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Adaptive tracking control for a class of uncertain switched nonlinear systems
Automatica 52 (2015) 185–191
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Brief paper
Adaptive tracking control for a class of uncertain switched nonlinear systems✩ Xudong Zhao a,b , Xiaolong Zheng a , Ben Niu c , Liang Liu a a
College of Engineering, Bohai University, Jinzhou 121013, Liaoning, China
b
Chongqing SANY High-intelligent Robots Co., Ltd., Chongqing, 401120, China
c
College of Mathematics and physics, Bohai University, Jinzhou 121013, Liaoning, China
article
info
Article history: Received 24 April 2014 Received in revised form 17 October 2014 Accepted 2 November 2014
Keywords: Switched nonlinear systems Adaptive backstepping Common Lyapunov function Tracking control
abstract This paper is concerned with the problem of tracking control for a class of switched nonlinear systems in lower triangular form with unknown functions and arbitrary switchings. Two classes of state feedback controllers are constructed by adopting the adaptive backstepping technique, and both of them are designed by using the common Lyapunov function (CLF) method. The first controller is designed under multiple adaptive laws. Then, the second one is designed based on constructing a maximum common adaptive parameter, which can overcome the problem of over-parameterization of the first controllers. It is shown that the designed state-feedback controllers can ensure that all the signals remain bounded and the tracking error converges to a small neighborhood of the origin. Finally, simulation results are presented to show the effectiveness of the proposed approaches. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction In the past decades, switched systems have attracted much attention since it can be used to describe a large number of physical and engineering systems, such as networked control systems (Zhao, Hill, & Liu, 2009), near space vehicle control systems (Bao, Li, Chang, Niu, & Yu, 2010), circuit and power systems (Homaee, Zakariazadeh, & Jadid, 2014), to list a few. As the most important issues in the study of switched linear or nonlinear systems, stability analysis and control synthesis are discussed extensively by a lot of researchers, and many excellent results have been obtained for various types of switched systems under arbitrary switching or constraint switching; see e.g., Wang, Wang, and Shi (2009), Xiong, Lam, Gao, and Ho (2005), Zhang and Shi (2009), Zhao and Hill (2008a,b), Zhao, Liu, Yin, and Li (2014), Zhao, Yin, Li, and Niu (2014), Zhao, Zhang, Shi, and Liu (2012) and references therein.
✩ This work was partially supported by the National Natural Science Foundation of China (61203123, 61304054, 61403041), the Liaoning Excellent Talents in University (LR2014035), and the Shandong Provincial Natural Science Foundation, China (ZR2012FQ019). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Andrey V. Savkin under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (X. Zhao), [email protected] (X. Zheng), [email protected] (B. Niu), [email protected] (L. Liu).
http://dx.doi.org/10.1016/j.automatica.2014.11.019 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
It is well known that the stability of a switched system under arbitrary switching can be guaranteed if a CLF exists for all subsystems (Vu & Liberzon, 2005). Therefore, CLF has been extensively used for control synthesis of switched linear systems (Briat & Seuret, 2012, 2013; Lian & Zhao, 2010; Liberzon, 2003; Margaliot & Langholz, 2003; Xiang & Xiao, 2014). Recently, there have been some results reported on the global stabilization problem for switched nonlinear systems in strict-feedback form under arbitrary switchings by using the backstepping technique (Ma & Zhao, 2010; Wu, 2009). Meanwhile, Long and Zhao (2012) investigated the global stabilization problem for a class of switched nonlinear systems in the p-normal form by the so-called power integrator backstepping design method. Unfortunately, uncertainty is not taken into account in the aforementioned papers, which widely exists in practical switched nonlinear systems. For general nonlinear systems in lower triangular form without switchings, it has been shown in the literature (Chen, Liu, Liu, & Lin, 2009; Krstić, Kokotović, & Kanellakopoulos, 1995) that the adaptive backstepping method is a powerful tool for controller design. As a base strategy in adaptive backstepping, the main objective is to cancel the unknown nonlinearity of system by constructing the virtual control functions and adaptive laws. This design method is of course expected to be useful to stabilize switched nonlinear systems with uncertainty. However, to the best of the authors’ knowledge, there is not any related results on
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X. Zhao et al. / Automatica 52 (2015) 185–191
the stability analysis and controller design for uncertain switched nonlinear systems in lower triangular form under arbitrary switchings. The reason lies in that, if we want to construct a CLF, the common virtual control functions need to be found first, but it is somewhat difficult to find these common virtual control functions for different subsystems. Based on the above illustrations, we know that the difficulty encountered in constructing a CLF is how to design a common virtual control function at each step when the adaptive backstepping technique is applied to switched nonlinear systems. One way to deal with this problem is to simply assume the existence and availability of a common virtual control function as was done in Wu (2009). Most recent, the authors in Ma and Zhao (2010) proposed some inequality constraints on the common virtual functions to improve previous results. However, it is required in Ma and Zhao (2010) that the common virtual functions should be first-order derivative, not related to switching laws, and satisfy some inequality constraints, which is somewhat too strict in practice to be applied. Furthermore, the uncertainty is not considered in Ma and Zhao (2010) and Wu (2009). Therefore, the design of the common virtual functions has not been fully solved in the existing literature when the system contains uncertainty, which motivates our present work. In this paper, the adaptive tracking problem is studied for a class of switched nonlinear systems with completely unknown uncertainties. An efficient approach of constructing common virtual control functions is proposed for the considered systems, and state-feedback controllers are designed via the adaptive backstepping technique where a Mamdani-type fuzzy logic system is utilized to approximate the redefined unknown functions. A design method with multiple adaptive laws is presented in first place. Furthermore, another method with only one adaptive law is proposed to avoid the problem of over-parameterization. Both controllers can guarantee that all the closed-loop signals remain bounded, and the system output tracks the reference signal while the computation burden is low. Notations: in this paper, the notations are standard. Rn denotes the n-dimensional Euclidean space, the notation ∥·∥ refers to the Euclidean vector norm. For positive integers 1 ≤ i ≤ n, 1 ≤ j ≤ m, we also denote Ξi,max = max{Ξi,j : 1 ≤ j ≤ m}, Ξi,min = min{Ξi,j : 1 ≤ j ≤ m}. 2. Problem formulation and preliminaries Consider a class of switched nonlinear system in the following form: x˙ i = gi,σ (t ) xi+1 + fi,σ (t ) (¯xi ), i = 1, 2, . . . , n − 1, x˙ n = gn,σ (t ) uσ (t ) + fn,σ (t ) (¯xn ), y = x1 ,
(1)
where x¯ i := (x1 , x2 , . . . , xi ) ∈ R , i = 1, 2, . . . , n is the system state, y is the system output; σ (t ) : [0, +∞) → M = {1, 2, . . . , m} is the switching signal; uk ∈ R is the control input of the kth subsystem; For any i = 1, 2 . . . , n and k = 1, 2, . . . , m, fi,k (¯xi ) is an unknown smooth nonlinear function representing the system uncertainty, and gi,k is a positive constant. Our control objective is to design state-feedback controllers such that the output of system (1) tracks a given time-varying signal yd (t ) within a bounded error and all the signals of the closedloop systems remain bounded under arbitrary switchings. T
i
Assumption 1. The tracking target yd (t ) and its time derivatives up to the nth order are continuous and bounded. Remark 1. System (1) will be reduced to system (1) in Ma and Zhao (2010) when the tracking control problem and the uncertain functions are not taken into account. Therefore, the systems considered in this paper are more general.
In the controller design and stability analysis procedure, fuzzy logic systems will be used to approximate the unknown functions. Therefore, the following useful concept and lemma are first recalled. Fuzzy logic systems include some IF–THEN rules, and the ith IF–THEN rule is written as Ri : If x1 is F1i and . . . and xn is Fni then y is Bi , where x = [x1 , x2 , . . . , xn ]T ∈ Rn , and y ∈ R are input and output of the fuzzy logic systems, respectively. F1i , F2i , . . . , Fni and Bi are fuzzy sets in R. By using the strategy of singleton fuzzification, the product inference and the center-average defuzzification, the fuzzy logic system can be formulated as N
y(x) =
wi
i=1
n
µF l ( x j ) j
j=1
N n i =1
,
µF l (xj )
j =1
j
where N is the number of IF–THEN rules, wi is the point n at which fuzzy membership function µBi (wi ) = 1. Let si (x) = j=1 µF l (xj )/
N
i=1
j [ nj=1 µF l (xj )], S (x) = [s1 (x), . . . , sN (x)]T and W = [w1 , w2 , j
. . . , wN ]T . Then the fuzzy logic system can be rewritten as y = W T S (x),
(2)
If all memberships are chosen as Gaussian functions, the following lemma holds. Lemma 1 (Wang & Mendel, 1992). Let f (x) be a continuous function defined on a compact set Ω . Then, for a given desired level of accuracy ε > 0, there exists a fuzzy logic system (2) such that sup f (x) − W T S (x) ≤ ε.
x∈Ω
Remark 2. Lemma 1 plays a key role in the following design procedure and it indicates that any given real continuous function f (x) can be represented by the linear combination of the basis function vector S (x) within a bounded error ε . That is, f (x) = W T S (x) + δ(ε), |δ(ε)| ≤ ε . It is noted that 0 < S T S ≤ 1. 3. Main Results In this section, we will present adaptive fuzzy control scheme for system (1) via the backstepping technique. In Section 3.1, a detailed design procedure will be given. In each step, a common virtual control function αi should be designed by using an appropriate common Lyapunov function Vi , and the control law uk will finally be designed. To avoid repetition, in Section 3.2, we only adopt a final common Lyapunov function to demonstrate the design procedure. 3.1. Adaptive control design under multiple adaptive laws In this subsection, a systemic control design procedure under multiple adaptive laws will be presented. Design the control laws as uk = −
1 gn,k
θˆn zn zn + λn zn + 2 2ζn2,min
,
(3)
where ζn,k and λn are positive design parameters, ζn,min = min 2 {ζn,k : k ∈ M }, θˆn is the estimation of θn = Wn,max , Wn,max = max{Wn,k : k ∈ M } and Wn,k is used in fuzzy logic system WnT,k Sn,k
X. Zhao et al. / Automatica 52 (2015) 185–191
(x) to approximate the unknown function fˆn,k (x). fˆn,k (x) will be specified in the proof of Theorem 1. The adaptive laws are defined as the solution to the following differential equations:
θ˙ˆ i =
ri 2ζi2,min
zi2 − βi θˆi ,
(4)
where ri , ζn,k and βi are positive design parameters, ζn,min
Theorem 1. Consider the closed-loop system (1) with the controllers (3) and the adaptive laws (4). For 1 ≤ i ≤ n, k ∈ M, there exists WiT,k Si,k (x) such that supx∈Ω fˆi,k (x) − WiT,k Si,k (x) ≤ εi,k in the sense that the approximation error εi,k is bounded, and all the initial values
of θˆi satisfy θˆi (0) ≥ 0. Then, the tracking error and closed-loop signals are bounded.
αi (Xi ) = −
1 gi,min
1 θˆi + λi + zi , 2 2 2ζi,min
2
(5)
where ζi,k is a positive design parameter, ζi,min = min{ζi,k : k ∈ M }, gi,min = min{gi,k : k ∈ M }, λi = gi,max + ci , gi,max = max
{gi,k : k ∈ M } and ci is a positive constant. θˆi is the estimation of 2 θi = Wi,max where Wi,max = max{Wi,k : k ∈ M } and Wi,k is used in fuzzy logic system WiT,k Si,k (x) to approximate the unknown ¯ ¯ (i) function fˆi,k (x). Xi = [¯xTi , θˆ i , y¯ d ]T with x¯ Ti = [x1 , x2 , . . . , xi ]T , θˆ i = [θˆ1 , θˆ2 , . . . , θˆi ]T , y¯ (di) = [yd , y˙ d , . . . , yd(i) ]T and y¯ d(i) being the ith
1
+ z12 ,
(9)
2
where ζ1,k is a positive design parameter. A feasible virtual control function can be constructed as
α1 = −
1 g1,min
θˆ1 1 z1 , + λ1 + 2 2ζ12,min
(10)
where λ1 = g1,max + c1 with c1 being a positive constant. By substituting (10) into (9), one has 2 1 z1
V˙ 1 ≤ −λ
Proof. For 1 ≤ i ≤ n − 1, we define the common virtual control functions as αi which are required to be the following form:
ζ12,k + ε12,k
=
required to satisfy θˆj (0) ≥ 0 such that θˆj ≥ 0. Now, we state one of our main results as follows.
Substituting (8) into (7), one gets V˙ 1 = g1,k z1 α1 + g1,k z1 z2 + z1 W1T,k S1,k (z1 ) + z1 δ(z1 ) 2 1 ≤ g1,k z1 α1 + g1,k z1 z2 + 2 z12 W1,k 2ζ1,k
+
min{ζn,k : k ∈ M }, and the choice of θˆj (0), j = 1, 2, . . . , n are
187
+
W1,k 2
ζ12,k + ε12,k
+
−
2ζ12,k
2
g1,k θˆ1
z12
2g1,min ζ12,min
+ g1,k z1 z2 .
(11)
Step 2. Let z3 = x3 − α2 , and choose 1
V2 = V1 +
2
z22 .
(12)
For any k ∈ M, the time derivative of V2 is given by V˙ 2 = V˙ 1 + z2 (g2,k α2 + g2,k z3 + f2,k − α˙ 1 )
= V˙ 1 + z2 (g2,k α2 + g2,k z3 + fˆ2,k ),
(13) ∂α
∂α
˙
∂α
(i+1)
.
derivative of yd .
where fˆ2,k = f2,k − α˙ 1 , α˙ 1 = ∂ x 1 x˙ 1 + ˆ 1 θˆ 1 + i=0 (1i) yd ∂ θ1 1 ∂ yd By Lemma 1, the following equation can be obtained
Remark 3. In Ma and Zhao (2010), αi should be a C 1 function
fˆ2,k = W2T,k S2,k (X2 ) + δ2,k (X2 ),
(14)
independent of k and satisfies αi ≤ −
1 gi,k
ˆ ( 2ζθi2 i,k
+ λi + )zi . Such 1 2
1
δ2,k (X2 ) ≤ ε2,k .
Substituting (14) into (13) yields that
a requirement is somewhat strict, and αi have to be constructed by some experiences since only an inequality is known. Considering that the state variables may be non-differentiable near the switch point, we relax the restriction that αi is a continuous and piecewise differentiable function. Furthermore, in our result, the common virtual control function αi can be directly given by (5).
V˙ 2 = V˙ 1 + g2,k z2 α2 + g2,k z2 z3 + z2 (W2T,k S (z2 ) + δ2,k (z2 ))
Step 1. Denote z1 = x1 − yd , z2 = x2 − α1 . Consider a Lyapunov function candidate as
where ζ2,k is a positive design parameter. Design the virtual control function α2 as
V1 =
1 2
z12 .
(6)
For any k ∈ M, the derivative of V1 is given by
α2 = −
(7)
where fˆ1,k = f1,k − y˙ d . By Lemma 1, the following equation can be obtained fˆ1,k = W1T,k S1,k (X1 ) + δ1,k (X1 ),
+
δ1,k (X1 ) ≤ ε1,k .
1 2ζ
z22 2 2,k
1 g2,min
ζ 2 + ε22,k 1 W2,k 2 + 2,k + z22 , 2
2
θˆ2 1 + λ2 + z2 , 2 2ζ22,min
(15)
(16)
where λ2 = g2,max + c2 with c2 being a positive constant. Then, one can get from (11), (15) and (16) that
V˙ 1 = z1 (g1,k α1 + g1,k z2 + f1,k − y˙ d )
= z1 (g1,k α1 + g1,k z2 + fˆ1,k ),
≤ V˙ 1 + g2,k z2 α2 + g2,k z2 z3
(8)
Remark 4. It should be pointed out that the fuzzy logic system is used to approximate the redefined unknown nonlinear function fˆ1,k that includes the unknown function f1,k and the derivative of the desired output rather than the unknown function f1,k only.
V˙ 2 ≤
2
−λj zj2 +
ζj2,k + εj2,k 2
j =1
+
Wj,k 2 2ζj2,k
−
+ gj,k zj zj+1
gj,k θˆj 2gj,min ζj2,min
zj2
.
(17)
Step i. Let zi+1 = xi+1 − αi , and assume that we have finished the first i − 1 (2 ≤ i ≤ n) steps. That is, for the following collection of auxiliary (z1 , . . . , zi−1 )-equations z˙j = gj,k xj+1 + φj,k (Xj ),
j = 1, . . . , i − 1,
(18)
188
X. Zhao et al. / Automatica 52 (2015) 185–191
where
Substituting (4) into (27) gives that j −1
∂αj−1
φj,k (Xj ) = fj,k (¯xj ) −
l =1
∂ xl
V˙ n ≤
x˙ l
(19)
We have a set of common virtual control functions as (5). A common Lyapunov function can be designed as i−1 1
zj2 .
2 j =1
(20)
i −1
−λj zj2 +
ζj2,k + εj2,k 2
j=1
+
Wj,k 2
−
2ζj2,k
1 2
Wj,k 2
n −1
+ gj,k zj zj+1
2gj,min ζj2,min
i
,
(21)
ζj2,k + εj2,k
Wj,k 2
2
gj,k θˆj − 2gj,min ζj2,min
2ζj2,k
Vn =
2
j=1
αn−1 = −
zl2
+
1 2rj
1
gn−1,min
1
.
(23)
˜θj2 ,
(24)
+
ζn2,k + εn2,k
2ζj2,k
2
−
cj zj2
−
j=1
1 2rj
βj θ˜j2
(30)
n ζj2,max + εj2,max
+
2
j=1
+
1 2rj
βθ
2 j j
.
(31)
n
1 j=1 2rj
{
βj θj2 +
}. One has
V˙ n ≤ −a0 Vn + b0 .
Vn (t ) ≤
(32)
Vn (0) −
θˆn−1 zn−1 zn−1 + λn−1 zn−1 + 2 2ζn2−1,min
lim |z1 | ≤
,
(25)
a0
e−a0 t +
b0 a0
,
t ≥ 0.
(33)
.
(34)
The proof is completed here.
rj
where λj = gj,max + cj , and cj is a positive constant.
2b0 a0
t →∞
3.2. Adaptive control design under one adaptive law In this subsection, a controller design approach with one adaptive law is presented. The control laws are chosen as uk = −
1
θˆ 2ζn2,min
gn,k
zn + λn zn +
zn 2
,
(35)
where ζn,k and λn are positive design parameters, ζn,min = min
2 {ζn,k : k ∈ M }, θˆ is the estimation of θ = ni=1 Wi,max , Wi,max = max{Wi,k : k ∈ M } and Wi,k is used in fuzzy logic system WiT,k Si,k (x) to approximate the unknown function fˆi,k (x).
gj,k θˆj − zj2 − λn zn2 2gj,min ζj2,min Wn,k 2 1 ˙ θˆn − θ˜n θˆ n + − 2 zn2 , rn 2ζn2,k 2ζn,min
b0
Inequality (33) indicates that all the signals in the closed-loop system are bounded. In particular, we have
2
+
1
According to the comparison principle. One gets
j =1
Wj,k 2
n
2
θˆn zn zn + λn zn + , (26) gn,k 2ζn2,min 2 2 where θj = Wj,max , θ˜j = θj − θˆj (j = 1, 2, . . . , n) are the error between θj and its estimation θˆj . For any k ∈ M, the time derivative of Vn satisfies n −1 ζj2,k + εj2,k 1 − θ˜j θ˙ˆ j V˙ n ≤ −λj zj2 + gj,k zj zj+1 + uk = −
(29)
1
ζj2,max +εj2,max
Step n. By repeatedly using the inductive argument above, a common Lyapunov function, a common virtual control function and state-feedback controllers are chosen, respectively, as n 1
zj2 ,
θ˜j θˆj = θ˜j (θj − θ˜j ) ≤ − θ˜j2 + θj2 .
zj2
n
Let a0 = min{2cj , βj : 1 ≤ j ≤ n}, b0 =
+ gj,k zj zj+1
j =1
j =1
(22)
−λj zj2 +
rj
2 2 One can get from (28)–(30) that
zi2 .
j =1
+
2
gj,k zj zj+1 ≤ gj,max
Analogous to the procedures above, the following inequality can be obtained V˙ i ≤
βj θ˜j θˆj
and
zj2
1 rj
gj,k θˆj
j =1
gj,k θˆj
+
2
θ˜j + − − 2 zj2 − λn zn2 2ζj2,k 2gj,min ζj2,min 2ζj,min Wn,k 2 ζn2,k + εn2,k 1 θn + + βn θ˜n θˆn + − 2 zn2 2 rn 2ζn2,k 2ζn,min n n −1 ζj2,k + εj2,k 1 2 ≤ −λj zj + + βj θ˜j θˆj + gj,k zj zj+1 . (28)
V˙ n ≤
ζj2,k + εj2,k
It is not difficult to see that
where ζj,k is a positive design parameter. Choose Vi = Vi−1 +
−λj zj2 + gj,k zj zj+1 +
j=1
For any k ∈ M, the time derivative of Vi−1 satisfies V˙ i−1 ≤
j =1
j −1 j −1 ∂αj−1 ˙ ∂αj−1 (l+1) − θˆ l − y . (l) d ˆ ∂ θl l =1 l =0 ∂ y d
Vi−1 =
n −1
(27)
The adaptive law is defined as the solution to the following differential equation:
θ˙ˆ =
n
r
2ζj2,min j=1
zj2 − β θˆ ,
(36)
X. Zhao et al. / Automatica 52 (2015) 185–191
where r , ζj,k and β are positive design parameters, ζj,min = min
{ζj,k : k ∈ M } and the choice of θˆ (0) is required to satisfy θˆ (0) ≥ 0 such that θˆ ≥ 0. Next, we give another main result of the paper.
Theorem 2. Consider the closed-loop system (1) with the controllers (35) and the adaptive laws (36). For 1 ≤ i ≤ n, k ∈ M, there exists
WiT,k Si,k (x) such that supx∈Ω fˆi,k (x) − WiT,k Si,k (x) ≤ εi,k in the sense that the approximation error εi,k is bounded, and the initial value of
θˆ satisfies θˆ (0) ≥ 0. Then, the tracking error and closed-loop signals are bounded.
Proof. For 1 ≤ i ≤ n − 1, define the common virtual control functions αi as:
αi (Xi ) = −
1
θˆ
+ λi +
2ζi2,min
gi,min
1
zi ,
2
(37)
where ζi,k is a positive design parameter, ζi,min = min{ζi,k : k ∈ M }, gi,min = min{gi,k : k ∈ M }, λi = gi,max + ci , gi,max =
max{gi,k : k ∈ M } and ci is a positive constant. θˆ is the estimation
2 n , Xi = [¯xT , θˆ , y¯ (i) ]T where x¯ T = [x1 , x2 , i d i i=1 Wi,max . . . , xi ]T , y¯ (di) = [yd , y˙ d , . . . , y(di) ]T and y¯ (di) being the ith derivative of θ =
of yd . Consider a common Lyapunov function V =
n 1
2 j =1
zj2 +
1 2r
θ˜ 2 ,
V˙ =
n −1
zi (gi,k αi + gi,k zi+1 + fi,k − α˙ i−1 )
r
zi (gi,k αi + gi,k zi+1 + fˆi,k ) 1
+ zn (gn,k uk + fˆn,k ) − θ˜ θ˙ˆ
(39)
r
where fˆi,k = fi,k − α˙ i−1 , α˙ i−1 = (l)
∂ yd
∂αj−1 x˙ ∂ xl l
j−1 l =1
+
∂αj−1 ˙ θˆ ∂ θˆ
+
j −1 l =0
.
For 1 ≤ i ≤ n, the following equation can be obtained by using Lemma 1. fˆi,k = WiT,k Si,k (Xi ) + δi,k (Xi ),
δ2,k (Xi ) ≤ εi,k .
V˙ ≤
β r
θ˜ θˆ +
2ζj2,k
−λ
r
−λj zj2 + gj,k zj zj+1 +
Wj,k 2
2 n zn
β
2 j ,k
θ˜ θˆ +
+
−
gj,k θˆ 2gj,min ζj2,min
ζn2,k + εn2,k
n j=1
2
+
−λj zj2 +
−
2ζj2,min
Wn,k 2 2ζn2,k
2
θ˜
ζj2,k + εj2,k
2 j ,k
g2,1 1
g2,2
z2 θˆ2 z2 + λ2 z2 + 2 2ζ22,1
z2 θˆ2 z2 + λ2 z2 + 2 2ζ22,1
, ,
−
+
u1 = −
α1 = −
zj2
θ 2ζn2,min n −1
1
g1,2
θˆ1 z1 z1 + λ1 z1 + 2 2ζ12,1
,
1
g2,1 1
g2,2
θˆ z2 z2 + λ2 z2 + 2 2 2ζ2,1
θˆ z2 z2 + λ2 z2 + 2 2 2ζ2,1
, ,
where z1 = x1 − yd , z2 = x2 − α1 , λ2 = c1 + g2,1 . The virtual control function α1 is given as
2
j=1
+
≤
1
where λ1 = c1 + g1,1 . The controller design based on Theorem 1 is completed here. In the next, another design according to Theorem 2 is presented. According to Theorem 2, an adaptive law θˆ and the control law u1 , u2 are chosen, respectively, as r r θ˙ˆ = 2 z12 + 2 z22 − β θˆ , 2ζ1,1 2ζ2,1
u2 = −
ζ +ε
(42)
where g1,1 = 2, g1,2 = 1, f1,1 = x1 , f1,2 = sin x1 , g2,1 = 2, g2,2 = 1, f2,1 = x1 x2 , f2,2 = x1 x22 . First, the controllers under multiple adaptive laws are designed by Theorem 1. The initial conditions are x1 (0) = 0.05, x2 (0) = 0.05, and θˆ1 (0) = θˆ2 (0) = 0. We choose c1 = 2, c2 = 1, r1 = 10, r2 = 3, β1 = β2 = 0.02, ς1,1 = 0.25, ς1,2 = 3, ς2,1 = 0.5, ς2,2 = 1.8. Second, the controllers under one adaptive law is designed by Theorem 2, and the initial conditions are x1 (0) = 0.05, x2 (0) = 0.05, θˆ (0) = 0. We choose c1 = 2, c2 = 1, r = 12, β = 0.025, ς1,1 = 0.25, ς1,2 = 3, ς2,1 = 1.5, ς2,2 = 1.8. The objective is to design the controllers uk such that y can track a desired trajectory yd under arbitrary switchings. According to Theorem 1, the adaptive laws θˆ1 , θˆ2 and the control law uk are chosen, respectively, as r1 r2 θ˙ˆ 1 = 2 z12 − β1 θˆ1 , θ˙ˆ 2 = 2 z22 − β2 θˆ2 , 2ζ1,1 2ζ2,1
(40)
Substituting (37)–(36) into (39), one has n −1
yd = sin t ,
α1 = −
i =1
yd
x˙ 2 = g2,σ (t ) uσ (t ) + f2,σ (t ) , y = x1 ,
where z1 = x1 − yd , z2 = x2 − α1 , λ2 = c2 + g2,1 . The virtual control function α1 is given by
1
∂αj−1 (l+1)
x˙ 1 = g1,σ (t ) x2 + f1,σ (t ) ,
u2 = −
+ zn (gn,k uk + fn,k − α˙ n−1 ) − θ˜ θ˙ˆ =
In this section, an example is presented to demonstrate the effectiveness of our main results. Consider the following two dimensional switched nonlinear system
u1 = −
i =1
n −1
4. A numerical example
(38)
where θ˜ = θ − θˆ is the error between θ and its estimation θˆ . For any k ∈ M, the time derivative of V satisfies
189
zn2
gj,k zj zj+1 .
(41)
j=1
The rest of proof is omitted here since it is similar to (29)–(34).
1 g1,2
θˆ z1 z1 + λ1 z1 + 2 2 2ζ1,1
,
where λ1 = c1 + g1,1 . The simulation results are shown in Figs. 1–4, respectively. Fig. 1 shows the system output y and reference signal yd . Fig. 2 depicts the response of the tracking error y − yd . Fig. 3 illustrates the trajectory of adaptive law. Fig. 4 demonstrates the evolution of switching signal. From Figs. 1–3, it can be seen that the output y of both controllers can track the target signal yd well, and all the closed-loop signals remain bounded.
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Fig. 4. Switching signal. Fig. 1. Tracking performances.
5. Conclusions The tracking control problem for a class of switched nonlinear systems with completely unknown nonlinear functions under arbitrary switchings is investigated in this paper. The application of adaptive backstepping technique is extended to a class of switched nonlinear systems with unknown uncertainties. Compared with some existing results, the main contributions of this paper lie in that: (1) the completely unknown uncertainties and the tracking problem are taken into account for the considered systems; (2) the computation burden is greatly reduced because the controllers do not involve the fuzzy basis function; (3) the method with only one adaptive law is proposed to solve the problem of over parameterization; (4) an improved approach of constructing common virtual control functions is proposed. The stability analysis in this paper guarantees that the designed controllers can ensure all the closed-loop signals remain bounded, and the system output converges to a small neighborhood of the reference signal. References Fig. 2. Responses of the tracking error y − yd .
Fig. 3. Responses of the adaptive laws.
Bao, Wen, Li, Bin, Chang, Juntao, Niu, Wenyu, & Yu, Daren (2010). Switching control of thrust regulation and inlet buzz protection for ducted rocket. Acta Astronautica, 67, 764–773. Briat, Corentin, & Seuret, Alexandre (2012). Convex dwell-time characterizations for uncertain linear impulsive systems. IEEE Transactions on Automatic Control, 57, 3241–3246. Briat, C., & Seuret, A. (2013). Affine minimal and mode-dependent dwell-time characterization for uncertain switched linear systems. IEEE Transactions on Automatic Control, 58, 1304–1310. Chen, Bing, Liu, Xiaoping, Liu, Kefu, & Lin, Chong (2009). Direct adaptive fuzzy control of nonlinear strict-feedback systems. Automatica, 45, 1530–1535. Homaee, Omid, Zakariazadeh, Alireza, & Jadid, Shahram (2014). Real-time voltage control algorithm with switched capacitors in smart distribution system in presence of renewable generations. International Journal of Electrical Power & Energy Systems, 54, 187–197. Krstić, Miroslav, Kokotović, Petar V., & Kanellakopoulos, Ioannis (1995). Nonlinear and adaptive control design. John Wiley & Sons, Inc.. Lian, Jie, & Zhao, Jun (2010). Adaptive variable structure control for uncertain switched delay systems. Circuits, Systems and Signal Processing, 29, 1089–1102. Liberzon, Daniel (2003). Switching in systems and control. Springer. Long, Lijun, & Zhao, Jun (2012). Control of switched nonlinear systems in p-normal form using multiple Lyapunov functions. IEEE Transactions on Automatic Control, 57, 1285–1291. Ma, Ruicheng, & Zhao, Jun (2010). Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings. Automatica, 46, 1819–1823. Margaliot, Michael, & Langholz, Gideon (2003). Necessary and sufficient conditions for absolute stability: the case of second-order systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 50, 227–234. Vu, Linh, & Liberzon, Daniel (2005). Common Lyapunov functions for families of commuting nonlinear systems. Systems & Control Letters, 54, 405–416.
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Wang, Lixin, & Mendel, Jerry M. (1992). Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Transactions on Neural Networks, 3, 807–814. Wang, Dong, Wang, Wei, & Shi, Peng (2009). Robust fault detection for switched linear systems with state delays. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 39, 800–805. Wu, Jenq-Lang (2009). Stabilizing controllers design for switched nonlinear systems in strict-feedback form. Automatica, 45, 1092–1096. Xiang, Weiming, & Xiao, Jian (2014). Stabilization of switched continuous-time systems with all modes unstable via dwell time switching. Automatica, 50, 940–945. Xiong, Junlin, Lam, James, Gao, Huijun, & Ho, Daniel W. C. (2005). On robust stabilization of Markovian jump systems with uncertain switching probabilities. Automatica, 41, 897–903. Zhang, Lixian, & Shi, Peng (2009). Stability, L2 -gain and asynchronous H∞ control of discrete-time switched systems with average dwell time. IEEE Transactions on Automatic Control, 54, 2192–2199. Zhao, Jun, & Hill, David J. (2008a). Dissipativity theory for switched systems. IEEE Transactions on Automatic Control, 53, 941–953. Zhao, Jun, & Hill, David J. (2008b). On stability, L2 -gain and H∞ control for switched systems. Automatica, 44, 1220–1232. Zhao, Jun, Hill, David J., & Liu, Tao (2009). Synchronization of complex dynamical networks with switching topology: a switched system point of view. Automatica, 45, 2502–2511. Zhao, Xudong, Liu, Xingwen, Yin, Shen, & Li, Hongyi (2014). Improved results on stability of continuous-time switched positive linear systems. Automatica, 50, 614–621. Zhao, Xudong, Yin, Shen, Li, Hongyi, & Niu, Ben (2014). Switching stabilization for a class of slowly switched systems. IEEE Transactions on Automatic Control, http://dx.doi.org/10.1109/TAC.2014.2322961. Zhao, Xudong, Zhang, Lixian, Shi, Peng, & Liu, Ming (2012). Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Transactions on Automatic Control, 57, 1809–1815.
International Journal of General Systems, and reviewer for various peer-reviewed journals, including Automatica and IEEE Transactions on Automatic Control. In addition, he has been granted as the outstanding reviewer for Automatica and Journal of the Franklin Institute. His research interests include hybrid systems, positive systems, multi-agent systems, fuzzy systems, Hinf control, filtering and their applications. His works have been widely published in international journals and conferences.
Xudong Zhao was born in Harbin, China, on July 7, 1982. He received the B.S. degree in Automation from Harbin Institute of Technology in 2005 and the Ph.D. degree from Control Science and Engineering from Space Control and Inertial Technology Center, Harbin Institute of Technology in 2010. Dr. Zhao was a lecturer and an associate professor at the China University of Petroleum, China. Since 2013, he joined Bohai University, China, where he is currently a professor. He is also a Postdoctoral Fellow in The University of Hong Kong from January 2014. Dr. Zhao serves as associate editor for Neurocomputing and
Liang Liu was born in Zaozhuang, Shandong Province, China, in 1985. He received his M.S. and Ph.D. degree from Qufu Normal University in 2010 and 2013. He is currently with College of Engineering, Bohai University. His research interests include decentralized adaptive control of complex systems and stochastic nonlinear control.
Xiaolong Zheng was born in Hubei Province, China, on October 8, 1990. He received the B.S. degree in automation from Yangtze University College of Technology and Engineering, Jingzhou, China, in 2013. He is pursuing for the M.S. degree in control theory and control engineering in Bohai University, Jinzhou, China. His research interests include adaptive control, switched nonlinear systems, intelligent control, stochastic nonlinear systems and their applications.
Ben Niu was born in Shandong Province, China, on May 2, 1982. He received his B.S. degree in Mathematics and Applied Mathematics from Liaocheng University, Liaocheng, China, in 2007, and his M.S. and Ph.D. degrees in Pure Mathematics and Control Theory and Applications in 2009 and 2012, respectively, both from Northeastern University, Shenyang, China. He is currently an associate professor of the College of Mathematics and Physics, Bohai University. His research interests are switched systems, stochastic systems, robust control, intelligent control and their applications.
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Adaptive tracking control for a class of uncertain switched nonlinear systems✩ Xudong Zhao a,b , Xiaolong Zheng a , Ben Niu c , Liang Liu a a
College of Engineering, Bohai University, Jinzhou 121013, Liaoning, China
b
Chongqing SANY High-intelligent Robots Co., Ltd., Chongqing, 401120, China
c
College of Mathematics and physics, Bohai University, Jinzhou 121013, Liaoning, China
article
info
Article history: Received 24 April 2014 Received in revised form 17 October 2014 Accepted 2 November 2014
Keywords: Switched nonlinear systems Adaptive backstepping Common Lyapunov function Tracking control
abstract This paper is concerned with the problem of tracking control for a class of switched nonlinear systems in lower triangular form with unknown functions and arbitrary switchings. Two classes of state feedback controllers are constructed by adopting the adaptive backstepping technique, and both of them are designed by using the common Lyapunov function (CLF) method. The first controller is designed under multiple adaptive laws. Then, the second one is designed based on constructing a maximum common adaptive parameter, which can overcome the problem of over-parameterization of the first controllers. It is shown that the designed state-feedback controllers can ensure that all the signals remain bounded and the tracking error converges to a small neighborhood of the origin. Finally, simulation results are presented to show the effectiveness of the proposed approaches. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction In the past decades, switched systems have attracted much attention since it can be used to describe a large number of physical and engineering systems, such as networked control systems (Zhao, Hill, & Liu, 2009), near space vehicle control systems (Bao, Li, Chang, Niu, & Yu, 2010), circuit and power systems (Homaee, Zakariazadeh, & Jadid, 2014), to list a few. As the most important issues in the study of switched linear or nonlinear systems, stability analysis and control synthesis are discussed extensively by a lot of researchers, and many excellent results have been obtained for various types of switched systems under arbitrary switching or constraint switching; see e.g., Wang, Wang, and Shi (2009), Xiong, Lam, Gao, and Ho (2005), Zhang and Shi (2009), Zhao and Hill (2008a,b), Zhao, Liu, Yin, and Li (2014), Zhao, Yin, Li, and Niu (2014), Zhao, Zhang, Shi, and Liu (2012) and references therein.
✩ This work was partially supported by the National Natural Science Foundation of China (61203123, 61304054, 61403041), the Liaoning Excellent Talents in University (LR2014035), and the Shandong Provincial Natural Science Foundation, China (ZR2012FQ019). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Andrey V. Savkin under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (X. Zhao), [email protected] (X. Zheng), [email protected] (B. Niu), [email protected] (L. Liu).
http://dx.doi.org/10.1016/j.automatica.2014.11.019 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
It is well known that the stability of a switched system under arbitrary switching can be guaranteed if a CLF exists for all subsystems (Vu & Liberzon, 2005). Therefore, CLF has been extensively used for control synthesis of switched linear systems (Briat & Seuret, 2012, 2013; Lian & Zhao, 2010; Liberzon, 2003; Margaliot & Langholz, 2003; Xiang & Xiao, 2014). Recently, there have been some results reported on the global stabilization problem for switched nonlinear systems in strict-feedback form under arbitrary switchings by using the backstepping technique (Ma & Zhao, 2010; Wu, 2009). Meanwhile, Long and Zhao (2012) investigated the global stabilization problem for a class of switched nonlinear systems in the p-normal form by the so-called power integrator backstepping design method. Unfortunately, uncertainty is not taken into account in the aforementioned papers, which widely exists in practical switched nonlinear systems. For general nonlinear systems in lower triangular form without switchings, it has been shown in the literature (Chen, Liu, Liu, & Lin, 2009; Krstić, Kokotović, & Kanellakopoulos, 1995) that the adaptive backstepping method is a powerful tool for controller design. As a base strategy in adaptive backstepping, the main objective is to cancel the unknown nonlinearity of system by constructing the virtual control functions and adaptive laws. This design method is of course expected to be useful to stabilize switched nonlinear systems with uncertainty. However, to the best of the authors’ knowledge, there is not any related results on
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the stability analysis and controller design for uncertain switched nonlinear systems in lower triangular form under arbitrary switchings. The reason lies in that, if we want to construct a CLF, the common virtual control functions need to be found first, but it is somewhat difficult to find these common virtual control functions for different subsystems. Based on the above illustrations, we know that the difficulty encountered in constructing a CLF is how to design a common virtual control function at each step when the adaptive backstepping technique is applied to switched nonlinear systems. One way to deal with this problem is to simply assume the existence and availability of a common virtual control function as was done in Wu (2009). Most recent, the authors in Ma and Zhao (2010) proposed some inequality constraints on the common virtual functions to improve previous results. However, it is required in Ma and Zhao (2010) that the common virtual functions should be first-order derivative, not related to switching laws, and satisfy some inequality constraints, which is somewhat too strict in practice to be applied. Furthermore, the uncertainty is not considered in Ma and Zhao (2010) and Wu (2009). Therefore, the design of the common virtual functions has not been fully solved in the existing literature when the system contains uncertainty, which motivates our present work. In this paper, the adaptive tracking problem is studied for a class of switched nonlinear systems with completely unknown uncertainties. An efficient approach of constructing common virtual control functions is proposed for the considered systems, and state-feedback controllers are designed via the adaptive backstepping technique where a Mamdani-type fuzzy logic system is utilized to approximate the redefined unknown functions. A design method with multiple adaptive laws is presented in first place. Furthermore, another method with only one adaptive law is proposed to avoid the problem of over-parameterization. Both controllers can guarantee that all the closed-loop signals remain bounded, and the system output tracks the reference signal while the computation burden is low. Notations: in this paper, the notations are standard. Rn denotes the n-dimensional Euclidean space, the notation ∥·∥ refers to the Euclidean vector norm. For positive integers 1 ≤ i ≤ n, 1 ≤ j ≤ m, we also denote Ξi,max = max{Ξi,j : 1 ≤ j ≤ m}, Ξi,min = min{Ξi,j : 1 ≤ j ≤ m}. 2. Problem formulation and preliminaries Consider a class of switched nonlinear system in the following form: x˙ i = gi,σ (t ) xi+1 + fi,σ (t ) (¯xi ), i = 1, 2, . . . , n − 1, x˙ n = gn,σ (t ) uσ (t ) + fn,σ (t ) (¯xn ), y = x1 ,
(1)
where x¯ i := (x1 , x2 , . . . , xi ) ∈ R , i = 1, 2, . . . , n is the system state, y is the system output; σ (t ) : [0, +∞) → M = {1, 2, . . . , m} is the switching signal; uk ∈ R is the control input of the kth subsystem; For any i = 1, 2 . . . , n and k = 1, 2, . . . , m, fi,k (¯xi ) is an unknown smooth nonlinear function representing the system uncertainty, and gi,k is a positive constant. Our control objective is to design state-feedback controllers such that the output of system (1) tracks a given time-varying signal yd (t ) within a bounded error and all the signals of the closedloop systems remain bounded under arbitrary switchings. T
i
Assumption 1. The tracking target yd (t ) and its time derivatives up to the nth order are continuous and bounded. Remark 1. System (1) will be reduced to system (1) in Ma and Zhao (2010) when the tracking control problem and the uncertain functions are not taken into account. Therefore, the systems considered in this paper are more general.
In the controller design and stability analysis procedure, fuzzy logic systems will be used to approximate the unknown functions. Therefore, the following useful concept and lemma are first recalled. Fuzzy logic systems include some IF–THEN rules, and the ith IF–THEN rule is written as Ri : If x1 is F1i and . . . and xn is Fni then y is Bi , where x = [x1 , x2 , . . . , xn ]T ∈ Rn , and y ∈ R are input and output of the fuzzy logic systems, respectively. F1i , F2i , . . . , Fni and Bi are fuzzy sets in R. By using the strategy of singleton fuzzification, the product inference and the center-average defuzzification, the fuzzy logic system can be formulated as N
y(x) =
wi
i=1
n
µF l ( x j ) j
j=1
N n i =1
,
µF l (xj )
j =1
j
where N is the number of IF–THEN rules, wi is the point n at which fuzzy membership function µBi (wi ) = 1. Let si (x) = j=1 µF l (xj )/
N
i=1
j [ nj=1 µF l (xj )], S (x) = [s1 (x), . . . , sN (x)]T and W = [w1 , w2 , j
. . . , wN ]T . Then the fuzzy logic system can be rewritten as y = W T S (x),
(2)
If all memberships are chosen as Gaussian functions, the following lemma holds. Lemma 1 (Wang & Mendel, 1992). Let f (x) be a continuous function defined on a compact set Ω . Then, for a given desired level of accuracy ε > 0, there exists a fuzzy logic system (2) such that sup f (x) − W T S (x) ≤ ε.
x∈Ω
Remark 2. Lemma 1 plays a key role in the following design procedure and it indicates that any given real continuous function f (x) can be represented by the linear combination of the basis function vector S (x) within a bounded error ε . That is, f (x) = W T S (x) + δ(ε), |δ(ε)| ≤ ε . It is noted that 0 < S T S ≤ 1. 3. Main Results In this section, we will present adaptive fuzzy control scheme for system (1) via the backstepping technique. In Section 3.1, a detailed design procedure will be given. In each step, a common virtual control function αi should be designed by using an appropriate common Lyapunov function Vi , and the control law uk will finally be designed. To avoid repetition, in Section 3.2, we only adopt a final common Lyapunov function to demonstrate the design procedure. 3.1. Adaptive control design under multiple adaptive laws In this subsection, a systemic control design procedure under multiple adaptive laws will be presented. Design the control laws as uk = −
1 gn,k
θˆn zn zn + λn zn + 2 2ζn2,min
,
(3)
where ζn,k and λn are positive design parameters, ζn,min = min 2 {ζn,k : k ∈ M }, θˆn is the estimation of θn = Wn,max , Wn,max = max{Wn,k : k ∈ M } and Wn,k is used in fuzzy logic system WnT,k Sn,k
X. Zhao et al. / Automatica 52 (2015) 185–191
(x) to approximate the unknown function fˆn,k (x). fˆn,k (x) will be specified in the proof of Theorem 1. The adaptive laws are defined as the solution to the following differential equations:
θ˙ˆ i =
ri 2ζi2,min
zi2 − βi θˆi ,
(4)
where ri , ζn,k and βi are positive design parameters, ζn,min
Theorem 1. Consider the closed-loop system (1) with the controllers (3) and the adaptive laws (4). For 1 ≤ i ≤ n, k ∈ M, there exists WiT,k Si,k (x) such that supx∈Ω fˆi,k (x) − WiT,k Si,k (x) ≤ εi,k in the sense that the approximation error εi,k is bounded, and all the initial values
of θˆi satisfy θˆi (0) ≥ 0. Then, the tracking error and closed-loop signals are bounded.
αi (Xi ) = −
1 gi,min
1 θˆi + λi + zi , 2 2 2ζi,min
2
(5)
where ζi,k is a positive design parameter, ζi,min = min{ζi,k : k ∈ M }, gi,min = min{gi,k : k ∈ M }, λi = gi,max + ci , gi,max = max
{gi,k : k ∈ M } and ci is a positive constant. θˆi is the estimation of 2 θi = Wi,max where Wi,max = max{Wi,k : k ∈ M } and Wi,k is used in fuzzy logic system WiT,k Si,k (x) to approximate the unknown ¯ ¯ (i) function fˆi,k (x). Xi = [¯xTi , θˆ i , y¯ d ]T with x¯ Ti = [x1 , x2 , . . . , xi ]T , θˆ i = [θˆ1 , θˆ2 , . . . , θˆi ]T , y¯ (di) = [yd , y˙ d , . . . , yd(i) ]T and y¯ d(i) being the ith
1
+ z12 ,
(9)
2
where ζ1,k is a positive design parameter. A feasible virtual control function can be constructed as
α1 = −
1 g1,min
θˆ1 1 z1 , + λ1 + 2 2ζ12,min
(10)
where λ1 = g1,max + c1 with c1 being a positive constant. By substituting (10) into (9), one has 2 1 z1
V˙ 1 ≤ −λ
Proof. For 1 ≤ i ≤ n − 1, we define the common virtual control functions as αi which are required to be the following form:
ζ12,k + ε12,k
=
required to satisfy θˆj (0) ≥ 0 such that θˆj ≥ 0. Now, we state one of our main results as follows.
Substituting (8) into (7), one gets V˙ 1 = g1,k z1 α1 + g1,k z1 z2 + z1 W1T,k S1,k (z1 ) + z1 δ(z1 ) 2 1 ≤ g1,k z1 α1 + g1,k z1 z2 + 2 z12 W1,k 2ζ1,k
+
min{ζn,k : k ∈ M }, and the choice of θˆj (0), j = 1, 2, . . . , n are
187
+
W1,k 2
ζ12,k + ε12,k
+
−
2ζ12,k
2
g1,k θˆ1
z12
2g1,min ζ12,min
+ g1,k z1 z2 .
(11)
Step 2. Let z3 = x3 − α2 , and choose 1
V2 = V1 +
2
z22 .
(12)
For any k ∈ M, the time derivative of V2 is given by V˙ 2 = V˙ 1 + z2 (g2,k α2 + g2,k z3 + f2,k − α˙ 1 )
= V˙ 1 + z2 (g2,k α2 + g2,k z3 + fˆ2,k ),
(13) ∂α
∂α
˙
∂α
(i+1)
.
derivative of yd .
where fˆ2,k = f2,k − α˙ 1 , α˙ 1 = ∂ x 1 x˙ 1 + ˆ 1 θˆ 1 + i=0 (1i) yd ∂ θ1 1 ∂ yd By Lemma 1, the following equation can be obtained
Remark 3. In Ma and Zhao (2010), αi should be a C 1 function
fˆ2,k = W2T,k S2,k (X2 ) + δ2,k (X2 ),
(14)
independent of k and satisfies αi ≤ −
1 gi,k
ˆ ( 2ζθi2 i,k
+ λi + )zi . Such 1 2
1
δ2,k (X2 ) ≤ ε2,k .
Substituting (14) into (13) yields that
a requirement is somewhat strict, and αi have to be constructed by some experiences since only an inequality is known. Considering that the state variables may be non-differentiable near the switch point, we relax the restriction that αi is a continuous and piecewise differentiable function. Furthermore, in our result, the common virtual control function αi can be directly given by (5).
V˙ 2 = V˙ 1 + g2,k z2 α2 + g2,k z2 z3 + z2 (W2T,k S (z2 ) + δ2,k (z2 ))
Step 1. Denote z1 = x1 − yd , z2 = x2 − α1 . Consider a Lyapunov function candidate as
where ζ2,k is a positive design parameter. Design the virtual control function α2 as
V1 =
1 2
z12 .
(6)
For any k ∈ M, the derivative of V1 is given by
α2 = −
(7)
where fˆ1,k = f1,k − y˙ d . By Lemma 1, the following equation can be obtained fˆ1,k = W1T,k S1,k (X1 ) + δ1,k (X1 ),
+
δ1,k (X1 ) ≤ ε1,k .
1 2ζ
z22 2 2,k
1 g2,min
ζ 2 + ε22,k 1 W2,k 2 + 2,k + z22 , 2
2
θˆ2 1 + λ2 + z2 , 2 2ζ22,min
(15)
(16)
where λ2 = g2,max + c2 with c2 being a positive constant. Then, one can get from (11), (15) and (16) that
V˙ 1 = z1 (g1,k α1 + g1,k z2 + f1,k − y˙ d )
= z1 (g1,k α1 + g1,k z2 + fˆ1,k ),
≤ V˙ 1 + g2,k z2 α2 + g2,k z2 z3
(8)
Remark 4. It should be pointed out that the fuzzy logic system is used to approximate the redefined unknown nonlinear function fˆ1,k that includes the unknown function f1,k and the derivative of the desired output rather than the unknown function f1,k only.
V˙ 2 ≤
2
−λj zj2 +
ζj2,k + εj2,k 2
j =1
+
Wj,k 2 2ζj2,k
−
+ gj,k zj zj+1
gj,k θˆj 2gj,min ζj2,min
zj2
.
(17)
Step i. Let zi+1 = xi+1 − αi , and assume that we have finished the first i − 1 (2 ≤ i ≤ n) steps. That is, for the following collection of auxiliary (z1 , . . . , zi−1 )-equations z˙j = gj,k xj+1 + φj,k (Xj ),
j = 1, . . . , i − 1,
(18)
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X. Zhao et al. / Automatica 52 (2015) 185–191
where
Substituting (4) into (27) gives that j −1
∂αj−1
φj,k (Xj ) = fj,k (¯xj ) −
l =1
∂ xl
V˙ n ≤
x˙ l
(19)
We have a set of common virtual control functions as (5). A common Lyapunov function can be designed as i−1 1
zj2 .
2 j =1
(20)
i −1
−λj zj2 +
ζj2,k + εj2,k 2
j=1
+
Wj,k 2
−
2ζj2,k
1 2
Wj,k 2
n −1
+ gj,k zj zj+1
2gj,min ζj2,min
i
,
(21)
ζj2,k + εj2,k
Wj,k 2
2
gj,k θˆj − 2gj,min ζj2,min
2ζj2,k
Vn =
2
j=1
αn−1 = −
zl2
+
1 2rj
1
gn−1,min
1
.
(23)
˜θj2 ,
(24)
+
ζn2,k + εn2,k
2ζj2,k
2
−
cj zj2
−
j=1
1 2rj
βj θ˜j2
(30)
n ζj2,max + εj2,max
+
2
j=1
+
1 2rj
βθ
2 j j
.
(31)
n
1 j=1 2rj
{
βj θj2 +
}. One has
V˙ n ≤ −a0 Vn + b0 .
Vn (t ) ≤
(32)
Vn (0) −
θˆn−1 zn−1 zn−1 + λn−1 zn−1 + 2 2ζn2−1,min
lim |z1 | ≤
,
(25)
a0
e−a0 t +
b0 a0
,
t ≥ 0.
(33)
.
(34)
The proof is completed here.
rj
where λj = gj,max + cj , and cj is a positive constant.
2b0 a0
t →∞
3.2. Adaptive control design under one adaptive law In this subsection, a controller design approach with one adaptive law is presented. The control laws are chosen as uk = −
1
θˆ 2ζn2,min
gn,k
zn + λn zn +
zn 2
,
(35)
where ζn,k and λn are positive design parameters, ζn,min = min
2 {ζn,k : k ∈ M }, θˆ is the estimation of θ = ni=1 Wi,max , Wi,max = max{Wi,k : k ∈ M } and Wi,k is used in fuzzy logic system WiT,k Si,k (x) to approximate the unknown function fˆi,k (x).
gj,k θˆj − zj2 − λn zn2 2gj,min ζj2,min Wn,k 2 1 ˙ θˆn − θ˜n θˆ n + − 2 zn2 , rn 2ζn2,k 2ζn,min
b0
Inequality (33) indicates that all the signals in the closed-loop system are bounded. In particular, we have
2
+
1
According to the comparison principle. One gets
j =1
Wj,k 2
n
2
θˆn zn zn + λn zn + , (26) gn,k 2ζn2,min 2 2 where θj = Wj,max , θ˜j = θj − θˆj (j = 1, 2, . . . , n) are the error between θj and its estimation θˆj . For any k ∈ M, the time derivative of Vn satisfies n −1 ζj2,k + εj2,k 1 − θ˜j θ˙ˆ j V˙ n ≤ −λj zj2 + gj,k zj zj+1 + uk = −
(29)
1
ζj2,max +εj2,max
Step n. By repeatedly using the inductive argument above, a common Lyapunov function, a common virtual control function and state-feedback controllers are chosen, respectively, as n 1
zj2 ,
θ˜j θˆj = θ˜j (θj − θ˜j ) ≤ − θ˜j2 + θj2 .
zj2
n
Let a0 = min{2cj , βj : 1 ≤ j ≤ n}, b0 =
+ gj,k zj zj+1
j =1
j =1
(22)
−λj zj2 +
rj
2 2 One can get from (28)–(30) that
zi2 .
j =1
+
2
gj,k zj zj+1 ≤ gj,max
Analogous to the procedures above, the following inequality can be obtained V˙ i ≤
βj θ˜j θˆj
and
zj2
1 rj
gj,k θˆj
j =1
gj,k θˆj
+
2
θ˜j + − − 2 zj2 − λn zn2 2ζj2,k 2gj,min ζj2,min 2ζj,min Wn,k 2 ζn2,k + εn2,k 1 θn + + βn θ˜n θˆn + − 2 zn2 2 rn 2ζn2,k 2ζn,min n n −1 ζj2,k + εj2,k 1 2 ≤ −λj zj + + βj θ˜j θˆj + gj,k zj zj+1 . (28)
V˙ n ≤
ζj2,k + εj2,k
It is not difficult to see that
where ζj,k is a positive design parameter. Choose Vi = Vi−1 +
−λj zj2 + gj,k zj zj+1 +
j=1
For any k ∈ M, the time derivative of Vi−1 satisfies V˙ i−1 ≤
j =1
j −1 j −1 ∂αj−1 ˙ ∂αj−1 (l+1) − θˆ l − y . (l) d ˆ ∂ θl l =1 l =0 ∂ y d
Vi−1 =
n −1
(27)
The adaptive law is defined as the solution to the following differential equation:
θ˙ˆ =
n
r
2ζj2,min j=1
zj2 − β θˆ ,
(36)
X. Zhao et al. / Automatica 52 (2015) 185–191
where r , ζj,k and β are positive design parameters, ζj,min = min
{ζj,k : k ∈ M } and the choice of θˆ (0) is required to satisfy θˆ (0) ≥ 0 such that θˆ ≥ 0. Next, we give another main result of the paper.
Theorem 2. Consider the closed-loop system (1) with the controllers (35) and the adaptive laws (36). For 1 ≤ i ≤ n, k ∈ M, there exists
WiT,k Si,k (x) such that supx∈Ω fˆi,k (x) − WiT,k Si,k (x) ≤ εi,k in the sense that the approximation error εi,k is bounded, and the initial value of
θˆ satisfies θˆ (0) ≥ 0. Then, the tracking error and closed-loop signals are bounded.
Proof. For 1 ≤ i ≤ n − 1, define the common virtual control functions αi as:
αi (Xi ) = −
1
θˆ
+ λi +
2ζi2,min
gi,min
1
zi ,
2
(37)
where ζi,k is a positive design parameter, ζi,min = min{ζi,k : k ∈ M }, gi,min = min{gi,k : k ∈ M }, λi = gi,max + ci , gi,max =
max{gi,k : k ∈ M } and ci is a positive constant. θˆ is the estimation
2 n , Xi = [¯xT , θˆ , y¯ (i) ]T where x¯ T = [x1 , x2 , i d i i=1 Wi,max . . . , xi ]T , y¯ (di) = [yd , y˙ d , . . . , y(di) ]T and y¯ (di) being the ith derivative of θ =
of yd . Consider a common Lyapunov function V =
n 1
2 j =1
zj2 +
1 2r
θ˜ 2 ,
V˙ =
n −1
zi (gi,k αi + gi,k zi+1 + fi,k − α˙ i−1 )
r
zi (gi,k αi + gi,k zi+1 + fˆi,k ) 1
+ zn (gn,k uk + fˆn,k ) − θ˜ θ˙ˆ
(39)
r
where fˆi,k = fi,k − α˙ i−1 , α˙ i−1 = (l)
∂ yd
∂αj−1 x˙ ∂ xl l
j−1 l =1
+
∂αj−1 ˙ θˆ ∂ θˆ
+
j −1 l =0
.
For 1 ≤ i ≤ n, the following equation can be obtained by using Lemma 1. fˆi,k = WiT,k Si,k (Xi ) + δi,k (Xi ),
δ2,k (Xi ) ≤ εi,k .
V˙ ≤
β r
θ˜ θˆ +
2ζj2,k
−λ
r
−λj zj2 + gj,k zj zj+1 +
Wj,k 2
2 n zn
β
2 j ,k
θ˜ θˆ +
+
−
gj,k θˆ 2gj,min ζj2,min
ζn2,k + εn2,k
n j=1
2
+
−λj zj2 +
−
2ζj2,min
Wn,k 2 2ζn2,k
2
θ˜
ζj2,k + εj2,k
2 j ,k
g2,1 1
g2,2
z2 θˆ2 z2 + λ2 z2 + 2 2ζ22,1
z2 θˆ2 z2 + λ2 z2 + 2 2ζ22,1
, ,
−
+
u1 = −
α1 = −
zj2
θ 2ζn2,min n −1
1
g1,2
θˆ1 z1 z1 + λ1 z1 + 2 2ζ12,1
,
1
g2,1 1
g2,2
θˆ z2 z2 + λ2 z2 + 2 2 2ζ2,1
θˆ z2 z2 + λ2 z2 + 2 2 2ζ2,1
, ,
where z1 = x1 − yd , z2 = x2 − α1 , λ2 = c1 + g2,1 . The virtual control function α1 is given as
2
j=1
+
≤
1
where λ1 = c1 + g1,1 . The controller design based on Theorem 1 is completed here. In the next, another design according to Theorem 2 is presented. According to Theorem 2, an adaptive law θˆ and the control law u1 , u2 are chosen, respectively, as r r θ˙ˆ = 2 z12 + 2 z22 − β θˆ , 2ζ1,1 2ζ2,1
u2 = −
ζ +ε
(42)
where g1,1 = 2, g1,2 = 1, f1,1 = x1 , f1,2 = sin x1 , g2,1 = 2, g2,2 = 1, f2,1 = x1 x2 , f2,2 = x1 x22 . First, the controllers under multiple adaptive laws are designed by Theorem 1. The initial conditions are x1 (0) = 0.05, x2 (0) = 0.05, and θˆ1 (0) = θˆ2 (0) = 0. We choose c1 = 2, c2 = 1, r1 = 10, r2 = 3, β1 = β2 = 0.02, ς1,1 = 0.25, ς1,2 = 3, ς2,1 = 0.5, ς2,2 = 1.8. Second, the controllers under one adaptive law is designed by Theorem 2, and the initial conditions are x1 (0) = 0.05, x2 (0) = 0.05, θˆ (0) = 0. We choose c1 = 2, c2 = 1, r = 12, β = 0.025, ς1,1 = 0.25, ς1,2 = 3, ς2,1 = 1.5, ς2,2 = 1.8. The objective is to design the controllers uk such that y can track a desired trajectory yd under arbitrary switchings. According to Theorem 1, the adaptive laws θˆ1 , θˆ2 and the control law uk are chosen, respectively, as r1 r2 θ˙ˆ 1 = 2 z12 − β1 θˆ1 , θ˙ˆ 2 = 2 z22 − β2 θˆ2 , 2ζ1,1 2ζ2,1
(40)
Substituting (37)–(36) into (39), one has n −1
yd = sin t ,
α1 = −
i =1
yd
x˙ 2 = g2,σ (t ) uσ (t ) + f2,σ (t ) , y = x1 ,
where z1 = x1 − yd , z2 = x2 − α1 , λ2 = c2 + g2,1 . The virtual control function α1 is given by
1
∂αj−1 (l+1)
x˙ 1 = g1,σ (t ) x2 + f1,σ (t ) ,
u2 = −
+ zn (gn,k uk + fn,k − α˙ n−1 ) − θ˜ θ˙ˆ =
In this section, an example is presented to demonstrate the effectiveness of our main results. Consider the following two dimensional switched nonlinear system
u1 = −
i =1
n −1
4. A numerical example
(38)
where θ˜ = θ − θˆ is the error between θ and its estimation θˆ . For any k ∈ M, the time derivative of V satisfies
189
zn2
gj,k zj zj+1 .
(41)
j=1
The rest of proof is omitted here since it is similar to (29)–(34).
1 g1,2
θˆ z1 z1 + λ1 z1 + 2 2 2ζ1,1
,
where λ1 = c1 + g1,1 . The simulation results are shown in Figs. 1–4, respectively. Fig. 1 shows the system output y and reference signal yd . Fig. 2 depicts the response of the tracking error y − yd . Fig. 3 illustrates the trajectory of adaptive law. Fig. 4 demonstrates the evolution of switching signal. From Figs. 1–3, it can be seen that the output y of both controllers can track the target signal yd well, and all the closed-loop signals remain bounded.
190
X. Zhao et al. / Automatica 52 (2015) 185–191
Fig. 4. Switching signal. Fig. 1. Tracking performances.
5. Conclusions The tracking control problem for a class of switched nonlinear systems with completely unknown nonlinear functions under arbitrary switchings is investigated in this paper. The application of adaptive backstepping technique is extended to a class of switched nonlinear systems with unknown uncertainties. Compared with some existing results, the main contributions of this paper lie in that: (1) the completely unknown uncertainties and the tracking problem are taken into account for the considered systems; (2) the computation burden is greatly reduced because the controllers do not involve the fuzzy basis function; (3) the method with only one adaptive law is proposed to solve the problem of over parameterization; (4) an improved approach of constructing common virtual control functions is proposed. The stability analysis in this paper guarantees that the designed controllers can ensure all the closed-loop signals remain bounded, and the system output converges to a small neighborhood of the reference signal. References Fig. 2. Responses of the tracking error y − yd .
Fig. 3. Responses of the adaptive laws.
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International Journal of General Systems, and reviewer for various peer-reviewed journals, including Automatica and IEEE Transactions on Automatic Control. In addition, he has been granted as the outstanding reviewer for Automatica and Journal of the Franklin Institute. His research interests include hybrid systems, positive systems, multi-agent systems, fuzzy systems, Hinf control, filtering and their applications. His works have been widely published in international journals and conferences.
Xudong Zhao was born in Harbin, China, on July 7, 1982. He received the B.S. degree in Automation from Harbin Institute of Technology in 2005 and the Ph.D. degree from Control Science and Engineering from Space Control and Inertial Technology Center, Harbin Institute of Technology in 2010. Dr. Zhao was a lecturer and an associate professor at the China University of Petroleum, China. Since 2013, he joined Bohai University, China, where he is currently a professor. He is also a Postdoctoral Fellow in The University of Hong Kong from January 2014. Dr. Zhao serves as associate editor for Neurocomputing and
Liang Liu was born in Zaozhuang, Shandong Province, China, in 1985. He received his M.S. and Ph.D. degree from Qufu Normal University in 2010 and 2013. He is currently with College of Engineering, Bohai University. His research interests include decentralized adaptive control of complex systems and stochastic nonlinear control.
Xiaolong Zheng was born in Hubei Province, China, on October 8, 1990. He received the B.S. degree in automation from Yangtze University College of Technology and Engineering, Jingzhou, China, in 2013. He is pursuing for the M.S. degree in control theory and control engineering in Bohai University, Jinzhou, China. His research interests include adaptive control, switched nonlinear systems, intelligent control, stochastic nonlinear systems and their applications.
Ben Niu was born in Shandong Province, China, on May 2, 1982. He received his B.S. degree in Mathematics and Applied Mathematics from Liaocheng University, Liaocheng, China, in 2007, and his M.S. and Ph.D. degrees in Pure Mathematics and Control Theory and Applications in 2009 and 2012, respectively, both from Northeastern University, Shenyang, China. He is currently an associate professor of the College of Mathematics and Physics, Bohai University. His research interests are switched systems, stochastic systems, robust control, intelligent control and their applications.