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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS

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A Novel Neural-Network-Based Adaptive Control Scheme for Output-Constrained Stochastic Switched Nonlinear Systems Ben Niu , Ding Wang , Huan Li, Xuejun Xie, Naif D. Alotaibi, and Fuad E. Alsaadi

Abstract—In this paper, a novel neural-network (NN)-based adaptive tracking controller design method is presented for the single-input/single-output nonlinear stochastic switched systems in lower triangular structures with an output constraint. First, a well-designed nonlinear mapping is introduced to transform the switched stochastic system to a new system without constraints, which implies the controller design of the transformed system is equivalent to that of the stochastic switched system. Then radial basis function NNs are applied to model the unknown nonlinearities and the adaptive backstepping technique is employed to construct two classes of adaptive neural controllers under different adaptive laws. It is proved that both controllers can assure all the signals in the closed-loop remain bounded in probability, and the tracking error finally converges to a neighborhood of the origin without violating the constraint. Furthermore, the use of the nonlinear mapping to deal with the asymmetric output constraint is also studied as a generalization result. Two illustrative examples with numerical data and simulation results are given to show the validity and performance of the proposed control schemes. Index Terms—Adaptive control, neural network (NN), nonlinear mapping, output constraints, stochastic switched systems.

I. I NTRODUCTION TOCHASTIC nonlinear models play a key role in many fields of science and technology applications and have been extensively investigated in the past years [1]–[5]. When the nonlinear stochastic systems are in lower triangular structures, the control design can be achieved by the backstepping

S

Manuscript received October 8, 2017; accepted November 20, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61673073 and Grant 61673242, and in part by the Liaoning Provincial Natural Science Foundation, China under Grant 201602009. This paper was recommended by Associate Editor W. He. (Corresponding author: Ben Niu.) B. Niu and H. Li are with the College of Mathematics and Physics, Bohai University, Jinzhou 121013, China (e-mail: niubenbhu@gmail.com; lihuanbhu@163.com). D. Wang is with the State Key Laboratory of Management and Control for Complex Systems Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China (e-mail: ding.wang@ia.ac.cn). X. Xie is with the Institute of Automation, Qufu Normal University, Qufu 273165, China (e-mail: xxj@qfnu.edu.cn). N. D. Alotaibi and F. E. Alsaadi are with the Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia (e-mail: ndalotaibi@kau.edu.sa; fuad-alsaadi@yahoo.com). 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/TSMC.2017.2777472

technique, which is a powerful tool to analyze stability and boundedness of such systems in probability [6]–[11]. On the other hand, the existence of various uncertainties in practical systems is inevitable. Due to the effect of uncertainty, the nonlinear systems may inhibit instability, hence the operating performance of such systems are hard to guarantee except under properly designed controls. Therefore, the stability analysis and the robust control design for uncertain nonlinear systems attracted a number of researchers over the past decades. Furthermore, adaptive neural (or fuzzy) control has attracted more and more attention if the nonlinear systems contain unknown nonlinearities [12]–[27], which has been regarded as a promising way to handle robust control problems. Until now, several important adaptive neural or fuzzy controller design techniques have been developed not only for the deterministic cases but also for the stochastic ones [28]–[34]. Especially by combining the backstepping synthesis design with radial basis function, neural network (NN) technique works [35]–[39] proposed effectively direct adaptive NN tracking controllers for nonlinear deterministic systems in lower triangular structures. Since then, by successfully extending the above-mentioned approaches to stochastic nonlinear systems, considerable results based on adaptive neural control were developed for stochastic nonlinear systems in lower triangular structures and their variants in [34] and [40]–[42]. Recently, control design of switched stochastic nonlinear systems has been widely developed [43]–[48]. Consequently, many significant accomplishments have been explored in these works on the properties of such systems under various circumstances. In particular, [49] considered the stability analysis problem for switched stochastic systems by using a comparison principle and multiple Lyapunov functions method, while [50] studied the global stabilization issue of singleinput/single-output (SISO) nonlinear strict-feedback stochastic switched systems under arbitrary switchings. However, to the best of the authors’ awareness, few results have been reported for stochastic nonlinear switched lower triangular systems with completely unknown system functions. Notice further, in cases of the mentioned switched stochastic nonlinear systems, different unknown uncertain nonlinearities may well appear as the system is switched among different subsystems. How to remove or compensate for those unknown uncertain nonlinearities is not a well understood topic. It is therefore understood that adaptive neural control seems an appropriate tool to solve

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this problem, however, how to expand it to the setting of switched systems and switching-based controls is a more than just challenging work. On the other hand, driven by the actual theoretical challenges and demand for innovative techniques, the control design problems of switched nonlinear systems with constraints has become a new research topic [51]–[56] of considerable interest. It appears that few works have been accomplished on the constrained controller design problems for nonlinear switched systems in lower triangular structures despite a number of outstanding developments in the area of switched systems and switching-based controls. In work [57], barrier Lyapunov functions (BLFs) are used to solve the control problems of the output-constrained (or state-constrained for that matter) switched nonlinear lower triangular systems under arbitrary switchings. However, an emanating problem appeared in the fact that the constructed asymmetric BLF is of a switching type, a C1 function. As [58] pointed out, the subsequently derived stabilizing controller functions may cause high power. This is somewhat undesirable effect because it could decrease the robustness of the control system and increase the control effort. Motivated by the aforementioned observations, this paper is focused on the adaptive neural tracking controller design problem for a class of SISO switched stochastic nonlinear lower triangular systems subject to an output constraint. The novel nonlinear mapping is successfully used to convert the design of the switched stochastic system with an output constraint into the design of the transformed system without a constraint. Furthermore, the proposed control method is further applied to design a tracking controller for a continuously stirred tank chemical reactor, known as a benchmark plant [50], and to an academic example in order to demonstrate the effectiveness and the feasibility of the results contributed in this paper. The main contributions are identified as follows. 1) It seems that this paper is the first one dedicated to adaptive neural tracking control problem of nonlinear lower triangular output-constrained stochastic switched systems by using the designed nonlinear mapping. 2) For the case of asymmetrical output constraint, the problem is turned into the one with symmetrical constraint by using a coordinate transformation, which overcomes the difficulties encountered in the control of using the asymmetrical BLF in work [57]. 3) The proposed control design method under single adaptive law needs only to consider one adaptive parameter, which has nothing to do with the complexity of the number of the NN nodes. Therefore, the burden of computation is significantly decreased and the developed results are thus more widely applicable to the practical system engineering problems. 4) Due to the value of the backstepping method as a nonlinear design tool and the wider applicability of the method developed here, it can be extended to various classes of nonlinear control systems and beyond, possibly to interconnected large-scale, multiagent systems, etc.

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II. P ROBLEM F ORMULATION AND P RELIMINARIES We consider the following class of switched lower triangular stochastic nonlinear systems: dxi = xi+1 + fσ (t),i (¯xi ) dt + gTσ (t),i (¯xi )dω i = 1, 2, . . . , n − 1 dxn = u + fσ (t),n (x) dt + gTσ (t),n (x)dω y = x1

(1)

where σ (t) : [0, ∞) → I = {1, 2, . . . , m} stands for the switching signal. x = (x1 , x2 , . . . , xn )T , u and y are the state, the control input, and the output of the system, respectively, x¯ i = (x1 , x2 , . . . , xi )T , i = 1, 2, . . . , n − 1. fk,i (·) and gk,i (·) are the unknown nonlinear smooth functions satisfying fk,i (0) = 0 and gk,i (0) = 0, ∀i = 1, 2, . . . , n, k = 1, 2, . . . , m. ω is a standard Wiener process with E{dω(t)} = 0. The system output y(t) is required to remain within the set y = {y ∈ R|E{|y|} < kc } ∀t ≥ 0

(2)

with kc being a positive constant. The control objective presented in this paper will design an adaptive neural tracking controller to guarantee that all the signals in the closed-loop system are bounded and the system output can follow the reference signal yd in probability with the output constraint (2) is never violated. Assumption 1: For any kc > 0, there exist positive constants B0 , B0 , ky , B1 , B2 , . . . , Bn satisfying max{B0 , B0 } ≤ ky < kc such that the desired trajectory yd (t) and its time derivatives satisfy −B0 ≤ yd (t) ≤ B0 , |˙yd (t)| < B1 , |ÿd (t)| < B2 , . . . , |y(n) d (t)| < Bn , ∀t ≥ 0. Remark 1: It seems that Assumption 1 is conservative for some practical problems because it requires the reference signal should be n-order continuously differentiable. The reason why we use Assumption 1 is that the backstepping technique used in this paper requires to get the ith derivative of the reference signal yd , i = 1, . . . , n. However, it should be pointed out that Assumption 1 is a common assumption when the backstepping technique is used to solve the tracking control issue for nonlinear lower triangular systems, and the similar assumptions can be found in [22] and [30]. In order to prevent the system output from transgressing the constraint, we introduce a specific nonlinear mapping transformation as defined below, which will be applied to derive our main results. Definition 1 [59]: A nonlinear mapping H : x1 → x1∗ is defined as follows: kc + x1 x1∗ = log . (3) kc − x1 It is fairly easy to check that H is a continuous one-to-one elementary function. Also, the curve of H is depicted in Fig. 1. From (3), we can deduce that 2 x1 = H−1 = kc 1 − x∗ (4) e 1 +1 and ∗ 2kc ex1 ∗ dx1 = (5) 2 dx1 . ∗ x1 e +1

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Proof: Due to log(·) is smooth function in its domain, we know that the elementary function H(yd ) ∈ Cn when yd ∈ y . Furthermore, yd (t) ∈ Cn and yd (t) ∈ y , ∀t ≥ 0 satisfying y∗d (t) = H(yd (t)) = (H ◦ yd )(t).

(10)

In the view of differential chain rules, the ith order time derivative of y∗d (t) is existed, ∀i = 1, 2, . . . , n, thus we get that y∗d (t) ∈ Cn . What is more important, on the grounds of the boundedness of y∗d (t) and the monotonicity of the mapping (3), we have kc + ky kc + ky ≤ y∗d (t) ≤ log (11) − log kc − ky kc − ky Fig. 1.

Sketch map of the H in Definition 1.

Substituting (5) into the x1 -equation of (1), we have dx1∗ = h1 x1∗ x2 + f˜σ (t),1 x1∗ dt + g˜ Tσ (t),1 x1∗ dω

(6)

∗ ∗ where h1 (x1∗ ) = (1/2kc )(ex1 + e−x1 + 2), f˜k,1 (x1∗ ) = ∗ ∗ ∗ (1/2kc )(ex1 + e−x1 + 2)fk,1 (kc − (2kc /[ex1 + 1])), g˜ Tk,1 (x1∗ ) = ∗ ∗ ∗ (1/2kc )(ex1 + e−x1 + 2)gTk,1 (kc − (2kc /[ex1 + 1])), k ∈ I. Furthermore, by denoting x2∗ = x2 , x3∗ = x3 , . . . , xn∗ = xn , and noticing 2kc ∗ ∗ fk,i kc − x∗ , x2 , . . . , xi = f˜k,i x¯ i∗ 1 e +1 2kc T , x2∗ , . . . , xi∗ = g˜ Tk,i x¯ i∗ gk,i kc − x∗ e 1 +1 i = 1, 2, . . . , n, k = 1, 2, . . . , m (7)

we can rewrite the considered class of switched nonlinear stochastic systems (1) in the form dx1∗ = h1 x1∗ x2∗ + f˜k,1 x1∗ dt + g˜ Tk,1 x1∗ dω ∗ dxi∗ = xi+1 + f˜k,i x¯ i∗ dt + g˜ Tk,i x¯ i∗ dω dxn∗ ∗

i = 2, 3, . . . ,n − 1 = u + f˜k,n x∗ dt + g˜ Tk,n x∗ dω

y = x1∗

(8) x1∗

−x1∗

where h1 (x1∗ ) = (1/2kc )(e + e + 2), x¯ i∗ = ∗ ∗ ∗ T ∗ (x1 , x2 , . . . , xi ) , i = 1, 2, . . . , n, and x = x¯ n∗ , k ∈ I. Apparently, for ∀k = 1, . . . , m, i = 1, . . . , n, the nonlinear functions f˜k,i (·) are unknown and smooth, g˜ k,i (·) are smooth functions with g˜ k,i (0) = 0. It is therefore that the desired trajectory to be tracked becomes

which in turn means y∗d (t) is bounded too. Remark 2: In fact, the boundedness of y∗d can be established from the boundedness of yd , because of y∗d = log([kc + yd ]/[kc − yd ]) and −ky ≤ yd ≤ ky . Lemma 2 [59]: Consider the mapping x1∗ = H(x1 ) in Definition 1, if x1∗ (t) → b(t), then it holds true x1 (t) → a(t) = H−1 (b(t)). Proof: Based on the elementary function H is continuous in its domain, we know that its inverse mapping H−1 is also a continuous elementary function. Let e∗ (t) = x1∗ (t) − b(t) and e(t) = x1 (t) − a(t), it follows from (3) that ⎡ ⎤ x1∗ b e − e ⎦ e = x1 − a = 2kc ⎣ (12) ∗ eb + 1 ex1 + 1 then

⎡

⎤ x1∗ b e − e ⎦ = 0 2kc ⎣ lim e = lim x ∗ e∗ →0 x1∗ →b b 1 e +1 e +1

(13)

which means if x1∗ (t) → b(t), then x1 (t) → a(t). Remark 3: In the view of Lemmas 1 and 2, it can be obtained that if x1∗ → y∗d , then x1 → yd . Thus, the controller design for system (8) is equivalent to the one for system (1). In this paper, the unknown nonlinear functions of the system (1) will be approximated by radial basis function NNs. Therefore, the following relevant details should be recalled. For any continuous unknown smooth function f (Z) defined in a compact set Z ⊂ Rq , there exists NN W ∗ T S(Z) as discussed in [17], [18], and [60], such that for a desired level of accuracy ε > 0 f (Z) = W ∗ T S(Z) + δ(Z), |δ(Z)| ≤ ε where

(14)

W∗

(9)

is the ideal constant weight vector and defined by

∗ T W = arg min sup f (Z) − W S(Z) (15)

From Definition 1, we know that x1∗ in the system (8) is a unconstrained variable and its domain is all real numbers (or R). Furthermore, the following two lemmas show that x1∗ → y∗d is equivalent to x1 → yd . Lemma 1 [59]: Consider the mapping y∗d = H(yd ) in Definition 1, if yd (t) ∈ Cn is bounded, then y∗d (t) ∈ Cn is also bounded.

δ(Z) is the approximation error, W = [w1 , . . . , wN ]T is the weight vector, and S(Z) = [s1 (Z), . . . , sN (Z)]T is the basis function vector with N being the number of the NN nodes and N > 1. Radial basis function si (Z) = exp [−(Z − ui )T (Z − ui )/ηi2 ], i = 1, 2, . . . , N, where ui = [ui1 , ui2 , . . . , uin ]T is the center of the receptive field and is the width of the Gaussian function.

y∗d = log

kc + yd . kc − yd

W∈RN

Z∈Z



Design the adaptive law θˆ˙i as follows:

III. C ONTROL D ESIGN In this section, we present an adaptive neural control design scheme for system (1) with the symmetric output constraint (2) that is based on the backstepping technique. Based on Remark 2, it is easy to know that if we are able to design a state-feedback controller such that all closed-loop signals in (8) are bounded without violation of the constraint, and the tracking error converges to a neighborhood of the origin, then by Remark 3 the control objective of the system (1) is achieved. First, we will give a detailed adaptive control design procedure under multiple adaptive laws. Thereafter, for the sake of avoiding double presentation of a similar development process, we briefly present the design procedure under single adaptive law. A. Adaptive Control Design Under Multiple Adaptive Laws For system (8), the virtual controls and the actual controller are designed as θˆ1 T 3 1 ξ1 z1 + 2 S1 (X1 )S1 (X1 )z31 − z1 (16) α1 (X1 ) = − h1 4 2a1 θî 3 zi − 2 SiT (Xi )Si (Xi )z3i αi (Xi ) = − ξi + 4 2ai i = 2, 3, . . . , n − 1 (17) θˆn T (18) u(Xn ) = −ξn zn − 2 Sn (Xn )Sn (Xn )z3n 2an where θî stands for the estimation of θi = Wmax,i 2 with Wmax,i 2 ≥ Wk,i 2 and the meaning of Wk,i will be specified in the later, k = 1, 2, . . . , m. ξi > 0, ai > 0 are ∗(i)T design parameters, i = 1, 2, . . . , n. Xi = [¯xi∗T , θ¯îT , y¯ d ]T , ∗(i) ∗(i) ∗(i) = [y∗d , y˙ ∗d , y¨ ∗d , . . . , yd ]T , yd θ¯î = [θˆ1 , θˆ2 , . . . , θî ]T , y¯ d ∗ represents the ith order time derivative of yd . Furthermore, zi satisfies the following change of coordinates: zi = xi∗ − αi−1 (X i−1 ), i = 1, 2, . . . , n

(19)

where αi are the virtual control signals with α0 = y∗d . Then based on Itô differentiation formula, we get, for ∀i = 1, 2, . . . , n dzi = Lzi dt + λzi dω

(20)

∗ Lzi = xi+1 + f˜σ (t),i −

j=1

−

i−1 j=1

−

∂αi−1 ˙ θˆj − ∂ θˆj

∂αi−1 ∗ xj+1 + f˜σ (t),j ∗ ∂xj

i−1 ∂αi−1

∗(j+1) y ∗(j) d j=0 ∂yd

i−1 1 ∂ 2 αi−1 T g˜ g˜ σ (t),q 2 ∂xp∗ ∂xq∗ σ (t),p p,q=1

λzi = g˜ Tσ (t),i −

i−1 ∂αi−1 j=1

with

∗ xn+1

= uσ (t) .

∂xj∗

g˜ Tσ (t),j

(22)

where λi and γi are positive design constants. Step 1: The tracking error is defined as z1 = x1∗ − y∗d , for ∀k ∈ I, one can obtain (23) dz1 = h1 x2∗ + f˜k,1 − y˙ ∗d dt + g˜ Tk,1 dω. Now, let us consider a Lyapunov function candidate as V1 =

1 4 z . 4 1

(24)

It can be readily verified from (23) and (24) that 3 LV1 = z31 h1 x2∗ + f˜k,1 − y˙ ∗d + z21 g˜ Tk,1 g˜ k,1 2 3 3 3 ∗ ∗ ≤ z1 h1 x2 + f˜k,1 − y˙ d + l1−2 z1 ˜gk,1 4 + l12 4 4 3 3 = z31 h1 x2∗ + f¯k,1 − z41 + l12 (25) 4 4 where f¯k,1 = f˜k,1 − y˙ ∗d + (3/4)l1−2 z1 ˜gk,1 4 + (3/4)z1 with l1 being a positive constant. In addition, by the estimation theory T S (X ), and then f¯ of NNs, f¯k,1 can be modeled by Wk,1 1 1 k,1 can be rewritten as follows: T f¯k,1 = Wk,1 S1 (X1 ) + δ1 (X1 ), |δ1 (X1 )| ≤ ε1

(26)

here δ1 (X1 ) stands for the approximation error and ε1 > 0 stands for a design constant. Substituting (26) into (25) results in the expression T LV1 = z31 h1 x2∗ + z31 Wk,1 S1 (X1 ) + z31 δ1 (X1 ) 3 3 − z41 + l12 4 4 Wk,1 2 3 ∗ ≤ z1 h1 x2 + S1T (X1 )S1 (X1 )z61 2a21 1 1 3 + a21 + ε14 + l12 2 4 4

(27)

where a1 is a positive design parameter too. Furthermore, by applying the coordinate transformation (19) with i = 2 and adding and subtracting α1 in (27), we obtain

where i−1

λi θ˙î = 2 SiT (Xi )Si (Xi )z6i − γi θî , i = 1, 2, . . . , n 2ai

(21)

1 1 3 LV1 = z31 h1 x2∗ − α1 + α1 + a21 + ε14 + l12 2 4 4 Wk,1 2 + S1T (X1 )S1 (X1 )z61 2a21 1 1 3 ≤ z31 h1 z2 + z31 h1 α1 + a21 + ε14 + l12 2 4 4 Wmax,1 2 + S1T (X1 )S1 (X1 )z61 2a21 where Wmax,1 2 ≥ Wk,1 2 , ∀k ∈ I.

(28)


5

And yet further, by using the virtual control in (16) and then substituting it into (28), we get 3 1 1 3 LV1 = z31 h1 z2 − ξ1 z41 − h1 z41 + a21 + ε14 + l12 4 2 4 4 Wmax,1 2 − θˆ1 + S1T (X1 )S1 (X1 )z61 2a21 3 1 3 1 1 ≤ h1 z41 + h1 z42 − ξ1 z41 − h1 z41 + a21 + ε14 4 4 4 2 4 2 3 2 Wmax,1 − θˆ1 T + l1 + S1 (X1 )S1 (X1 )z61 4 2a21 1 1 3 1 = −ξ1 z41 + a21 + ε14 + l12 + h1 z42 2 4 4 4 Wmax,1 2 − θˆ1 + S1T (X1 )S1 (X1 )z61 . (29) 2a21

where f¯k,2 = f˜k,2 −Lα1 +(3/4)l2−2 z2 ˜gk,2 −(∂α1 /∂x1∗ )˜gk,1 4 + (1/4)h1 z2 + (3/4)z2 with l2 > 0 being a constant. Similar to T S (X ), and then f¯ step 1, f¯k,2 can be modeled by Wk,2 2 2 k,2 can be expressed as T S2 (X2 ) + δ2 (X2 ), |δ2 (X2 )| ≤ ε2 f¯k,2 = Wk,2

where δ2 (X2 ) represents the approximation error and ε2 represents a given positive design parameter. By adding and subtracting αi with i = 2 in (17) and combining it with (34), the following inequality can be obtained: LV2 ≤

The term (1/4)h1 z42 in there will be handled in the next step of the procedure. Step 2: Upon defining z2 = x2∗ − α1 , it follows that: T ∂α1 dz2 = x3∗ + f˜k,2 − Lα1 dt + g˜ k,2 − ∗ g˜ k,1 dω (30) ∂x1

Wmax,1 2 − θˆ1 2a21

S1T (X1 )S1 (X1 )z61

1 1 3 − ξ1 z41 + a21 + ε14 + l12 + z32 z3 2 4 4 1 2 1 4 32 3 + z2 α2 + a2 + ε2 + l2 2 4 4 Wmax,2 2 + S2T (X2 )S2 (X2 )z62 2a22

where ∂α1 ∗(j+1) ∂α1 ∗ ˜ ∂α1 ˙ x2 + fk,1 + θˆ1 + y ∗ ∗(j) d ∂x1 ∂ θˆ1 j=0 ∂yd

(34)

(35)

1

Lα1 =

+

1 ∂ 2 α1 T g˜ g˜ k,1 . 2 ∂x1∗2 k,1

(31)

Consider a Lyapunov function candidate in the form 1 (32) V2 = V1 + z42 . 4 It follows from (30) and (32) that: LV2 = LV1 + z32 x3∗ + f˜k,2 − Lα1 T 3 ∂α1 ∂α1 + z22 g˜ k,2 − ∗ g˜ k,1 g˜ k,2 − ∗ g˜ k,1 2 ∂x1 ∂x1 3 ∗ ≤ LV1 + z2 x3 + f˜k,2 − Lα1 4 3 −2 4 ∂α1 3 + l2 z2 g˜ k,2 − ∗ g˜ k,1 + l22 4 ∂x1 4 Wk,1 2 − θˆ1 ≤ −ξ1 z41 + S1T (X1 )S1 (X1 )z61 2a21 1 1 3 + a21 + ε14 + l12 2 4 4 3 ∂α1 + z32 x3∗ + f˜k,2 + l2−2 z2 ˜gk,2 − ∗ g˜ k,1 4 − Lα1 4 ∂x1 1 32 + h1 z2 + l2 4 4 2 Wk,1 − θˆ1 = S1T (X1 )S1 (X1 )z61 2a21 1 1 3 − ξ1 z41 + a21 + ε14 + l12 2 4 4 3 3 + z32 x3∗ + f¯k,2 − z42 + l22 (33) 4 4

where Wmax,2 2 ≥ Wk,2 2 , ∀k ∈ I and a2 > 0 stands for a design constant. Upon using the virtual control in (17) with i = 2 and then substituting it into (35), we infer that LV2 ≤

2 Wmax,j 2 − θˆj j=1

2a2j

SjT

6 Xj Sj Xj zj

1 1 3 − ξ1 z41 + a21 + ε14 + l12 + z32 z3 − ξ2 z42 2 4 4 3 4 1 2 1 4 32 − z2 + a2 + ε2 + l2 4 2 4 4 2 2 Wmax,j − θˆj T 6 ≤ Sj Xj Sj Xj zj 2a2j j=1 1 1 3 3 1 − ξ1 z41 + a21 + ε14 + l12 + z42 + z43 2 4 4 4 4 3 4 1 2 1 4 32 4 − ξ2 z2 − z2 + a2 + ε2 + l2 4 4 4 2 2 2 Wmax,j − θˆj = SjT (Z)Sj (Z)z6j 2 2a j j=1 1 1 1 3 − ξj z4j + a2j + εj4 + lj2 + z43 . 2 4 4 4

(36)

Step i: Similarly as before, let zi = xi∗ − αi−1 . Then according to Itô’s formula we have ∗ + f˜k,i − Lαi−1 dt dzi = xi+1 ⎛ ⎞T i−1 ∂αi−1 + ⎝g˜ k,i − g˜ k,j ⎠ dω ∂xj∗ j=1

(37)



i−1 where f¯k,i = f˜k,i + (3/4)li−2 zi ˜gk,i − j=1 ([∂αi−1 ]/ [∂xj∗ ])˜gk,j 4 − Lαi−1 + zi with lj > 0 being constants. T S (X ), k ∈ I are used to approximate f¯ Furthermore, Wk,i i i k,i such that (43) f¯k,i = W T Si (Xi ) + δi (Xi ), |δi (Xi )| ≤ εi

where Lαi−1 =

i−1 ∂αi−1 ∗ ˜k,j x + f j+1 ∂xj∗ j=1

i−1 i−1 ∂αi−1 ˙ ∂αi−1 ∗(j+1) y θˆj + ∗(j) d ∂ θˆj ∂y

+

j=1

1 2

+

j=0

i−1 p,q=1

k,i

d

∂ 2 αi−1 T g˜ g˜ k,q . ∂xp∗ ∂xq∗ k,p

(38)

Now, consider a stochastic Lyapunov function candidate as follows: 1 Vi = Vi−1 + z4i . (39) 4 With regards to (37) and (39), it follows that: ∗ LVi = LVi−1 + z3i xi+1 + f˜k,i − Lαi−1 ⎛ ⎞T i−1 ∂αi−1 3 2⎝ g˜ k,j ⎠ + zi g˜ k,i − 2 ∂xj∗ j=1 ⎛ ⎞ i−1 ∂αi−1 × ⎝g˜ k,i − g˜ k,j ⎠. (40) ∂xj∗ j=1

From the procedures in the formerly derived steps, the expression of LVi−1 in (40) can be corrected to be i−1 Wmax,j 2 − θˆj LVi−1 ≤ SjT Xj Sj Xj z6j 2 2a j j=1 1 1 3 1 − ξj z4j + a2j + εj4 + lj2 + z4i (41) 2 4 4 4 where Wmax,j 2 ≥ Wk,j 2 and aj > 0 are design parameters, k ∈ I, j = 1, 2, . . . , i − 1. Thus it can be obtained that i−1 Wmax,j 2 − θˆj LVi ≤ SjT Xj Sj Xj z6j 2 2a j j=1 1 2 1 4 32 4 − ξj zj + aj + εj + lj 2 4 4 ⎛ 4 i−1 ∂α 3 i−1 ⎜ −2 3 ∗ ˜ g˜ k,j + zi ⎝xi+1 + fk,i + li zi g˜ k,i − ∗ 4 ∂xj j=1 ⎞ 1 ⎟ 3 − Lαi−1 + zi ⎠ + li2 4 4 =

i−1 Wmax,j 2 − θˆj j=1

2a2j

SjT Xj Sj Xj z6j

1 1 3 − + a2j + εj4 + lj2 2 4 4 3 3 ∗ + z3i xi+1 + f¯k,i − z4i + li2 4 4

where δi (Xi ) stands for the approximation error and εi > 0 stands for a given design constant. Then following the same deduction above, we can obtain the following inequality: i Wmax,j 2 − θˆj SjT Xj Sj Xj z6j LVi ≤ 2 2a j j=1 1 2 1 4 32 1 4 − ξj zj + aj + εj + lj + z4i+1 (44) 2 4 4 4 where Wmax,j 2 ≥ Wk,j 2 and aj > 0 are positive design constant, k ∈ I, j = 1, 2, . . . , i. Step n: Again define zn = xn∗ −αn−1 and then by Itô formula, we have dzn = uk + f˜k,n − Lαn−1 dt ⎛ ⎞T n−1 ∂α n−1 + ⎝g˜ k,n − g˜ k,j ⎠ dω (45) ∂xj∗ j=1

where Lαn−1 =

n−1 ∂αn−1 ˜k,j u + f k ∂xj∗ j=1

+

n−1 n−1 ∂αn−1 ˙ ∂αn−1 ∗(j+1) θˆj + y ∗(j) d ∂ θˆj ∂y j=1

+

1 2

j=0

n−1 p,q=1

d

∂ 2 αn−1 T g˜ g˜ k,q . ∂xp∗ ∂xq∗ k,p

(46)

Now, consider the following stochastic Lyapunov function candidate: n 1 4 1 2 ˜ (47) z + θ Vn = 4 j 2λj j j=1

where θ˜j = θj − θˆj and j = 1, 2, . . . , n. With regard to (45) and (47), and then taking (44) with i = n − 1 into account yields the following inequality: n−1 Wmax,j 2 − θˆj SjT Xj Sj Xj z6j LVn ≤ 2 2a j j=1 1 1 3 1 ˙ − ξj z4j + a2j + εj4 + lj2 − θ˜j θˆj 2 4 4 λj ⎛ ⎜ + z3n ⎝uk + f˜k,n − Lαn−1

ξj z4j

(42)

⎞ 4 n−1 ∂αn−1 3 1 ⎟ + ln−2 zn ˜ k,j ∗ g g˜ k,n − + 4 zn ⎠ 4 ∂x j j=1


32 1 ln − θñ θ˙ˆn 4 λn n−1 Wmax,j 2 − θˆj

7

+ =

2a2j

j=1

By combining (51) with (52), we can conclude the resulting inequality as follows: SjT Xj Sj Xj z6j

1 1 3 1 − + a2j + εj4 + lj2 − θ˜j θ˙ˆj 2 4 4 λj 3 1 3 + z3n uk + f¯k,n − z4n + ln2 − θñ θ˙ˆn 4 4 λn

LVn ≤

ξj z4j

(48)

f¯k,n = f˜k,n − Lαn−1 + − ∗ ])˜ 4 + z and l are positive con([∂α ]/[∂x g n−1 k,j n j j j=1 stants, j = 1, 2, . . . , n. Similar to the above steps, f¯k,n can be rewritten as (3/4)ln−2 zn ˜gk,n

where n−1

T f¯k,n = Wk,n Sn (Xn ) + δn (Xn ), |δn (Xn )| ≤ εn

(49)

where δn (Xn ) stands for the approximation error and εn > 0 is a design constant. By means of designing the actual controller in (18) and substituting (49) into (48), the following inequality can be obtained: n−1 Wmax,j 2 − θˆj SjT Xj Sj Xj z6j LVn ≤ 2 2a j j=1 1 2 1 4 32 1 ˙ 4 ˜ ˆ − ξj zj + aj + εj + lj − θj θj 2 4 4 λj Wk,n 2 − θˆn T Sn (Xn )Sn (Xn )z6n 2a2n 1 1 3 1 + a2n + εn4 + ln2 − θñ θ˙ˆn 2 4 4 λn n Wmax,j 2 − θˆj T 6 = Sj Xj Sj Xj zj 2a2j j=1 − ξn z4n +

1 1 3 1 − ξj z4j + a2j + εj4 + lj2 − θ˜j θ˙ˆj 2 4 4 λj

(53)

j=1

which means Vn is a common Lyapunov function for the switched system (1). It is apparent at this point that we have completed the design of the adaptive neural control via the backstepping technique for the considered stochastic systems. Thus, we provide the first main result of this paper by the following theorem. Theorem 1: Consider the whole closed-loop system consists of (1), the controller (18) and the adaptive laws (22). Assume that the packaged functions f¯k,i (Xi ), for 1 ≤ i ≤ n, k ∈ I, can be approximated by NNs in the sense that the approximation error εi is bounded, the initial condition satisfies θî (0) ≥ 0, and the initial output y(0) satisfies −kc < y(0) < kc . Then, under arbitrary switchings, the following closed-loop system properties are guaranteed. 1) The signals zj and θ˜j , j = 1, . . . , n remain within the compact set X defined as ⎧

⎨

n

4

b0 ˜ X = zj , θj

E zj ≤ 4Vn (0) + 4 ⎩ a 0

j=1 ⎫ ! ⎬

2λj b0

˜ , j = 1, 2, . . . , n Vn (0) +

θj ≤ ⎭ b a0 (54)

(50)

where Wmax,j 2 ≥ Wk,j 2 and aj > 0 are design constants, k ∈ I, j = 1, 2, . . . , n. Thereafter, substitution of (22) into (50) yields n Wmax,j 2 − θˆj T 6 Sj Xj Sj Xj zj LVn ≤ 2a2j j=1 1 1 3 1 −ξj z4j + a2j + εj4 + lj2 + γj θ˜j θˆj 2 4 4 λj n 1 2 1 4 32 1 4 ˜ ˆ ≤ −ξj zj + aj + εj + lj + γj θj θj . 2 4 4 λj j=1

(51) It should be further noted that the following inequality holds: 1 1 θ˜j θˆj ≤ − θ˜j2 + θj2 . 2 2

n 1 −ξj z4j − γj θ˜j2 2λj j=1 n 1 2 1 4 32 1 aj + εj + lj + γj θj2 + 2 4 4 2λj

(52)

and eventually shall converge to the compact set X1 defined as ⎧

n ⎨

4 b0 E zj ≤ 4 X1 = zj , θ˜j

⎩ a 0

j=1 ⎫ !

⎬ 2λ b j 0

˜ , j = 1, 2, . . . , n . (55)

θj ≤ ⎭ b a0 2) The output constraint (2) is never violated, i.e., the output y(t) remains within the set y = {y| − kc < E{y} < kc }, ∀t ≥ 0. 3) The boundedness of all closed-loop signals of the control system can be ensured in probability. 4) The quartic mean square tracking error z1 eventually shall converge to the following compact set y∗ :

%

4 & b0 y∗ = y∗ (t) ∈ R

E y∗ − y∗d ≤ 8 ∀t > T1 a0 where T1 denotes a finite time and is specified as T1 = max{0, (1/a0 ) ln(a0 V(0)/b0 )}. Furthermore, the desired tracking performance of y − yd can be obtained by properly adjusting the design parameters in a0 and b0 .



Proof: 1) First, we choose a Lyapunov function V = Vn . Based on (53), we get n 1 −ξj z4j − γj θ˜j2 LV ≤ 2λj j=1 n 1 2 1 4 32 1 2 a + εj + lj + γj θ . + (56) 2 j 4 4 2λj j j=1

let a0 = min{4ξj , γj | j = 1, . . . , n} and b0 = Further, n 2 4 2 2 j=1 {(1/2)aj + (1/4)εj + (3/4)lj + (1/2λj )γj θj }. Then, (56) can be rewritten as LV ≤ −a0 V + b0 ∀t ≥ 0.

(57)

It can be inferred from (57) that dE{V(t)} ≤ −a0 E{V(t)} + b0 dt and thus (58) satisfies b0 −a0 t b0 e + E{V(t)} ≤ V(0) − a0 a0

(58)

(59)

which also means b0 . (60) a0 Thus, with reference to inequality (60), the chosen definition of Lyapunov function V, and [61, Th. 2.1], we conclude that under arbitrary switchings, zj , j = 1, 2, . . . , n are semiglobally, uniformly, and ultimately bounded in the sense of the fourmoment and θ˜j , j = 1, 2, . . . , n are semiglobally, uniformly, and ultimately bounded in mean square, and all the signals in the closed-loop system remain within the set X for all times. Moreover, we can conclude from (59) that

employed coordinate transformations zi = xi∗ − αi−1 that all signals xj∗ remain bounded in probability, j = 1, . . . , n. On the ground of expression (18), it is concluded that u(t) is bounded too. Furthermore, it can be fairly easily verified from the boundedness of xj∗ (j = 1, 2, . . . , n), equality (4) and x2∗ = x2 , x3∗ = x3 , . . . , xn∗ = xn , that xj (j = 1, 2, . . . , n) is bounded in probability. Therefore, we deduced that all the closed-loop signals are and remain bounded. 4) Based on (61), we can deduce that there exists a finite time T1 = max{0, (1/a0 ) ln(a0 V(0)/b0 )} such that %

4 & 8b0 ∀t > T1 . (64) E y∗ − y∗d ≤ 4E[V(t)] ≤ a0 Hence, the desired tracking performance of z1 can be achieved by properly adjusting the design parameters in a0 and b0 . Due to the fact that y → yd as y∗ → y∗d proved in Lemma 2, it is further verified that the desired tracking performance of y − yd can also be obtained by appropriately adjusting the design parameters in a0 and b0 . Remark 4: It is easy to check that the control input (19) ' . In fact, the control input in in the paper [33] depends on W ' (18) is also dependent on W as it contains the variable ' θn and the formula Wmax,n 2 = θn . Thus, the design method in this paper is similar to the one in the paper [33].

E{V(t)} ≤ V(0) +

E{V(t)} ≤ V(0)e−a0 t +

b0 ∀t ≥ 0 a0

(61)

b0 . a0

(62)

and as t −→ ∞, we have lim E{V(t)} ≤

t→+∞

Thus, the signals zj and θ˜j , j = 1, 2, . . . , n eventually shall converge to the compact set X1 . 2) Due to the formula z1 = x1∗ − y∗d and the boundedness of z1 (t) and y∗d (t), we obtain that the state x1∗ (t) is also bounded in probability, ∀t ≥ 0. Then, it can be inferred from (4) that the following formula is true: 2 E{x1 (t)} = E kc 1 − x∗ (t) e 1 + 1 2 ∀t ≥ 0. (63) = kc 1 − E{x1∗ (t)} +1 e Hence, we can conclude from the above equation and the boundedness of x1∗ (t) that −kc < E{x1 (t)} = E{y(t)} < kc , which means y(t) ∈ y , ∀t ≥ 0. 3) As θj is a constant, θˆj is also bounded in probability, ∀j = 1, 2, . . . , n. Since αj is defined by zj and θˆj , therefore, readily it follow that αj is also bounded in probability, ∀j = 1, 2, . . . , n − 1. Thus, we can infer from the

B. Adaptive Control Design Under Single Adaptive Law In this section, we propose a novel procedure for adaptive neural control design under single adaptive law. For system (8), the feasible virtual controls and the actual controller are designed as given by the following equalities: θˆ T 1 3 α1 (X1 ) = − ξ1 z1 + 2 S1 (X1 )S1 (X1 )z31 − z1 (65) h1 4 2a1 θˆ 3 zi − 2 SiT (Xi )Si (Xi )z3i αi (Xi ) = − ξi + 4 2ai i = 2, 3, . . . , n − 1 (66) θˆ T (67) u(Xn ) = −ξn zn − 2 Sn (Xn )Sn (Xn )z3n 2an where θˆ is the estimation of θ = Wmax,max 2 with ∗(i)T Wmax,max 2 ≥ Wmax,i 2 , k ∈ I, Xi = [¯xi∗T , θˆ , y¯ d ]T . ξi > 0, ai > 0 are design constants, i = 1, 2, . . . , n. ∗(i) ∗(i) ∗(i) y¯ d = [y∗d , y˙ ∗d , y¨ ∗d , . . . , yd ]T , and yd is the ith order time ∗ derivative of yd . The adaptive law is defined as follows: λ θ˙ˆ = ST (Xi )Si (Xi )z6i − γ θˆ 2 i 2a i i=1 n

(68)

where quantities λ and γ are positive design parameters. The design procedure is rather similar to one derived in Theorem 1, and therefore, we option only to present the sage with a final stochastic Lyapunov function candidate as V = Vn =

n 1 j=1

ˆ where θ˜ = θ − θ.

4

z4j +

1 2 θ˜ 2λ

(69)


9

According to (19), we have ⎛ ⎞T i−1 ∂αi−1 ∗ dzi = xi+1 + f˜k,i − Lαi−1 dt + ⎝g˜ k,i − g˜ k,j ⎠ dω ∂xj∗

j=1

(70) where Lαi−1 =

i−1 ∂αi−1 j=1

+

∂xj∗

∗ xj+1 + f˜k,j +

i−1 ∂αi−1

∗(j+1) y ∗(j) d j=0 ∂yd

i−1 ∂αi−1 ˙ 1 ∂ 2 αi−1 T g˜ g˜ k,q . θˆ + 2 ∂xp∗ ∂xq∗ k,p ∂ θˆ

(71)

p,q=1

Taking (69) and (70) into account, we can get 4 3 LV ≤ z31 h1 z2 + h1 α1 + f˜k,1 − y˙ ∗d + l1−2 z1 g˜ k,1 + 4 ⎧ ⎛ n−1 ⎪ ⎨ ⎜ + z3j ⎝zj+1 + αj + f˜k,j − Lαj−1 ⎪ j=2 ⎩ 4 ⎞ j−1 ⎟ ∂αj−1 3 −2 + lj zj g˜ k,j − g˜ k,l ∗ ⎠+ 4 ∂xl l=1 ⎛

32 l 41

⎫ ⎪ 3 2⎬ l 4 j⎪ ⎭

+ z3n ⎝uk + f˜k,n − Lαn−1 4 ⎞ n−1 ∂α 3 −2 n−1 ⎠ + ln zn g˜ k,n − g ˜ k,l 4 ∂xl∗ l=1

3 1 + ln2 − θ˜ θ˙ˆ λ 4 3 3 3 3 ≤ z1 h1 z1 + h1 α1 + f¯k,1 − z41 + l12 4 4 4 n−1 3 3 3 z3j zj + αj + f¯k,j − z4j + lj2 + 4 4 4 j=2

3 3 1 + z3n uk + f¯k,n − z4n + ln2 − θ˜ θ˙ˆ 4 4 λ

(72)

where f¯k,1 = f˜k,1 − y˙ ∗d + (3/4)l1−2 z1 ˜gk,1 4 + (3/4)z1 , f¯k,2 = f˜k,2 − Lα1 + (3/4)l2−2 z2 ˜gk,2 − (∂α1 /∂x1∗ )˜gk,1 4 + (1/4)h1 z2 + (3/4)z2 , f¯k,j = f˜k,j − Lαj−1 + (3/4)lj−2 zj ˜gk,j − j−1 ∗ g 4 + z , j = 3, . . . , n with l > 0, k,l j j l=1 ([∂αj−1 ]/[∂xl ])˜ T S (X ) j = 1, 2, . . . , n being a constant. Furthermore, NNs Wk,j j j are employed to approximate f¯k,j , and thus f¯k,j can be rewritten as follows: T Sj Xj + δj Xj , δj Xj ≤ εj f¯k,j = Wk,j

(73)

where again δj (Xj ) stands for the approximation error and εj stands for a given positive constant, j = 1, 2, . . . , n.

Taking into consideration (73) and substituting (65)–(70) into (72), we obtain the following inequality: n Wmax,j 2 − θˆ SjT Xj Sj Xj z6j LV ≤ 2 2a j j=1 1 1 3 1 (74) − ξj z4j + a2j + εj4 + lj2 − θ˜ θ˙ˆ 2 4 4 λ where set of aj expresses positive design parameters. Now, by taking the adaptive law (68) into account we obtain the following resulting inequality: n Wmax,j 2 − θˆ T 6 Sj Xj Sj Xj zj LV ≤ 2a2j j=1 1 1 3 1 − ξj z4j + a2j + εj4 + lj2 + γ θ˜ θˆ 2 4 4 λ n 1 1 2 1 4 32 4 −ξj zj + aj + εj + lj + γ θ˜ θˆ ≤ 2 4 4 λ j=1

n 1 ≤ γ θ˜ 2 −ξj z4j − 2λ j=1 n 1 2 1 4 32 1 + aj + εj + lj + γ θ2 2 4 4 2λ

(75)

j=1

where the inequality θ˜ θˆ ≤ −(1/2)θ˜ 2 + (1/2)θ 2 was used in the same sense as before. Thus, given these circumstances like in Theorem 1, the next main result can be summarized as presented in Theorem 2 below. Theorem 2: Consider the closed-loop system consists of the system (1), the controller (67), and the adaptive law (68). Suppose that the packaged functions f¯k,i (Xi ), for 1 ≤ i ≤ n, k ∈ I, can be approximated by NNs in the sense that the approximation error εi is bounded, the initial condition θˆ satisfies ˆ θ(0) ≥ 0, and the initial output y(0) satisfies −kc < y(0) < kc . Then, under arbitrary switchings, the following properties of the resulting closed-loop system are guaranteed. 1) The error signals zj , j = 1, 2, . . . , n and θ˜ remain within the compact set X defined by ⎧

⎨

n

4

b0 ˜ X = zj , θ

E zj ≤ 4V(0) + 4 ⎩ a 0

j=1 !

b0 2λ

˜ , j = 1, 2, . . . , n V(0) +

θ ≤ b a0 (76) and shall eventually converge to the compact set X1 defined by ⎧

⎨

n

4

b0 ˜ X1 = zj , θ

E zj ≤ 4 ⎩ a 0

j=1 !

2λ b0

˜ , j = 1, 2, . . . , n . (77)

θ ≤ b a0



2) The output constraint is never violated, i.e., the output y(t) remains within the set y = {y| − kc < E{y} < kc }, ∀t ≥ 0. 3) The boundedness of all closed-loop signals of the control system can be guaranteed in probability. 4) The quartic mean square tracking error z1 eventually shall converge to the following compact set y∗ :

%

∗

& b0 ∗ ∗ 4

∗ ≤8 ∀t > T1 y = y (t) ∈ R E y − yd a0 where T1 denotes a finite time and is specified as T1 = max{0, (1/a0 ) ln(a0 V(0)/b0 )}. Furthermore, the desired tracking performance of y − yd can be obtained by properly adjusting the design parameters in a0 and b0 . Proof: In the next step, we present solely the stability analysis of closed-loop system (8). For the chosen stochastic Lyapunov function V that satisfies (75), we set a0 = min{4ξj , γ | j = 1, . . . , n} and b0 = nj=1 {(1/2)a2j + (1/4)εj4 + (3/4)lj2 } + (1/2λ)γ θ 2 . Then we readily have

Then, the system (1) is converted into the following form:

LV ≤ −a0 V + b0 , ∀t ≥ 0.

dς = x2 + f¯k,1 (ς ) dt + g¯ Tk,1 (ς )dω dxi = xi+1 + f¯k,i (ς, x¯ 2:i ) dt + g¯ Tk,i (ς, x¯ 2:i )dω i = 2, 3, . . . , n − 1 dxn = u + f¯k,n (ς, x¯ 2:n ) dt + g¯ Tk,n (ς, x¯ 2:n )dω y=ς

(82)

where x¯ 2:i = (x2 , x3 , . . . , xi )T , f¯k,1 (ς ) = fk,1 (ς + ([kc1 + kc2 ]/2)), g¯ Tk,1 (ς ) = gTk,1 (ς + ([kc1 + kc2 ]/2)), f¯k,i (ς, x¯ 2:i ) = fk,i (ς + ([kc1 + kc2 ]/2), x¯ 2:i ) and g¯ Tk,i (ς, x¯ 2:i ) = gTk,i (ς + ([kc1 + kc2 ]/2), x¯ 2:i ), k ∈ I, i = 2, 3, . . . , n. The desired trajectory becomes yd = yd − ([kc1 + kc2 ]/2). Furthermore, now the nonlinear mapping H : ς → x1∗ is designed as follows: x1∗ = log

It is therefore understood that we can rewrite the system (1) as dx1∗ = h1 x1∗ x2∗ + f˜k,1 x1∗ dt + g˜ Tk,1 x1∗ dω ∗ + f˜k,i x¯ i∗ dt + g˜ Tk,i x¯ i∗ dω dxi∗ = xi+1 dxn∗

i = 2, 3, . . . , n − 1 = u + f˜k,n x∗ dt + g˜ Tk,n x∗ dω

y∗ = x1∗

(84)

where we have x2∗ = x2 , x3∗ = x3 , . . . , xn∗ = xn , f¯k,i (kc − ∗ ∗ (2kc /[ex1 + 1]), x2∗ , . . . , xi∗ ) = f˜k,i (¯xi∗ ), g¯ Tk,i (kc − (2kc /[ex1 + ∗ ∗ 1]), x2∗ , . . . , xi∗ ) = g˜ Tk,i (¯xi∗ ), h1 (x1∗ ) = (1/2kc )(ex1 + e−x1 + 2), x¯ i∗ = (x1∗ , x2∗ , . . . , xi∗ )T , i = 1, 2, . . . , n, x∗ = x¯ n∗ , k ∈ I. Obviously, for ∀i = 1, 2, . . . , n, k = 1, 2, . . . , m, the nonlinear functions f˜k,i (·) are unknown and smooth, while g˜ k,i (·) are smooth functions with g˜ k,i (0) = 0. Accordingly, now the desired trajectory becomes

IV. A SYMMETRIC O UTPUT C ONSTRAINT In this section, we will deal with the case of the system (1) when an asymmetric constraint is imposed on its output. In particular, the system output y(t) is required to remain within the set (79)

where kc1 > 0, kc2 > 0 and kc1 = kc2 . First, the following coordinate transformation is introduced: kc1 + kc2 . (80) ς = x1 − 2 Notice that here, we regard ς as the plant system output, and also define kc = |[(kc2 − kc1 )/2]|. Thus the output constraint (79) can be rewritten as follows: y1 = {ς (t) ∈ R|−kc < ς (t) < kc } ∀t ≥ 0.

(83)

(78)

The rest of proof procedure is omitted here since it is rather similar to the proof in Theorem 1. Remark 5: For the design processes of Theorems 1 and 2, the major difference is that one unknown parameter needs to be defined in each step of the backstepping procedure when control design with multiple adaptive laws is used, which means n adaptive laws should be designed to develop the controller (18); while control design under single adaptive law is adopted, only one unknown parameter needs to be defined in the final step of the backstepping procedure, thus the controller (67) can be obtained by designing only one adaptive law. Remark 6: Compared with control design with multiple adaptive laws, the advantage of control design under single adaptive law is that it can overcome the problem of overparameterization, which means when it is used, the computational burden can be obviously reduced. However, when different unknown parameters need to be estimated, it is taken for granted that control design with multiple adaptive laws should be used.

y1 = {y ∈ R|kc1 < E{y} < kc2 } ∀t ≥ 0

kc + ς . kc − ς

(81)

y∗d = log

kc + yd kc − yd

.

(85)

It should be noted that now the situation is the same as for the system (1) with symmetric output constraint. Thus, we present the third main result without repeating the details of the proposed control design procedure. Theorem 3: Consider the system (1) with the asymmetric output constraint (79) and suppose that the packaged functions f¯k,i (Xi ), for 1 ≤ i ≤ n, k ∈ I can be approximated by NNs with the approximation error εi is bounded. Then the corresponding controller and the adaptive law can be constructed similarly to (18) and (22) such that under arbitrary switchings and when the initial condition θî satisfies θî (0) ≥ 0 as well as the initial output y(0) satisfies kc1 < y(0) < kc2 , the following properties


11

of the resulting closed-loop system are guaranteed. 1) The signals zj and θ˜j , j = 1, . . . , n remain within the compact set X defined by ⎧

n ⎨

4 b0 E zj ≤ 4Vn (0) + 4 X = zj , θ˜j

⎩ a0

j=1 !

2λj b0

˜ Vn (0) + , j = 1, 2, . . . , n

θj ≤ b a0 (86) and eventually shall converge to the compact set X1 defined by ⎧

⎨

n

4

b0 E zj ≤ 4 X1 = zj , θ˜j

⎩ a 0

j=1 !

2λj b0

˜ , j = 1, 2, . . . , n . (87)

θj ≤ b a0 2) The asymmetric output constraint is never violated, i.e., the output y(t) remains within the set y1 = {y|kc1 < E{y} < kc2 }, ∀t ≥ 0. 3) The boundedness of all closed-loop signals of the control system can be assured in probability. 4) The quartic mean square tracking error z1 eventually shall converge to the following compact set y∗ :

%

∗

4 & b0 ∗ ∗

≤8 ∀t > T1 y∗ = y (t) ∈ R E y − yd a 0

where T1 denotes a finite time and is specified as T1 = max{0, (1/a0 ) ln(a0 V(0)/b0 )}. Furthermore, the desired tracking performance of y − yd can be obtained by properly adjusting the design parameters in a0 and b0 . Proof: The proof is considerably similar to the one of Theorem 1, hence omitted. Remark 7: We agree that the use of RBF NNs is not new and other types of estimation techniques may be employed to model the existing unknown nonlinear functions, such as the biological NNs in [62], the multidimensional Taylor network in [63], etc. However, it should be emphasized that RBF NNs have been verified to be an effective and popular estimation tool in [17], [18], [32], and [34], and it is why RBF NNs are utilized in this paper. Furthermore, the main concern of this paper is the handling of the output constraints, thus the conservativeness caused by the use of RBF NNs can be neglected to some degree. Remark 8: It needs to be highlighted that quartic Lyapunov functions (QLFs) has been used to solve the tracking control problem in [30]. However, the presented approach in [30] can not ensure that the output constraint is never violated. We will confirm this fact by simulation study in Example 1. V. I LLUSTRATIVE E XAMPLES In this section, two examples along with the respective numerical data and simulation results are presented to demonstrate the correctness and feasibility of the proposed control designs as well as to illustrate the system performances that can be achieved.

Fig. 2.

Sketch map of the process of CSTR.

Example 1: As is shown in [50], a type of closed, continuously stirred tank, chemical reactor (CSTR) with two modes of feed stream (see Fig. 2) can be modeled as the following stochastic nonlinear switched system model: 1 dx1 = x2 + fσ (t),1 (x1 ) dt + fσ (t),1 (x1 )dω 4 dx2 = udt y = x1 (88) where f1,1 = (1/2)x1 , f2,1 = 2x1 . This exothermic chemical reactor process represents a kind of benchmark problem. The goal is to enforces the output y to track the desired trajectory yd = 0.2 + 0.3 sin(t), subject to the constraint interval y : |E{y}| < kc = 0.56. Define H : x1 → x1∗ as x1∗ = log([0.56 + x1 ]/[0.56 − x1 ]) and x2∗ = x2 , the system (88) is transformed into ∗ 25 x∗ ∗ −x1∗ ∗ ˜ 1 e +e + 2 x2 + fσ (t),1 x1 dt dx1 = 28 1 + f˜σ (t),1 x1∗ dω 4 dx2∗ = udt y∗ = x1∗

(89) x1∗

−x1∗

x1∗

+ 2)(1 − (2/[e + 1])), f˜2,1 = where f˜1,1 = (1/4)(e + e ∗ ∗ ∗ (ex1 + e−x1 + 2)(1 − (2/[ex1 + 1])) and y∗d = log([0.76 + 0.3 sin(t)]/[0.36 − 0.3 sin(t)]). According to results in Theorem 1, the feasible virtual control signal, the actual control law, and the adaptive laws are chosen as follows: θˆ1 T 1 3 ξ1 z1 + 2 S1 (X1 )S1 (X1 )z31 − z1 α1 = − x∗ −x1∗ 4 1 2a1 e +e +2 u = −ξ2 z2 −

θˆ2 T S2 (X2 )S2 (X2 )z32 2a22

λi θ˙î = 2 SiT (Xi )Si (Xi )z6i − γi θî , i = 1, 2. 2ai

(90)

Here, variables denote deviations z1 = x1∗ − y∗d , z2 = x2∗ − α1 . The computer simulation experiment was run with the initial conditions [x1∗ (0), x2∗ (0)]T = [3.2958, 0.0523]T , and [θˆ1 (0), θˆ2 (0)]T = [1, 1]T . Notice next that ξ1 = ξ2 = 4, a1 = 0.1, a2 = 0.5, λ1 = 5, λ2 = 1, and γ1 = γ2 = 0.5.



Fig. 3. Tracking performances of the closed-loop system (88) under BLF and QLF.

Fig. 5.

Responses of the adaptive laws.

) 0.5 ln( 1 + x32 ) sin(x12 x2 ), gT2,3 = 0.5x32 cos(x12 x22 ). The object is let the output y track the desired trajectory yd = 1 + 0.48 sin(t), subject to the asymmetric constraint interval y1 : 0.5 < E{y} < 1.5. First, based on (80), the following coordinate transformation is chosen: ς = x1 − 1

(92)

in which ς is regarded as the new plant system output. By defining kc = 0.5, thus the output constraint of (91) can be rewritten as follows: y1 = {ς (t) ∈ R| − 0.5 < ς (t) < 0.5} ∀t ≥ 0. Fig. 4.

Tracking performances of the closed-loop system (89).

The obtained simulation results, which illustrated the achieved system performance of the CSTR process in the closed loop, are shown in Figs. 3–5. Furthermore, it can be inferred from Fig. 3 that the desired tracking performance is achieved by using the proposed BLF method in this paper; while the output constraint is destroyed by using the proposed QLF method in [30]. Finally, we can conclude that all the figures clearly demonstrate a high quality, controlled performance under the proposed design for the adaptive neural control. Example 2: In order to illustrate further the effectiveness of the proposed control designs, a somewhat more academic, stochastic switched nonlinear system has been investigated. The plant system is represented by the following model: dx1 = x2 + fσ (t),1 (x1 ) dt + gTσ (t),1 (x1 )dω dx2 = x3 + fσ (t),2 (x1 , x2 ) dt + gTσ (t),2 (x1 , x2 )dω dx3 = u + fσ (t),3 (x1 , x2 , x3 ) dt + gTσ (t),3 (x1 , x2 , x3 )dω (91) y = x1 where f1,1 = (1/2)x1 , f2,1 = 2x12 , gT1,1 = x12 , gT2,1 = (1/5)x12 , f1,2 = ([2(1−e−x1 x2 )]/[1+e−x1 x2 ]), f2,2 = (x1 x2 /[1+ x12 + x22 ]), gT1,2 = x12 x2 , gT2,2 = (0.2 + x12 )x2 , f1,3 = x1 x2 x32 , f2,3 = ([x12 + x32 ]/[2(1 + x12 + x22 + x32 )]), gT1,3 =

(93)

Furthermore, the system (91) is changed into the following form: dς = x2 + f¯k,1 (ς ) dt + g¯ Tk,1 (ς )dω dx2 = x3 + f¯k,2 (ς, x2 ) dt + g¯ Tk,2 (ς, x2 )dω dx3 = u + f¯k,3 (ς, x2 , x3 ) dt + g¯ Tk,3 (ς, x2 , x3 )dω y=ς

(94)

where f1,1 = (1/2)(ς + 1), f2,1 = 2(ς + 1)2 , gT1,1 = (ς + 1)2 , gT2,1 = (1/5)(ς + 1)2 , f1,2 = ([2(1 − e−(ς+1)x2 )]/[1 + e−(ς+1)x2 ]), f2,2 = ([(ς + 1)x2 ]/[1 + (ς + 1)2 + x22 ]), gT1,2 = (ς +1)2 x2 , gT2,2 = (0.2+(ς +1)2 )x2 , f1,3 = (ς +1)x2 x32 , f2,3 = ([(ς + 1)2 + x32 ]/[2(1 + (ς + 1)2 + x22 + x32 )]), gT1,3 = ) 0.5 ln( 1 + x32 ) sin((ς + 1)2 x2 ), gT2,3 = 0.5x32 cos((ς + 1)2 x22 ). The desired trajectory of the system (94) becomes yd = yd − 1 = 0.48 sin(t). By defining H : ς → x1∗ as x1∗ = log([0.5 + ς ]/[0.5 − ς ]), ∗ x2 = x2 , and x3∗ = x3 , then the system (94) is converted into the following system: ∗ ∗ dx1∗ = ex1 + e−x1 + 2 x2∗ + f˜σ (t),1 x1∗ dt + g˜ Tσ (t),1 x1∗ dω dx2∗ = x3∗ + f˜σ (t),2 x1∗ , x2∗ dt + g˜ Tσ (t),2 x1∗ , x2∗ dω dx3∗ = u + f˜σ (t),3 x1∗ , x2∗ , x3∗ dt + g˜ Tσ (t),3 x1∗ , x2∗ , x3∗ dω y∗ = x1∗ .

(95)


13

Here, the individual functions and variables denote 1 1 1 x∗ ∗ f˜1,1 = e 1 + e−x1 + 2 − x∗ +1 2 2 e 1 +1 2 ∗ ˜f2,1 = 2 ex1 + e−x1∗ + 2 1 − ∗ 1 +1 2 ex1 + 1 2 ∗ 1 1 T x1 −x1∗ − ∗ +1 g˜ 1,1 = e + e +2 2 ex1 + 1 2 1 1 1 x∗ ∗ e 1 + e−x1 + 2 − x∗ +1 g˜ T2,1 = 5 2 e 1 +1 ⎛ ⎞ 1 1 +1 2 − x∗ e 1 +1

−

2⎝1 − e f˜1,2 =

− 12 −

1 +e

1 2

f˜2,2 =

1+

1 2

−

=

g˜ T2,2 = f˜1,3 =

f˜2,3 =

1 x∗ e 1 +1

1 x∗ e 1 +1

⎠

x2∗

Fig. 6.


Fig. 7.


Fig. 8.


+ 1 x2∗

2 2 + 1 + x2∗

2 1 1 + 1 x2∗ − ∗ 2 ex1 + 1 2 1 1 − ∗ +1 0.2 + x2∗ 2 ex1 + 1 2 1 1 − x∗ + 1 x2∗ x3∗ 2 e 1 +1 2 2 1 1 − + 1 + x3∗ x1∗ 2 e +1 2 ∗ 2 ∗ 2 1 1 2 1 + 2 − x∗ + 1 + x2 + x3

g˜ T1,2

−

1 +1 x∗ e 1 +1

x2∗

e 1 +1

2 ∗ 2 1 1 sin − ∗ + 1 x2∗ = 0.5 ln 1 + x3 2 ex1 + 1 2 ∗ 2 ∗ 2 1 1 − ∗ +1 x2 = 0.5 cos x3 . 2 ex1 + 1 )

g˜ T1,3 g˜ T2,3

The desired trajectory of the system (95) becomes y∗d = log([0.5 + 0.48 sin(t)]/[0.5 − 0.48 sin(t)]). According to the results of Theorem 3, the feasible virtual control signal, the actual control law, and the adaptive laws are constructed as follows: θˆ1 T 1 3 ξ1 z1 + 2 S1 (X1 )S1 (X1 )z31 − z1 α1 = − x∗ −x1∗ 4 1 2a1 e +e +2 θˆ2 3 z2 − 2 S2T (X2 )S2 (X2 )z32 α2 = − ξ2 + 4 2a2 θˆ3 T u = −ξ3 z3 − 3 S3 (X3 )S3 (X3 )z33 2a3 λ i θ˙î = 2 SiT (Xi )Si (Xi )z6i − γi θî , i = 1, 2, 3 (96) 2ai where z1 = x1∗ − y∗d , z2 = x2∗ − α1 , z3 = x3∗ − α2 . The simulation experiment was run under the initial conditions [x1∗ (0), x2∗ (0), x3∗ (0)]T = [0.15, 0.16, −0.03]T , and

[θˆ1 (0), θˆ2 (0), θˆ3 (0)]T = [0, 0, 0]T . Notice further the following specifications taken: ξ1 = ξ2 = 3, ξ3 = 4, a1 = 0.1, a2 = 0.3, a3 = 0.5, λ1 = 3, λ2 = λ3 = 1, γ1 = γ2 = 0.3, and γ3 = 0.1. The simulation results for this plant example when controlled by the proposed design of adaptive neural control are depicted in Figs. 6–9. These simulation


Fig. 9.


Responses of the adaptive laws.

results also performance.

demonstrate

a

remarkable

closed-loop

VI. C ONCLUSION In this paper, we have presented a constructive method to handle the tracking controller design problem by means of an adaptive NN-based control design for a class of uncertain switched stochastic nonlinear lower triangular systems in the presence of an output constraint. In the design process, first a new nonlinear mapping is employed in order to transform the problem of controlling the output-constrained switched stochastic system into the tracking control problem of the transformed system without a constraint but still preserving the lower triangular structures. The backstepping technique for nonlinear systems and NNs approximating the unknown nonlinear system functions are combined to derive the proposed tracking control design, which also employs arbitrary switchings, carried out on the transformed system representation. The proposed control design is shown to insure that, simultaneously, all signals in the resulting switched closed-loop system remain bounded in probabilistic sense and the tracking error converges to a neighborhood of the origin while never violating the output constraint. Two illustrative examples, one being the benchmark CSTR process and the other an academic one, along with complete sets of simulation results are provided to demonstrate the validity of the proposed control schemes and feasibility of high quality control performance in the closed loop. There still exist several significant issues to be explored. One is to improve the constraint (2) such that for t ≥ 0, the result Prob{|y(t)| > kc } = 0 can be realized. Another is to solve the control goal of this paper by designing switching signals. R EFERENCES [1] L. Arnold, Stochastic Differential Equations: Theory and Applications. New York, NY, USA: Wiley, 1972. [2] P. Shi, Y. Q. Xia, G. P. Liu, and D. Rees, “On designing of slidingmode control for stochastic jump systems,” IEEE Trans. Autom. Control, vol. 51, no. 1, pp. 97–103, Jan. 2006.

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