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Procedia Engineering
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 15 (2011) 1181 – 1185 www.elsevier.com/locate/procedia
Advanced in Control Engineeringand Information Science Sliding Mode Control for a Class of Uncertain Neutral Delay Systems He Youguoa , zhaoleib, Zhang Muyongb a* a
b
Faculty of Information Engineering, Shenyang University, Shenyang 110044 , China Electric Detection Department, Shenyang Artillery Academy, Shenyang 110044 , China
Abstract The sliding mode control problem of a class of uncertain neutral delay systems is considered. By selecting a sliding surface depending on the current state and delayed state, the paper gives delay-independent sufficient condition and delay-dependent sufficient condition in terms of linear matrix inequalities such that the closed-loop system is guaranteed to be asymptotically stable. According to reaching condition, the sliding mode control method is proposed. The control method guarantees that the trajectory of system arrives at the sliding surface in finite time interval and is kept here thereafter. © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license.
Selection and/or peer-review under responsibility of [CEIS 2011]
Keywords:sliding mode control; neutral delay systems; linear matrix inequalities
1. Introduction A neutral time-delay system is one depending not only on state delay, but also on derivative of the delay state. There have been a number of developments in the search for system stability and control methods for uncertain neutral time-delay systems. In another active research area, sliding mode control(SMC) is an effective method of robustness control[1-4]. Y Niu et al proposed sliding-mode control problem for uncertain neutral delay systems[5]. But the external disturbances are not considered in this paper. This motivated us to study the delay-dependent sliding mode control problem of a class of uncertain neutral delay systems. In this note, the sliding mode control problem of a class of uncertain neutral delay systems is considered. Mismatched uncertainties and external disturbances are introduced into the neutral systems. The delay-independent sufficient condition and delay-dependent sufficient condition in terms of linear matrix inequalities are derived to guarantee the asymptotic stability of the closed loop system. When
* Corresponding author. Tel.: 024-62268531. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2011.08.218
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LMIs are feasible, the sliding mode control method is proposed. The control method guarantees that the trajectory of system arrives at the sliding surface in finite time interval and is kept here thereafter. 2. Problem formulation and preliminaries Consider the following uncertain neutral delay system:
x& (t ) − Ad x& (t − d ) = ( A + ΔA) x(t ) + ( Ah + ΔAh ) x(t − h) + B[u (t ) + f (t )] ,
(1)
, θ ∈ [−τ ,0] . (2) where ΔA , ΔAh are unknown time-varying system parameter uncertainties, f (t ) is external disturbance, ϕ (θ ) is a continuous vector valued initial function, τ = max(d , h) . For the convenience of proof, the following lemma and assumptions are given: Lemma 1: Let X, Y be real matrices of appropriate dimensions, for any ε > 0 , x(θ ) = ϕ (θ )
XY + Y T X T ≤ ε −1 XX T + εY T Y
Lemma 2: Let D, E and F(t) be real matrices of appropriate dimensions with F(t) satisfying F T (t ) F (t ) ≤ I then, for any ε > 0 , EF (t ) H + H T F T (t ) E T ≤ ε −1 EE T + εH T H . Assumption 1: The admissible parameter uncertainties are of the norm-bounded form
[ΔA
ΔAh ] = [EF (t ) H
E h F (t ) H h ]
where H, Hh, E and Eh are known constant matrices, F(t) is unknown time-varying matrices with Lebesgue measurable elements bounded by F T (t ) F (t ) ≤ I . Assumption 2: The external disturbance is bounded as f (t ) ≤ ρ 0 . The objective of this paper is to design a sliding mode controller such that the closed-loop system is stochastically stable. 3. Delay-independent sliding surface design Let us choose the sliding surface s (t ) = B T P[ x (t ) − Ad x (t − d )]
(3)
u = − Kx(t ) + u r (t )
(4) (5)
where P is a positive definite matrix to be designed. Let the SMC strategy be given as follows: s(t ) ⎧ T ⎪ − B P[ Ax(t ) + Ah x(t − h)] − ρ s(t ) u r (t ) = ⎨ ⎪− B T P[ Ax(t ) + Ah x(t − h)] ⎩
where K is chosen such that ( A − BK ) is stable, and
ρ = B T PA x(t ) + B T PAh x(t − h) + γ + ρ 0
s(t ) ≠ 0 s(t ) = 0
(6)
with γ > 0 constant. Then we obtain the following result. Theorem 1: For uncertain neutral systems (1), (2), if there exist matrices P > 0 , Q1 > 0 , Q2 > 0; and scalars ε 1 > 0 and ε 2 > 0 satisfying the following LMIs (7) W1 = Q1 − AdT (Q1 + Q2 + ε 1 H T H ) Ad > 0
(8)
T
W 2 = Q2 − ε 2 H h H h > 0 ⎡ Θ ⎢ AT Π T ⎢ d ⎢ AhT P ⎢ T ⎢E P T ⎣⎢ E h P
with Θ = P ( A − BK ) + ( A − BK ) T P + Q1 + Q2 + ε 1 H T H ,
ΠAd
PAh
PE
- W1
0
0
0 0
- W2 0
0 - ε1I
0
0
0
PE h ⎤ 0 ⎥⎥ 0 ⎥ 0 , Q1 > 0 , Q2 > 0;
and scalars ε 1 > 0 and ε 2 > 0 satisfying the following
LMIs W1 > 0 , W2 > 0 , Σ < 0 , then we can have V& < 0 . So the VSC law (4)-(6) with sliding surface (3) can guarantee that the closed-loop system is globally asymptotically stable. Finally, by Schur's complement, it is easily shown that the matrix inequality Σ < 0 is equivalent to LMI (9). Theorem 2: Suppose that there exist matrices P > 0 , Q1 > 0 , Q2 > 0; and scalars ε 1 > 0 and ε 2 > 0 satisfying LMIs (7)-(9), and the sliding surface is given by (3). Then, for neutral delay system (1), (2) with SMC (4)-(6), every state trajectory is attracted towards the sliding surface in a finite time and once the trajectory hits the sliding surface it remains there for all subsequent time. Proof: By using (1), (3) and (4), one can obtain s& = B T P[ x& (t ) − Ad x& (t − d )] = B T P[( A + ΔA) x(t ) + ( Ah + ΔAh ) x(t − h) + B(u + f )]
⎡ s⎤ = B T P[( A − BK + ΔA) x(t ) + ( Ah + ΔAh ) x(t − h)] − B T PB ⎢ B T P( Ax(t ) + Ah x(t − h) ) + ρ ⎥ + B T PBf s ⎥⎦ ⎢⎣ Considering a candidate Lyapunov function V = 1 s T ( B T PB ) −1 s 2
Differentiating V (t ) along the solution of equation (3) yields V& = s T ( B T PB ) −1 s& = s T ( B T PB ) −1 B T P ( A − BK + ΔA) x (t ) + s T ( B T PB ) −1 B T P ( Ah + ΔAh ) x(t − h) − s T ( B T PAx(t ) + B T PAh x(t − h)) − ρ s + s T f
From assumption1、2 we can have
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V& ≤ s ( ( B T PB) −1 B T P( A − BK ) + ( B T PB) −1 B T PE H ) x(t ) + s ( ( B T PB) −1 BT PAh + ( B T PB) −1 B T PEh H h ) x(t − h) − s γ
Let ( B T PB) −1 B T P = R , so above inequality can be rewritten V& ≤ s ( R( A − BK ) + RE H ) x(t ) + s ( RAh + RE h H h ) x(t − h) − s γ In the state space, define the following domain: Ω γ = {x (t ) : ζ x (t ) + ξ x (t − h) ≤ 0.5γ } . Where ζ = R( A − BK ) + RE H , ξ = RAh + REh H h . It is seen that the reaching condition of sliding mode is satisfied within the domain Ωγ , the state trajectories of closed-loop system will enter domain Ωγ in a finite time and remain there. 4. Delay-dependent sliding surface design
In fact, the delay-independent condition is relatively conservative. Next, we discuss the delaydependent condition. so we have the following theorem. Theorem 3: For uncertain neutral systems (1), (2), if there exist matrices P , Q1 , Q2, Q3 , Q4, and scalars ε 1 > 0 and ε 2 > 0 satisfying the following LMIs (11) W1 = Q1 − AdT (Q1 + Q2 + d Q3 + h Q4 + ε 1 H T H ) Ad > 0 T
(12)
W2 = Q 2 − ε 2 H h H h > 0 ΠAd - W1
PE h ⎤ 0 ⎥⎥ (13) 0 - W2 0 0 ⎥ 0 and ε 2 > 0 satisfying the following matrix inequalities W1 > 0 , W2 > 0 and Σ < 0 , then we can have V& < 0 , So the VSC law (4)-(6) with sliding surface (3) can guarantee that the closed-loop system is globally asymptotically stable. Finally, by Schur’s complement, it is easily shown that the matrix inequality Σ < 0 is equivalent to LMI (11-13). 5. Conclusion The sliding mode control problem of a class of uncertain neutral delay systems is considered in this paper. The neutral delay systems under consideration may have parameter uncertainties in the state matrices as well as external disturbance. The delay-independent sufficient condition and delay-dependent sufficient conditions in terms of linear matrix inequalities are derived to guarantee the asymptotic stability of the closed-loop system. When LMIs are feasible, the sliding mode control method is proposed to guarantee that the trajectory of system arrives at the sliding surface in finite time interval and is kept here thereafter. References [1] Choi H H, “On the existence of linear sliding surface for a class of uncertain dynamic systems with mismatched uncertainties,” Automatica, vol.35(10), 1999, pp 1707-1715. [2] Pen shi, Yuanqing Xia, and G P Liu et, “On Designing of Sliding Mode Control for Stochastic Jump Systems,”IEEE Trans on Automatic control, vol. 51(1), 2006, pp 97-103. [3] Niu Y, Lam J, and Wang X et al., “Sliding-mode control for nonlinear state-delayed systems using neural network approximation,” IEE Proc., Control Theory Appl., vol.150(3), 1999, pp 233-239. [4] Hu J, Chu J, Su H, “SMVSC for a class of time-delay uncertain systems with mismatching uncertainties,” IEE Proc., Control Theory Appl., vol.147(6), 2000, pp 687-693. [5] Y Niu,J Lam and X Wang, “Sliding-mode control for uncertain neutral delay systems,” IEE Proc., Control Theory Appl.,vol. 151(1), 2004, pp 38-44.
