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H∞ non-fragile observer-based sliding mode control for uncertain time-delay systems
ARTICLE IN PRESS
Journal of the Franklin Institute 347 (2010) 567–576 www.elsevier.com/locate/jfranklin
H1 non-fragile observer-based sliding mode control for uncertain time-delay systems$ Leipo Liua,, Zhengzhi Hana, Wenlin Lib a
School of Electrical and Information Engineering, Shanghai Jiao Tong University, Shanghai 200240, China b College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China Received 22 November 2007; accepted 6 October 2009
Abstract The problem of H1 sliding mode control for uncertain time-delay systems subjected to input nonlinearity is investigated. Using the sliding mode control, a robust law is established such that the reachability of the sliding surface in the state-estimation space is guaranteed, and the sufficient condition for asymptotic stability of the error system and sliding mode dynamics with disturbance attenuation level is derived via linear matrix inequality (LMI). Finally, a simulation example is presented to verify the validity of the proposed method. & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Sliding mode control; Time-delay systems; Input nonlinearity; Linear matrix inequality
1. Introduction The analysis of the time-delay systems is one of the difficult but hot points in control theory and control engineering domain. Since time delay often occurs in various engineering systems and causes serious deterioration of the stability and performance of the system, considerable research has been done to the control of time-delay systems [1–3]. In the past, H1 control concept was proposed to reduce the effect of the disturbance input on the measured output to within a prescribed level [4,5,14]. Because sliding mode control (SMC) has attractive features such as fast response and good transient response, it is also
$
Supported by National Science Foundation of China (60674024).
Corresponding author. Tel.: þ86 21 34202028; fax: þ86 21 62932083.
E-mail address: [email protected] (L. Liu). 0016-0032/$32.00 & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2009.10.021
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insensitive to variations in system parameters and external disturbances. SMC is an effective robust control approach for uncertain time-delay systems [6,7,9,12]. On the other hand, input nonlinearity is often found in practical systems. It has been shown that input nonlinearity can cause a serious degradation of the system performance if the controller is not well designed [8]. Therefore, it is clear that the effects of input nonlinearity must be taken into account when analyzing and implementing a SMC scheme. In recent years, some papers about input nonlinearity have been presented [8–14], but there are few works undertaken on SMC for time-delay systems about unknown state subjected to input nonlinearity, and the existence of the perturbations also adds the complexity of SMC design. To date, the problem of H1 non-fragile observer-based SMC for the time-delay systems subjected to input nonlinearity remains open. Motivated by the aforementioned reasons, the purpose of this paper lies in the development of an H1 SMC for the uncertain time-delay systems subjected to input nonlinearity. Utilizing a non-fragile observer, a robust law is established based on the state estimate such that the reachability of the sliding surface is guaranteed. The sufficient condition for the asymptotic stability of the overall closed-loop system with disturbance attenuation level is derived via LMI. Finally, an example illustrates the validity of the proposed method. Notations. The notations in this paper are quite standard. Rn and Rnm denote, respectively, the n-dimensional Euclidean space and the set of n m real matrices. The superscript T denotes the transpose and the notation X ZY , (respectively, X 4Y ) where X and Y are symmetric matrices, means that X Y is positive semi-definite (respectively, positive definite). L2 stands for the space of square integral vector functions. J J will refer to the Euclidean vector norm, represents the symmetric form of matrix. 2. Preliminaries and problem formulation Consider a class of uncertain time-delay systems subjected to input nonlinearity described by 8 _ ¼ ðA þ DAðtÞÞxðtÞ þ ðA1 þ DA1 ðtÞÞxðt hÞ þ BfðuÞ þ f ðxÞ þ GvðtÞ; > < xðtÞ yðtÞ ¼ CxðtÞ; ð1Þ > : xðtÞ ¼ fðtÞ; t 2 ½h; 0; where xðtÞ 2 Rn is the state, f ðÞ is the nonlinear disturbance input, vðtÞ 2 Rq is the exogenous noise, yðtÞ 2 Rl is the measured output, fðtÞ is a continuous vector-valued initial function, fðuÞ is continuous function vector. Here A; A1 ; B; G and C are real constant matrices of appropriate dimensions, B is of column full rank. DAðtÞ and DA1 ðtÞ are uncertainties which are assumed to be the form of ½DA DA1 ¼ DF ðtÞ½E E1 ;
ð2Þ
where D, E and E1 are real constant matrices and F ðÞ : R-Rkl is an unknown time-varying matrix function satisfying F T ðtÞF ðtÞrI: In dealing with this study, the following assumptions and lemmas are necessary for the sake of convenience.
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Assumption 1. ðA; BÞ is completely controllable, ðA; CÞ is completely observable. Assumption 2. For 8x1 ; x2 , f ðxÞ satisfies Jf ðx1 Þ f ðx2 ÞJrrJx1 x2 J; where r is a positive constant. Assumption 3. The nonlinear input fðuÞ applied to the system satisfies the following property: uT fðuÞZauT u; where a is a nonzero positive constant, and fð0Þ ¼ 0. A nonlinear function fðuðtÞÞ satisfying uT fðuÞZauT u is illustrated in Fig. 1. Lemma 1 (Lein [2]). Let Q ¼ QT ; S; R ¼ RT be real matrices of appropriate dimensions, then Q S o0 ST R is equivalent to Ro0; Q SR1 ST o0. Lemma 2 (Lien [2]). Let D, E and F ðtÞ be real matrices of appropriate dimensions with F ðtÞ satisfying F T ðtÞF ðtÞrI and scalar e40, the following inequality 1 DF ðtÞE þ E T F T ðtÞDT reDDT þ E T E e is always satisfied.
20 (0.6+0.3sinu)u 15 10 5 0
o
u
−5 −10 −15 −20 −20
−15
−10
−5
0
5
10
Fig. 1. Nonlinear function fðuÞ ¼ ð0:6 þ 0:3sinðuÞÞu.
15
20
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3. Sliding mode control 3.1. Non-fragile observer design First, the following non-fragile state observer is utilized to estimate the state of uncertain time-delay systems (1): ^ þ A1 xðt ^ hÞ þ BfðuÞ þ f ðxÞ ^ þ ðL þ DLðtÞÞðy C xÞ; ^ x^_ ðtÞ ¼ AxðtÞ
ð3Þ
where L 2 Rnq is the observer gain to be designed later and DLðtÞ is a nonlinear function matrix satisfying JDLðtÞJrd; where d is a positive constant. ^ Define the error eðtÞ ¼ xðtÞ xðtÞ, then it follows from systems (1) and (3) that e_ ðtÞ ¼ ðA LC þ DAÞeðtÞ DLðtÞCeðtÞ þ ðA1 þ DA1 Þeðt hÞ ^ hÞ þ f ðxÞ f ðxÞ ^ þ DAxðtÞ ^ þ GvðtÞ; þDA1 xðt ye ðtÞ ¼ CeðtÞ:
ð4Þ
where ye(t) is the output of error system. Then we introduce H1 performance measure as follows: Z 1 J¼ ½yTe ðsÞye ðsÞ g2 vT ðsÞvðsÞ dt: 0
Therefore, the problem is to determine the error eðtÞ within the upper bound, i.e. Jye ðtÞJ2 sup og: vðtÞ2L2 JvðtÞJ2 A novel switching function is chosen as ^ sðtÞ ¼ sðtÞ þ BT xðtÞ;
ð5Þ
with ^ hÞ BT f ðxÞ; ^ BT AxðtÞ ^ BT A1 xðt ^ _ ¼ BT BK xðtÞ sðtÞ where the matrix K is to be chosen later, obviously, BT B is nonsingularity. Control input uðtÞ in system (1) should be appropriately designed such that the estimated state in system (3) can be driven to the sliding surface even when the input nonlinearity is presented. The SMC law is derived as follows: uðtÞ ¼
sðtÞ ^ cðxÞ; JsðtÞJ
ð6Þ
T T ^ ¼ ð1=aÞðJK xðtÞJþJðB ^ ^ ^ þ bÞ, b is where cðxÞ BÞ1 BT Lðy C xÞJþdJðB BÞ1 BT J Jy C xJ an arbitrarily positive scalar. This proposed control scheme above will drive the estimate state to approach the sliding mode surface sðtÞ ¼ 0 in a finite time, and it is stated in the following theorem.
Theorem 1. If the control input u(t) is designed as Eq. (6), then the trajectories of the observer system (3) converge to the sliding surface sðtÞ ¼ 0 in a finite time.
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Proof. From system (3) and Eq. (5), we have ^ þ BT BfðuÞ þ BT ðL þ DLÞðy C xÞ: ^ s_ ðtÞ ¼ BT BK xðtÞ
ð7Þ
Let V1 ¼ 12sT ðtÞðBT BÞ1 sðtÞ: It follows from Eq. (7) that ^ þ BT BfðuÞ þ BT ðL þ DLðtÞÞðy C xÞÞ; ^ V_ 1 ¼ sT ðtÞðBT BÞ1 ðBT BK xðtÞ ^ þ sT ðtÞfðuÞ ¼ sT ðtÞK xðtÞ ^ rsT ðtÞfðuÞ þ sT ðtÞðBT BÞ1 BT ðL þ DLðtÞÞðy C xÞ; ^ ^ þ dJðBT BÞ1 BT J Jy C xJÞ: ^ þ JsðtÞJðJK xðtÞJ þ JðBT BÞ1 BT Lðy C xÞJ
ð8Þ
Using Eq. (6) and Assumption 3, we have uT fðuÞ ¼
sT ðtÞ 2 ^ ^ ðxÞ; cðxÞfðuÞZac JsðtÞJ
then ^ sT ðtÞfðuÞr acðxÞJsðtÞJ:
ð9Þ
Substituting Eq. (9) into Eq. (8) yields V_ r bJsJo0
for JsðtÞJa0:
ð10Þ
From Eq. (10), we prove the finite time convergence of system (3) toward the surface sðtÞ ¼ 0. Then the proof is completed. & From s_ ðtÞ ¼ 0, the following equivalent control law can be obtained ^ ðBT BÞ1 BT ðL þ DLðtÞÞðy C xÞ: ^ feq ¼ K xðtÞ
ð11Þ
Substituting Eq. (11) into the observer system (3) and noting B ¼ I BðBT BÞ1 BT , the sliding mode dynamics in the state estimation space can be obtained as follows: ^ hÞ þ BðL þ DLðtÞÞCeðtÞ þ f ðxÞ: ^ ^ þ A1 xðt x^_ ðtÞ ¼ ðA BKÞxðtÞ
ð12Þ
Hence, the stability of the overall closed-loop system with Eqs. (1) and (4) will be analyzed through the error system (4) and the sliding mode dynamics (12). 3.2. Analysis of asymptotic stability In the following theorem, the sufficient condition for the asymptotic stability of the overall closed-loop system with disturbance attenuation level is given via LMI. Theorem 2. Consider that the systems (4) and (12). Given a scalar g40, the switching function is chosen as Eq. (5), and the SMC law is chosen as Eq. (6). If there exist matrices Q1 40, Q2 40, L, K, and scalars ei 40 ði ¼ 1; 2; 3; 4Þ satisfying the following linear matrix
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inequality (LMI) 2 G11 A1 þ e1 E T E1 6 6 Q1 þ e1 E1T E1 6 6 6 6 6 6 6 4
C T LT B
T
0
G
0 G33
0 A1 þ e2 E T E1
0 0
Q2 þ e2 E1T E1
0
g2 I
N1
3
7 0 7 7 N3 7 7 7o0; 0 7 7 0 7 5 N6
ð13Þ
with G11 ¼ ðA LCÞ þ ðA LCÞT þ Q1 þ e1 E T E þ e3 d2 C T C þ 2rI þ e4 d2 C T C þ C T C; G33 ¼ ðA BKÞ þ ðA BKÞT þ e2 E T E þ 2rI þ Q2 ; N1 ¼ ½D; D; I; 0; N3 ¼ ½0; 0; 0; B; N6 ¼ diagfe1 I; e2 I; e3 I; e4 Ig; the overall closed-loop system with Eqs. (4) and (12) is asymptotically stable with disturbance attenuation level g. Proof. Choose the following Lyapunov functional candidate: Z t Z t T T T ^ ds: ^ ^ V2 ðtÞ ¼ e ðtÞeðtÞ þ x^ T ðsÞQ2 xðsÞ e ðsÞQ1 eðsÞ ds þ x ðtÞxðtÞ þ th
ð14Þ
th
First, we consider the overall closed-loop system with vðtÞ ¼ 0, we have V_ 2 ðtÞ ¼ eT ðtÞ½ðA LCÞ þ ðA LCÞT eðtÞ 2eT ðtÞDLðtÞCeðtÞ þ 2eT ðtÞA1 eðt hÞ ^ þ 2eT ðtÞDAeðtÞ þ 2eT ðtÞDA1 eðt hÞ þ 2eT ðtÞDAxðtÞ ^ þ 2eT ðtÞ½f ðxÞ f ðxÞ ^ hÞ þ eT ðtÞQ1 eðtÞ eT ðt hÞQ1 eðt hÞ þ x^ T ðtÞ½ðA BKÞ þ 2eT ðtÞDA1 xðt ^ hÞ þ 2x^ T ðtÞBDLCeðtÞ ^ þ 2x^ T ðtÞA1 xðt þðA BKÞT xðtÞ ^ x^ T ðt hÞQ2 xðt ^ hÞ: ^ þ x^ T ðtÞQ2 xðtÞ þ2x^ T ðtÞBLCeðtÞ þ 2x^ T ðtÞf ðxÞ
ð15Þ
Using Lemma 2 and Assumption 2, we have 2eT ðtÞDF ðtÞ½EeðtÞ þ E1 eðt hÞ 1 r eT ðtÞDDT eðtÞ þ e1 ½EeðtÞ þ E1 eðt hÞT ½EeðtÞ þ E1 eðt hÞ; e1 ^ hÞ ^ þ E1 xðt 2eT ðtÞDF ðtÞ½E xðtÞ 1 T ^ hÞT ½E xðtÞ ^ hÞ; ^ þ E1 xðt ^ þ E1 xðt r e ðtÞDDT eðtÞ þ e2 ½E xðtÞ e2 2eT ðtÞDLðtÞCeðtÞr2dJeT ðtÞJ JCeðtÞJr
1 T e ðtÞeðtÞ þ e3 d2 eT ðtÞC T CeðtÞ; e3
ð16Þ
ð17Þ ð18Þ
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2x^ T ðtÞBDLCeðtÞr2Jx^ T ðtÞBJ dJCeðtÞJr
573
1 T T ^ þ e4 d2 eT ðtÞC T CeðtÞ; x^ ðtÞBB xðtÞ e4
ð19Þ
T ^ 2eT ðtÞ½f ðxÞ f ðxÞr2Je ðtÞJ rJeðtÞJ ¼ 2reT ðtÞeðtÞ;
ð20Þ
^ ^ ^ 2x^ T ðtÞf ðxÞr2J x^ T ðtÞJ rJxðtÞJ ¼ 2rx^ T ðtÞxðtÞ:
ð21Þ
Substituting Eqs. (16)–(21) into Eq. (15) results in V_ 2 ðtÞrwT ðtÞXwðtÞ; where wðtÞ ¼ ½eT ðtÞ eT ðt hÞ x^ T ðtÞ x^ T ðt hÞT , and 2 3 T 0 X1 A1 þ e1 E T E1 C T LT B 6 7 6 Q1 þ e1 E1T E1 7 0 0 6 7; X¼6 T A1 þ e2 E E1 7 X3 4 5 Q2 þ e2 E1T E1 with X1 ¼ ðA LCÞ þ ðA LCÞT þ Q1 þ e1 E T E þ þ
1 DDT e1
1 1 DDT þ I þ e3 d2 C T C þ 2rI þ e4 d2 C T C; e2 e3
X3 ¼ ðA BKÞ þ ðA BKÞT þ e2 E T E þ
1 T BB þ 2rI þ Q2 : e4
It can be shown that if LMI (13) is satisfied, Xo0 is held by Lemma 1, then V_ 2 ðtÞr wT ðtÞXwðtÞo0 (for wðtÞa0), which shows the closed-loop system is asymptotically stable. Next, when vðtÞ 2 L2 , assuming zero conditions for the closed-loop system, the performance index is Z 1 ½yTe ðsÞye ðsÞ g2 vT ðsÞvðsÞ dt JðtÞ ¼ Z
0
1
½yTe ðsÞye ðsÞ g2 vT ðsÞvðsÞ þ V_ 2 ðsÞ dt þ V2 ðtÞjt¼0 V2 ðtÞjt-1 ;
¼ 0
r
Z
1
Z0
1
¼
½yTe ðsÞye ðsÞ g2 vT ðsÞvðsÞ þ V_ 2 ðsÞ dt qT ðsÞPqðsÞ dt;
0
where qðtÞ ¼ ½eT ðtÞ eT ðt hÞ x^ T ðtÞ x^ T ðt hÞ vT ðtÞT and 2 T X1 þ C T C A1 þ e1 E T E1 C T LT B 0 6 T 6 Q1 þ e1 E1 E1 0 0 6 6 P¼6 A1 þ e2 E T E1 X3 6 6 Q2 þ e2 E1T E1 4
G
3
7 7 7 7 7: 7 0 7 5 g2 I 0 0
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By utilizing Lemma 1, it is seen that Po0 is equivalent to Eq. (13). This means that JðtÞo0 (for qðtÞa0), so the overall closed-loop system is asymptotically stable with disturbance attenuation g. Then the proof is obtained. &
6 x1 x2
4
x1,x2
2
0
−2
−4
−6
0
0.5
1
1.5
2
2.5 t/sec
3
3.5
4
4.5
5
Fig. 2. Trajectories of state xðtÞ.
3 the estimate of x1 the estimate of x2
2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2
0
0.5
1
1.5
2
2.5 t/sec
3
3.5
^ Fig. 3. Trajectories of state estimate xðtÞ.
4
4.5
5
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4. Simulation results 0 0:2 0 0 1 We consider the system (1)–(3) with: A ¼ ½1 3 0, A1 ¼ ½0:5 0, B ¼ ½1, D ¼ ½0, 0:1sinðx1 Þ 0 t E ¼ ½0:1 0, E1 ¼ ½0:1 0, F ðtÞ ¼ 1, G ¼ ½1, f ðx; tÞ ¼ ½ 0 , vðtÞ ¼ 1=ðe þ 100Þ,
3.5
s
3 2.5 2 1.5 1 0.5 0 −0.5
0
0.5
1
1.5
2
2.5 t/sec
3
3.5
4
4.5
5
Fig. 4. Trajectories of sliding mode variable sðtÞ.
50 u
0 −50 −100 −150 −200 −250 −300
0
0.5
1
1.5
2
2.5 t/sec
3
Fig. 5. Control signal uðtÞ.
3.5
4
4.5
5
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0 DL ¼ ½0:01sinðtÞ , fðuÞ ¼ ð0:6 þ 0:3sinðuÞÞu, C ¼ ½0 1, a ¼ 0:3, b ¼ 5; r ¼ 0:1; d ¼ 0:01; h ¼ 0:1 and g ¼ 0:3162. 1:0306 0:7460 Solving LMI (13) yields L ¼ ½1:0370 28:5922 ; K ¼ ½3:0001 27:5294, Q1 ¼ ½0:7460 27:9018, 1:0852 0:0000 Q2 ¼ ½0:0000 27:4274, e1 ¼ 18:3865, e2 ¼ 18:1899, e3 ¼ 23:5444, e4 ¼ 26:9663. ^ And then, the initial states are xð0Þ ¼ ½5 6T and xð0Þ ¼ ½2 3T , the simulation results are given in Figs. 2–5, which show the validity of the proposed method.
5. Conclusion The problem of H1 sliding mode control for uncertain time-delay systems subjected to input nonlinearity has been presented. Based on the sliding mode control technique, the sufficient condition for asymptotic stability of the error system and sliding mode dynamics with disturbance attenuation level is derived via LMI. Finally, a simulation example is presented to verify the validity of the proposed method. Acknowledgement The authors are grateful for the support of the National Natural Science Foundation of China (Grant no. 60674024). References [1] Y.-J. Sun, Stability criterion for a class of descriptor systems with discrete and distributed time delays, Chaos, Solitons & Fractals 33 (2007) 986–993. [2] C.-H. Lien, Delay-dependent and delay-independent guaranteed cost control for uncertain neutral systems with time-varying delays via LMI approach, Chaos, Solitons & Fractals 33 (2007) 1017–1027. [3] M. De la Sen, On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays, Applied Mathematics and Computation 190 (2007) 382– 401. [4] G.-H. Yang, J. Liang Wang, Y. Chai Soh, Reliable H1 controller design for linear systems, Automatica 37 (2001) 717–725. [5] A.N. Madiwale, W.M. Haddad, D.S. Bernstein, Robust H1 control design for systems with structured parameter uncertainty, Systems & Control Letters 12 (1989) 393–407. [6] Y. Niu, D.W.C. Ho, Robust observer design for Itoˆ stochastic time-delay systems via sliding mode control, Systems & Control Letters 55 (2006) 781–793. [7] G.-Y. Tang, S.-S. Lu, R. Dong, Optimal sliding mode control for linear time-delay systems with sinusoidal disturbances, Journal of Sound and Vibration 304 (2007) 263–271. [8] K.-C. Hsu, Variable structure control design for uncertain dynamic systems with sector nonlinearities, Automatica 34 (1998) 505–508. [9] M.-L. Hung, J.-J. Yan, Decentralized model-reference adaptive control for a class of uncertain large-scale time-varying delayed systems with series nonlinearities, Chaos, Solitons & Fractals 33 (2007) 1558–1568. [10] J.-S. Lin, J.-J. Yan, T.-L. Liao, Chaotic synchronization via adaptive sliding mode observers subject to input nonlinearity, Chaos, Solitons & Fractals 24 (2005) 371–381. [11] W.-D. Chang, Robust adaptive single neural control for a class of uncertain nonlinear systems with input nonlinearity, Information Sciences 171 (2005) 261–271. [12] F.-M. Yu, H.-Y. Chung, S.-Y. Chen, Fuzzy sliding mode controller design for uncertain time-delayed systems with nonlinear input, Fuzzy Set & Systems 140 (2003) 359–374. [13] T. Fliegner, H. Logemann, E.P. Ryan, Low-gain integral control of continuous-time linear systems subject to input and output nonlinearities, Automatica 39 (2003) 455–462. [14] W. Ligang, W. Changhong, G. Huijun, Z. Lixian, Sliding mode H1 control for a class of uncertain nonlinear state-delayed systems, Journal of Systems Engineering and Electronics 17 (2006) 576–585.
Journal of the Franklin Institute 347 (2010) 567–576 www.elsevier.com/locate/jfranklin
H1 non-fragile observer-based sliding mode control for uncertain time-delay systems$ Leipo Liua,, Zhengzhi Hana, Wenlin Lib a
School of Electrical and Information Engineering, Shanghai Jiao Tong University, Shanghai 200240, China b College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China Received 22 November 2007; accepted 6 October 2009
Abstract The problem of H1 sliding mode control for uncertain time-delay systems subjected to input nonlinearity is investigated. Using the sliding mode control, a robust law is established such that the reachability of the sliding surface in the state-estimation space is guaranteed, and the sufficient condition for asymptotic stability of the error system and sliding mode dynamics with disturbance attenuation level is derived via linear matrix inequality (LMI). Finally, a simulation example is presented to verify the validity of the proposed method. & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Sliding mode control; Time-delay systems; Input nonlinearity; Linear matrix inequality
1. Introduction The analysis of the time-delay systems is one of the difficult but hot points in control theory and control engineering domain. Since time delay often occurs in various engineering systems and causes serious deterioration of the stability and performance of the system, considerable research has been done to the control of time-delay systems [1–3]. In the past, H1 control concept was proposed to reduce the effect of the disturbance input on the measured output to within a prescribed level [4,5,14]. Because sliding mode control (SMC) has attractive features such as fast response and good transient response, it is also
$
Supported by National Science Foundation of China (60674024).
Corresponding author. Tel.: þ86 21 34202028; fax: þ86 21 62932083.
E-mail address: [email protected] (L. Liu). 0016-0032/$32.00 & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2009.10.021
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insensitive to variations in system parameters and external disturbances. SMC is an effective robust control approach for uncertain time-delay systems [6,7,9,12]. On the other hand, input nonlinearity is often found in practical systems. It has been shown that input nonlinearity can cause a serious degradation of the system performance if the controller is not well designed [8]. Therefore, it is clear that the effects of input nonlinearity must be taken into account when analyzing and implementing a SMC scheme. In recent years, some papers about input nonlinearity have been presented [8–14], but there are few works undertaken on SMC for time-delay systems about unknown state subjected to input nonlinearity, and the existence of the perturbations also adds the complexity of SMC design. To date, the problem of H1 non-fragile observer-based SMC for the time-delay systems subjected to input nonlinearity remains open. Motivated by the aforementioned reasons, the purpose of this paper lies in the development of an H1 SMC for the uncertain time-delay systems subjected to input nonlinearity. Utilizing a non-fragile observer, a robust law is established based on the state estimate such that the reachability of the sliding surface is guaranteed. The sufficient condition for the asymptotic stability of the overall closed-loop system with disturbance attenuation level is derived via LMI. Finally, an example illustrates the validity of the proposed method. Notations. The notations in this paper are quite standard. Rn and Rnm denote, respectively, the n-dimensional Euclidean space and the set of n m real matrices. The superscript T denotes the transpose and the notation X ZY , (respectively, X 4Y ) where X and Y are symmetric matrices, means that X Y is positive semi-definite (respectively, positive definite). L2 stands for the space of square integral vector functions. J J will refer to the Euclidean vector norm, represents the symmetric form of matrix. 2. Preliminaries and problem formulation Consider a class of uncertain time-delay systems subjected to input nonlinearity described by 8 _ ¼ ðA þ DAðtÞÞxðtÞ þ ðA1 þ DA1 ðtÞÞxðt hÞ þ BfðuÞ þ f ðxÞ þ GvðtÞ; > < xðtÞ yðtÞ ¼ CxðtÞ; ð1Þ > : xðtÞ ¼ fðtÞ; t 2 ½h; 0; where xðtÞ 2 Rn is the state, f ðÞ is the nonlinear disturbance input, vðtÞ 2 Rq is the exogenous noise, yðtÞ 2 Rl is the measured output, fðtÞ is a continuous vector-valued initial function, fðuÞ is continuous function vector. Here A; A1 ; B; G and C are real constant matrices of appropriate dimensions, B is of column full rank. DAðtÞ and DA1 ðtÞ are uncertainties which are assumed to be the form of ½DA DA1 ¼ DF ðtÞ½E E1 ;
ð2Þ
where D, E and E1 are real constant matrices and F ðÞ : R-Rkl is an unknown time-varying matrix function satisfying F T ðtÞF ðtÞrI: In dealing with this study, the following assumptions and lemmas are necessary for the sake of convenience.
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Assumption 1. ðA; BÞ is completely controllable, ðA; CÞ is completely observable. Assumption 2. For 8x1 ; x2 , f ðxÞ satisfies Jf ðx1 Þ f ðx2 ÞJrrJx1 x2 J; where r is a positive constant. Assumption 3. The nonlinear input fðuÞ applied to the system satisfies the following property: uT fðuÞZauT u; where a is a nonzero positive constant, and fð0Þ ¼ 0. A nonlinear function fðuðtÞÞ satisfying uT fðuÞZauT u is illustrated in Fig. 1. Lemma 1 (Lein [2]). Let Q ¼ QT ; S; R ¼ RT be real matrices of appropriate dimensions, then Q S o0 ST R is equivalent to Ro0; Q SR1 ST o0. Lemma 2 (Lien [2]). Let D, E and F ðtÞ be real matrices of appropriate dimensions with F ðtÞ satisfying F T ðtÞF ðtÞrI and scalar e40, the following inequality 1 DF ðtÞE þ E T F T ðtÞDT reDDT þ E T E e is always satisfied.
20 (0.6+0.3sinu)u 15 10 5 0
o
u
−5 −10 −15 −20 −20
−15
−10
−5
0
5
10
Fig. 1. Nonlinear function fðuÞ ¼ ð0:6 þ 0:3sinðuÞÞu.
15
20
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3. Sliding mode control 3.1. Non-fragile observer design First, the following non-fragile state observer is utilized to estimate the state of uncertain time-delay systems (1): ^ þ A1 xðt ^ hÞ þ BfðuÞ þ f ðxÞ ^ þ ðL þ DLðtÞÞðy C xÞ; ^ x^_ ðtÞ ¼ AxðtÞ
ð3Þ
where L 2 Rnq is the observer gain to be designed later and DLðtÞ is a nonlinear function matrix satisfying JDLðtÞJrd; where d is a positive constant. ^ Define the error eðtÞ ¼ xðtÞ xðtÞ, then it follows from systems (1) and (3) that e_ ðtÞ ¼ ðA LC þ DAÞeðtÞ DLðtÞCeðtÞ þ ðA1 þ DA1 Þeðt hÞ ^ hÞ þ f ðxÞ f ðxÞ ^ þ DAxðtÞ ^ þ GvðtÞ; þDA1 xðt ye ðtÞ ¼ CeðtÞ:
ð4Þ
where ye(t) is the output of error system. Then we introduce H1 performance measure as follows: Z 1 J¼ ½yTe ðsÞye ðsÞ g2 vT ðsÞvðsÞ dt: 0
Therefore, the problem is to determine the error eðtÞ within the upper bound, i.e. Jye ðtÞJ2 sup og: vðtÞ2L2 JvðtÞJ2 A novel switching function is chosen as ^ sðtÞ ¼ sðtÞ þ BT xðtÞ;
ð5Þ
with ^ hÞ BT f ðxÞ; ^ BT AxðtÞ ^ BT A1 xðt ^ _ ¼ BT BK xðtÞ sðtÞ where the matrix K is to be chosen later, obviously, BT B is nonsingularity. Control input uðtÞ in system (1) should be appropriately designed such that the estimated state in system (3) can be driven to the sliding surface even when the input nonlinearity is presented. The SMC law is derived as follows: uðtÞ ¼
sðtÞ ^ cðxÞ; JsðtÞJ
ð6Þ
T T ^ ¼ ð1=aÞðJK xðtÞJþJðB ^ ^ ^ þ bÞ, b is where cðxÞ BÞ1 BT Lðy C xÞJþdJðB BÞ1 BT J Jy C xJ an arbitrarily positive scalar. This proposed control scheme above will drive the estimate state to approach the sliding mode surface sðtÞ ¼ 0 in a finite time, and it is stated in the following theorem.
Theorem 1. If the control input u(t) is designed as Eq. (6), then the trajectories of the observer system (3) converge to the sliding surface sðtÞ ¼ 0 in a finite time.
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Proof. From system (3) and Eq. (5), we have ^ þ BT BfðuÞ þ BT ðL þ DLÞðy C xÞ: ^ s_ ðtÞ ¼ BT BK xðtÞ
ð7Þ
Let V1 ¼ 12sT ðtÞðBT BÞ1 sðtÞ: It follows from Eq. (7) that ^ þ BT BfðuÞ þ BT ðL þ DLðtÞÞðy C xÞÞ; ^ V_ 1 ¼ sT ðtÞðBT BÞ1 ðBT BK xðtÞ ^ þ sT ðtÞfðuÞ ¼ sT ðtÞK xðtÞ ^ rsT ðtÞfðuÞ þ sT ðtÞðBT BÞ1 BT ðL þ DLðtÞÞðy C xÞ; ^ ^ þ dJðBT BÞ1 BT J Jy C xJÞ: ^ þ JsðtÞJðJK xðtÞJ þ JðBT BÞ1 BT Lðy C xÞJ
ð8Þ
Using Eq. (6) and Assumption 3, we have uT fðuÞ ¼
sT ðtÞ 2 ^ ^ ðxÞ; cðxÞfðuÞZac JsðtÞJ
then ^ sT ðtÞfðuÞr acðxÞJsðtÞJ:
ð9Þ
Substituting Eq. (9) into Eq. (8) yields V_ r bJsJo0
for JsðtÞJa0:
ð10Þ
From Eq. (10), we prove the finite time convergence of system (3) toward the surface sðtÞ ¼ 0. Then the proof is completed. & From s_ ðtÞ ¼ 0, the following equivalent control law can be obtained ^ ðBT BÞ1 BT ðL þ DLðtÞÞðy C xÞ: ^ feq ¼ K xðtÞ
ð11Þ
Substituting Eq. (11) into the observer system (3) and noting B ¼ I BðBT BÞ1 BT , the sliding mode dynamics in the state estimation space can be obtained as follows: ^ hÞ þ BðL þ DLðtÞÞCeðtÞ þ f ðxÞ: ^ ^ þ A1 xðt x^_ ðtÞ ¼ ðA BKÞxðtÞ
ð12Þ
Hence, the stability of the overall closed-loop system with Eqs. (1) and (4) will be analyzed through the error system (4) and the sliding mode dynamics (12). 3.2. Analysis of asymptotic stability In the following theorem, the sufficient condition for the asymptotic stability of the overall closed-loop system with disturbance attenuation level is given via LMI. Theorem 2. Consider that the systems (4) and (12). Given a scalar g40, the switching function is chosen as Eq. (5), and the SMC law is chosen as Eq. (6). If there exist matrices Q1 40, Q2 40, L, K, and scalars ei 40 ði ¼ 1; 2; 3; 4Þ satisfying the following linear matrix
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inequality (LMI) 2 G11 A1 þ e1 E T E1 6 6 Q1 þ e1 E1T E1 6 6 6 6 6 6 6 4
C T LT B
T
0
G
0 G33
0 A1 þ e2 E T E1
0 0
Q2 þ e2 E1T E1
0
g2 I
N1
3
7 0 7 7 N3 7 7 7o0; 0 7 7 0 7 5 N6
ð13Þ
with G11 ¼ ðA LCÞ þ ðA LCÞT þ Q1 þ e1 E T E þ e3 d2 C T C þ 2rI þ e4 d2 C T C þ C T C; G33 ¼ ðA BKÞ þ ðA BKÞT þ e2 E T E þ 2rI þ Q2 ; N1 ¼ ½D; D; I; 0; N3 ¼ ½0; 0; 0; B; N6 ¼ diagfe1 I; e2 I; e3 I; e4 Ig; the overall closed-loop system with Eqs. (4) and (12) is asymptotically stable with disturbance attenuation level g. Proof. Choose the following Lyapunov functional candidate: Z t Z t T T T ^ ds: ^ ^ V2 ðtÞ ¼ e ðtÞeðtÞ þ x^ T ðsÞQ2 xðsÞ e ðsÞQ1 eðsÞ ds þ x ðtÞxðtÞ þ th
ð14Þ
th
First, we consider the overall closed-loop system with vðtÞ ¼ 0, we have V_ 2 ðtÞ ¼ eT ðtÞ½ðA LCÞ þ ðA LCÞT eðtÞ 2eT ðtÞDLðtÞCeðtÞ þ 2eT ðtÞA1 eðt hÞ ^ þ 2eT ðtÞDAeðtÞ þ 2eT ðtÞDA1 eðt hÞ þ 2eT ðtÞDAxðtÞ ^ þ 2eT ðtÞ½f ðxÞ f ðxÞ ^ hÞ þ eT ðtÞQ1 eðtÞ eT ðt hÞQ1 eðt hÞ þ x^ T ðtÞ½ðA BKÞ þ 2eT ðtÞDA1 xðt ^ hÞ þ 2x^ T ðtÞBDLCeðtÞ ^ þ 2x^ T ðtÞA1 xðt þðA BKÞT xðtÞ ^ x^ T ðt hÞQ2 xðt ^ hÞ: ^ þ x^ T ðtÞQ2 xðtÞ þ2x^ T ðtÞBLCeðtÞ þ 2x^ T ðtÞf ðxÞ
ð15Þ
Using Lemma 2 and Assumption 2, we have 2eT ðtÞDF ðtÞ½EeðtÞ þ E1 eðt hÞ 1 r eT ðtÞDDT eðtÞ þ e1 ½EeðtÞ þ E1 eðt hÞT ½EeðtÞ þ E1 eðt hÞ; e1 ^ hÞ ^ þ E1 xðt 2eT ðtÞDF ðtÞ½E xðtÞ 1 T ^ hÞT ½E xðtÞ ^ hÞ; ^ þ E1 xðt ^ þ E1 xðt r e ðtÞDDT eðtÞ þ e2 ½E xðtÞ e2 2eT ðtÞDLðtÞCeðtÞr2dJeT ðtÞJ JCeðtÞJr
1 T e ðtÞeðtÞ þ e3 d2 eT ðtÞC T CeðtÞ; e3
ð16Þ
ð17Þ ð18Þ
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2x^ T ðtÞBDLCeðtÞr2Jx^ T ðtÞBJ dJCeðtÞJr
573
1 T T ^ þ e4 d2 eT ðtÞC T CeðtÞ; x^ ðtÞBB xðtÞ e4
ð19Þ
T ^ 2eT ðtÞ½f ðxÞ f ðxÞr2Je ðtÞJ rJeðtÞJ ¼ 2reT ðtÞeðtÞ;
ð20Þ
^ ^ ^ 2x^ T ðtÞf ðxÞr2J x^ T ðtÞJ rJxðtÞJ ¼ 2rx^ T ðtÞxðtÞ:
ð21Þ
Substituting Eqs. (16)–(21) into Eq. (15) results in V_ 2 ðtÞrwT ðtÞXwðtÞ; where wðtÞ ¼ ½eT ðtÞ eT ðt hÞ x^ T ðtÞ x^ T ðt hÞT , and 2 3 T 0 X1 A1 þ e1 E T E1 C T LT B 6 7 6 Q1 þ e1 E1T E1 7 0 0 6 7; X¼6 T A1 þ e2 E E1 7 X3 4 5 Q2 þ e2 E1T E1 with X1 ¼ ðA LCÞ þ ðA LCÞT þ Q1 þ e1 E T E þ þ
1 DDT e1
1 1 DDT þ I þ e3 d2 C T C þ 2rI þ e4 d2 C T C; e2 e3
X3 ¼ ðA BKÞ þ ðA BKÞT þ e2 E T E þ
1 T BB þ 2rI þ Q2 : e4
It can be shown that if LMI (13) is satisfied, Xo0 is held by Lemma 1, then V_ 2 ðtÞr wT ðtÞXwðtÞo0 (for wðtÞa0), which shows the closed-loop system is asymptotically stable. Next, when vðtÞ 2 L2 , assuming zero conditions for the closed-loop system, the performance index is Z 1 ½yTe ðsÞye ðsÞ g2 vT ðsÞvðsÞ dt JðtÞ ¼ Z
0
1
½yTe ðsÞye ðsÞ g2 vT ðsÞvðsÞ þ V_ 2 ðsÞ dt þ V2 ðtÞjt¼0 V2 ðtÞjt-1 ;
¼ 0
r
Z
1
Z0
1
¼
½yTe ðsÞye ðsÞ g2 vT ðsÞvðsÞ þ V_ 2 ðsÞ dt qT ðsÞPqðsÞ dt;
0
where qðtÞ ¼ ½eT ðtÞ eT ðt hÞ x^ T ðtÞ x^ T ðt hÞ vT ðtÞT and 2 T X1 þ C T C A1 þ e1 E T E1 C T LT B 0 6 T 6 Q1 þ e1 E1 E1 0 0 6 6 P¼6 A1 þ e2 E T E1 X3 6 6 Q2 þ e2 E1T E1 4
G
3
7 7 7 7 7: 7 0 7 5 g2 I 0 0
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By utilizing Lemma 1, it is seen that Po0 is equivalent to Eq. (13). This means that JðtÞo0 (for qðtÞa0), so the overall closed-loop system is asymptotically stable with disturbance attenuation g. Then the proof is obtained. &
6 x1 x2
4
x1,x2
2
0
−2
−4
−6
0
0.5
1
1.5
2
2.5 t/sec
3
3.5
4
4.5
5
Fig. 2. Trajectories of state xðtÞ.
3 the estimate of x1 the estimate of x2
2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2
0
0.5
1
1.5
2
2.5 t/sec
3
3.5
^ Fig. 3. Trajectories of state estimate xðtÞ.
4
4.5
5
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4. Simulation results 0 0:2 0 0 1 We consider the system (1)–(3) with: A ¼ ½1 3 0, A1 ¼ ½0:5 0, B ¼ ½1, D ¼ ½0, 0:1sinðx1 Þ 0 t E ¼ ½0:1 0, E1 ¼ ½0:1 0, F ðtÞ ¼ 1, G ¼ ½1, f ðx; tÞ ¼ ½ 0 , vðtÞ ¼ 1=ðe þ 100Þ,
3.5
s
3 2.5 2 1.5 1 0.5 0 −0.5
0
0.5
1
1.5
2
2.5 t/sec
3
3.5
4
4.5
5
Fig. 4. Trajectories of sliding mode variable sðtÞ.
50 u
0 −50 −100 −150 −200 −250 −300
0
0.5
1
1.5
2
2.5 t/sec
3
Fig. 5. Control signal uðtÞ.
3.5
4
4.5
5
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0 DL ¼ ½0:01sinðtÞ , fðuÞ ¼ ð0:6 þ 0:3sinðuÞÞu, C ¼ ½0 1, a ¼ 0:3, b ¼ 5; r ¼ 0:1; d ¼ 0:01; h ¼ 0:1 and g ¼ 0:3162. 1:0306 0:7460 Solving LMI (13) yields L ¼ ½1:0370 28:5922 ; K ¼ ½3:0001 27:5294, Q1 ¼ ½0:7460 27:9018, 1:0852 0:0000 Q2 ¼ ½0:0000 27:4274, e1 ¼ 18:3865, e2 ¼ 18:1899, e3 ¼ 23:5444, e4 ¼ 26:9663. ^ And then, the initial states are xð0Þ ¼ ½5 6T and xð0Þ ¼ ½2 3T , the simulation results are given in Figs. 2–5, which show the validity of the proposed method.
5. Conclusion The problem of H1 sliding mode control for uncertain time-delay systems subjected to input nonlinearity has been presented. Based on the sliding mode control technique, the sufficient condition for asymptotic stability of the error system and sliding mode dynamics with disturbance attenuation level is derived via LMI. Finally, a simulation example is presented to verify the validity of the proposed method. Acknowledgement The authors are grateful for the support of the National Natural Science Foundation of China (Grant no. 60674024). References [1] Y.-J. Sun, Stability criterion for a class of descriptor systems with discrete and distributed time delays, Chaos, Solitons & Fractals 33 (2007) 986–993. [2] C.-H. Lien, Delay-dependent and delay-independent guaranteed cost control for uncertain neutral systems with time-varying delays via LMI approach, Chaos, Solitons & Fractals 33 (2007) 1017–1027. [3] M. De la Sen, On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays, Applied Mathematics and Computation 190 (2007) 382– 401. [4] G.-H. Yang, J. Liang Wang, Y. Chai Soh, Reliable H1 controller design for linear systems, Automatica 37 (2001) 717–725. [5] A.N. Madiwale, W.M. Haddad, D.S. Bernstein, Robust H1 control design for systems with structured parameter uncertainty, Systems & Control Letters 12 (1989) 393–407. [6] Y. Niu, D.W.C. Ho, Robust observer design for Itoˆ stochastic time-delay systems via sliding mode control, Systems & Control Letters 55 (2006) 781–793. [7] G.-Y. Tang, S.-S. Lu, R. Dong, Optimal sliding mode control for linear time-delay systems with sinusoidal disturbances, Journal of Sound and Vibration 304 (2007) 263–271. [8] K.-C. Hsu, Variable structure control design for uncertain dynamic systems with sector nonlinearities, Automatica 34 (1998) 505–508. [9] M.-L. Hung, J.-J. Yan, Decentralized model-reference adaptive control for a class of uncertain large-scale time-varying delayed systems with series nonlinearities, Chaos, Solitons & Fractals 33 (2007) 1558–1568. [10] J.-S. Lin, J.-J. Yan, T.-L. Liao, Chaotic synchronization via adaptive sliding mode observers subject to input nonlinearity, Chaos, Solitons & Fractals 24 (2005) 371–381. [11] W.-D. Chang, Robust adaptive single neural control for a class of uncertain nonlinear systems with input nonlinearity, Information Sciences 171 (2005) 261–271. [12] F.-M. Yu, H.-Y. Chung, S.-Y. Chen, Fuzzy sliding mode controller design for uncertain time-delayed systems with nonlinear input, Fuzzy Set & Systems 140 (2003) 359–374. [13] T. Fliegner, H. Logemann, E.P. Ryan, Low-gain integral control of continuous-time linear systems subject to input and output nonlinearities, Automatica 39 (2003) 455–462. [14] W. Ligang, W. Changhong, G. Huijun, Z. Lixian, Sliding mode H1 control for a class of uncertain nonlinear state-delayed systems, Journal of Systems Engineering and Electronics 17 (2006) 576–585.