]. There is also another important means to deal with the issue of explosion of complexity, which is the so-called command filter technique first proposed in [35]. It should be stressed that the advantages of the command filter-based backstepping design are twofold. 1) The controller design and the backstepping iterations are decoupled. 2) The requirement for analytic computation of command signal derivatives is avoided.

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On the other hand, there is an increasing interest on the modeling, analysis, synthesis, and control of switched systems [36]–[50]. As we know a switched system subject to arbitrary switchings is asymptotically stable if and only if all subsystems share a common Lyapunov function (CLF) [51]–[53]. However, utilizing this design method to adaptively track switched nonlinear lower-triangular systems in the presence of arbitrary switchings or unpredictable switchings is a challenging task in adaptive control field. The reason is that it is often hard to construct (or find) the common virtual control functions for different subsystems in the backstepping design procedure. In recent years, some backstepping-based adaptive design schemes have been investigated for uncertain nonlinear switched systems. For example, using backstepping control design and the CLF method, the problem of adaptive stabilization for a class of uncertain switched lower-triangular nonlinear systems has been considered in [54]; a novel switched adaptive design approach for the adaptive stabilization problem of switched nonlinearly parameterized systems was proposed by exploiting the generalized multiple Lyapunov functions method and the parameter separation technique in [55]. Nevertheless, the nonlinear functions of the considered systems in the abovementioned results are not completely unknown. Moreover, the proposed methods above cannot solve the issue of explosion of complexity. Inspired by the aforementioned considerations, a novel adaptive neural tracking controller design scheme is presented in this paper for a class of nonlinear switched lowertriangular systems with completely unknown nonlinear function and output constraints. First, a new type of BLF is employed to translate the switched system with output constraints into a new system without constraints, which implies the design for the transformed system is essentially equivalent to the one for the original system. The desired controller is then obtained by combining the command filter technique in the backstepping design procedure with the approximating capacity of radial basis function (RBF) NNs. Unlike the existing results on nonlinear switched systems, the main contributions of this paper are listed as follows. 1) It seems that this paper is the first one dedicated to adaptive neural tracking controller design problem of switched nonlinear output-constrained lower-triangular systems by using the introduced BLFs. 2) When the asymmetric output constraint is considered, the problem is easily handled by using an asymmetric BLF, which overcomes the difficulties encountered in the control of using the asymmetric BLF in [7] and [53]. 3) In addition, the explosion of complexity problem in the classic backstepping approach is avoided by the skillful use of the command filter technique. The proposed design scheme can efficiently reduce the computational burden of the controller design process and can be conveniently implemented in practical applications.

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II. P ROBLEM D ESCRIPTION AND P RELIMINARIES A. Output-Constrained Control Problem Consider the following uncertain nonlinear switched lowertriangular systems:   ξ˙i = ξi+1 + fσ (t),i ξ¯i + dσ (t),i (ξ, t), 1 ≤ i ≤ n − 2   ξ˙n−1 = ξn + fσ (t),n−1 ξ¯n−1 + dσ (t),n−1 (ξ, t) ξ˙n = u + fσ (t),n (ξ ) + dσ (t),n (ξ, t),

n≥3

y = ξ1

(1) def

where σ (t) : [0, ∞) → E = {1, 2, . . . , m} is the switching signal. ξ = ξ n , ξ i = [ξ1 , ξ2 , . . . , ξi ]T ∈ Ri , i = 1, 2, . . . , n, u ∈ R and y ∈ R are the state variables, control input and control output of the system, respectively. For i = 1, 2, . . . , n, p = 1, 2, . . . , m, fp,i (ξ¯i ) are unknown smooth nonlinear functions, and dp,i (ξ, t) are the unknown external disturbances satisfying |dp,i (ξ, t)| ≤ di with di being constants. The output y(t) is restricted to the set sy = {y ∈ R| − lc < y < lc }, ∀t ≥ 0

(2)

where lc > 0 is a prescribed constant. Remark 1: The constraint (2) is regarded as a symmetric output constraint. When the output y(t) is subjected to the limitation   ay = y ∈ R| − lc1 < y < lc2 , ∀t ≥ 0 (3) with lc1 > 0, lc2 > 0 and lc1 = lc2 are given constants, the constraint (3) is regarded as an asymmetric output constraint. For the target signal yd (t), our control aim is to propose an adaptive tracking controller design approach for system (1) by using NNs, such that all the closed-loop variables remain bounded, meanwhile, the output can track the target signal without the violation of the output constraint (2). Assumption 1: For the limitation lc > 0, suppose that the following inequalities: −N 0 ≤ yd (t) ≤ N 0 , |˙yd (t)| < (n) N1 , . . . , |yd (t)| < Nn hold, where yd (t) is the target signal, and N 0 , N 0 , ly , N1 , . . . , Nn are positive constants with max{N 0 , N 0 } ≤ ly < lc , ∀t ≥ 0. Remark 2: It seems that Assumption 1 is a little conservative. In fact, Assumption 1 has been commonly made when the output-constrained tracking control problem is studied (see [7], [53]). To handle the output constraint (2), a new BLF is introduced by the following definition. Definition 1 [53]: The following nonlinear mapping H : ξ1 → ξ1∗ is called a symmetric BLF: ξ1∗ = log

lc + ξ 1 lc − ξ 1

(4)

where H is a continuous logarithmic function and is illustrated in Fig. 1 when lc = 0.3. It follows from (4) that:   2 −1 ξ 1 = H = lc 1 − ξ ∗ (5) e 1 +1

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B. NN Approximation Since RBF NNs can be used as universal approximators (see [5], [24] for details), we employ this technique to model the uncertain nonlinear functions of the system (1). Therefore, the following relevant details should be invoked. Suppose that h(Z) is any unknown continuous function defined in a compact set Z ⊂ Rq , then there exists NN W ∗ T S(Z), such that for ∀ε > 0 h(Z) = W ∗ T S(Z) + δ(Z), Fig. 1.

with W ∗ is the ideal constant weight vector satisfying

 ∗ T W = arg min sup h(Z) − W S(Z) .

Schematic of a symmetric BLF with lc = 0.3. ∗

2lc eξ1 ξ˙1∗ ξ˙1 =  2 . ∗ eξ1 + 1 Substituting (6) into (1) yields       ξ˙1∗ = g1 ξ1∗ ξ2 +

fp,1 ξ1∗ + g1 ξ1∗ dp,1 (ξ, t)

W∈RN

(6)

(7)

∗ ∗ where g1 (ξ1∗ ) = (1/2lc )(eξ1 + e−ξ1 + 2),

fp,1 (ξ1∗ ) = g1 (ξ1∗ ) ∗ ξ1 fp,1 (lc − (2lc /e + 1)). fp,i = fp,i ,

dp,1 = g1 dp,1 ,

dp,i = dp,i Further, let ξi∗ = ξi ,

with    2lc ∗ ∗ ∗ ∗

fp,i ξ i = fp,i lc − ξ ∗ , ξ2 , ξ3 , . . . , ξi e1 + 1   ∗   ∗ 2lc ∗ ∗ ∗

dp,1 ξ , t = g1 ξ1 dp,1 lc − ξ ∗ , ξ2 , ξ3 , . . . , ξ n , t e 1 +1     2lc

, ξ2∗ , ξ3∗ , . . . , ξn∗ , t dp,i ξ ∗ , t = dp,i lc − ξ ∗ e 1 +1 p ∈ E, i = 2, . . . , n

then the system (1) can be translated into the following form:       ξ˙1∗ = g1 ξ1∗ ξ2∗ + f˜p,1 ξ1∗ + d˜ p,1 ξ ∗ , t     ∗ ξ˙i∗ = ξi+1 + f˜p,i ξ¯i∗ + d˜ p,i ξ ∗ , t , i = 2, 3, . . . , n − 2  ∗    ∗ ξ˙n−1 + dσ (t),n−1 ξ ∗ , t = ξn∗ + fσ (t),n−1 ξ¯n−1     ξ˙n∗ = u + f˜p,n ξ ∗ + d˜ p,n ξ ∗ , t y∗ = ξ1∗

|δ(Z)| ≤ ε

(8)

with ξ¯i∗ = [ξ1∗ , ξ2∗ , . . . , ξi∗ ]T ∈ Ri , i = 1, 2, . . . , n, ξ¯n∗ = ξ ∗ are the system states, u ∈ R is the system input, and y∗ ∈ R is the system output, respectively. The output y∗ is now unconstrained and the target signal is transformed into y∗d = log(lc + yd /lc − yd ). The control aim is thus to construct an adaptive tracking controller for system (8) by using NNs, such that all the variables of resulting the close-loop system are bounded and the output y∗ (t) can follow the target signal y∗d (t). Remark 3: Based on the results in [53], we know that the design of the system (8) is equivalent to the one of the system (1). Hence, we will first develop an approach to achieve the control aim of the systems (8), which implies the control aim of the system (1) is also finished.

(9)

(10)

Z∈Z

W = [w1 , w2 , . . . , wN ]T is the weight vector, S(Z) = [s1 (Z), s2 (Z), . . . , sN (Z)]T is the basis function vector with N being the number of the NN nodes and N > 1, and δ(Z) is the approximation error. RBF si (Z) = exp [−(Z − ui )T (Z − ui )/ηi2 ], where ηi is the width of the Gaussian function and ui = [ui1 , ui2 , . . . , uin ]T is the center of the receptive field, i = 1, 2, . . . , N. III. C ONTROLLER D ESIGN AND S TABILITY A NALYSIS The objective of this section is to develop an adaptive neural tracking control design approach for system (1). The main idea lies in that if we can construct a feedback controller to achieve the control aim of the system (8), then the designed controller is also effective to accomplish the control aim of the system (1). First, we will give a detailed adaptive control design procedure under the symmetric constraint (2). Thereafter, we present briefly the design procedure under the asymmetric constraint (4) as an extended result. A. Controller Design In this section, a parameter adaptive law and an adaptive neural tracking controller will be presented for systems (8) via the backstepping technique, which means the desired controller of the systems (1) is simultaneously obtained. Moreover, the command filter technique will be applied to deal with the explosion of complexity problem in the conventional backstepping procedure. The adaptive backstepping-based neural control design contains n steps. Before proceeding with the control design, the unknown constants θ is first defined, which is specified as     ∗ 2 θ = max Wp,i  : i = 1, 2, . . . , n, p = 1, 2, . . . , m . (11) ∗ are Apparently, θ is an unknown constant since Wp,i unknown constants, i = 1, 2, . . . , n, p ∈ E. Furthermore, define

θ = θ − θ , where  θ is the estimate of θ . Step 1: First, we define the tracking error vector as z∗1 = ξ1∗ − y∗d and the compensated tracking error signal as

z¯∗1 = z∗1 − η1 = ξ1∗ − y∗d − η1 .

(12)

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The time derivative of z˙¯∗1 can thus be obtained as z˙¯∗1 = z˙∗1 − η˙ 1 = ξ˙1∗ − y˙ ∗d − η˙ 1       = g1 ξ1∗ ξ2∗ + f˜p,1 ξ1∗ + d˜ p,1 ξ ∗ , t − y˙ ∗d − η˙ 1 . Let

z∗2

=

(13)

ξ2∗

− α2,v and choose      η˙ 1 = −k1 η1 + g1 ξ1∗ α2,v − α2,a + g1 ξ1∗ η2

(14)

where k1 > 0 is the design parameter, α2,v passes a twoorder command filter and the dynamic of the command filter is defined as ζ˙1,1 = ωζ1,2   ζ˙1,2 = −2ς ωζ1,2 − ω ζ1,1 − α2,a (15) with α2,v (t) = ζ1,1 and α˙ 2,v (t) = ωζ1,2 are the outputs of the filter, ω > 0 and ς ∈ (0, 1] are the design parameters, and the initial values are chosen as ζ1,1 (0) = α2,a (0) and ζ1,2 (0) = 0. Substituting (14) into (13) yields       z˙¯∗1 = g1 ξ1∗ ξ2∗ + f˜p,1 ξ1∗ + d˜ p,1 ξ ∗ , t − y˙ ∗d + k1 η1  ∗  ∗   − g1 ξ1 α2,v + g1 ξ1 α2,a − g1 ξ1∗ η2 . (16) ∗ S (Z ), p ∈ E to approximate By employing RBF NNs Wp,1 1 1 f˜p,1 (ξ1∗ ) with Z1 = [ξ1∗ ]T , and let

z¯∗2

=

z∗2

− η2

then the pseudocontrol signal α2,a is chosen as  1 1     −k1 z∗1 + y˙ ∗d − g21 ξ1∗ z∗1 α2,a = 2 g1 ξ1∗  1 ∗ T ˆ − z1 θS1 (Z1 )S1 (Z1 ) . 2

(17)

1 ¯∗ 2 1 1 2 1 z1 + εp,1 2 ≤ z¯∗1 + ε1∗ 2 2 2 2 2 1 ¯∗ 2 T 1 ∗ ∗ z¯1 Wp,1 S1 (Z1 ) ≤ z1 θ S1 (Z1 )S1 (Z1 ) + 2 2  ∗  ∗  1 2 ∗ ∗2 1 2  ∗  ∗



z¯1 g1 ξ1 dp,1 ξ , t ≤ g1 ξ1 z1 + dp,1 ξ , t 2 2 1 2 ∗ ∗2 1 2 ≤ g1 ξ1 z1 + d1 2 2 where ε1∗ = max{εp,1 , p ∈ E}.

z¯∗2 = z∗2 − η2 .

(18)

(22)

∗ z˙¯2 = ξ˙2∗ − α˙ 2,v − η˙ 2     = ξ3∗ + f˜p,2 ξ¯2∗ + d˜ p,2 ξ ∗ , t − α˙ 2,v − η˙ 2 .

(27)

Let z∗3 = ξ3∗ − α3,v and choose η˙ 2 = −k2 η2 + α3,v − α3,a + η3

(28)

where k2 > 0 is the design parameter, α3,v passes a twoorder command filter and the dynamic of the command filter is defined as (29)

where α3,v (t) = ζ2,1 and α˙ 3,v (t) = ωζ2,2 are the outputs of the filter, the design parameters ω and ς are the same as in (15), and the initial values are chosen as ζ2,1 (0) = α3,a (0) and ζ2,2 (0) = 0. Furthermore, we can infer that     ∗ z˙¯2 = z∗3 + f˜p,2 ξ¯2∗ + d˜ p,2 ξ ∗ , t − α˙ 2,v + k2 η2 + α3,a . (30) ∗ S (Z ), p ∈ E to approximate By employing RBF NNs Wp,2 2 2 ∗ ∗ ∗ T f˜p,2 (ξ¯2 ) with Z2 = [ξ1 , ξ2 ] , and let z¯∗3 = z∗3 − η3 , then we choose the pseudocontrol signal α3,a as

  1 α3,a = −k2 z∗2 + α˙ 2,v − z¯∗2 θˆ S2T (Z2 )S2 (Z2 ) − g1 ξ1∗ z¯∗1 . 2 Substituting (31) into (30), we have    ∗ ∗ z˙¯2 = z¯∗3 + f˜p,2 ξ 2 + d˜ p,2 ξ ∗ , t − k2 z¯∗2

(31)

  1 ∗ T z1 θˆ S2 (Z2 )S2 (Z2 ) − g1 ξ1∗ z¯∗1 2   ∗ = z¯∗3 + Wp,2 S2 (Z2 ) + εp,2 + d˜ p,2 ξ ∗ , t − k2 z¯∗2   1 − z∗2 θˆ S2T (Z2 )S2 (Z2 ) − g1 ξ1∗ z¯∗1 . (32) 2 Consider the Lyapunov function candidate as follows: −

1 2 V2 = V1 + z¯∗2 2

(23) then we have (24)

(26)

The time derivative of z¯∗2 is

ζ˙2,1 = ωζ2,2   ζ˙2,2 = −2ς ωζ2,2 − ω ζ2,1 − α3,a

Combining (16) with (18), it can be obtained that 1   ∗ z˙¯∗1 = −k1 z¯∗1 − g21 ξ1∗ z∗1 + Wp,1 S1 (Z1 ) + εp,1 2     1 + d˜ p,1 ξ ∗ , t + g1 ξ1∗ z¯∗2 − z∗1 θˆ S1T (Z1 )S1 (Z1 ). (19) 2 Further, we consider the Lyapunov function candidate as follows: 1 (20) V1 = z¯∗2 2 1 then the time derivative of V1 along with (19) is 1 2  ∗  ∗2 ∗ ∗ ∗ V˙ 1 = −k1 z¯∗2 1 − g1 ξ1 z¯ 1 + z¯ 1 Wp,1 S1 (Z1 ) + z¯ 1 εp,1 2       dp,1 ξ ∗ , t + g1 ξ1∗ z¯∗1 z¯∗2 + z¯∗1 g1 ξ1∗

1 θˆ ST (Z1 )S1 (Z1 ). − z¯∗2 (21) 2 1 1 By utilizing Young’s inequality, we get z¯∗1 εp,1 ≤

Substituting (22)–(24) into (21), we can infer that   1 2 2 V˙ 1 ≤ −k1 z¯∗1 + g1 ξ1∗ z¯∗1 z¯∗2 + z¯∗1 θ˜ S1T (Z1 )S1 (Z1 ) 2 1 1 2 1 1 2 + + z¯∗1 + ε1∗ 2 + d1 2 2 2 2   1 2 2 ≤ −k1 z¯∗1 + g1 ξ1∗ z¯∗1 z¯∗2 + z¯∗1 θ˜ S1T (Z1 )S1 (Z1 ) 2 1 1 1 2 (25) + + z¯∗1 + ε1∗ 2 + d¯ 12 . 2 2 2 Step 2: Let z∗2 = ξ2∗ − α2,v , and the compensated tracking error signal is chosen as

(33)

  2 ∗ V˙ 2 = V˙ 1 + z¯∗2 Wp,2 S2 (Z2 ) + z¯∗2 εp,2 + z¯∗2

dp,2 ξ ∗ , t − k2 z¯∗2   1 2 T ˆ 2 (Z2 )S2 (Z2 ) − g1 ξ1∗ z¯∗1 z¯∗2 . (34) + z¯∗2 z¯∗3 − z¯∗2 θS 2

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By employing the Young’s inequality, we infer that 1 ¯∗ 2 T 1 z θ S2 (Z2 )S2 (Z2 ) + 2 2 2 1 ¯∗ 2 1 2 1 ¯∗ 2 1 ∗ 2 ∗ z¯2 εp,2 ≤ z2 + εp,2 ≤ z2 + ε2 2 2 2 2  ∗  1 ∗2 1 2  ∗  1 ∗2 1 2 ∗

z¯2 dp,2 ξ , t ≤ z¯2 +

d ξ , t ≤ z¯2 + d¯ 2 2 2 p,2 2 2 ∗ where ε2 = max{εp,2 , p ∈ E}. Substituting (35)–(37) into (34), we obtain ∗ S2 (Z2 ) ≤ z¯∗2 Wp,2

(35) (36) (37)

1 2 1 1 V˙ 2 ≤ V˙ 1 + z¯∗2 θ˜ S2T (Z2 )S2 (Z2 ) + + d¯ 22 2 2 2 1 2 2 2 ∗ ∗ ∗ ∗ + z¯2 + ε2 − k2 z¯2 + z¯3 − z¯∗1 z¯∗2 2  2  2    1 ∗2 T −kl z¯∗l 2 + z¯∗l 2 + ≤ z¯l θ˜ Sl (Zl )Sl (Zl ) 2 l=1 l=1  2   1 ∗2 1 2 εl + dl + 1 + z¯∗2 z¯∗3 . + (38) 2 2 l=1

Step i (3 ≤ i ≤ n − 1): Let z∗i = ξi∗ − αi,v , and the compensated tracking error signal is defined as z¯∗i = z∗i − ηi . The time derivative of

∗ z˙¯i

(39)

is given by

∗ z¯˙i = ξ˙i∗ − α˙ i,v − η˙ i     ∗ = ξi+1 + f˜p,i ξ¯i∗ + d˜ p,i ξ ∗ , t − α˙ i,v − η˙ i .

(40)

∗ −α Let z∗i+1 = ξi+1 i+1,v and choose

η˙ i = −ki ηi + αi+1,v − αi+1,a + ηi+1

(42)

with αi+1,v (t) = ζi,1 and α˙ i+1,v (t) = ωζi,2 are the outputs of the filter, ω and ς are the same as in (15), and the initial values are chosen as ζi,1 (0) = αi+1,a (0) and ζi,2 (0) = 0. Then, substituting (41) into (40), one gets     ∗ z˙¯i = z∗i+1 + f˜p,i ξ¯i∗ + d˜ p,i ξ ∗ , t − α˙ i,v + ki ηi + αi+1,a . (43) ∗ S (Z ), p ∈ E to approximate By employing RBF NNs Wp,i i i f˜p,i (ξ¯i∗ ) with Zi = [ξ1∗ , . . . , ξi∗ ]T , and let z¯∗i+1 = z∗i+1 − ηi+1 , then we choose the pseudocontrol signal αi+1,a as

1 αi+1,a = −ki z∗i + α˙ i,v − z∗i θˆ SiT (Zi )Si (Zi ) − z∗i−1 . 2 Combining (43) with (44), we have     ∗ z˙¯i = z¯∗i+1 + f˜p,i ξ¯i∗ + d˜ p,i ξ ∗ , t − ki z¯∗i 1 − z¯∗1 θˆ S1T (Z1 )S1 (Z1 ) − z¯∗i−1 2   ∗ = z¯∗i+1 + Wp,i Si (Zi ) + εp,i + d˜ p,i ξ ∗ , t − ki z¯∗i 1 − z∗i θˆ SiT (Zi )Si (Zi ) − z¯∗i−1 . 2

Furthermore, the Lyapunov function candidate of this step is defined as 1 2 (46) Vi = Vi−1 + z¯∗i 2 then the time derivative of Vi along with (45) is   2 ∗ V˙ i = V˙ i−1 + z¯∗i Wp,i dp,i ξ ∗ , t − ki z¯∗i Si (Zi ) + z¯∗i εp,i + z¯∗i

1 2 + z¯∗i z¯∗i+1 − z¯∗i θˆ SiT (Zi )Si (Zi ) − z¯∗i−1 z¯∗i . (47) 2 By applying the Young’s inequality, we conclude that 1 ¯∗ 2 T 1 zi θ Si (Zi )Si (Zi ) + 2 2 1 ¯∗ 2 1 2 1 ¯∗ 2 1 ∗ 2 ∗ z¯i εp,i ≤ zi + εp,i ≤ zi + εi 2 2 2 2  ∗  1 ∗2 1 2  ∗  1 ∗2 1 2 ∗

¯

z¯i dp,i ξ , t ≤ zi + dp,i ξ , t ≤ z¯i + d¯ i 2 2 2 2

∗ Si (Zi ) ≤ z¯∗i Wp,i

(48) (49) (50)

where εi∗ = max{εp,i , p ∈ E}. Substituting (48)–(50) into (47), it can be inferred that 1 2 T 1 1 ˜ i (Zi )Si (Zi ) + + d¯ i2 V˙ i ≤ V˙ i−1 + z¯∗i θS 2 2 2 1 2 2 2 ∗ ∗ ∗ ∗ + z¯i + εi − ki z¯i + z¯i+1 − z¯∗i−1 z¯∗i 2  i  i    1 ∗2 T −kl z¯∗l 2 + z¯∗l 2 + ≤ z¯l θ˜ Sl (Zl )Sl (Zl ) 2 l=1 l=1  i   1 ∗2 1 ¯ 2 i + εl + dl + + z¯∗i z¯∗i+1 . (51) 2 2 2 l=1

(41)

where ki > 0 is the design parameter, αi+1,v passes a twoorder command filter and the state equation of the command filter is defined as ζ˙i,1 = ωζi,2   ζ˙i,2 = −2ς ωζi,2 − ω ζi,1 − αi+1,a

5

(44)

(45)

Step n: Define z∗n = ξn∗ − αn,v and the compensated tracking error signal as z¯∗n = z∗n − ηn then the time derivative of z¯∗n is given by     z˙¯∗n = u + f˜p,n x¯ n∗ + dp,n x∗ , t − α˙ n,v − η˙ n .

(52)

(53)

∗ S (Z ), p ∈ E to approxiBy employing RBF NNs Wp,n n n mate f˜p,n (ξ¯n∗ ) with Zi = [ξ1∗ , . . . , ξn∗ ]T , and let η˙ n = −kn ηn with kn > 0 is the design parameter. Therefore, (53) can be expressed as   ∗ ∗ z˙¯n = u + Wp,n Sp (Zn ) + εp,n +

dp,n ξ ∗ , t − α˙ n,v + kn zn . (54)

Now, the actual controller is designed as follows: 1 (55) u = −kn z∗n + α˙ n,v − z¯∗n θˆ SnT (Zn )Sn (Zn ) − z¯∗n−1 . 2 Substituting (55) into (54), we can get   ∗ ∗ z˙¯n = Wp,n Sn (Zn ) + εp,n +

dp,n ξ ∗ , t − kn z¯∗n 1 − z∗n θˆ SnT (Zn )Sn (Zn ) − z¯∗n−1 . (56) 2 In the final step, the total Lyapunov function candidate is defined as 1 2 1 Vn = Vn−1 + z¯∗n + θ˜ 2 (57) 2 2r

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with r > 0. Then differentiating Vn yields 1 ˙ V˙ n = V˙ n−1 + z¯∗n z¯∗n − θ˜ θ˙˜ r 1 ∗ ∗ = V˙ n−1 + z¯n Wp,n Sn (Zn ) − z¯∗n z¯∗n−1 + z¯∗n εp,n − θ˜ θ˙˜ r  ∗  1 2 2 ∗

T + z¯n dp,n ξ , t − kn z¯∗n − z¯∗n θˆ Sn (Zn )Sn (Zn ). 2 Similar to (35)–(37), we have 1 ∗2 T 1 z¯n θ Sn (Zn )Sn (Zn ) + 2 2 1 ∗2 1 1 ∗2 1 ∗2 ∗ 2 ¯ ¯ z¯n εp,n ≤ zn + εp,n ≤ zn + εn 2 2 2 2  ∗  1 2 1 2  ∗  1 2 1 2 ∗

∗ ∗

¯ ¯ z¯n dp,n ξ , t ≤ zn + dp,n ξ , t ≤ zn + dn 2 2 2 2 where εn∗ = max{εp,n , p ∈ E}. Substituting (59)–(61) into (58), we obtain ∗ Sn (Zn ) ≤ z¯∗n Wp,n

(58)

(59) (60) (61)

1 2 T 1 1 ˜ n (Zn )Sn (Zn ) + + d¯ n2 V˙ n ≤ V˙ n−1 + z¯∗n θS 2 2 2 1 2 2 2 2 + z¯∗n + εn∗ − kn z¯∗n + z¯∗n − z¯∗n−1 z¯∗n 2  n  n    1 ∗2 1 ¯ 2 εl + dl −kl z¯∗l 2 + z¯∗l 2 + ≤ 2 2 l=1

l=1

n n θ˜   r ∗ 2 T z¯l θ˜ Sl (Zl )Sl (Zl ) − θ˙ˆ + . + r 2 2

(62)

l=1

Define the adaptive law as follows: n   r ∗2 T θ˙ˆ = z¯l Sl (Zl )Sl (Zl ) − k0 θˆ 2

(63)

l=1

with k0 > 0. Nothing that k0 k0 k0 θ˜ θˆ ≤ − θ˜ 2 + θ 2 . (64) r 2r 2r Consider (63) and (64), then (62) can be transformed into  n  n  k   1 ∗2 1 ¯ 2 0 ˜2 ∗2 ∗2 ˙ ε + dl Vn ≤ −kl z¯l + z¯l − θ + 2r 2 l 2 l=1

l=1

k0 2 n θ + ≤ −αVn + β + (65) 2r 2 where α = min{k0 , 2ki − 2, i = 1, 2, . . . , n} and β = n ((1/2)ε∗ 2 + (1/2)d¯ 2 ) + (k0 /2r)θ 2 + (n/2). l=1

l

l

B. Stability Analysis The stability of the proposed closed-loop system (1) with the designed controller (55) is analyzed in this section. Theorem 1: Consider the switched closed-loop system consisting of (1), (55), and (63). Suppose that Assumption 1 holds. Then under arbitrary switching signal, the following features of the closed-loop system can be proven. 1) The symmetric output constraint (2) is never transgressed, i.e., the output y(t) remains within the set sy = {y(t) ∈ R| − lc < y < lc }, ∀t ≥ 0. 2) The boundedness of all the variables of the closed-loop system (1) can be assured.

3) The output tracking error z∗1 can be reduced arbitrarily small by appropriate choice of the design parameters, which implies y(t) can follow the target signal yd (t). Proof: 1) For stability analysis of the closed-loop system (1), we define the Lyapunov function V = Vn , then it can be inferred from (65) that   β −α(t−t0 ) β e . (66) V(t) ≤ + V(t0 ) − α α Further, based on (66) and [35, Th. 1], we can infer that all the variables in V, i.e., z∗i , z¯∗i , i = 1, 2, . . . , n, and

θ are uniformly ultimately bounded. The boundedness of z∗1 (t) and y∗d (t) implies that the state ξ1∗ (t) is also bounded. Hence, from the boundedness of ξ1∗ (t) and (5), we can conclude that ξ1 (t) = y(t) ∈ sy , ∀t ≥ 0. 2) Because θ is a constant and

θ is bounded, therefore θˆ is also bounded. From the coordinate transformations z¯∗i = z∗i − ηi , and z∗i , z¯∗i , i = 1, 2, . . . , n are bounded, it is easy to check that ηi are bounded. Since α1,a is defined by ξ1∗ , y˙ d , z∗1 , z¯∗1 and θˆ , therefore it follows that α1,a is bounded, and the boundedness of α˙ 1,v can also be assured based on the command filter in (15), which means ξ2∗ is bounded. Further, we can progres∗ are sively infer that the signals αi,a , αi,v , α˙ i,v and ξj+1 bounded, i = 2, 3, . . . , n, j = 2, 3, . . . , n − 1. In addition, it is concluded from (55) that u(t) is bounded too. Finally, it can be easily verified, from ξ2∗ = ξ2 , ξ3∗ = ξ3 , ξ4 , . . . , ξn∗ = ξn that ξj (j = 2, 3, . . . , n) are bounded. Therefore, we deduce that all the closed-loop variables of (1) are bounded. 3) Furthermore, from (66), we can obtain that V(t) ≤ V(0)e−αt +

β , ∀t ≥ 0 α

(67)

and as t −→ ∞, we conclude that β α

(68)

2β . α

(69)

lim V(t) ≤

t→+∞

which implies lim z¯∗2 t→+∞ 1

≤

From (69) and [35, Th. 2], we know that, by properly adjusting the design parameters, the output tracking error z∗1 = ξ1∗ −y∗d (t) can be made arbitrarily small. In addition, due to the output y∗ (t) can follow the target signal y∗d (t), we can also infer that y(t) can follow the target signal yd (t). Detailedly, let e∗ (t) = ξ1∗ (t) − y∗d (t) and e(t) = ξ1 (t) − yd (t), it follows from (4) and (5) that: e = ξ1 − yd (t)     2 2 = lc 1 − ξ ∗ − lc 1 − y∗ (t) 1 d +1 ⎡ e +1 ⎤e y∗d (t) ξ1∗ e −e  ∗ ⎦ = 2lc ⎣  ∗ (70) yd (t) ξ1 e +1 e +1

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7

where ξ2∗ = ξ2 , ξ3 , . . . , ξn∗ = ξn , h1 (ξ1∗ ) = (1/lc1 + ∗ ∗ lc2 )(eξ1 + e−ξ1 + 2), f˜p,1 (ξ1∗ ) = h1 (ξ1∗ )fp,1 (lc2 − ((lc1 + ∗ ∗ lc2 )/eξ1 + 1)), f˜p,i (ξ¯i∗ ) = fp,i (lc2 − ((lc1 + lc2 )/eξ1 + = h1 (ξ1∗ ) dp,1 (lc2 − 1), ξ2∗ , ξ3∗ , . . . , ξi∗ ), d˜ p,1 (ξ ∗ , t) ξ1∗ ((lc1 + lc2 )/e + 1), ξ2∗ , ξ3∗ , . . . , ξn∗ , t), d˜ p,i (ξ ∗ , t) = dp,i ∗ (lc2 − ((lc1 + lc2 )/eξ1 + 1), ξ2∗ , ξ3∗ , . . . , ξn∗ , t), i = 1, 2, . . . , n, p = 1, 2, . . . , m. Accordingly, the desired trajectory becomes y∗d = log Fig. 2.

Schematic of an asymmetric BLF with lc = 0.4 and lc2 = 0.3.

then

⎡

⎤ y∗d (t) ξ1∗ − e e  ∗ ⎦ = 0 lim e = ∗ lim∗ 2lc ⎣  ∗ e∗ →0 ξ1 →yd (t) eyd (t) + 1 eξ1 + 1 (71)

which means if

ξ1∗ (t)

→

y∗d (t),

then ξ1 (t) → yd (t).

C. Control Design for Asymmetric Output Constraint In this section, we will deal with the case of the system (1) when the output is restricted to the asymmetric constraint (3). Our control aim here is to construct an adaptive neural tracking controller for system (1) such that the target signal yd (t) can be tracked within a small bounded error and all the closedloop variables remain bounded with the asymmetric output constraint (3) is not violated. Assumption 2: For the limitation lc1 < 0, lc2 > 0, suppose that the following inequalities: −N 0 ≤ yd (t) ≤ N 0 , |˙yd (t)| < (n) N1 , . . . , |yd (t)| < Nn hold, where yd (t) is the target signal, and N 0 , N 0 , ly , N1 , . . . , Nn are positive constants with max{N 0 , N 0 } ≤ ly < lci , i = 1, 2, ∀t ≥ 0. For the output constraint (3) is asymmetric, we first define the following asymmetric BLF: ξ1∗ = log

lc 1 + ξ 1 . lc 2 − ξ 1

(72)

A schematic of an asymmetric BLF with lc1 = 0.4 and lc2 = 0.3 is shown in Fig. 2. It follows from (72) that:   l c 1 + lc 2 (73) ξ1 = lc 2 − ξ ∗ e 1 +1   ξ∗ ∗ lc + lc2 e 1 ξ˙1 ξ˙1 = 1 (74) 2 . ∗ eξ1 + 1 Based on (74), it is therefore that the system (1) can be rewritten as       ξ˙1∗ = h1 ξ1∗ ξ2∗ + f˜p,1 ξ1∗ + d˜ p,1 ξ ∗ , t     ∗ ξ˙i∗ = ξi+1 + f˜p,i ξ¯i∗ + d˜ p,i ξ ∗ , t , i = 2, 3, . . . , n − 2     ∗ ∗ +

dσ (t),n−1 ξ ∗ , t = ξn∗ + fσ (t),n−1 ξ¯n−1 ξ˙n−1     ξ˙n∗ = u + f˜p,n ξ ∗ + d˜ p,n ξ ∗ , t y∗ = ξ1∗

(75)

lc1 + yd . lc2 − yd

(76)

It should be emphasized that the situation is the same as for the system (1) with symmetric output constraint. Therefore, we present the second main result without repeating the details of the control design procedure. Theorem 2: Under Assumption 2, consider the system (1) with the asymmetric output constraint (3) and suppose that all the unknown nonlinear functions fp,i (ξ¯i ) for 1 ≤ i ≤ n, p ∈ E can be approximated by NNs. Then the corresponding controller and the adaptive law can be constructed similar to (55) and (63) such that under arbitrary switchings, the following properties of the closed-loop system are guaranteed. 1) The asymmetric output constraint (3) is never transgressed, i.e., the output y(t) remains within the set ay = {y(t) ∈ R| − lc1 < y < lc2 }, ∀t ≥ 0. 2) The boundedness of all the closed-loop variables of (1) can be assured. 3) The output tracking error z∗1 can be reduced arbitrarily small by appropriate choice of the design parameters, which implies y(t) can follow the target signal yd (t). Proof: The method of proof is quite similar to that of Theorem 1 and is omitted. Remark 4: Compared with the proposed control algorithms in [24], the advantages of the proposed adaptive neural control algorithm in this paper are twofold. First, our control algorithms are useful for both switched nonlinear systems and nonswitched nonlinear systems. However, the control algorithms proposed in [24] are only suitable for nonlinear systems without switchings. Second, the control algorithms in [24] cannot be applied to the systems with output constraints. Instead, our proposed control algorithms can handle the output-constrained systems effectively. IV. I LLUSTRATIVE E XAMPLES To check the effectiveness and availability of the proposed control schemes, two simulation examples will be provided in this section. Example 1: We consider the following numerical example: ⎧ ξ˙1 = ξ2 + f1,σ (t) (ξ ⎪ ⎪  1 ) + d1,σ (t) (ξ, t) ⎨˙ ξ2 = ξ3 + f2,σ (t) ξ¯2 + d2,σ (t) (ξ, t) (77) ξ˙ = u + f3,σ (t) ξ¯3 + d3,σ (t) (ξ, t) ⎪ ⎪ ⎩ 3 y = ξ1 where σ (t) : [0, +∞) → E = {1, 2} with f1,1 (ξ1 ) = e−ξ1 ξ1 , f1,2 (ξ1 ) = 0.5ξ1 , f2,1 (ξ¯2 ) = ξ1 sin(ξ2 ), f2,2 (ξ¯2 ) = cos(ξ1 )ξ2 , f3,1 (ξ¯3 ) = (ξ2 /1 + ξ22 ξ32 ), f3,2 (ξ¯3 ) = ξ22 + ξ1 ξ3 , d1,1 (ξ, t) = 0.1 sin(t) sin(ξ1 ξ2 ), d1,2 (ξ, t) = 0.3 sin(t)

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Trajectories of y∗ (t) and y∗d (t).

sin(ξ2 ξ3 ), d2,1 (ξ, t) = 0.3 cos(t) cos(ξ2 ξ3 ), d2,2 (ξ, t) = 0.4 cos(t) cos(ξ1 ξ3 ), d3,1 (ξ, t) = 0.2 sin(t) cos(t) cos(ξ1 ξ2 ) and d3,2 (ξ, t) = 0.2sin2 (t) sin(ξ2 ξ3 ). It is assumed that these nonlinearities are unknown. The target signal is given as yd = 0.3 sin(t), and the symmetric output constraint is defined as |y| < lc = 0.4. By defining the BLF ξ1∗ = log(0.4 + ξ1 /0.4 − ξ1 ) and ξ2∗ = ξ2 ,ξ3∗ = ξ3 , the system (77) is transformed into     ⎧ ∗ ξ˙1 = g1 ξ1∗ ξ2∗+ f˜p,1 ξ1∗ + d˜ p,1 (ξ ∗ , t) ⎪ ⎪ ⎨ ˙∗ ξ2 = ξ3∗ + f˜p,2 ξ¯2∗ + d˜ p,2 (ξ ∗ , t) (78) ⎪ ξ˙ ∗ = u + f˜p,3 ξ¯3∗ + d˜ p,3 (ξ ∗ , t) ⎪ ⎩ 3∗ ∗ y = ξ1 ξ1∗

Fig. 4.

Trajectories of y(t) and yd (t).

Fig. 5.

Given switching signal σ (t).

Fig. 6.

ˆ Trajectory of the adaptive law θ.

−ξ1∗

where g1 (ξ1∗ ) = (1/0.8)(e + e + 2), f˜p,1 (ξ1∗ ) = ξ1∗ ∗ ∗ ˜ g1 (ξ1 )fp,1 (0.4 − (0.8/e + 1)), dp,1 (ξ1 , t) = g1 (ξ1∗ )dp,1 (t), ∗ f˜p,i (ξ¯i∗ ) = fp,i (0.4 − (0.8/eξ1 + 1), . . . , ξi∗ ) and d˜ p,i (ξ¯i∗ , t) = ∗ dp,i (0.4 − (0.8/eξ1 + 1), . . . , ξi∗ , t), p = 1, 2, i = 2, 3. Further, the transformed target signal becomes y∗d = log(0.4 + 0.3 sin(t)/0.4 − 0.3 sin(t)). According to Theorem 1, we construct the actual control law as 1 (79) u = −k3 z∗3 + α˙ 3,v − z∗3 θˆ S3T (Z3 )S3 (Z3 ) − z∗2 2 the virtual control laws as  1 1   α2,a =  ∗  −k1 z∗1 + y˙ ∗d − g21 ξ1∗ z∗1 2 g1 ξ1  1 ∗ T (80) − z1 θˆ S1 (Z1 )S1 (Z1 ) 2 1 α3,a = −k2 z∗2 + α˙ 2,v − z∗2 θˆ S2T (Z2 )S2 (Z2 ) 2   − g1 x1∗ z∗1 (81) and the adaptive law as 3   r ∗2 T θ˙ˆ = z¯l Sl (Zl )Sl (Zl ) − k0 θˆ . 2

(82)

l=1

In the simulation, the design parameters of (79)–(82) are chosen as k1 = 25, k2 = 50, k3 = 75, ω = 105, ς = 0.96, r = 0.4, and k0 = 0.06, and the initial values are set ξ0∗ = [0.5, 0, 0]T and θˆ (0) = 0. Finally, the simulation results are shown in Figs. 3–7. Example 2: To further verify the applicability of the proposed design method, we consider an electromechanical

system shown by Fig. 8, and the model of the electromechanical system is given by [56]  D¨q + B˙q + N sin(q) = τ (83) M τ˙ + Hτ = V − Km q˙ . From [56], one can find the physical meaning of all the parameters in the system (83). Moreover, in this example, the values of all the parameter in (83) are chosen as the same as in [56]. As is shown in [57], the electromechanical system (83) can be modeled as the following switched nonlinear system model: ξ˙1 = fk,1 + ξ2   ξ3 ξ˙2 = fk,2 ξ¯2 + M 1 ˙ξ3 = fk,3 + u L y = ξ1 , k ∈ {1, 2}.

(84)

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Fig. 7.

Trajectory of the control input u.

Fig. 8.

Schematic of the electromechanical system.

Fig. 9.

Trajectories of y∗ (t) and y∗d (t).

When k = 1, it describes an electromechanical system, where f1,1 = 0, f1,2 = −(B/M)ξ2 − (N/M) sin(ξ1 ) and f1,3 = −(R/L)ξ3 − (KB /L)ξ2 ; when k = 2, it describes a general nonlinear system, where f2,1 = 0, f2,2 = sin(ξ1 ξ2 ) and f2,3 = cos(ξ1 ξ2 ξ3 ). All the nonlinear functions here are assumed to be unknown. The target signal is given as yd (t) = 0.1(sin(t) + sin(0.1t)), and the output is subject to the asymmetric constraint −0.3 < y < 0.4, t ≥ 0. For the output constraint is asymmetric, we define the asymmetric BLF as ξ1∗ = log

0.3 + ξ1 0.4 − ξ1

(85)

and the transformed target signal is thus y∗d = log [0.3+ (0.1(sin(t) + sin(0.1t)))/0.4 − (0.1(sin(t) + sin(0.1t)))]. Similar to Example 1, the system (84) can be transformed into a unconstrained system based on (75) and (85). In order

9

Fig. 10.

Trajectories of y(t) and yd (t).

Fig. 11.

Given switching signal σ (t).

Fig. 12.

Trajectory of the adaptive law θˆ .

to simplify the design process, we do not give the transformed equation here. Similarly, based on Theorem 1, we construct the actual control law as  1 ∗ T ∗ ∗ ˆ (86) u = L −k3 z3 + α˙ 3,v − z3 θS3 (Z3 )S3 (Z3 ) − z2 2 the virtual control laws as 1 α2,a = −k1 z∗1 + y˙ ∗d − z∗1 θˆ S1T (Z1 )S1 (Z1 ) 2   1 ∗ T ∗ ∗ ˆ α3,a = M −k2 z2 + α˙ 2,v − z2 θ S2 (Z2 )S2 (Z2 ) − z¯1 2 and the adaptive law as 3   r ∗2 T θ˙ˆ = z¯l Sl (Zl )Sl (Zl ) − k0 θˆ . 2 l=1

(87) (88)

(89)

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Fig. 13.

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Trajectory of the control input u.

In the simulation, the design parameters in the equations (86)–(89) are chosen as k1 = 25, k2 = 50, k3 = 75, ω = 85, ς = 0.9, r = 0.4, and k0 = 0.06, and the initial values are set ξ0∗ = [2, 1, −0.5]T and θˆ (0) = 0. Finally, the simulation results are shown by Figs. 9–13. V. C ONCLUSION In this paper, we have proposed the controller design method for a class of uncertain output-constrained nonlinear switched lower-triangular systems. The application of the adaptive backstepping technique is generalized to a class of uncertain nonlinear switched systems with command filter technique and the CLF method. The key to the success of the proposed controller design method is the introduction of a novel BLF that is used to convert the issue of controlling the output-constrained system into the issue of regulating the transformed system without a constraint. Furthermore, the presented approach assures that all the variables of the resulting closed-loop system keep bounded and the system output can follow the desired target signal while never violating the output constraint. In addition, the novel asymmetric BLF is also proposed, which can be used to handle the asymmetric output constraint. In the end, two simulation examples are provided to illustrate the applicability of the developed results. R EFERENCES [1] P. Mhaskar, N. H. El-Farra, and P. D. Christofides, “Stabilization of nonlinear systems with state and control constraints using Lyapunovbased predictive control,” Syst. Control Lett., vol. 55, no. 8, pp. 650–659, 2006. [2] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, vol. 36, no. 6, pp. 789–814, 2000. [3] Y.-J. Liu and S. C. Tong, “Barrier Lyapunov functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints,” Automatica, vol. 64, pp. 70–75, Feb. 2016. [4] Y.-J. Liu and S. C. Tong, “Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems,” Automatica, vol. 76, pp. 143–152, Feb. 2017. [5] W. He, Y. H. Chen, and Z. Yin, “Adaptive neural network control of an uncertain robot with full-state constraints,” IEEE Trans. Cybern., vol. 46, no. 3, pp. 620–629, Mar. 2016. [6] K. D. Do, “Control of nonlinear systems with output tracking error constraints and its application to magnetic bearings,” Int. J. Control, vol. 83, no. 6, pp. 1199–1216, 2010.

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Ben Niu was born in Shandong Province, China, in 1982. He received the B.S. degree in mathematics and applied mathematics from Liaocheng University, Liaocheng, China, in 2007, and the M.S. and Ph.D. degrees in pure mathematics and control theory and applications from Northeastern University, Shenyang, China, in 2009 and 2012, respectively. He was a Post-Doctoral Fellow with the College of Control Science and Engineering, Dalian University of Technology, Dalian, China, from 2014 to 2016. He joined Bohai University, Jinzhou, China, in 2013, where he is currently an Associate Professor. His current research interests include switched systems, stochastic systems, robust control, intelligent control, and their applications.

Yanjun 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, and the Ph.D. degree in control theory and control engineering from the Dalian University of Technology, Dalian, China, in 2007. He is currently a Professor with the College of Science, Liaoning University of Technology, Jinzhou, China. He is currently an Associate Editor of the IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS : S YSTEMS . His current research interests include adaptive fuzzy control, nonlinear control, neural network control, reinforcement learning, and optimal control.

Guangdeng Zong received the Ph.D. degree in control theory and application from Southeast University, Nanjing, China, in 2005. He is currently a Full Professor with the School of Engineering, Qufu Normal University, Jining, China. His current research interests include time-delay systems, switched systems, and robust control.

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Zhaoyu Han was born in Liaoning Province, China, in 1992. She received the B.S. degree in mathematics and applied mathematics from Bohai University, Jinzhou, China, in 2014, where she is currently pursuing the M.S. degree in operation science and control theory. Her current research interests include switched nonlinear systems and adaptive control.

Jun Fu (SM’12) received the Ph.D. degree in mechanical engineering from Concordia University, Montreal, QC, Canada, in 2009. He is a Post-Doctoral Fellow/Associate with the Department of Mechanical Engineering, Massachusetts Institute of Technology (MIT), Cambridge, MA, USA, from 2010 to 2014. He is a Full Professor with Northeastern University, Shenyang, China. He has authored/co-authored over 60 publications which appeared in journals, conference proceedings, and book chapters. His current research interests include mathematical programming, dynamic optimization, switched systems, control of hysteretic systems, and robust control of nonlinear systems. Dr. Fu was a recipient of the Chinese Government Award for outstanding self-financed Ph.D. student abroad. He has actively participated in considerable number of conference organizations as the Session Chair, including the 2011 American Control Conference. He was a Founding Member and the Chair of the Information Flow and IT Committee of the MIT Post-Doctoral Association.