Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays

Automatica 47 (2011) 1838–1841

Contents lists available at ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Technical communique

Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays✩ Phan T. Nam a,∗ , Pubudu N. Pathirana b a

Department of Mathematics, Quynhon University, Binhdinh, Viet Nam

b

School of Engineering, Deakin University, Geelong, VIC 3217, Australia

article

info

Article history: Received 16 September 2010 Received in revised form 4 January 2011 Accepted 10 April 2011 Available online 1 June 2011 Keywords: Reachable set Interval time-varying delays Lyapunov–Krasovskii functional

abstract The problem of reachable set estimation of linear uncertain polytopic time-varying delay systems subject to bounded peak inputs is studied in this paper. The delays considered in this paper are assumed to be non-differentiable and vary within an interval where the lower and upper bounds are known. Based on the Lyapunov–Krasovskii method and delay decomposition technique, a sufficient condition for the existence of a ball that bounds the reachable set of the system is proposed in terms of matrix inequalities containing only one scalar which can be solved by using an one-dimensional search method and Matlab’s LMI Toolbox and allow us to find the smallest radius. A numerical example is given to illustrate the effectiveness of the proposed result. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Reachable set for a dynamic system is the set of all the states starting from the origin by inputs with a peak value (Zuo, Ho, & Wang, 2010). Reachable set estimation is an important problem in the robust control theory (Fridman & Dambrine, 2009; Fridman & Shaked, 2003; Kim, 2008; Zuo et al., 2010, and the references therein). By using a Lyapunov function applying the S-procedure, Boyd, Ghaoui, Feron, and Balakrishnam (1994) derived a matrix inequality condition for an ellipsoid bounding the reachable set of linear systems without any time delay. Based on the Lyapunov–Razumikhin method, Fridman and Shaked (2003) first extended this problem to linear systems with time-varying delays. Later, based on the modified Lyapunov–Krasovskii type functional, an improved result is reported in Kim (2008). Very recently, by constructing a maximal Lyapunov–Krasovskii functional, Zuo et al. (2010) have proposed another result for linear uncertain polytopic time-varying delay systems. To the best of our knowledge, only the above three results currently exist for linear time-varying delay systems. The delays considered in the above papers are from zero

to an upper bound. In practice, however, these delays may vary in a range for which the lower bound is not necessary to be zero. Therefore, in this paper, we consider linear uncertain polytopic systems with non-differentiable interval time-varying delays: x˙ (t ) = Ax(t ) + Ad x(t − τ (t )) + Bω(t ), x( t ) ≡ 0 ,

0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.05.003

(1)

where x(t ) ∈ R is the state, matrices A, Ad and B are uncertain but belong to a given polytope n

 Φ=

[A, Ad , B] :=

p −

ξi [Ai , Adi , Bi ],

i=1

p −

 ξi = 1, ξi ≥ 0 ,

i =1

with Ai , Adi , Bi , (i = 1, . . . , p) are constant matrices with appropriate dimensions and ξi , (i = 1, . . . , p) are time-invariant uncertainties. The time-varying delay τ (t ) is bounded and is not necessarily differentiable 0 ≤ τm ≤ τ (t ) ≤ τM < ∞,

(2)

and ω(t ) ∈ R is the disturbance satisfying m

2 ωT (t )ω(t ) ≤ ωm ,

✩ This work was supported by the Australian Research Council under the Discovery grant DP0667181, Endeavour Awards, Australia and the National Foundation for Science and Technology Development, Viet Nam. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Keqin Gu under the direction of Editor André L. Tits. ∗ Corresponding author. Tel.: +84 907 416 946; fax: +84 563 846 089. E-mail addresses: [email protected] (P.T. Nam), [email protected] (P.N. Pathirana).

t ≥0

t ∈ [−τM , 0]

(3)

where τm , τM , ωn are positive constants. Recently, the delay decomposition technique (Fridman, Shaked, & Liu, 2009; Gouaisbaut & Peaucelle, 2006; Lam, Gao, & Wang, 2007; Yue, Tian, & Zhang, 2009) and the free-weighting matrix technique (Gao & Chen, 2007; He, Wang, Lin, & Wu, 2007) have been used widely since they have significantly improved the stability analysis of time-delay systems. In this paper, we propose to use Lyapunov–Krasovskii functional for exponential stability

P.T. Nam, P.N. Pathirana / Automatica 47 (2011) 1838–1841

−τ

∫

1839

∫

0

(Mondie & Kharitonov, 2005) together with the delay decomposition technique and the free-weighting matrix technique to derive a ball that bounds the reachable set of system (1)–(3). Our derived ball has a smaller radius than the existing results available in Fridman and Shaked (2003), Kim (2008) and Zuo et al. (2010). Our condition is expressed in terms of matrix inequalities containing only one scalar which can be solved easily by using an one-dimensional search method and Matlab’s LMI Toolbox. Moreover, we use the bounding technique as in Ramos and Peres (2002), He, Wu, She, and Liu (2004) and Phat and Nam (2007) to relax the requirement of common matrix variables in order to satisfy the complete sets of matrix inequalities. A numerical example is provided to show the superiority of our result. The following lemmas are useful for our main result.

V3 = (τM − τ )

Lemma 1 (Gu, 2000). Given a matrix Q > 0, the following integral inequality holds

V˙ 1 = 2xT (t )P x˙ (t ) = −α V1 + 2xT (t )P x˙ (t ) + α xT (t )Px(t ),

(9)

V˙ 2 = −α V2 + xT (t )[Q + R + S ]x(t ) − xT (t − τM )e−ατM × Qx(t − τM ) − xT (t − τ )e−ατ Rx(t − τ ) − xT (t − τm )e−ατm Sx(t − τm ),

(10)

− τM

t

∫

t −τ (t )

x˙ T (s)M x˙ (s)ds

x(t ) x(t − τ (t ))

 ≤

T 

−M

M

M

−M

v

−τ 0

∫

V5 = τm

0

∫ v

−τm

eα u x˙ T (t + u)Y x˙ (t + u)dα dv,

eα u x˙ T (t + u)Z x˙ (t + u)dudv.

Using the spectral properties of symmetric positive-definite matrices Pi , we have min

i=1,2,...,p

λmin (Pi )‖x(t )‖2 ≤ V (t , xt ).

(8)

Taking the derivative of Vi , i = 1, 2, . . . , 5 in t, we have

V˙ 3 = −α V3 + (τM − τ ) x˙ (t )X x˙ (t ) 2 T

x(t ) . x(t − τ (t ))



V4 = (τ − τm )

eα u x˙ T (t + u)X x˙ (t + u)dudv,

−τM v ∫ −τm ∫ 0



− (τM − τ )

t −τ

∫

eα(s−t ) x˙ T (s)X x˙ (s)ds

t −τM

≤ −α V3 + (τM − τ )2 x˙ T (t )X x˙ (t ) ∫ t −τ x˙ T (s)X x˙ (s)ds. − (τM − τ )e−ατM

Lemma 2 (Fridman & Shaked, 2003). Let V be a Lyapunov function for system (1)–(3). If V˙ + α V − α2 ωT (t )ω(t ) ≤ 0 then V ≤ 1. ωm

(11)

t −τM

Similarly, we have

2. Main result Theorem 3. If there exist n × n-matrices Pi > 0, Qi > 0, Ri > 0, Si > 0, Xi > 0, Yi > 0, Zi > 0, Ci , Di , Ei , Fi , Gi , Hi , Ii , Ji , Ki , Li , Mi , Ni , i = 1, . . . , p, (6n + m) × (6n + m)-matrices U > 0, V > 0 and a scalar α > 0 such that the following matrix inequalities hold:

Ωii < −U ,

Σii < −V ,

Ωij + Ωji <

2 p−1

U,

i = 1, 2, . . . , p,

Σij + Σji <

2 p−1

V,

(4)

(5)

where Ωij , Σij are denoted in Appendix. Then the reachable sets of system (1)–(3) is bounded by a ball B(0, r ) = {x ∈ Rn |‖x‖ ≤ r }, with

min

i=1,2,...,p

λmin (Pi )

.

(6)

τ +τ

Proof. Denote τ = m 2 M . Noting that, for any t ∈ R+ , τ (t ) ∈ [τm , τ ] or τ (t ) ∈ (τ , τM ], define two sets:

Π1 = {t : τ (t ) ∈ (τ , τM ]},

and

Π2 = {t : τ (t ) ∈ [τm , τ ]}.

With matrices Pi , Qi , Ri , Si , Xi , Yi , Zi , C∑ i , Di , Ei , Fi , Gi , H i , Ii , J i , Ki , ∑ p p Li , Mi , Ni , i = 1, . . . , p, we denote P = ξ i Pi , Q = i = 1 i=1 ξi Qi , ∑p . . . , N = i=1 ξi Ni . Consider the following Lyapunov–Krasovskii functionals V = V1 + V2 + V3 + V4 + V5

(7)

V˙ 5 ≤ −α V5 + τm2 x˙ T (t )Z x˙ (t )

t

eα(s−t ) xT (s)Qx(s)ds +

V2 = t −τM t

∫

+ t −τm

∫

t

∫

(13)

x˙ T (s)Z x˙ (s)ds.

t −τ

x˙ T (s)X x˙ (s)ds

− t −τM

∫

t −τ (t )

x˙ T (s)X x˙ (s)ds −

t −τM

t −τ

∫

t −τ (t )

x˙ T (s)X x˙ (s)ds.

(14)

Denoting ∆T = [C T , DT , E T , F T , GT , H T ], ηT (t ) = [xT (t ) xT (t − τm ) xT (t − τ ) xT (t − τM ) xT (t − τ (t )) x˙ (t )], and ζ (t ) = [ηT (t ) ωT (t )], we have 2ηT (t )∆[−˙x(t ) + Ax(t ) + Ad x(t − τ (t )) + Bω(t )] = 0.

(15)

By adding (9)–(15), using Lemma 1, and then employing the technique in Ramos and Peres (2002), we obtain

α T ω (t )ω(t ) 2 ωm   p p−1 − p − − ≤ ζ T (t ) ξi2 Ωii + ξi ξj [Ωij + Ωji ] ζ (t )

V˙ + α V −

≤ −ζ (t )

i =1  p − i =1

t −τ

eα(s−t ) xT (s)Sx(s)ds,

t

Case I. For t ∈ Π1 , i.e. τ (t ) ∈ (τ , τM ], we have

T

eα(s−t ) xT (s)Rx(s)ds

∫

t −τm

where V1 = xT (t )Px(t ),

∫

(12)

x˙ T (s)Y x˙ (s)ds,

t −τ

=−

1

t −τm

∫

− (τ − τm )e−ατ

− τm e−ατm

i = 1, 2, . . . , p − 1; j = i + 1, . . . , p,

r = 

V˙ 4 ≤ −α V4 + (τ − τm )2 x˙ T (t )Y x˙ (t )

 = −ζ (t ) T

1

i=1 j=i+1

ξ

2 i U

−

p−1 − p − i=1 j=i+1

p−1 − p −

ξi ξj

2 p−1

 U

ζ (t )

 (ξi − ξj ) U ζ (t ) < 0.

p − 1 i=1 j=i+1

2

(16)

1840

P.T. Nam, P.N. Pathirana / Automatica 47 (2011) 1838–1841

Case II. For t ∈ Π2 , we have t −τm

∫ −

Table 1 Computed radiuses.

x˙ T (s)Y x˙ (s)ds

t −τ

∫

t −τ (t )

=−

x˙ T (s)Y x˙ (s)ds −

t −τm

∫

t −τ (t )

t −τ

x˙ T (s)Y x˙ (s)ds,

and

with Γ T = [I T , J T , K T , LT , M T , N T ]. By the same way as above, we also obtain

α T ω (t )ω(t ) < 0. 2 ωm

(17)

By Lemma 2, we have V (t , xt ) ≤ 1. This implies that ‖x(t )‖ ≤ r , ∀t ≥ 0 due to (8). The proof is completed.  Remark 4. It is easy to see that radius r is smallest if δ = mini=1,2,...,p λmin (Pi ) is largest. So, one can consider Pi , i = 1, 2, . . . , p as decision variable and the additional requirement that

δ I ≤ Pi ,

i = 1, 2, . . . , p.

(18)

Remark 5. Note that the matrix inequalities (4), (5) cannot be simplified to LMIs. However, when α is fixed, then the matrix inequalities (4), (5) reduce to LMIs. Therefore, we can combine an one-dimensional search method with Matlab’s LMI Toolbox to solve the matrix inequalities (4), (5). Remark 6. This result may further be improved if Jensen’s inequality is replaced by the following less conservative inequality (see in Kwon & Park, 2009)

∫

t2

−

x˙ T (s)Rx˙ (s)ds ≤ 2ζ T (t )F T

t1

∫

t2

x˙ (s)ds

t1

+ (t2 − t1 )ζ T (t )F T R−1 F ζ (t ) with matrix F ∈ Rn×(6n+m) . 3. A numerical example Example 7. Consider the following system, which is considered in Fridman and Shaked (2003), Kim (2008) and Zuo et al. (2010), with

 A1 =

0

 A2 =

−2 −2 0

0



−0.7

,

 Ad1 =



0 , −1.1

 Ad2 =

−1 −1 −1 −1

0



−0.9 

,

0 , −1.1

B1 =

Fridman03

Kim08

Zuo10

Ours

√ √19.70 √19.70 19.70

√

√ √1.98 √5.3 5.3

√ √1.2 √1.2 1.1

2.97

– –

radius than the ones in Fridman03, Kim08, Zuo10, even if time delay is a constant. Also the radius in case c is smaller than the ones in cases a and b.

2ηT (t )Γ [−˙x(t ) + Ax(t ) + Ad x(t − τ (t )) + Bω(t )] = 0,

V˙ + α V −

Methods Case a Case b Case c

  −0.5 1

,

B2 = B1 ,

and the time-delay function τ (t ) is considered in 3 cases: (a) τ (t ) = 0.7, i.e. delay is a constant. (b) τ (t ) is non-differentiable and 0 ≤ τ (t ) ≤ 0.7. (c) τ (t ) is non-differentiable and 0.1 ≤ τ (t ) ≤ 0.7. Based on Theorem 3, we can find a smaller radius r as listed in Table 1. Note that Kim (2008) approach cannot be applied for cases b and c, since τ (t ) is non-differentiable. From Table 1, it can be seen that the proposed approach presented in this paper provides a smaller

4. Conclusion In this paper, a sufficient condition for the existence of a ball that bounds the reachable set of linear uncertain polytopic systems with interval time-varying delays has been proposed in terms of matrix inequalities which can be solved by using Matlab’s LMI Toolbox and allow us to find the smallest radius of the ball. The effectiveness of the proposed result is illustrated via a numerical example Acknowledgments The authors would like to thank Professor Hieu Trinh and anonymous Reviewers for their comments and suggestions, which improved the paper. Appendix

 11 Ωij  ⋆    ⋆    ⋆ Ωij =   ⋆    ⋆   ⋆  11 Σij  ⋆    ⋆    ⋆ Σij =   ⋆    ⋆   ⋆

Ωij12

ATj EiT

ATj FiT

Ωij15

Ω 16

Ci Bj

Ωij22

e−ατ Yi

0

Di Adj

−D i

⋆

Ωij33

0

Ωij35

Di Bj  

−Ei

⋆

⋆

Ωij44

Ωij45

−F i

Ei B j  

⋆

⋆

⋆

Ωij55

Ωij56

⋆

⋆

⋆

⋆

Ωij66

⋆

⋆

⋆

⋆

⋆

ATj LTi

Σij15

Σij16

−J i

Ji Bj  

−K i

Ki Bj  

Σij12

ATj KiT

Σij22

0

0

⋆

Σij33

e−ατM Xi

Σij25 Σij35

⋆

⋆

Σij44

Li Adj

−L i

⋆

⋆

⋆

Σij55

Σij56

⋆

⋆

⋆

⋆

Σij66

⋆

⋆

⋆

⋆

⋆



Fi Bj 

 , Gi Bj    Hi Bj   α  − 2 ωm  Ii Bj



Ωij15 = ATj GTi + Ci Adj ,

Ωij16 = Pi − Ci + ATj HiT , Ωij22 = −e−ατm (Zi + Si ) − e−ατ Yi , Ωij33 = −e−ατ (Yi + Ri ) − e−ατM Xi , Ωij35 = e−ατM Xi + Ei Adj , Ωij44 = −e−ατM (Qi + Xi ),

Ωij45 = e−ατM Xi + Fi Adj ,

Ωij55 = −2e−ατM Xi + Gi Adj + ATdj GTi ,

Li B j 

 , Mi Bj    Ni Bj   α  − 2 ωm

Ωij11 = α Pi + Qi + Ri + Si − e−ατm Zi + Ci Aj + ATj CiT , Ωij12 = e−ατm Zi + ATj DTi ,



P.T. Nam, P.N. Pathirana / Automatica 47 (2011) 1838–1841

Ωij56 = −Gi + ATdj HiT , Ωij66 Σij11 Σij12 Σij16 Σij22

= (τM − τ ) Xi + (τ − τm ) Yi + τm Zi − Hi − 2

2

= α Pi + Qi + Ri + Si − e =e

−ατm

Zi +

= P i − Ii + = −e

−ατm

ATj JiT

ATj NiT

,

ατm

Σij15

Z i + Ii A j +

=

ATj MiT

ATj IiT

HiT

,

,

+ Ii Adj ,

,

(Zi + Si ) − e−ατ Yi ,

Σij25 = e−ατ Yi + Ji Adj Σij33 = −e−ατ (Yi + Ri ) − e−ατM Xi , Σij35 = e−ατ Yi + Ki Adj ,

Σij44 = −e−ατM (Qi + Xi ),

Σij55 = −2e−ατ Yi + Mi Adj + ATdj MiT , Σij56 = −Mi + ATdj NiT , Σij66 = (τM − τ )2 Xi + (τ − τm )2 Yi + τm Zi − Ni − NiT . References Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnam, V. (1994). Linear matrix inequalities in systems and control theory. Philadelphia, PA: SIAM. Fridman, E., & Dambrine, M. (2009). Control under quantization, saturation and delay: a LMI approach. Automatica, 45(10), 2258–2264. Fridman, E., & Shaked, U. (2003). On reachable sets for linear systems with delay and bounded peak inputs. Automatica, 39(11), 2005–2010. Fridman, E., Shaked, U., & Liu, K. (2009). New conditions for delay-derivativedependent stability. Automatica, 45(11), 2723–2727.

1841

Gao, H., & Chen, T. (2007). New results on stability of discrete-time systems with tim e-varying state delay. IEEE Transactions on Automatic Control, 52(2), 328–334. Gouaisbaut, F., & Peaucelle, D. (2006). Delay-dependent stability analysis of linear time delay systems. In IFAC workshop time delay systems, Aquila, Italy. Gu, K. (2000). An integral inequality in the stability problem of time-delay systems. In Proceedings of 39th IEEE conf. on decision and control. Sydney, Australia (pp. 2805–2810). He, Y., Wang, Q., Lin, C., & Wu, M. (2007). Delay-range-dependent stability for systems with time-varying delay. Automatica, 43(2), 371–376. He, Y., Wu, M., She, J. H., & Liu, G. P. (2004). Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties. IEEE Transactions on Automatic Control, 49(5), 828–832. Kim, J. H. (2008). Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances. Automatica, 44(11), 2940–2943. Kwon, O. M., & Park, J. H. (2009). Exponential stability analysis for uncertain neural networks with interval time-varying delays. Applied Mathematics and Computation, 212(2), 530–541. Lam, J., Gao, H., & Wang, C. (2007). Stability analysis for continuous systems with two additive time-varying delay components. Systems & Control Letters, 56(1), 16–24. Mondie, S., & Kharitonov, V. L. (2005). Exponential estimates for retarded timedelay systems: an LMI approach. IEEE Transactions on Automatic Control, 50(2), 268–273. Phat, V. N., & Nam, P. T. (2007). Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function. International Journal of Control, 80(8), 1333–1341. Ramos, D. C. W., & Peres, P. L. D. (2002). An LMI condition for the robust stability of uncertain continuous time linear sysems. IEEE Transactions on Automatic Control, 47(4), 675–678. Yue, D., Tian, E., & Zhang, Y. (2009). A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay. International Journal of Robust and Nonlinear Control, 18(13), 1493–1518. Zuo, Z., Ho, D. W. C., & Wang, Y. (2010). Reachable set bounding for delayed systems with polytopic uncertainties: the maximal Lyapunov–Krasovskii functional approach. Automatica, 46(5), 949–952.