Adaptive neural network tracking control for a class ... - Semantic Scholar
Neurocomputing 173 (2016) 2121–2128
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Adaptive neural network tracking control for a class of switched strict-feedback nonlinear systems with input delay$ Ben Niu n, Lu Li College of Mathematics and Physics and Automation Research Institute, Bohai University, Jinzhou, Liaoning Province 121013, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 10 July 2015 Received in revised form 21 September 2015 Accepted 19 October 2015 Communicated by Xudong Zhao Available online 31 October 2015
In this paper, a neural-network-based control scheme is developed for the tracking control problem of a class of switched strict-feedback nonlinear systems with uncertain input delay and external time-varying disturbances. First, the auxiliary signals are obtained by masterly constructing a filter and a virtual observer. Then the adaptive backstepping technique and neural network (NN) are employed to construct a common Lyapunov function (CLF) and a state feedback controller for all subsystems. It is proved that all signals of the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB), and that the tracking error ultimately converges to an adequately small compact set. Finally, a simulation example is given to illustrate the effectiveness of the proposed control approach. & 2015 Elsevier B.V. All rights reserved.
Keywords: Switched nonlinear system Adaptive control Neural network Input delay
1. Introduction In the past decades, there has been increasing interest in the study of switched systems [1–8], as such systems can describe the behaviour of a large number of practical plants resulting from the interactions of continuous dynamics, discrete dynamics, and logic decisions, etc. From a theoretical viewpoint, the main contributions of studying switched systems are twofold: on one side several researchers have paid attention to state/mode observability, that is, to the possibility of reconstructing from the measured data, the continuous state, or the discrete mode; on the other side, the main interest has been devoted to stability analysis and control synthesize problems. Correspondingly, several nonlinear control design approaches such as Lyapunov stability theory and backstepping technique were extended to the case of switched nonlinear systems, and many novel results have been reported in the literature, see e.g., [9–16] and references therein. On the other hand, backstepping-based adaptive control technique has become one of the most popular control approaches for nonlinear systems in strict-feedback form, and some important results have been reported [17–23]. Especially, neural-network-based adaptive control technique for uncertain nonlinear systems in strict☆ This work was partially supported by the National Natural Science Foundation of China [Grant number 61304054, 61403041 and 61403354], the Program for Liaoning Provincial Excellent Talents in University, China [Grant number LJQ2014122]. n Corresponding author at: College of Control Science and Engineering, Dalian University of Technology, Dalian, Liaoning Province 116024, People's Republic of China. E-mail addresses: [email protected] (B. Niu), [email protected] (L. Li).
http://dx.doi.org/10.1016/j.neucom.2015.10.059 0925-2312/& 2015 Elsevier B.V. All rights reserved.
feedback form has been widely studied by researchers [24–30]. For example, robust adaptive tracking control is considered for a class of uncertain MIMO nonlinear systems with input nonlinearities by neural network in [31]; and in [32], an adaptive control scheme is proposed for a class of uncertain nonlinear system in strict-feedback form with backstepping method and NN, just name a few. However, the aforementioned results on adaptive fuzzy or NN control design are only for non-switched nonlinear systems in strict-feedback form, few results have appeared for the one of switched nonlinear systems in strict-feedback form [33–36]. Therefore, the study on adaptive control of switched uncertain nonlinear systems in strict-feedback form by employing NN is of great significance. Further, input delays are often encountered in many practical control systems, such as hydraulic rolling mill systems, networked control systems, manufacturing processes, and so on. Very careful attention should be put on input delays since it may result in oscillation or even instability of the overall closed-loop systems. In dealing with the adaptive control of nonlinear systems with input delays, various methods have been proposed in the past years [37– 40]. For instance, an adaptive tracking control method for Multiple Input Multiple Output (MIMO) nonlinear systems with input delay is proposed in [41]. By introducing a definition of the design variable with the input integral term, the input delayed system is converted to the non-delayed system and then the effective state feedback controller is obtained. Ref. [42] considered the stabilization problems for nonlinear systems under sampled and delayed measurements, and with inputs subject to delay and zero-order hold. However, to the best of the authors' knowledge, there is no result to deal with input delays for switched uncertain nonlinear
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systems in strict-feedback form, and this motivates the present work. In this paper, we investigate the adaptive NN tracking control problem for a class of switched strict-feedback nonlinear systems subject to uncertain input delay and external time-varying disturbances. The control scheme is proposed based on backstepping, NNs and adaptive control. The proposed controller ensures that all signals in the closed-loop system are SGUUB, and that the tracking error of the system converges to a small neighbourhood. Compared with the previous works, the main contributions of this paper can be concluded as follows: (i) This work is the first result to deal with the adaptive NN tracking control problem for switched strict-feedback nonlinear systems with uncertain input delay. (ii) The switched input delayed system is tactfully transformed into the non-delayed system by designing a new variable with the input integral term. (iii) In the design procedure, the common virtue control function at each step is constructed explicitly; subsequently, the CLF and the state-feedback controller can be obtained explicitly. The paper is structured as follows. In Section 2 the NN tracking control problem for a class of switched nonlinear systems is presented. Section 3 is devoted to the main contribution of the paper, namely the design of the state feedback controller. The effectiveness of the proposed controller is shown via simulation results in Section 4. The paper finishes with a conclusion in Section 5.
2. System description and problem formulation Consider the following switched nonlinear time delay systems in strict-feedback form: 8 _ > > > ξ i ¼ f σ ðtÞ;i ðξi Þ þ ξi þ 1 þlσ ðtÞ;i ðξ; tÞ; 1 r i rn 1; > > < ξ_ ¼ f n σ ðtÞ;n ðξÞ þg σ ðtÞ ðξÞuðt τ ðtÞÞ þ lσ ðtÞ;n ðξ; tÞ; ð1Þ >y¼ξ ; > 1 > > > : ξðtÞ ¼ ξ ðtÞ; τ r t r0; 0
where ξ ¼ ½ξ1 ; …; ξn T A R and y A R are the system state and output, respectively, ξi ¼ ½ξ1 ; …; ξi T ; ξn ¼ ξ. uðt τ ðtÞÞ A R is the control input, uðtÞ ¼ 0; t o 0. σ ðtÞ : ½0; 1Þ-Ξ ¼ def f1; 2; …; Ng is a piecewise constant function called switching signal (or law), which takes values in the compact set Ξ . If σ ðtÞ ¼ p, then we say the pth subsystem is active and the remaining subsystems are inactive. For p A Ξ ; i ¼ 1; 2; …; n, g p ðξÞ is the uncertain control gain function, f p;i n
ðξi Þ are unknown continuous function, lp;i ðξ; tÞ are time-varying disturbances, ξ0 ðtÞ is a continuous vector function. The objective of this paper is to make the system output to track the reference signal yr ðtÞ.
Remark 1. For switched systems under arbitrary switchings, it has been proved that global asymptotic stability is equivalent to the existence of a CLF. Thus, this paper tries to solve the adaptive NN tracking control problem of the switched systems (1) under arbitrary switchings by adopting the CLF design method. The following assumptions and lemma are needed throughout the paper. Assumption 1. The input delay satisfies the following formulas: τðtÞ ¼ τ þ Δτ ðtÞ; Δτ ðtÞ o δτ ; τðtÞ Z 0; τ Z 0, where τ is a known constant and δt is an unknown small constant. Assumption 2. The control gain function satisfies the following formulas: g p ðξÞ ¼ g n ðξ n 1 Þ þ Δg p ðξÞ; Δg p ðξÞ o δg ; g p ðξÞ Z g 0 ; g n ðξ n 1 Þ Zg 0; p A Ξ , where g n ðξ n 1 Þ is a known continuous function and δg ; g 0
are unknown small positive constants. In the paper, substitute g n ðξÞ for g n ðξ n 1 Þ. Assumption 3. The unknown disturbances lp;i ðÞ; ð1 r ir n; p A Ξ Þ satisfy lp;i ðÞ r l i with l i being unknown constants. Assumption 4. The reference signal yr ðtÞ is n times differentiable, that is, yr ðtÞ r y r ; yðiÞ r ðtÞ r y r ð1 r i r nÞ, where y r is an unknown constant. Lemma 1 (Long and Zhao [36]). For the continuous function Q i ðβ i Þ and the bounded closed set C βi , there is a neural network satisfying Q i ðβ i Þ ¼ W Ti Si ðβ i Þ þ εwi ðβ i Þ; 8 βi A C βi , where Si ðβ i Þ ¼ ½Si;1 ðβ i Þ; …; Si;mi ðβ i ÞT . The Gaussian basis function is selected as Si;j ðβ i Þ ¼ 2 βi μi;j =2ς2i;j , W i A Rmi is the weight vector of the neural nete work. εwi ðβ i Þ r ε wi is the estimation error. Denote the best weight vector as 8 9 = < T n sup W i Si ðβi Þ li ðβi Þ : W i ≔arg minm ; wi A R i :β A C β i i
4
Define W i the estimation of W ni . The conclusion can be easily promoted to the vector functions. For the continuous vector function f ðξÞ and the bounded closed set C ξ , by the estimate ability of Radial basis function (RBF) neural networks, there is a perfect RBF neural network which satisfies f p ðξÞ ¼ φp ðξÞθp þ εf ;p ðξÞ 8 ξ A C ξ , where φp ðξÞ ¼ ½φTp;1 ðξÞ; …; φTp;n ðξÞT , φp;i ðξÞ ¼ ½φp;i;1 ðξÞ; …; φp;i;q ðξÞ. The Gaussian basis function is selected as φp;i;j ðξÞ ¼ 2 ξ μi;j =2ς2i;j . θ A Rq is the weight vector of the neural network. e εf ;p ðξÞ r ε f is the estimation error. Denote the best weight vector as ( ) n θp ≔arg minq sup φp ðξÞθP f p ðξÞ : θp A R
ξ A Cξ
For convenience θ is used to denote θ in some equations below without confusion, and W is used to denote W n in some equations below too. Further, notation kk denotes the 2-norm of a vector or the Frobenius norm of a matrix. n
3. Main result Based on Lemma 1, system (1) can be rewritten as (
ξ_ ¼ Aξ þ Bgp ðξÞuðt τðtÞÞ þ φp ðξÞθp þ εf ;p ðξÞ þlp ðξ; tÞ; y ¼ C ξ; p A Ξ ;
where 2 0 6⋮ 6 A¼6 40 0
3
1
0
⋱
7 7 7; 15
⋯
0
2
4
⋮
φp;n ðξn Þ
7; 7 5
C ¼ ½1 0 ⋯ 0;
1
3
φp;1 ðξ1 Þ 6 7 6 φp;2 ðξ2 Þ 7
φp ðξÞ ¼ 6 6
2 3 0 6⋮7 6 7 B ¼ 6 7; 405
ð2Þ
2
3
εf ;p1 ðξ1 Þ 6 ε ðξ Þ 7 6 f ;p2 2 7
εf ;p ðξÞ ¼ 6 6 4
⋮
εf ;pn ðξÞ
2
7; 7 5
θp;1
3
7 θp ¼ 6 4 ⋮ 5; θp;q
φp;i ðξ i Þ A R1q ; lp ðξ; tÞ ¼ lp;1 ðξ; tÞ; …; lp;n ðξ; tÞ T ; εf ;p ðξÞ r ε f :
ð3Þ
In the next, we substitute lp;i ; εf ;p for lp;i ðξ; tÞ; εf ;p ðξ; tÞ for simplification, respectively, throughout the paper.
B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
(
Then we design a filter as
where
η_ ¼ A0 η þ ky þ Bg ðξÞuðt τÞ; _ ¼ A Ω þ φ ðξÞ; p A Ξ ; Ω 0 p n
ð4Þ
β
Q 0p;i ð i Þ ¼
8
> < 1 zi ¼ ξi αi 1 ; 2 r i r n 1; ð9Þ R > > : zn ¼ ξn αn 1 þ g n ðξÞ 0 uðt þ χ Þ dχ : τ Further, define assistant functions
8 > Q p;1 ðβ 1 Þ ¼ φp;1 ðξ1 Þθp þ tanhðz1 =ς1 Þðε f þ l 1 Þ; > > > > 0 > > Q > p;i ðβi Þ ¼ Q p;i ðβi Þ; 2 r ir n 1; > > 2 > > 2 0 n2 zn 1 þ d2 > Q p;n 1 ðβ > n 1 Þ ¼ Q p;n 1 ðβ n 1 Þ þ dn 1;5 θ n 1;4 g ðξÞzn 1 ; > > > > n 1 > X > ∂α n 1 > > ðξj þ 1 þ φp;j ðξÞθp Þ > Q p;n ðβ n Þ ¼ kn ðy η1 Þ > > ∂ξj > j¼1 > > > < !2 n 1 1 n 1 : X X ∂αn 1 ðjÞ nX ∂αn 1 ^_ ∂ αn 1 2 > > y θ þ r zn w > 1 j j1 r > ^ ∂ ξ > j > j ¼ 1 ∂yr j ¼ 1 ∂ θ wj j¼1 > > > > Z > n 1 0 n X ∂g > > > > þ uðt þ χ Þdχ ðξj þ 1 þ φp;j ðξÞθp Þ > > ∂ξj τ > > j¼1 > > > !2 > Z 0 > n 1 X > ∂g n > 2 > > þ r uðt þ χ Þ d χ zn ; 2 > : ∂ξ j¼1
j
i1 X
½ξj þ 1 þ φj ðξ j Þθp
pi 1;j tanh ðzi pi 1;j =ςi Þðε f þ l i Þ;
∂α i ; 1 r i rn: ∂ ξj
ð11Þ
ð12Þ
τ
ð10Þ
1 r i rn;
ð13Þ
where ai , bi are the design parameters, θ^ wi is the approximation of θnwi , and 2 θnwj ¼ max W np;i ; h iT T β1 ¼ yr ; ξ1 T ; βi ¼ y Tr;i ; ξ i ; αi 1 ; ψ wi ; 2 ri r n 1; " Z ∂g n 0 βn ¼ y Tr;n ; ξT ; αn 1 ; ψ wn ; zn ; uðt þ χ Þ dχ ; …; ∂ ξ1 τ #T Z 0 ∂g n uðt þ χ Þ dχ ; ∂ ξn 1 τ h iT y r;i ¼ yr ; ⋯yðir 1Þ ;
where
Δu ¼ gp ðξÞuðt τðtÞÞ gn ðξÞuðt τðtÞÞ:
∂ξj
Define the virtual control αi ¼ a2i zi b2i θ^ wi Si ðβi Þ2 zi þ yðiÞ r ;
Ωi ðtÞ A R1q ; i ¼ 1; 2; …; n;
¼ A0 eþ BΔu þ εf ;p ðξÞ þlp ðξ; tÞ;
j¼1
j¼1
7 ξ^ ¼ 6 4 ⋮ 5; ^ξ n
e ¼ ξ ξ^ ;
i1 X ∂αi 1
9 i1 i1 X X ∂αi 1 ^_ ∂αi 1 ðjÞ = θ wj y þtanhðzi =ςi Þðε f þ l i Þ j1 r ; ^ j ¼ 1 ∂ θ wj j ¼ 1 ∂yr
ð5Þ
where 2 3 2 3 η1 k1 6 7 6 7 k ¼ 4 ⋮ 5; η ¼ 4 ⋮ 5; ηn kn 2 3
φp;i ðξ i Þθp
:
and a virtual observer as
ξ^ ¼ η þ Ωθp ;
2123
ψ wi ¼
i1 X ∂αi 1 ^_ θ wj : ^ j ¼ 1 ∂ θ wj
ð14Þ
From Lemma 1, we employ the RBF neural network to approximate the continuous function Q p;i ðβi Þ such that Q p;i ðβ i Þ ¼ W Tp;i Si ðβi Þ þ εwp;i ðβ i Þ;
8 βi A C βi :
ð15Þ
Now, the main result can be summarized in terms of the following theorem. Theorem 1. Suppose that the switched system (1) satisfies Assumptions 1–4. If the virtual control αi is defined by (13), the control law is selected as uðtÞ ¼ αn =g n ðξÞ;
ð16Þ
and the adaptive law is selected as
_
θ^ wi ¼ r 2i ðSi ðβi Þ2 z2i δ2i θ^ wi Þ;
1 r i rn;
ð17Þ
then the tracking error is bounded and ultimately converges to an adequately small compact set: pffiffiffiffiffiffiffi y y r 2V ; lim y y r kn υn ; ð18Þ r r t-1
where k is the design parameter, V and υn are constants. Further, the closed-loop system is SGUUB. n
Proof. First, choose the Lyapunov function V(t) as VðtÞ ¼
n X i¼1
2
V i ðtÞ;
1 b 2 V i ðtÞ ¼ z2i þ i 2 θ~ wi ; 1 r i r n; 2 2r i
ð19Þ
n where θ~ wi ¼ θ^ wi θwi . Step 1. For the z1-equation, from (2), (4) and (9)?, we have
z_ 1 ¼ y_ y_ r
¼ ξ2 þ φp;1 ðyÞθp þ εf p;1 þ lp;1 y_ r ¼ z2 þ α1 þ φp;1 ðyÞθp þ εf p;1 þ lp;1 y_ r :
ð20Þ
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B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
Using Young's inequality, it can be obtained that 2 2 T 1 2 z1 W np;1 S1 ðβi Þ r b1 W np;1 S1 ðβi Þ z21 þ 2 4b1 2 2 1 2 r b1 W n1 S1 ðβ i Þ z21 þ 2 ; 4b1 z1 εwp;1 r d1;1 z21 þ
1
2
2
4d1;1
ε2wp;1 r d21;1 z21 þ
1 2
4d1;1
þ αi þ φp;i ðξ i Þθp zi
ð22Þ
1
1
2 4b1
1
þ
ε
δ
ς ε
θ
j¼1
2 bi ðiÞ i Þ yr þ 2 r2
_
θ~ w2 θ^ w2 þ d2i;1 z2i þ
z2i þ 1 2
4di;1
kςi ðε f þ l i Þ r kvi V i þ bvi þ Δvi ;
kvi ¼ minf2m2i ; r 2i δi g, 2
where
z2i
2
4di 1;1
ð27Þ
þ
z2i þ 1 2
4di;1
,
b δ bvi ¼ 12 þ i2 i 4b
2 2
i
2
2
m2i ¼ a2i di;1 di;2
n
ðθ w i Þ2 þ
1 2 4d1;2
ε þ 2 wi
Pi
j¼1
1 , 2 4di 1;1
Δvi ¼
kςi ðε f þ l i Þ.
Step n 1. By integral medium theorem, it is easy to obtain Z 0 uðt þ χ Þ dχ ¼ τuðζ Þ: ð28Þ ( ζ A ½t τ; t s:t:
¼ ξn þ φp;n 1 ðξ n 1 Þθp þ εf p;n 1 ðξ n 1 Þ þ lp;n 1 ðξ; tÞ α_ n 2
ð24Þ
¼ ηn þ Ωn θp þ en þ φp;n 1 ðξÞθp þ εf p;n 1 Z þ lp;n 1 α_ n 2 ¼ zn þ αn 1 g n ðξÞ
1 z2 , 4d21;1 2
2 n b2 δ bv1 ¼ 12 þ 12 1 ð w1 Þ2 þ 12 2w1 4b1 4d1;2
θ
ςi
z_ n 1 ¼ ξ_ n 1 α_ n 2
ε
þ kς1 ðε f þ l 1 Þ r kv1 V 1 þbv1 þ Δv1 ; n o 2 2 2 where kv1 ¼ min 2m21 ; r 21 δ1 ; m21 ¼ a21 d1;1 d1;2 ; Δv1 ¼
i1 h X zi pi 1;j k ς i ðε f þ l i Þ r z i α i ðε f þ l i Þ þ
For the zn 1 -equation, ð2 ri r n 2Þ, from (5), (6), and (9), we have
δ θ θ
δ
θ
2 bi ~ ^_ zi ðε þ l Þ þ kςi ðε f þl i Þ θ θ þ z tanh w w i i i ςi f i r 2i
τ
1 2 2 2 2 z2 þk 1 ð f þl 1 Þ b1 1 ~ w1 ^ w1 r w1 þ 2 2 4d1;2 4d1;1 2 2 2 2 2 b 1 1 b 1 n m21 z21 1 1 ~ w1 þ 2 z22 þ 2 þ 1 1 ð w1 Þ2 þ 2 2w1 2 2 4d1;1 4b1 4d1;2 þ
ξj þ 1 þ φp;j ðξ j Þθp þ εf p;j ðξ j Þ
j¼1
ε 2w1 þ 2 z21 þ kς1 ðε f þ l 1 Þ 2 4d1;2 4d1;1 2 2 2 2~ 2 2 ^ þ b1 θ w1 S1 ðβ1 Þ z1 δ1 θ w1 r ða21 d1;1 d1;2 Þz21 2
þ d1;2 z21 þ
i X
þ
2
2
zi pi 1;j tanh
T
2 b _ 2 2 þ kς1 ðε f þ l 1 Þ þ 21 θ~ w1 θ^ w1 r ða21 d1;1 Þz21 b1 θ~ w1 S1 ðβ 1 Þ z21 r1 4b1
i1 X
þ
2
4di;1
z2i þ 1 þ
þW ni Si ðβi Þ þ εwi ðβ
2
þ
1
2
þ di;1 z21 þ
j¼1
T b1 ~ ^_ 1 θ w θ w r z1 ½α1 þ W n1 S1 ðβ1 Þ þ εw1 y_ r þ d21;1 z21 þ 2 z22 r 21 1 1 4d1;1
1
∂ ξj
9 i1 i X X ∂αi 1 ^_ ∂αi 1 ðjÞ = ½ξj þ 1 þ φp;j ðξ j Þθp θ wj y ðj 1Þ r ; ^ ∂ ξj j ¼ 1 θ wj j ¼ 1 ∂yr
j¼1
b2 _ þ lp;1 y_ r þ 21 θ~ w1 θ^ w1 r z1 ðα1 þ φp;1 ðξ1 Þθp y_ r Þ þz1 z2 r1 2 h i b _ 2 þ jz1 jðε f þ l 1 Þ þ 21 θ~ w1 θ^ w1 rz1 α1 þ φp;1 ðξ1 Þθp y_ r þ d1;1 z21 r1 1 þ 2 z22 þ z1 tanhðz1 =ς1 Þðε f þ l 1 Þ þ kς1 ðε f þ l 1 Þ 4d1;1
1
j¼1
h
i1 X ∂α i 1
1 2 1 n ð23Þ θ~ w1 θ^ w1 r θ~ w1 þ ðθw1 Þ2 ; 2 2 2 2 2 n 2 where εw1 ¼ max εwp;1 , W 1 ¼ max W p;1 ðξÞ ; p A Ξ . Differentiating V 1 in (19) along the (22) and substituting virtual control (13), adaptive law (17) and (21)–(23) into it, we obtain V_ 1 ¼ z1 z2 þ α1 þ φp;1 ðξ1 Þθp þ εf p;1
þ
:
i1
) i1 i X X ∂αi 1 ^_ ∂αi 1 ðjÞ θ y þlp;j ðξ; tÞ : þ þ j zi jðε f i þ l i Þ wj þ ðj 1Þ r ^ j ¼ 1 θ wj j ¼ 1 ∂yr 2 i1 n X b _ ∂ ξi 1 þ zi ðε f i þ l i Þ þ 2i θ~ wi θ^ wi r zi αi þ φp;i ðξ i Þθp ∂ ξj ri j¼1
ð21Þ
ε2w1 ;
8 i1 <X ∂α
ε þ kς1 ðε f þ l 1 Þ. Notice that we use jzj r
z tanhðz=ςÞ þ kς; k ¼ 0:2785 to deal with jz1 jðε f þ l 1 Þ. Step i. For the zi-equation ð2 r ir n 2Þ, from (2), (4) and (9)?, we can infer that
0 τ
uðt þ χ Þ dχ þ Ωn θp þen
þ φp;n 1 ðξÞθp þ εf p;n 1 þ lp;n 1 α_ n 2 ¼ zn þ αn 1 τg n ðξÞuðηÞ þ Ωn θp þen þ φp;n 1 ðξÞθp þ εf p;n 1 þ lp;n 1 α_ n 2 :
ð29Þ
z_ i ¼ ξ_ i α_ i 1 ¼ ξi þ 1 þ φp;i ðξ i Þθp þ εf p;i ðξ i Þ þ lp;i ðξ; tÞ α_ i 1 ¼ zi þ 1 þ αi þ φp;i ðξ i Þθp þ εf p;i þlp;i α_ i 1 :
ð25Þ
Differentiating V n 1 in (19) along (29) and substituting (12), (16), (26) into it, we get V_ n 1 ¼ zn 1 ½zn þ αn 1 τg n ðξÞuðηÞ þ Ωn θp þ en þ φp;n 1 ðξÞθp þ εf p;n 1 þlp;n 1 α_ n 2 þ
According to the control (9) and (13), we get
α_ i 1 ¼
i1 X ∂ αi 1 h j¼1
þ
i1 X ∂α i 1 j¼1
∂θ^
i
r kvn 1 V n 1 þ bvn 1 þ Δvn 1 ;
ξj þ 1 þ φp;j ðξ j Þθp þ εf p;j ðξ j Þ þ lp;j ðξ; tÞ
∂ξj
wj
þ
kvn 1 ¼
where
i X ∂αi 1
yðjÞ : j1 r j ¼ 1 ∂yr
ð26Þ
2 dn 1;j 21 4d
n 2;1
n
Differentiating V 2 in (19) along (25) and substituting (13), (26) into it, we obtain 2
2
bn 1 ~ _ θ w θ^ w r 2n 1 n 1 n 1
b _ V_ i ¼ zi ðzi þ 1 þ αi þ φp;i ðξ i Þθp þ εf p;i þ lp;i α_ i 1 Þ þ 2i θ~ wi θ^ wi r zi ½zi þ 1 ri
n 1Þ
2
þ
Ωn 2
4dn 1;5
δ
, Δvn 1 ¼
2
þ
ε
ð30Þ
2 minf2m2n 1 ; r 2n 1 n 1 g,
2 wn 1 2 4dn 1;2
þ
z2n 1 2
4dn 2;1
þ
z2n
2
4dn 1;1
2 2 e þ ðτuð2 ηÞ Þ þ 2 4dn 1;4 4dn 1;3
m2n 1 ¼ a2n 1
, bvn 1 ¼
Pn 1 j¼1
1
2
4bn 1
þ
P4
j¼1
bn 1 δn 1 ð w 2 2
kςn 1 ðε f þ l j Þ.
Step n. For the zn-equation, from Assumption 2, we get Z 0 d z_ n ¼ η_ n α_ n 1 þ ðg n ðξÞ uðt þ χ Þ dχ Þ dt τ
2
θ
B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
2125
2
þ φp;j ðξÞθp þ r 22 z2n
y
1.5
n 1 X j¼1
yr
∂g n ∂ ξj
Z
!2
0 τ
uðt þ χ Þdχ
þ
1 1 nX ðε f þ l j Þ2 ; 2 4r 2 j ¼ 1
ð33Þ 1 Output y and targety y
r
where r 1 ; r 2 are the design parameters in (9). Differentiating V n in (19) along (31) and substituting (16), (17), (32) and (33) into it, we get 2 3 Z 0 n 1 X ∂g n _ n _ 4 ξ uðt þ χ Þ dχ þg ðξÞuðtÞ5 V n ¼ zn kn ðy η1 Þ α_ n 1 þ ∂ ξj j τ j¼1
0.5 0 −0.5
2
−1
þ
bn ~ ^_ θ w θ w r kvn V n þ bvn þ Δvn ; r 2n n n
ð34Þ
−1.5
where kvn ¼ minf2m2n ; r 2n δn g, m2n ¼ a2n dn;1 2
−2 0
10
20
30
40
50
60
t/s
ε þ r4r1 þ2 rr22
θ
2
2
Pn 1
1 2
j¼1
2
n 1;1
ðε f þ l j Þ2 :
ϑvn ; ϑvn ¼ bvn =kvn ; V n0 ¼ V n ð0Þ. Define
3
z
V n ¼ maxfV n0 ; ϑvn g;
1
2
V n r V n ; lim V n ¼ ϑvn ;
ð35Þ
t-1
r
we have qffiffiffiffiffiffiffiffiffi r qffiffiffiffiffiffiffiffiffi r qffiffiffiffiffiffiffiffiffi n n n jzn j r 2V n ; θ~ wn r 2V n ; θ^ wn r 2V n þ θwn : bn bn
1
1
Tracking error z =y−y
2 2 bn δn n 2 2 1 1 2 þ 2 ð wn Þ þ 2 4bn 4dn;1 wn
Δvn ¼ 4d2 zn ,
1 , 2 4dn 1;1
Thus, we can conclude that 0 rV n ðtÞ r ðV n0 ϑvn Þe kvn t þ
Fig. 1. Tracking performances of the closed-loop system (52).
0
ð36Þ
By the property of Gaussian basis function, we get pffiffiffiffiffiffi Si;j ðβi Þ r 1; Si ðβi Þ r mi ; 1 ri r n:
−1
−2
−3
bvn ¼
2
0
10
20
30
40
50
60
t/s Fig. 2. Responses of the tracking error z1 ¼ y yr .
¼ kn ðy η1 Þ α_ n 1 þ
n 1 X
∂g n _ ξ ∂ ξj j j¼1
Z
0 τ
uðt þ χ Þ dχ þ g n uðtÞ:
ð31Þ
ð37Þ
From Assumption 4, virtual control (13) and controller (16), we have 8" 1 < 2 2^ 2 ðnÞ 2 uðtÞ ¼ 1 g n ðξÞ an zn bn θ wn Sn ðβn Þ zn þ yr r g : an 0 9
qffiffiffiffiffiffiffiffiffi n ðnÞ = rn 2 þ bn mn ð38Þ 2V n þ θwn jzn j þ yr r u; ; bn where
qffiffiffiffiffiffiffiffiffi n qffiffiffiffiffiffiffiffiffi 1 rn 2 u¼ a2n þ bn mn 2V n þ y r : 2V n þ θwn g0 bn
ð39Þ
From (26), we have zn α_ n 1 r zn
Thus, we have bvn 1 r b vn 1 , where
n 1 X
n 1 X ∂α n 1 ∂αn 1 ^_ ðξj þ 1 þ φp;j ðξÞθp Þ zn θ wj ^ ∂ ξ j j¼1 j ¼ 1 ∂θ w
b vn 1 ¼
j
zn
þ
n X ∂αn 1
n 1 X
yðjÞ þ r 21 z2n ði 1Þ r j ¼ 1 ∂yr j¼1
∂α n 1 ∂ ξj
!2
1 1 nX ðε f þ l j Þ2 : 2 4r 1 j ¼ 1
ð32Þ
zn
∂g _ ξ ∂ξj j j¼1
Z
0 τ
uðt þ χ Þ dχ ¼ zn
n 1 X
n
∂g ∂ξj j¼1
Z
n 1 X
0 τ
uðt þ χ Þ dχ ½ξj þ 1
∂g n þ φp;j ðξÞθp þ εf p;j ðξÞ þ lp;j ðξ; tÞ r zn ∂ξj j¼1
Z
0 τ
uðt þ χ Þ dχ ½ξj þ 1
2
n 1;5
τ2 u 2 2
2
bn 1 δn 1 n Ω ðθwn 1 Þ2 þ 2 n 2 4d 2
þ
4dn 1;4
þ
n 1 X
kςn 1 ðε f þ l j Þ:
þ
ε 2wn 1 2 4dn 1;2
þ
e2 2 4dn 1;3
ð40Þ
j¼1
Repeating the backstepping design process, we can obtain the following results: V i r V i;
n
2 4bn 1
þ
Based on (10), we can obtain n 1 X
1
jzi j r
lim V i ¼ ϑvi ; 1 ri r n 1;
t-1
qffiffiffiffiffiffiffiffi 2V i ;
qffiffiffiffiffiffiffiffi ~ ri 2V i ; 1 ri r n 1; θ wi r bi
qffiffiffiffiffiffiffiffi ^ ri n 2V i þ θwi ; θ wi r bi jαi j r α i
1 r ir n 2;
1 r i r n 1;
ð41Þ
ð42Þ
ð43Þ ð44Þ
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B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
qffiffiffiffiffiffiffiffi ξ r 2V i þ α i ;
ð45Þ
i
2
1 r i r n 1:
ð46Þ
Step n þ 1. Overall, differentiating the Lyapunov function V in (19), we get V_ ¼
n X
V_ i ðtÞ r kv VðtÞ þ bv ;
ð47Þ
Switching signal σ(t)
where
qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi r 2V i þy r ; α i ¼ a2i þ b21 mi i 2V i þ θnwi bi
i¼1
2 2 P 2 2 b δ where kv ¼ minf2m21 ; …; 2m2n ; r 21 δ1 ; …; r 2n δn g, bv ¼ ni¼ 1 12 þ i2 i 4bi 2 P P 2 ε 2w 2 2 r 21 þ r 22 n Ω ðθwi Þ2 þ 2i þ ni ¼ 11 ij ¼ 1 kςi ðε f þl j Þ þ 2 n þ 2e þ τ2 u þ 4r 2 r2 Pn 1
4di;2
4dn 1;5
ðε f þ l j Þ . Thus, we V 0 ¼ Vð0Þ.
4dn 1;3
4dn 1;4
1
1 2
0
2
j¼1
have
0 r VðtÞ r ðV 0 ϑv Þe kv t þ ϑv ,
^ θ w i r respec-
tively. It is easy to obtain that ξ, Ω, η, e are all bounded, which means the closed-loop system is SGUUB. Moreover, lim VðtÞ ¼ ϑv ;
30
40
50
60
Fig. 4. Switching signal for the system (52).
subsequently, the CLF and adaptive NN controller can be obtained explicitly. However, the work in [10,11] supposes that there exists a common virtual control at each step under a so-called simultaneous domination assumption. Obviously, the result in this paper can be regarded as a generalization of the works in [10,11].
ð48Þ
t-1
4. Simulation results
then we have pffiffiffiffiffiffiffi y yr r 2V ;
ð49Þ
pffiffiffiffiffiffiffiffi lim y yr r 2ϑv ¼
t-1
where sffiffiffiffiffi 2 n ; k ¼ kv
20
t/s
ϑv ¼ bv =kv ,
pffiffiffiffiffiffiffi Define V ¼ maxfV 0 ; ϑv g. Then, we have kzk r 2V , pffiffiffiffiffiffiffi n ðr i =bi Þ 2V þ θwi . Further, z; θ^ i ; αi and ξ n 1 are bounded
VðtÞ rV ;
10
ϑn ¼
sffiffiffiffiffi 2 pffiffiffiffiffi n n bv ¼ k ϑ ; kv
ð50Þ
pffiffiffiffiffi bv :
ð51Þ
In this section, we given a numerical example to illustrate the effectiveness of the proposed control scheme. Example 1. Consider the following switched nonlinear system with uncertain input delay: 8 _ > > > ξ 1 ¼ f σ ðtÞ;1 ðξ1 Þ þ ξ2 þ lσ ðtÞ;1 ðξ; tÞ; > > < ξ_ ¼ f σ ðtÞ;2 ðξÞ þ g σ ðtÞ ðξÞuðt τ ðtÞÞ þ lσ ðtÞ;2 ðξ; tÞ; 2 ð52Þ > > y ¼ ξ1 ; σ ðt Þ : ½0; 1Þ-f1; 2g; > > > : ξðtÞ ¼ ξ ðtÞ; τ rt r 0; 0
It implies that the tracking error is bounded and ultimately converges to an adequately small compact set which can be n adjusted by the design parameter k . The proof is thus completed.□ Remark 3. In the control design procedure of this paper, the common virtual control at each step is constructed explicitly; 3 u 2
Control input u
1
T 3 where ξ ¼ ξ1 ; ξ2 ; f 1;1 ðξ1 Þ ¼ 0:1ξ1 , f 1;2 ðξÞ ¼ 0:3ξ1 ξ2 , g 1 ðξÞ ¼ 2 þ 2 sin ðξ1 Þ þ 0:05 sin ðξ2 Þ, l1;1 ðξ; tÞ ¼ 0:1 sin ðξ1 Þ, l1;2 ðξ; tÞ ¼ 0:2 sin 2 2 ð0:2tÞ, f 2;1 ðξ1 Þ ¼ 0:2ξ1 , f 2;2 ðξÞ ¼ 0:1ξ1 ξ2 , g 2 ðξÞ ¼ 2 þ sin ðξ1 Þ þ 0:05 cos ðξ1 Þ, l2;1 ðξ; tÞ ¼ 0:05 cos 2 ðξ1 ξ2 Þ, l2;2 ðξ; tÞ ¼ 0:3 cos ð0:1tÞ sin ð0:2tÞ, τðtÞ ¼ 0:3 þ 0:1 sin ðtÞ, τ ¼ 0:3, yr ðtÞ ¼ sin ð0:3tÞ sin ð0:6tÞ, A ¼ 00 10 , B ¼ 01 . 3 1 Further, we choose parameters: k ¼ ½3 2T , A0 ¼ 2 0 , a1 ¼ 1:2, a2 ¼ 1:3, b1 ¼ 0:1, b2 ¼ 0:1, γ 1 ¼ 1:2, γ 2 ¼ 1:2, σ 1 ¼ 0:05, σ 2 ¼ 0:05. Construct a virtual observer ξ^ for system (52) as 8 η_ ¼ A0 η þ ky þ Bg1 ðξ1 Þuðt 0:3Þ; > > < _ ¼ A Ω þ φ ðξÞ; Ω 0 p > > : ξ^ ¼ η þ Ωθ ; p
where "
0
Ω¼
−1
(
−2
−3
#
Ω1 ; Ω2
10
20
30
40
t/s Fig. 3. Responses of the control input u.
50
60
#
p ¼ 1; 2:
Define variables z1 ¼ ξ1 yr ; z 2 ¼ ξ2 α 1 þ
0
"
η1 η¼ η ; 2
R0 0:3
uðt þ θÞ dθ:
and define the virtual control αi ¼ a2i zi b2i θ^ wi Si ðλi Þ2 zi þ yðiÞ r ;
1 r i r2;
ð53Þ
B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
2127
where
[11] B. Niu, J. Zhao, Output tracking control for a class of switched non-linear systems
λ1 ¼ yr ; x1 T ;
with partial state constraints, IET Control Theory Appl. 7 (2013) 623–631. [12] Y.E. Wang, X.M. Sun, B.W. Wu, Lyapunov Krasovskii functionals for switched nonlinear input delay systems under asynchronous switching, Automatica 61
2
∂α _ ∂g λ2 ¼ 4yr ; y_ r ; x1 ; x2 ; α1 ; ^ 1 θ^ w1 ; z2 ; 1 ∂x1 ∂ θ w1
Z
0 τ
#T uðt þ θÞ dθ
:
ð54Þ
Moreover, the corresponding assistant functions are as follows:
(2015) 126–133. [13] L. Long, J. Zhao, H 1 control of switched nonlinear systems in p-normal form using multiple Lyapunov functions, IEEE Trans. Autom. Control 57 (2012) 1285–1291. [14] L. Long, J. Zhao, Output-feedback stabilisation for a class of switched nonlinear systems with unknown control coefficients, Int. J. Control 86 (2013) 386–395.
8 2 2 2 2 > > < Q p;1 ðλ1 Þ ¼ φp;1 ðξ1 Þθp þ tanhðz1 =ς1 Þðε f þ h 1 Þ þ d1:5 θ z1 þ d1;4 g 1 ðx1 Þz1 ;
2 ∂α 1 ∂α 1 ∂α1 ^_ ∂g R 0 2 ∂α 1 > _ Q ð λ Þ ¼ k ðy η Þ ð ξ þ φ ðxÞ θ Þ θ þr z2 þ 1 τ uðt þ θÞ dθðξ2 þ φ1 ðξÞθp Þ y > p w 2 2 1 1 1 : p;2 1 ∂yr r ∂θ^ w ∂ ξ1 2 ∂ ξ1 ∂ ξ1 1 !2 Z 0 ∂g 1 þ r 22 uðt þ θÞ dθ z2 ; p ¼ 1; 2: ∂ξ1 τ
Choose control law as uðtÞ ¼ α2 =g 1 ðx1 Þ and adaptive law as _
2
θ^ w1 ¼ γ 2i ðkSi ðλi Þk z2i σ 2i θ^ wi Þ;
1 ri r 2; ð56Þ P 2 and S1 A R125 , S2 A R16561 , S1;i ¼ exp j ¼ 1 ðλ1;j ui;j Þ2 , 1 r i r P 25, S2;i ¼ exp 8j ¼ 1 ðλ2;j vi;j Þ2 , 1 r i r 6561, ui;j A f 1 0:5 0 0:5 1g, v1;j A f 1 0 1g. Based on Theorem 1, the simulation results of the proposed control scheme for system (52) are shown in Figs. 1–4.
5. Conclusions This paper has proposed an adaptive tracking control scheme for a class of disturbed nonlinear switched systems in strick-feedback form with uncertain input delay by backstepping and NN. A filter and a virtual observer are constructed to substitute the system state. The designed controller ensures that all signals of the closed-loop system are SGUUB, and the tracking error converges to an adequately small compact set. The theoretical result is illustrated through a simulation example. The future research directions conclude how to deal with the switched system with state delays; how to design an observer for output feedback of the switched system.
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ð55Þ
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[36] L.J. Long, J. Zhao, Adaptive output-feedback neural control of switched uncertain nonlinear systems with average dwell time, IEEE Trans. Neural Netw. Learn. Syst. 26 (2015) 1350–1362. [37] B. Zhou, Input delay compensation of linear systems with both state and input delays by adding integrators, Syst. Control Lett. 82 (2015) 51–63. [38] S.Y. Yoon, Z.L. Lin, Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay, Syst. Control Lett. 62 (2013) 837–844. [39] Z.Y. Zhang, C. Lin, Bing Chen, Complete LKF approach to stabilization for linear systems with time-varying input delay, J. Frankl. Inst. 352 (2015) 2425–2440. [40] Raafia Majeed, Sohaira Ahmad, Muhammad Rehan, Delay-range-dependent observer-based control of nonlinear systems under input and output timedelays, Appl. Math. Comput. 262 (2015) 145–159. [41] Q. Zhu, T.P. Zhang, S.M. Fei, Adaptive tracking control for input delayed MIMO nonlinear systems, Neurocomputing 74 (2010) 472–480. [42] I. Karafyllis, M. Krstic, Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold, IEEE Trans. Autom. Control 57 (2012) 1141–1154.
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.
Lu Li was born in Liaoning Province, China, on October 1, 1992. She received the B.S. degree in mathematics and applied mathematics from Bohai University, Jinzhou, China, in 2014. She is working towards the M.S. degree in operation science and control theory from Bohai University, Jinzhou, China. Her current research interests include switched nonlinear systems and adaptive control.
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Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Brief Papers
Adaptive neural network tracking control for a class of switched strict-feedback nonlinear systems with input delay$ Ben Niu n, Lu Li College of Mathematics and Physics and Automation Research Institute, Bohai University, Jinzhou, Liaoning Province 121013, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 10 July 2015 Received in revised form 21 September 2015 Accepted 19 October 2015 Communicated by Xudong Zhao Available online 31 October 2015
In this paper, a neural-network-based control scheme is developed for the tracking control problem of a class of switched strict-feedback nonlinear systems with uncertain input delay and external time-varying disturbances. First, the auxiliary signals are obtained by masterly constructing a filter and a virtual observer. Then the adaptive backstepping technique and neural network (NN) are employed to construct a common Lyapunov function (CLF) and a state feedback controller for all subsystems. It is proved that all signals of the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB), and that the tracking error ultimately converges to an adequately small compact set. Finally, a simulation example is given to illustrate the effectiveness of the proposed control approach. & 2015 Elsevier B.V. All rights reserved.
Keywords: Switched nonlinear system Adaptive control Neural network Input delay
1. Introduction In the past decades, there has been increasing interest in the study of switched systems [1–8], as such systems can describe the behaviour of a large number of practical plants resulting from the interactions of continuous dynamics, discrete dynamics, and logic decisions, etc. From a theoretical viewpoint, the main contributions of studying switched systems are twofold: on one side several researchers have paid attention to state/mode observability, that is, to the possibility of reconstructing from the measured data, the continuous state, or the discrete mode; on the other side, the main interest has been devoted to stability analysis and control synthesize problems. Correspondingly, several nonlinear control design approaches such as Lyapunov stability theory and backstepping technique were extended to the case of switched nonlinear systems, and many novel results have been reported in the literature, see e.g., [9–16] and references therein. On the other hand, backstepping-based adaptive control technique has become one of the most popular control approaches for nonlinear systems in strict-feedback form, and some important results have been reported [17–23]. Especially, neural-network-based adaptive control technique for uncertain nonlinear systems in strict☆ This work was partially supported by the National Natural Science Foundation of China [Grant number 61304054, 61403041 and 61403354], the Program for Liaoning Provincial Excellent Talents in University, China [Grant number LJQ2014122]. n Corresponding author at: College of Control Science and Engineering, Dalian University of Technology, Dalian, Liaoning Province 116024, People's Republic of China. E-mail addresses: [email protected] (B. Niu), [email protected] (L. Li).
http://dx.doi.org/10.1016/j.neucom.2015.10.059 0925-2312/& 2015 Elsevier B.V. All rights reserved.
feedback form has been widely studied by researchers [24–30]. For example, robust adaptive tracking control is considered for a class of uncertain MIMO nonlinear systems with input nonlinearities by neural network in [31]; and in [32], an adaptive control scheme is proposed for a class of uncertain nonlinear system in strict-feedback form with backstepping method and NN, just name a few. However, the aforementioned results on adaptive fuzzy or NN control design are only for non-switched nonlinear systems in strict-feedback form, few results have appeared for the one of switched nonlinear systems in strict-feedback form [33–36]. Therefore, the study on adaptive control of switched uncertain nonlinear systems in strict-feedback form by employing NN is of great significance. Further, input delays are often encountered in many practical control systems, such as hydraulic rolling mill systems, networked control systems, manufacturing processes, and so on. Very careful attention should be put on input delays since it may result in oscillation or even instability of the overall closed-loop systems. In dealing with the adaptive control of nonlinear systems with input delays, various methods have been proposed in the past years [37– 40]. For instance, an adaptive tracking control method for Multiple Input Multiple Output (MIMO) nonlinear systems with input delay is proposed in [41]. By introducing a definition of the design variable with the input integral term, the input delayed system is converted to the non-delayed system and then the effective state feedback controller is obtained. Ref. [42] considered the stabilization problems for nonlinear systems under sampled and delayed measurements, and with inputs subject to delay and zero-order hold. However, to the best of the authors' knowledge, there is no result to deal with input delays for switched uncertain nonlinear
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B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
systems in strict-feedback form, and this motivates the present work. In this paper, we investigate the adaptive NN tracking control problem for a class of switched strict-feedback nonlinear systems subject to uncertain input delay and external time-varying disturbances. The control scheme is proposed based on backstepping, NNs and adaptive control. The proposed controller ensures that all signals in the closed-loop system are SGUUB, and that the tracking error of the system converges to a small neighbourhood. Compared with the previous works, the main contributions of this paper can be concluded as follows: (i) This work is the first result to deal with the adaptive NN tracking control problem for switched strict-feedback nonlinear systems with uncertain input delay. (ii) The switched input delayed system is tactfully transformed into the non-delayed system by designing a new variable with the input integral term. (iii) In the design procedure, the common virtue control function at each step is constructed explicitly; subsequently, the CLF and the state-feedback controller can be obtained explicitly. The paper is structured as follows. In Section 2 the NN tracking control problem for a class of switched nonlinear systems is presented. Section 3 is devoted to the main contribution of the paper, namely the design of the state feedback controller. The effectiveness of the proposed controller is shown via simulation results in Section 4. The paper finishes with a conclusion in Section 5.
2. System description and problem formulation Consider the following switched nonlinear time delay systems in strict-feedback form: 8 _ > > > ξ i ¼ f σ ðtÞ;i ðξi Þ þ ξi þ 1 þlσ ðtÞ;i ðξ; tÞ; 1 r i rn 1; > > < ξ_ ¼ f n σ ðtÞ;n ðξÞ þg σ ðtÞ ðξÞuðt τ ðtÞÞ þ lσ ðtÞ;n ðξ; tÞ; ð1Þ >y¼ξ ; > 1 > > > : ξðtÞ ¼ ξ ðtÞ; τ r t r0; 0
where ξ ¼ ½ξ1 ; …; ξn T A R and y A R are the system state and output, respectively, ξi ¼ ½ξ1 ; …; ξi T ; ξn ¼ ξ. uðt τ ðtÞÞ A R is the control input, uðtÞ ¼ 0; t o 0. σ ðtÞ : ½0; 1Þ-Ξ ¼ def f1; 2; …; Ng is a piecewise constant function called switching signal (or law), which takes values in the compact set Ξ . If σ ðtÞ ¼ p, then we say the pth subsystem is active and the remaining subsystems are inactive. For p A Ξ ; i ¼ 1; 2; …; n, g p ðξÞ is the uncertain control gain function, f p;i n
ðξi Þ are unknown continuous function, lp;i ðξ; tÞ are time-varying disturbances, ξ0 ðtÞ is a continuous vector function. The objective of this paper is to make the system output to track the reference signal yr ðtÞ.
Remark 1. For switched systems under arbitrary switchings, it has been proved that global asymptotic stability is equivalent to the existence of a CLF. Thus, this paper tries to solve the adaptive NN tracking control problem of the switched systems (1) under arbitrary switchings by adopting the CLF design method. The following assumptions and lemma are needed throughout the paper. Assumption 1. The input delay satisfies the following formulas: τðtÞ ¼ τ þ Δτ ðtÞ; Δτ ðtÞ o δτ ; τðtÞ Z 0; τ Z 0, where τ is a known constant and δt is an unknown small constant. Assumption 2. The control gain function satisfies the following formulas: g p ðξÞ ¼ g n ðξ n 1 Þ þ Δg p ðξÞ; Δg p ðξÞ o δg ; g p ðξÞ Z g 0 ; g n ðξ n 1 Þ Zg 0; p A Ξ , where g n ðξ n 1 Þ is a known continuous function and δg ; g 0
are unknown small positive constants. In the paper, substitute g n ðξÞ for g n ðξ n 1 Þ. Assumption 3. The unknown disturbances lp;i ðÞ; ð1 r ir n; p A Ξ Þ satisfy lp;i ðÞ r l i with l i being unknown constants. Assumption 4. The reference signal yr ðtÞ is n times differentiable, that is, yr ðtÞ r y r ; yðiÞ r ðtÞ r y r ð1 r i r nÞ, where y r is an unknown constant. Lemma 1 (Long and Zhao [36]). For the continuous function Q i ðβ i Þ and the bounded closed set C βi , there is a neural network satisfying Q i ðβ i Þ ¼ W Ti Si ðβ i Þ þ εwi ðβ i Þ; 8 βi A C βi , where Si ðβ i Þ ¼ ½Si;1 ðβ i Þ; …; Si;mi ðβ i ÞT . The Gaussian basis function is selected as Si;j ðβ i Þ ¼ 2 βi μi;j =2ς2i;j , W i A Rmi is the weight vector of the neural nete work. εwi ðβ i Þ r ε wi is the estimation error. Denote the best weight vector as 8 9 = < T n sup W i Si ðβi Þ li ðβi Þ : W i ≔arg minm ; wi A R i :β A C β i i
4
Define W i the estimation of W ni . The conclusion can be easily promoted to the vector functions. For the continuous vector function f ðξÞ and the bounded closed set C ξ , by the estimate ability of Radial basis function (RBF) neural networks, there is a perfect RBF neural network which satisfies f p ðξÞ ¼ φp ðξÞθp þ εf ;p ðξÞ 8 ξ A C ξ , where φp ðξÞ ¼ ½φTp;1 ðξÞ; …; φTp;n ðξÞT , φp;i ðξÞ ¼ ½φp;i;1 ðξÞ; …; φp;i;q ðξÞ. The Gaussian basis function is selected as φp;i;j ðξÞ ¼ 2 ξ μi;j =2ς2i;j . θ A Rq is the weight vector of the neural network. e εf ;p ðξÞ r ε f is the estimation error. Denote the best weight vector as ( ) n θp ≔arg minq sup φp ðξÞθP f p ðξÞ : θp A R
ξ A Cξ
For convenience θ is used to denote θ in some equations below without confusion, and W is used to denote W n in some equations below too. Further, notation kk denotes the 2-norm of a vector or the Frobenius norm of a matrix. n
3. Main result Based on Lemma 1, system (1) can be rewritten as (
ξ_ ¼ Aξ þ Bgp ðξÞuðt τðtÞÞ þ φp ðξÞθp þ εf ;p ðξÞ þlp ðξ; tÞ; y ¼ C ξ; p A Ξ ;
where 2 0 6⋮ 6 A¼6 40 0
3
1
0
⋱
7 7 7; 15
⋯
0
2
4
⋮
φp;n ðξn Þ
7; 7 5
C ¼ ½1 0 ⋯ 0;
1
3
φp;1 ðξ1 Þ 6 7 6 φp;2 ðξ2 Þ 7
φp ðξÞ ¼ 6 6
2 3 0 6⋮7 6 7 B ¼ 6 7; 405
ð2Þ
2
3
εf ;p1 ðξ1 Þ 6 ε ðξ Þ 7 6 f ;p2 2 7
εf ;p ðξÞ ¼ 6 6 4
⋮
εf ;pn ðξÞ
2
7; 7 5
θp;1
3
7 θp ¼ 6 4 ⋮ 5; θp;q
φp;i ðξ i Þ A R1q ; lp ðξ; tÞ ¼ lp;1 ðξ; tÞ; …; lp;n ðξ; tÞ T ; εf ;p ðξÞ r ε f :
ð3Þ
In the next, we substitute lp;i ; εf ;p for lp;i ðξ; tÞ; εf ;p ðξ; tÞ for simplification, respectively, throughout the paper.
B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
(
Then we design a filter as
where
η_ ¼ A0 η þ ky þ Bg ðξÞuðt τÞ; _ ¼ A Ω þ φ ðξÞ; p A Ξ ; Ω 0 p n
ð4Þ
β
Q 0p;i ð i Þ ¼
8
> < 1 zi ¼ ξi αi 1 ; 2 r i r n 1; ð9Þ R > > : zn ¼ ξn αn 1 þ g n ðξÞ 0 uðt þ χ Þ dχ : τ Further, define assistant functions
8 > Q p;1 ðβ 1 Þ ¼ φp;1 ðξ1 Þθp þ tanhðz1 =ς1 Þðε f þ l 1 Þ; > > > > 0 > > Q > p;i ðβi Þ ¼ Q p;i ðβi Þ; 2 r ir n 1; > > 2 > > 2 0 n2 zn 1 þ d2 > Q p;n 1 ðβ > n 1 Þ ¼ Q p;n 1 ðβ n 1 Þ þ dn 1;5 θ n 1;4 g ðξÞzn 1 ; > > > > n 1 > X > ∂α n 1 > > ðξj þ 1 þ φp;j ðξÞθp Þ > Q p;n ðβ n Þ ¼ kn ðy η1 Þ > > ∂ξj > j¼1 > > > < !2 n 1 1 n 1 : X X ∂αn 1 ðjÞ nX ∂αn 1 ^_ ∂ αn 1 2 > > y θ þ r zn w > 1 j j1 r > ^ ∂ ξ > j > j ¼ 1 ∂yr j ¼ 1 ∂ θ wj j¼1 > > > > Z > n 1 0 n X ∂g > > > > þ uðt þ χ Þdχ ðξj þ 1 þ φp;j ðξÞθp Þ > > ∂ξj τ > > j¼1 > > > !2 > Z 0 > n 1 X > ∂g n > 2 > > þ r uðt þ χ Þ d χ zn ; 2 > : ∂ξ j¼1
j
i1 X
½ξj þ 1 þ φj ðξ j Þθp
pi 1;j tanh ðzi pi 1;j =ςi Þðε f þ l i Þ;
∂α i ; 1 r i rn: ∂ ξj
ð11Þ
ð12Þ
τ
ð10Þ
1 r i rn;
ð13Þ
where ai , bi are the design parameters, θ^ wi is the approximation of θnwi , and 2 θnwj ¼ max W np;i ; h iT T β1 ¼ yr ; ξ1 T ; βi ¼ y Tr;i ; ξ i ; αi 1 ; ψ wi ; 2 ri r n 1; " Z ∂g n 0 βn ¼ y Tr;n ; ξT ; αn 1 ; ψ wn ; zn ; uðt þ χ Þ dχ ; …; ∂ ξ1 τ #T Z 0 ∂g n uðt þ χ Þ dχ ; ∂ ξn 1 τ h iT y r;i ¼ yr ; ⋯yðir 1Þ ;
where
Δu ¼ gp ðξÞuðt τðtÞÞ gn ðξÞuðt τðtÞÞ:
∂ξj
Define the virtual control αi ¼ a2i zi b2i θ^ wi Si ðβi Þ2 zi þ yðiÞ r ;
Ωi ðtÞ A R1q ; i ¼ 1; 2; …; n;
¼ A0 eþ BΔu þ εf ;p ðξÞ þlp ðξ; tÞ;
j¼1
j¼1
7 ξ^ ¼ 6 4 ⋮ 5; ^ξ n
e ¼ ξ ξ^ ;
i1 X ∂αi 1
9 i1 i1 X X ∂αi 1 ^_ ∂αi 1 ðjÞ = θ wj y þtanhðzi =ςi Þðε f þ l i Þ j1 r ; ^ j ¼ 1 ∂ θ wj j ¼ 1 ∂yr
ð5Þ
where 2 3 2 3 η1 k1 6 7 6 7 k ¼ 4 ⋮ 5; η ¼ 4 ⋮ 5; ηn kn 2 3
φp;i ðξ i Þθp
:
and a virtual observer as
ξ^ ¼ η þ Ωθp ;
2123
ψ wi ¼
i1 X ∂αi 1 ^_ θ wj : ^ j ¼ 1 ∂ θ wj
ð14Þ
From Lemma 1, we employ the RBF neural network to approximate the continuous function Q p;i ðβi Þ such that Q p;i ðβ i Þ ¼ W Tp;i Si ðβi Þ þ εwp;i ðβ i Þ;
8 βi A C βi :
ð15Þ
Now, the main result can be summarized in terms of the following theorem. Theorem 1. Suppose that the switched system (1) satisfies Assumptions 1–4. If the virtual control αi is defined by (13), the control law is selected as uðtÞ ¼ αn =g n ðξÞ;
ð16Þ
and the adaptive law is selected as
_
θ^ wi ¼ r 2i ðSi ðβi Þ2 z2i δ2i θ^ wi Þ;
1 r i rn;
ð17Þ
then the tracking error is bounded and ultimately converges to an adequately small compact set: pffiffiffiffiffiffiffi y y r 2V ; lim y y r kn υn ; ð18Þ r r t-1
where k is the design parameter, V and υn are constants. Further, the closed-loop system is SGUUB. n
Proof. First, choose the Lyapunov function V(t) as VðtÞ ¼
n X i¼1
2
V i ðtÞ;
1 b 2 V i ðtÞ ¼ z2i þ i 2 θ~ wi ; 1 r i r n; 2 2r i
ð19Þ
n where θ~ wi ¼ θ^ wi θwi . Step 1. For the z1-equation, from (2), (4) and (9)?, we have
z_ 1 ¼ y_ y_ r
¼ ξ2 þ φp;1 ðyÞθp þ εf p;1 þ lp;1 y_ r ¼ z2 þ α1 þ φp;1 ðyÞθp þ εf p;1 þ lp;1 y_ r :
ð20Þ
2124
B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
Using Young's inequality, it can be obtained that 2 2 T 1 2 z1 W np;1 S1 ðβi Þ r b1 W np;1 S1 ðβi Þ z21 þ 2 4b1 2 2 1 2 r b1 W n1 S1 ðβ i Þ z21 þ 2 ; 4b1 z1 εwp;1 r d1;1 z21 þ
1
2
2
4d1;1
ε2wp;1 r d21;1 z21 þ
1 2
4d1;1
þ αi þ φp;i ðξ i Þθp zi
ð22Þ
1
1
2 4b1
1
þ
ε
δ
ς ε
θ
j¼1
2 bi ðiÞ i Þ yr þ 2 r2
_
θ~ w2 θ^ w2 þ d2i;1 z2i þ
z2i þ 1 2
4di;1
kςi ðε f þ l i Þ r kvi V i þ bvi þ Δvi ;
kvi ¼ minf2m2i ; r 2i δi g, 2
where
z2i
2
4di 1;1
ð27Þ
þ
z2i þ 1 2
4di;1
,
b δ bvi ¼ 12 þ i2 i 4b
2 2
i
2
2
m2i ¼ a2i di;1 di;2
n
ðθ w i Þ2 þ
1 2 4d1;2
ε þ 2 wi
Pi
j¼1
1 , 2 4di 1;1
Δvi ¼
kςi ðε f þ l i Þ.
Step n 1. By integral medium theorem, it is easy to obtain Z 0 uðt þ χ Þ dχ ¼ τuðζ Þ: ð28Þ ( ζ A ½t τ; t s:t:
¼ ξn þ φp;n 1 ðξ n 1 Þθp þ εf p;n 1 ðξ n 1 Þ þ lp;n 1 ðξ; tÞ α_ n 2
ð24Þ
¼ ηn þ Ωn θp þ en þ φp;n 1 ðξÞθp þ εf p;n 1 Z þ lp;n 1 α_ n 2 ¼ zn þ αn 1 g n ðξÞ
1 z2 , 4d21;1 2
2 n b2 δ bv1 ¼ 12 þ 12 1 ð w1 Þ2 þ 12 2w1 4b1 4d1;2
θ
ςi
z_ n 1 ¼ ξ_ n 1 α_ n 2
ε
þ kς1 ðε f þ l 1 Þ r kv1 V 1 þbv1 þ Δv1 ; n o 2 2 2 where kv1 ¼ min 2m21 ; r 21 δ1 ; m21 ¼ a21 d1;1 d1;2 ; Δv1 ¼
i1 h X zi pi 1;j k ς i ðε f þ l i Þ r z i α i ðε f þ l i Þ þ
For the zn 1 -equation, ð2 ri r n 2Þ, from (5), (6), and (9), we have
δ θ θ
δ
θ
2 bi ~ ^_ zi ðε þ l Þ þ kςi ðε f þl i Þ θ θ þ z tanh w w i i i ςi f i r 2i
τ
1 2 2 2 2 z2 þk 1 ð f þl 1 Þ b1 1 ~ w1 ^ w1 r w1 þ 2 2 4d1;2 4d1;1 2 2 2 2 2 b 1 1 b 1 n m21 z21 1 1 ~ w1 þ 2 z22 þ 2 þ 1 1 ð w1 Þ2 þ 2 2w1 2 2 4d1;1 4b1 4d1;2 þ
ξj þ 1 þ φp;j ðξ j Þθp þ εf p;j ðξ j Þ
j¼1
ε 2w1 þ 2 z21 þ kς1 ðε f þ l 1 Þ 2 4d1;2 4d1;1 2 2 2 2~ 2 2 ^ þ b1 θ w1 S1 ðβ1 Þ z1 δ1 θ w1 r ða21 d1;1 d1;2 Þz21 2
þ d1;2 z21 þ
i X
þ
2
2
zi pi 1;j tanh
T
2 b _ 2 2 þ kς1 ðε f þ l 1 Þ þ 21 θ~ w1 θ^ w1 r ða21 d1;1 Þz21 b1 θ~ w1 S1 ðβ 1 Þ z21 r1 4b1
i1 X
þ
2
4di;1
z2i þ 1 þ
þW ni Si ðβi Þ þ εwi ðβ
2
þ
1
2
þ di;1 z21 þ
j¼1
T b1 ~ ^_ 1 θ w θ w r z1 ½α1 þ W n1 S1 ðβ1 Þ þ εw1 y_ r þ d21;1 z21 þ 2 z22 r 21 1 1 4d1;1
1
∂ ξj
9 i1 i X X ∂αi 1 ^_ ∂αi 1 ðjÞ = ½ξj þ 1 þ φp;j ðξ j Þθp θ wj y ðj 1Þ r ; ^ ∂ ξj j ¼ 1 θ wj j ¼ 1 ∂yr
j¼1
b2 _ þ lp;1 y_ r þ 21 θ~ w1 θ^ w1 r z1 ðα1 þ φp;1 ðξ1 Þθp y_ r Þ þz1 z2 r1 2 h i b _ 2 þ jz1 jðε f þ l 1 Þ þ 21 θ~ w1 θ^ w1 rz1 α1 þ φp;1 ðξ1 Þθp y_ r þ d1;1 z21 r1 1 þ 2 z22 þ z1 tanhðz1 =ς1 Þðε f þ l 1 Þ þ kς1 ðε f þ l 1 Þ 4d1;1
1
j¼1
h
i1 X ∂α i 1
1 2 1 n ð23Þ θ~ w1 θ^ w1 r θ~ w1 þ ðθw1 Þ2 ; 2 2 2 2 2 n 2 where εw1 ¼ max εwp;1 , W 1 ¼ max W p;1 ðξÞ ; p A Ξ . Differentiating V 1 in (19) along the (22) and substituting virtual control (13), adaptive law (17) and (21)–(23) into it, we obtain V_ 1 ¼ z1 z2 þ α1 þ φp;1 ðξ1 Þθp þ εf p;1
þ
:
i1
) i1 i X X ∂αi 1 ^_ ∂αi 1 ðjÞ θ y þlp;j ðξ; tÞ : þ þ j zi jðε f i þ l i Þ wj þ ðj 1Þ r ^ j ¼ 1 θ wj j ¼ 1 ∂yr 2 i1 n X b _ ∂ ξi 1 þ zi ðε f i þ l i Þ þ 2i θ~ wi θ^ wi r zi αi þ φp;i ðξ i Þθp ∂ ξj ri j¼1
ð21Þ
ε2w1 ;
8 i1 <X ∂α
ε þ kς1 ðε f þ l 1 Þ. Notice that we use jzj r
z tanhðz=ςÞ þ kς; k ¼ 0:2785 to deal with jz1 jðε f þ l 1 Þ. Step i. For the zi-equation ð2 r ir n 2Þ, from (2), (4) and (9)?, we can infer that
0 τ
uðt þ χ Þ dχ þ Ωn θp þen
þ φp;n 1 ðξÞθp þ εf p;n 1 þ lp;n 1 α_ n 2 ¼ zn þ αn 1 τg n ðξÞuðηÞ þ Ωn θp þen þ φp;n 1 ðξÞθp þ εf p;n 1 þ lp;n 1 α_ n 2 :
ð29Þ
z_ i ¼ ξ_ i α_ i 1 ¼ ξi þ 1 þ φp;i ðξ i Þθp þ εf p;i ðξ i Þ þ lp;i ðξ; tÞ α_ i 1 ¼ zi þ 1 þ αi þ φp;i ðξ i Þθp þ εf p;i þlp;i α_ i 1 :
ð25Þ
Differentiating V n 1 in (19) along (29) and substituting (12), (16), (26) into it, we get V_ n 1 ¼ zn 1 ½zn þ αn 1 τg n ðξÞuðηÞ þ Ωn θp þ en þ φp;n 1 ðξÞθp þ εf p;n 1 þlp;n 1 α_ n 2 þ
According to the control (9) and (13), we get
α_ i 1 ¼
i1 X ∂ αi 1 h j¼1
þ
i1 X ∂α i 1 j¼1
∂θ^
i
r kvn 1 V n 1 þ bvn 1 þ Δvn 1 ;
ξj þ 1 þ φp;j ðξ j Þθp þ εf p;j ðξ j Þ þ lp;j ðξ; tÞ
∂ξj
wj
þ
kvn 1 ¼
where
i X ∂αi 1
yðjÞ : j1 r j ¼ 1 ∂yr
ð26Þ
2 dn 1;j 21 4d
n 2;1
n
Differentiating V 2 in (19) along (25) and substituting (13), (26) into it, we obtain 2
2
bn 1 ~ _ θ w θ^ w r 2n 1 n 1 n 1
b _ V_ i ¼ zi ðzi þ 1 þ αi þ φp;i ðξ i Þθp þ εf p;i þ lp;i α_ i 1 Þ þ 2i θ~ wi θ^ wi r zi ½zi þ 1 ri
n 1Þ
2
þ
Ωn 2
4dn 1;5
δ
, Δvn 1 ¼
2
þ
ε
ð30Þ
2 minf2m2n 1 ; r 2n 1 n 1 g,
2 wn 1 2 4dn 1;2
þ
z2n 1 2
4dn 2;1
þ
z2n
2
4dn 1;1
2 2 e þ ðτuð2 ηÞ Þ þ 2 4dn 1;4 4dn 1;3
m2n 1 ¼ a2n 1
, bvn 1 ¼
Pn 1 j¼1
1
2
4bn 1
þ
P4
j¼1
bn 1 δn 1 ð w 2 2
kςn 1 ðε f þ l j Þ.
Step n. For the zn-equation, from Assumption 2, we get Z 0 d z_ n ¼ η_ n α_ n 1 þ ðg n ðξÞ uðt þ χ Þ dχ Þ dt τ
2
θ
B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
2125
2
þ φp;j ðξÞθp þ r 22 z2n
y
1.5
n 1 X j¼1
yr
∂g n ∂ ξj
Z
!2
0 τ
uðt þ χ Þdχ
þ
1 1 nX ðε f þ l j Þ2 ; 2 4r 2 j ¼ 1
ð33Þ 1 Output y and targety y
r
where r 1 ; r 2 are the design parameters in (9). Differentiating V n in (19) along (31) and substituting (16), (17), (32) and (33) into it, we get 2 3 Z 0 n 1 X ∂g n _ n _ 4 ξ uðt þ χ Þ dχ þg ðξÞuðtÞ5 V n ¼ zn kn ðy η1 Þ α_ n 1 þ ∂ ξj j τ j¼1
0.5 0 −0.5
2
−1
þ
bn ~ ^_ θ w θ w r kvn V n þ bvn þ Δvn ; r 2n n n
ð34Þ
−1.5
where kvn ¼ minf2m2n ; r 2n δn g, m2n ¼ a2n dn;1 2
−2 0
10
20
30
40
50
60
t/s
ε þ r4r1 þ2 rr22
θ
2
2
Pn 1
1 2
j¼1
2
n 1;1
ðε f þ l j Þ2 :
ϑvn ; ϑvn ¼ bvn =kvn ; V n0 ¼ V n ð0Þ. Define
3
z
V n ¼ maxfV n0 ; ϑvn g;
1
2
V n r V n ; lim V n ¼ ϑvn ;
ð35Þ
t-1
r
we have qffiffiffiffiffiffiffiffiffi r qffiffiffiffiffiffiffiffiffi r qffiffiffiffiffiffiffiffiffi n n n jzn j r 2V n ; θ~ wn r 2V n ; θ^ wn r 2V n þ θwn : bn bn
1
1
Tracking error z =y−y
2 2 bn δn n 2 2 1 1 2 þ 2 ð wn Þ þ 2 4bn 4dn;1 wn
Δvn ¼ 4d2 zn ,
1 , 2 4dn 1;1
Thus, we can conclude that 0 rV n ðtÞ r ðV n0 ϑvn Þe kvn t þ
Fig. 1. Tracking performances of the closed-loop system (52).
0
ð36Þ
By the property of Gaussian basis function, we get pffiffiffiffiffiffi Si;j ðβi Þ r 1; Si ðβi Þ r mi ; 1 ri r n:
−1
−2
−3
bvn ¼
2
0
10
20
30
40
50
60
t/s Fig. 2. Responses of the tracking error z1 ¼ y yr .
¼ kn ðy η1 Þ α_ n 1 þ
n 1 X
∂g n _ ξ ∂ ξj j j¼1
Z
0 τ
uðt þ χ Þ dχ þ g n uðtÞ:
ð31Þ
ð37Þ
From Assumption 4, virtual control (13) and controller (16), we have 8" 1 < 2 2^ 2 ðnÞ 2 uðtÞ ¼ 1 g n ðξÞ an zn bn θ wn Sn ðβn Þ zn þ yr r g : an 0 9
qffiffiffiffiffiffiffiffiffi n ðnÞ = rn 2 þ bn mn ð38Þ 2V n þ θwn jzn j þ yr r u; ; bn where
qffiffiffiffiffiffiffiffiffi n qffiffiffiffiffiffiffiffiffi 1 rn 2 u¼ a2n þ bn mn 2V n þ y r : 2V n þ θwn g0 bn
ð39Þ
From (26), we have zn α_ n 1 r zn
Thus, we have bvn 1 r b vn 1 , where
n 1 X
n 1 X ∂α n 1 ∂αn 1 ^_ ðξj þ 1 þ φp;j ðξÞθp Þ zn θ wj ^ ∂ ξ j j¼1 j ¼ 1 ∂θ w
b vn 1 ¼
j
zn
þ
n X ∂αn 1
n 1 X
yðjÞ þ r 21 z2n ði 1Þ r j ¼ 1 ∂yr j¼1
∂α n 1 ∂ ξj
!2
1 1 nX ðε f þ l j Þ2 : 2 4r 1 j ¼ 1
ð32Þ
zn
∂g _ ξ ∂ξj j j¼1
Z
0 τ
uðt þ χ Þ dχ ¼ zn
n 1 X
n
∂g ∂ξj j¼1
Z
n 1 X
0 τ
uðt þ χ Þ dχ ½ξj þ 1
∂g n þ φp;j ðξÞθp þ εf p;j ðξÞ þ lp;j ðξ; tÞ r zn ∂ξj j¼1
Z
0 τ
uðt þ χ Þ dχ ½ξj þ 1
2
n 1;5
τ2 u 2 2
2
bn 1 δn 1 n Ω ðθwn 1 Þ2 þ 2 n 2 4d 2
þ
4dn 1;4
þ
n 1 X
kςn 1 ðε f þ l j Þ:
þ
ε 2wn 1 2 4dn 1;2
þ
e2 2 4dn 1;3
ð40Þ
j¼1
Repeating the backstepping design process, we can obtain the following results: V i r V i;
n
2 4bn 1
þ
Based on (10), we can obtain n 1 X
1
jzi j r
lim V i ¼ ϑvi ; 1 ri r n 1;
t-1
qffiffiffiffiffiffiffiffi 2V i ;
qffiffiffiffiffiffiffiffi ~ ri 2V i ; 1 ri r n 1; θ wi r bi
qffiffiffiffiffiffiffiffi ^ ri n 2V i þ θwi ; θ wi r bi jαi j r α i
1 r ir n 2;
1 r i r n 1;
ð41Þ
ð42Þ
ð43Þ ð44Þ
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B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
qffiffiffiffiffiffiffiffi ξ r 2V i þ α i ;
ð45Þ
i
2
1 r i r n 1:
ð46Þ
Step n þ 1. Overall, differentiating the Lyapunov function V in (19), we get V_ ¼
n X
V_ i ðtÞ r kv VðtÞ þ bv ;
ð47Þ
Switching signal σ(t)
where
qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi r 2V i þy r ; α i ¼ a2i þ b21 mi i 2V i þ θnwi bi
i¼1
2 2 P 2 2 b δ where kv ¼ minf2m21 ; …; 2m2n ; r 21 δ1 ; …; r 2n δn g, bv ¼ ni¼ 1 12 þ i2 i 4bi 2 P P 2 ε 2w 2 2 r 21 þ r 22 n Ω ðθwi Þ2 þ 2i þ ni ¼ 11 ij ¼ 1 kςi ðε f þl j Þ þ 2 n þ 2e þ τ2 u þ 4r 2 r2 Pn 1
4di;2
4dn 1;5
ðε f þ l j Þ . Thus, we V 0 ¼ Vð0Þ.
4dn 1;3
4dn 1;4
1
1 2
0
2
j¼1
have
0 r VðtÞ r ðV 0 ϑv Þe kv t þ ϑv ,
^ θ w i r respec-
tively. It is easy to obtain that ξ, Ω, η, e are all bounded, which means the closed-loop system is SGUUB. Moreover, lim VðtÞ ¼ ϑv ;
30
40
50
60
Fig. 4. Switching signal for the system (52).
subsequently, the CLF and adaptive NN controller can be obtained explicitly. However, the work in [10,11] supposes that there exists a common virtual control at each step under a so-called simultaneous domination assumption. Obviously, the result in this paper can be regarded as a generalization of the works in [10,11].
ð48Þ
t-1
4. Simulation results
then we have pffiffiffiffiffiffiffi y yr r 2V ;
ð49Þ
pffiffiffiffiffiffiffiffi lim y yr r 2ϑv ¼
t-1
where sffiffiffiffiffi 2 n ; k ¼ kv
20
t/s
ϑv ¼ bv =kv ,
pffiffiffiffiffiffiffi Define V ¼ maxfV 0 ; ϑv g. Then, we have kzk r 2V , pffiffiffiffiffiffiffi n ðr i =bi Þ 2V þ θwi . Further, z; θ^ i ; αi and ξ n 1 are bounded
VðtÞ rV ;
10
ϑn ¼
sffiffiffiffiffi 2 pffiffiffiffiffi n n bv ¼ k ϑ ; kv
ð50Þ
pffiffiffiffiffi bv :
ð51Þ
In this section, we given a numerical example to illustrate the effectiveness of the proposed control scheme. Example 1. Consider the following switched nonlinear system with uncertain input delay: 8 _ > > > ξ 1 ¼ f σ ðtÞ;1 ðξ1 Þ þ ξ2 þ lσ ðtÞ;1 ðξ; tÞ; > > < ξ_ ¼ f σ ðtÞ;2 ðξÞ þ g σ ðtÞ ðξÞuðt τ ðtÞÞ þ lσ ðtÞ;2 ðξ; tÞ; 2 ð52Þ > > y ¼ ξ1 ; σ ðt Þ : ½0; 1Þ-f1; 2g; > > > : ξðtÞ ¼ ξ ðtÞ; τ rt r 0; 0
It implies that the tracking error is bounded and ultimately converges to an adequately small compact set which can be n adjusted by the design parameter k . The proof is thus completed.□ Remark 3. In the control design procedure of this paper, the common virtual control at each step is constructed explicitly; 3 u 2
Control input u
1
T 3 where ξ ¼ ξ1 ; ξ2 ; f 1;1 ðξ1 Þ ¼ 0:1ξ1 , f 1;2 ðξÞ ¼ 0:3ξ1 ξ2 , g 1 ðξÞ ¼ 2 þ 2 sin ðξ1 Þ þ 0:05 sin ðξ2 Þ, l1;1 ðξ; tÞ ¼ 0:1 sin ðξ1 Þ, l1;2 ðξ; tÞ ¼ 0:2 sin 2 2 ð0:2tÞ, f 2;1 ðξ1 Þ ¼ 0:2ξ1 , f 2;2 ðξÞ ¼ 0:1ξ1 ξ2 , g 2 ðξÞ ¼ 2 þ sin ðξ1 Þ þ 0:05 cos ðξ1 Þ, l2;1 ðξ; tÞ ¼ 0:05 cos 2 ðξ1 ξ2 Þ, l2;2 ðξ; tÞ ¼ 0:3 cos ð0:1tÞ sin ð0:2tÞ, τðtÞ ¼ 0:3 þ 0:1 sin ðtÞ, τ ¼ 0:3, yr ðtÞ ¼ sin ð0:3tÞ sin ð0:6tÞ, A ¼ 00 10 , B ¼ 01 . 3 1 Further, we choose parameters: k ¼ ½3 2T , A0 ¼ 2 0 , a1 ¼ 1:2, a2 ¼ 1:3, b1 ¼ 0:1, b2 ¼ 0:1, γ 1 ¼ 1:2, γ 2 ¼ 1:2, σ 1 ¼ 0:05, σ 2 ¼ 0:05. Construct a virtual observer ξ^ for system (52) as 8 η_ ¼ A0 η þ ky þ Bg1 ðξ1 Þuðt 0:3Þ; > > < _ ¼ A Ω þ φ ðξÞ; Ω 0 p > > : ξ^ ¼ η þ Ωθ ; p
where "
0
Ω¼
−1
(
−2
−3
#
Ω1 ; Ω2
10
20
30
40
t/s Fig. 3. Responses of the control input u.
50
60
#
p ¼ 1; 2:
Define variables z1 ¼ ξ1 yr ; z 2 ¼ ξ2 α 1 þ
0
"
η1 η¼ η ; 2
R0 0:3
uðt þ θÞ dθ:
and define the virtual control αi ¼ a2i zi b2i θ^ wi Si ðλi Þ2 zi þ yðiÞ r ;
1 r i r2;
ð53Þ
B. Niu, L. Li / Neurocomputing 173 (2016) 2121–2128
2127
where
[11] B. Niu, J. Zhao, Output tracking control for a class of switched non-linear systems
λ1 ¼ yr ; x1 T ;
with partial state constraints, IET Control Theory Appl. 7 (2013) 623–631. [12] Y.E. Wang, X.M. Sun, B.W. Wu, Lyapunov Krasovskii functionals for switched nonlinear input delay systems under asynchronous switching, Automatica 61
2
∂α _ ∂g λ2 ¼ 4yr ; y_ r ; x1 ; x2 ; α1 ; ^ 1 θ^ w1 ; z2 ; 1 ∂x1 ∂ θ w1
Z
0 τ
#T uðt þ θÞ dθ
:
ð54Þ
Moreover, the corresponding assistant functions are as follows:
(2015) 126–133. [13] L. Long, J. Zhao, H 1 control of switched nonlinear systems in p-normal form using multiple Lyapunov functions, IEEE Trans. Autom. Control 57 (2012) 1285–1291. [14] L. Long, J. Zhao, Output-feedback stabilisation for a class of switched nonlinear systems with unknown control coefficients, Int. J. Control 86 (2013) 386–395.
8 2 2 2 2 > > < Q p;1 ðλ1 Þ ¼ φp;1 ðξ1 Þθp þ tanhðz1 =ς1 Þðε f þ h 1 Þ þ d1:5 θ z1 þ d1;4 g 1 ðx1 Þz1 ;
2 ∂α 1 ∂α 1 ∂α1 ^_ ∂g R 0 2 ∂α 1 > _ Q ð λ Þ ¼ k ðy η Þ ð ξ þ φ ðxÞ θ Þ θ þr z2 þ 1 τ uðt þ θÞ dθðξ2 þ φ1 ðξÞθp Þ y > p w 2 2 1 1 1 : p;2 1 ∂yr r ∂θ^ w ∂ ξ1 2 ∂ ξ1 ∂ ξ1 1 !2 Z 0 ∂g 1 þ r 22 uðt þ θÞ dθ z2 ; p ¼ 1; 2: ∂ξ1 τ
Choose control law as uðtÞ ¼ α2 =g 1 ðx1 Þ and adaptive law as _
2
θ^ w1 ¼ γ 2i ðkSi ðλi Þk z2i σ 2i θ^ wi Þ;
1 ri r 2; ð56Þ P 2 and S1 A R125 , S2 A R16561 , S1;i ¼ exp j ¼ 1 ðλ1;j ui;j Þ2 , 1 r i r P 25, S2;i ¼ exp 8j ¼ 1 ðλ2;j vi;j Þ2 , 1 r i r 6561, ui;j A f 1 0:5 0 0:5 1g, v1;j A f 1 0 1g. Based on Theorem 1, the simulation results of the proposed control scheme for system (52) are shown in Figs. 1–4.
5. Conclusions This paper has proposed an adaptive tracking control scheme for a class of disturbed nonlinear switched systems in strick-feedback form with uncertain input delay by backstepping and NN. A filter and a virtual observer are constructed to substitute the system state. The designed controller ensures that all signals of the closed-loop system are SGUUB, and the tracking error converges to an adequately small compact set. The theoretical result is illustrated through a simulation example. The future research directions conclude how to deal with the switched system with state delays; how to design an observer for output feedback of the switched system.
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ð55Þ
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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.
Lu Li was born in Liaoning Province, China, on October 1, 1992. She received the B.S. degree in mathematics and applied mathematics from Bohai University, Jinzhou, China, in 2014. She is working towards the M.S. degree in operation science and control theory from Bohai University, Jinzhou, China. Her current research interests include switched nonlinear systems and adaptive control.
