Discrete-Time Sliding-Mode Control Of Uncertain Systems With . - Hindawi

Maode Yan et al.3Lemma 1 (see [17], Moon’s inequality). Assume that a Rna , b Rnb , and N Rna nb . Then for any matrices X Rna na , Y Rna nb , and Z Rnb nb , the following holds: T 2aT Nb YTbY NXa NTΘ1 L2 LT2 dM W 1 , abZwhere ϑ denotes L1 H1T K T H3T , σ denotes (1 κ)Ad U1 , and χdenotes (dM dm 1) 1 U1 ,,(5)Θ2 L3 L1 (AT κATd I) K T AT12 LT2 dM W 2 ,Θ3 L3 LT3 dM W 3 where (10)(6)Lemma 2 (see [18]). For any matrices D Rn p , E R p n ,F R p p with F T (k)F(k) I and scalars ε 0,Tδi Ei EiT .i 1 X Y 0.Y ZT T2 1TTDFE E F D ε DD εE E.Then, the system (4) with Assumption 1 is asymptoticallystable, and the sliding surface of (3) is given byS(k) KL1 1 x1 (k) x2 (k) 0.(11)(7)The main objective of this paper is to design the slidingsurface S(k) and a DT-SMC law u(k) such thatProof. Let η(k) x1 (k 1) x1 (k), the system (4) can berewritten in an equivalent descriptor form:x1 (k 1) x1 (k) η(k),(1) the sliding surface is asymptotically stable;(2) the DT-SMC law can guarantee discrete-time slidingmode reaching condition.0 (A11 (k) Ad11 (k) I)x1 (k) η(k) Ad11 (k)k 1η(i),i k d(k)(12)3.MAIN RESULTSwhereIn this section, the results of slidng surface design and robustDT-SMC law will be presented for a class of uncertain linearsystems with time-varying delays. Let us first consider theproblem of sliding surface design. In order to reduce theoverdesign entailed in the stability analysis methods for timedelay systems, we employ the descriptor approach to derivethe delay-dependent sufficient conditions for the existence ofthe sliding surface in terms of LMIs. The first result on theasymptotic stability of designing sliding surface is presentedin the following theorem.Theorem 1. If, for certain prescribed positive number κ, thereexist positive scalars δ1 0, δ2 0, (n m) (n m), matricesL1 0, L2 , L3 , U1 0, U2 0, W 1 , W 2 , W 3 , and m (n m) matrix K such that the following LMIs hold: Θ1 Θ200L1ϑLTLTW1 W2 0 U1(13)Choose the Lyapunov-Krasovskii functional candidate asV (k) V1 (k) V2 (k) V3 (k) V4 (k).(14)Here,V1 (k) x1T (k)P1 x1 (k),V2 (k) k 1x1T (i)Qx1 (i),i k d(k)V3 (k) dm 1 k 1x1T (i)Qx1 (i),(15)j dM 2 i k j 1(8)V4 (k) 1 k 1ηT (i)Gη(i),j dM i k j W 3 κAd U1 0,Ad11 (k) Ad11 ΔAd11 . 22 ΘTσ000L3LT3 3 000 U1 U1 H2T 0 δI0000 2 0, χ000 δI001 L01 1 dM U2 A11 (k) A11 A12 C ΔA11 ΔA12 C,(9)where P1 , Q, G R(n m) (n m) are symmetric positive definite matrices to be determined. The LyapunovKrasovskii functional candidate V (k) is positive definite forall x1 (k) / 0. Now, in order to evaluate the forward differenceΔV (k) V (k 1) V (k), in what follows we calculateΔV1 , ΔV2 , ΔV3 , and ΔV4 , respectively. For ΔV1 (k), by usingthe descriptor system form (12), we obtain

4Journal of Control Science and EngineeringΔV1 (k) ηTFor ΔV3 , we get(k)P1 η(k) 2x1T (k)P1 η(k) ηT (k)P1 η(k) 2xT1 (k)P Tη(k) η (k)P1 η(k) 2x1 (k)P 1k 2xT1 (k)P Tj dM 2 η(k)T(20)For ΔV4 , we obtainKn m Ad11 (k)η(i),P1 0ΔV1 (k) ηT (k)P1 η(k) 2xT1 (k)P T dM η (k)Gη(k) i k d(k)x1 (k)Wη(i) M PTK Based on the above results in (17), (19)–(21), and usingLemma 2, we can derive x1T (k d(k))(δ2 1 H2T H2 Q)x1 (k d(k)) x1T (k) ηT (k) x1T (k d(k)) η(k) (A11 (k) Ad11 (k) I)x1 (k) η(k)k 1 dM xT1 (k)Wx1 (k) x1 (k) (P T Kn m Ad11 M)x1 (k d(k)) η(i)G ηT(k)P1 η(k) 2xT1(k)P Tη ( j)Gη( j). xT1 (k)Ψx1 (k) 2xT1 (k)n m Ad11 (k) Tj k dM(A11 (k) Ad11 (k) I)x1 (k) η(k) T (21)k 1TΔV (k) ΔV1 (k) ΔV2 (k) ΔV3 (k) ΔV4 (k) η(k)[ηT (k)Gη(k) ηT (k j)Gη(k j)]j dMT[ P P ]. If identifying N : P T Kn m Ad11 (k), a : x1 (k), and2 3b : η(i) in (16), then applying Lemma 1 gives rise to thefollowing inequality: 1 ΔV4 where xT1 (k) [x1T (k) ηT (k)], Kn m [0 I ] , and P x1T ( j)Qx1 ( j).j k dM 1(A11 (k) Ad11 (k) I)x1 (k) η(k)(16)k 1k dm (dM dm )x1T (k)Qx1 (k) i k d(k) [x1T (k)Qx1 (k) x1T (k j 1)Qx1 (k j 1)]0 T dm 1 ΔV3 Ψ ηT (i)Gη(i) 0 x1T (k)Ad11(22)i k dM 2xT1 (k)[M P T Kn m Ad11 (k)][x1 (k) x1 (k d(k))],(17)where W, M R(n m) (n m) are constant matrices withappropriate dimensions satisfying W Mwhere Ψ P 0. ΔV2 k 1x1T (i)Qx1 (i) i k d(k 1) 1k 1i k d(k 1) 1k dmMT0 ωP dM W M 0 0 0 P1 dM Gk 1 2 δi P Ti 10Ei0 EiT Pwhere ω denotes δ1 1 (H1T C T H3T )(H1 H3 C) (dM dm 1)Q and δ1 , δ2 are any positive scalars. Let ξ [ x1T (k) ηT (k) x1T (k d(k)) ] andx1T (i)Qx1 (i) i k d(k) 1 ΨΦ x1T (k)Qx1 (k) x1T (k d(k))Qx1 (k d(k)) I(23)i k d(k)x1T (i)Qx1 (i) x1T (i)Qx1 (i) x1T (k)Qx1 (k) x1T (k d(k))Qx1 (k d(k)) 0 AT11 C T AT12 II For ΔV2 , we have A11 A12 C I I (18)I0T Gk M ηT (k) , 1 TTδ2 H2 H2 Qx1 (k d(k))PT x1T (i)Qx1 (i).0 M . δ2 1 H2T H2 QPTAd11(24)Theni k dM 1(19)ΔV (k) ξ T (k)Φξ(k).(25)

Maode Yan et al.5In order to obtain a convenient LMI, we restrict the choice ofM to be M κPT0 Ad11 (k).(26)Proof. The design of DT-SMC law guarantees that the reaching condition must be satisfied when there exist uncertaintiesand time-varying delays in systems. We consider the reachingcondition for the DT-SMC proposed in [21] as follows:Define P 1 L : L1 0L2 L3 Q 1 U1 ,K CL1 ,, W (P ) W(P 1 ) 1 TG 1 U2 ,W1 W2TW2 W3 CΔAd11 x1 (k d(k)) bound and lower bound of β(k) 2CAd11 x1 (k d(k)) i 1 [Ad2i xi (k d(k))], γU and γL arethe upper bound and lower bound of γ (k) f (k, xk ), q 0,ε 0, 1 qτ 0, and τ 0 is the sampling period.ΔS(k) S(k 1) S(k) . τ ·sgn(S(k)) qτS(k),(27)ΔV (k) 0, ξ(k) / 0.(28)It follows from Lyapunov stability theory [20] that the system(4) is asymptotically stable. Moreover, based on the abovedefinition of the sliding surface in (3), it can be obtained withC KL1 1 .Once the sliding surface is appropriately designedaccording to Theorem 1, the next result on the design of DTSMC is summarized in the following theorem.Theorem 2. Consider the system (2) with Assumptions 1–3.If the linear sliding surface is designed as in (11), then thefollowing DT-SMC lawu(k) B 12 [(A2i CA1i )xi (k)] (1 qτ)S(k)i 1 τ ·sgn(S(k)) (α0 β0 γ0 )(29) (ρα ρβ ργ )·sgn(S(k))can guarantee the discrete-time sliding mode reaching condition. Here,S(k) S1 (k) S2 (k) · · · Sm (k) ,(31)ΔS(k) S(k 1) S(k) τ ·sgn(S(k)) qτS(k),TMultiply (18) by diag[(P 1 ) , U2 ] and diag[(P 1 ), U2 ] onthe left- and right-hand sides, respectively. Alternatively,Tmultiply (24) by diag[(P 1 ) , U1 ] and diag[(P 1 ), U1 ] onthe left- and right-hand sides, respectively. By using the Schurcomplement [19], we obtain that (8) is equivalent to Φ 0,and (18) is equivalent to (9), which yieldsif S(k) 0,if S(k) 0.From the designed sliding surfaceS(k) Cx(k) [C I]x(k),(32)we haveΔS(k) (CA11 CΔA11 )x1 (k) (CAd11 CΔAd11 )x1 (k d(k)) (CA12 CΔA12 )x2 (k) 2 [A2i xi (k) Ad2i xi (k d(k))] Bu(k) f (k, xk ).i 1(33)Since it is assumed that ΔA11 , ΔAd11 , ΔA12 , and f (k, xk )are bounded, using the discrete-time version of improvedRazumikhin theorem in [22], for any solution x(k d(k))of (2), there exists a constant θ such that the followinginequality is satisfied: x(k d(k)) θ x(k) , k, dm d(k) dM .(34)We can also obtain that α (k) CΔA11 x1 (k) CΔA12 x2 (k), 2 β(k) C(Ad11 ΔAd11 )x1 (k d(k)) i 1 [Ad2i xi (k d(k))],and γ (k) f (k, xk ) will be bounded with upper and lowerbounds. Let the bounds beαL α (k) CΔA11 x1 (k) CΔA12 x2 (k) αU , βL β(k) C(Ad11 ΔAd11 )x1 (k d(k)) 2 [Ad2i xi (k d(k))] βU ,i 1γL γ f (k, xk ) γU .(35)sgn(S(k)) sgn(S1 ) sgn(S2 ) · · · sgn(Sm ) ,α αα0 U L ,2βU βLβ0 ,2γU γLγ0 ,2α αρα U L ,2βU βLρβ ,2γU γLργ ,2(30)C KL1 1 , αU and αL are the upper bound and lower boundof α (k) CΔA11 x1 (k) CΔA12 x2 (k), βU and βL are the upperHere, the inequalityαL αL,1 αL,2 · · · αL,mT 1 α 2 · · · α m α (k) αT αU αU,1 αU,2 · · · αU,m(36)T,implies that αL,i α i (k) αU,i , i 1, 2, . . . , m; similar and γ (k).notations are used for β(k)

6Journal of Control Science and Engineering4Then, substituting (29) into (33), we have3.5 (α0 β0 γ0 ) (ρα ρβ ργ )·sgn(S(k)).3(37)According to previous discussion, the following relationshold:α (k) α0 ρα ·sgn(S(k)) if S(k) 0,α (k) α0 ρα ·sgn(S(k)) if S(k) 0,Time-varying delay d(k) γ (k) qτS(k) τ ·sgn(S(k))ΔS(k) α (k) β(k)2.521.510.5 β(k) β0 ρβ ·sgn(S(k)) if S(k) 0,(38) β(k) β0 ρβ ·sgn(S(k)) if S(k) 0,γ (k) γ0 ργ ·sgn(S(k)) if S(k) 0,0510152025Time (k)303540Figure 1: Time-varying delays d(k).γ (k) γ0 ργ ·sgn(S(k)) if S(k) 0.Thus, the sign change of ΔS(k) in (37) is opposite to thatof S(k), irrespective of the value of the uncertainties α (k), β(k),and γ (k). Moreover, the closed-loop control system (2)satisfies the reaching condition (31).Remark 1. The descriptor system approach [13, 14, 16]can greatly reduce the overdesign entailed in the existingmethods because the descriptor model leads to a system thatis equivalent to the original one (from the point of viewof stability) and requires bounding of fewer cross-terms.Therefore, in this work, we apply the descriptor systemapproach to the SMC design for discrete-time systems withuncertainties and time-varying delays.Remark 2. SMC approach is an effective method to achievethe robustness and invariant property to matched uncertainties and disturbance on the sliding surface [9, 21]. Inthis paper, the results of Theorem 1 guarantee that thesliding motion dynamics will be robustly asymptoticallystable. Furthermore, the descriptor-system-approach-basedDT-SMC in Theorem 2 provides a solution to satisfy the sliding mode reaching condition of the designed discrete-timesliding surface for discrete-time systems with uncertaintiesand time-varying delays.4.In this section, an illustrative example is given to verify thedesign method proposed in this paper. Consider the discretetime system (2) with the following description:x1 (k) 0.50A11 , 0.20 x12 (k)Ad11 A21 x11 (k)0 0.1 ,Ad21 Ad22 0.05,,0.1ΔA11 B 1,0.02 sin(0.01kπ) 0.02 sin(0.01kπ) ,0.01 cos(0.01kπ) 0.04 sin(0.01kπ) ΔA12 0.050.04 ΔAd11 0 ,0.01 sin(0.01kπ) 0.04 sin(0.01kπ) 0.01 cos(0.01kπ) 0.05 sin(0.01kπ),f (k, x(k)) 0.4 sin(x11 (k)).(39)TInitial states of the system are x1 (k) [1.0 0.5] , x2 (k) 0.6, for k [ dM 0]. Using Theorem 1 and choosing κ 0.5, we obtain C [0.9305 0.5168]. Then, the linear slidingsurface is S(k) [C 1]x(k). According to Theorem 2, therobust DT-SMC law is designed asu(k) B 12 [(A2i CA1i )xi (k)] (1 qτ)S(k)i 1 ( τ ρα ρβ ργ )·sgn(S(k)) ,NUMERICAL EXAMPLE 0.7,0.05 0.8 , 0A12 A22 2.0,0.30 ,(40)where ρα 0.0238 x11 (k) 0.0393 x12 (k) 0.0672 x2 (k) ,ρβ 0.5014 x11 (k) 0.2785 x12 (k) 0.125 x2 (k) , ργ 0.4 sin x11 (k) , τ 0.1, q 5, and 0.1. The timevarying delay 1 d(k) 3 is shown in Figure 1. Systemstate trajectories are illustrated in Figure 2. The resultingsliding surface is in Figure 3. Figure 4 depicts the controlinput signal.It is observed from Figure 2 that the state trajectories ofthe system all converge to the origin quickly. The system canbe stabilized quickly by the proposed method and the reaching motion satisfies the sliding reaching condition in spite of

7130.82.50.620.41.5Control input u(k)System states x11 , x12 , and x2Maode Yan et al.0.20 0.2 0.410.50 0.5 0.6 1 0.8 1.5 1510152025Time (k)303540 2510152025Time (k)303540Figure 4: Control input u(k).x11x12x2Figure 2: System states x11 , x12 , and x2 .for uncertain systems with time-varying delays. An exampleshows the validity and effectiveness of the proposed DT-SMCdesign method.10.8ACKNOWLEDGMENTSSliding surface S(k)0.6The authors wish to thank the associate editor and anonymous reviewers for providing constructive suggestions whichhave improved the presentation of the paper. This researchwas supported by the Natural Sciences and EngineeringResearch Council of Canada and the Canada Foundation ofInnovation.0.40.20 0.2 0.4 0.6REFERENCES 0.8 1510152025Time (k)303540Figure 3: Sliding surface S(k).the time-varying delays and uncertainties. Simulation resultsillustrate that the proposed approach in this paper is feasibleand effective for discrete-time uncertain linear systems withtime-varying delays.5.CONCLUSIONIn this paper, the problem of robust DT-SMC for uncertainsystems with time-varying delays has been studied. By usingthe descriptor model transformation and a recent resulton bounding cross products of vectors, a delay-dependentsufficient condition for the existence of stable sliding surfacesis constructed for all admissible uncertainties. Based onthis existence condition, the corresponding reaching motioncontroller is developed such that the reaching motionsatisfies the discrete-time sliding mode reaching condition[1] E.-K. Boukas and Z.-K. Liu, Deterministic and Stochastic TimeDelay Systems, Birkhäuser, Boston, Mass, USA, 2002.[2] K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-DelaySystems, Birkhäuser, Boston, Mass, USA, 2003.[3] M. Wu, Y. He, J.-H. She, and G.-P. Liu, “Delay-dependentcriteria for robust stability of time-varying delay systems,”Automatica, vol. 40, no. 8, pp. 1435–1439, 2004.[4] P. Park and J. W. Ko, “Stability and robust stability for systemswith a time-varying delay,” Automatica, vol. 43, no. 10, pp.1855–1858, 2007.[5] W.-H. Chen, Z.-H. Guan, and X. Lu, “Delay-dependentguaranteed cost control for uncertain discrete-time systemswith delay,” IEE Proceedings: Control Theory and Applications,vol. 150, no. 4, pp. 412–416, 2003.[6] L. Zhang, Y. Shi, T. Chen, and B. Huang, “A new methodfor stabilization of networked control systems with randomdelays,” IEEE Transactions on Automatic Control, vol. 50, no. 8,pp. 1177–1181, 2005.[7] H. Gao, J. Lam, C. Wang, and Y. Wang, “Delay-dependentoutput-feedback stabilization of discrete-time systems withtime-varying state delay,” IEE Proceedings: Control Theory andApplications, vol. 151, no. 6, pp. 691–698, 2004.[8] H. Gao and T. Chen, “New results on stability of discrete-timesystems with time-varying state delay,” IEEE Transactions onAutomatic Control, vol. 52, no. 2, pp. 328–334, 2007.

8[9] V. Utkin, “Variable structure systems with sliding modes,”IEEE Transactions on Automatic Control, vol. 22, no. 2, pp.212–222, 1977.[10] A. J. Koshkouei and A. S. I. Zinober, “Sliding mode time delaysystems,” in Proceedings of the IEEE International Workshop onVariable Structure Systems (VSS ’96), pp. 97–101, Tokyo, Japan,December 1996.[11] Y. Xia and Y. Jia, “Robust sliding-mode control for uncertaintime-delay systems: an LMI approach,” IEEE Transactions onAutomatic Control, vol. 48, no. 6, pp. 1086–1092, 2003.[12] Y. Niu, J. Lam, X. Wang, and D. W. C. Ho, “Observer-basedsliding mode control for nonlinear state-delayed systems,”International Journal of Systems Science, vol. 35, no. 2, pp. 139–150, 2004.[13] E. Fridman, “New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems,” Systems &Control Letters, vol. 43, no. 4, pp. 309–319, 2001.[14] E. Fridman and U. Shaked, “A descriptor system approach toH control of linear time-delay systems,” IEEE Transactions onAutomatic Control, vol. 47, no. 2, pp. 253–270, 2002.[15] E. Fridman and U. Shaked, “A new H filter design for lineartime delay systems,” IEEE Transactions on Signal Processing,vol. 49, no. 11, pp. 2839–2843, 2001.[16] E. Fridman, F. Gouaisbaut, M. Dambrine, and J.-P. Richard,“Sliding mode control of systems with time-varying delays viadescriptor approach,” International Journal of Systems Science,vol. 34, no. 8-9, pp. 553–559, 2003.[17] Y. S. Moon, P. G. Park, W. H. Kwon, and Y. S. Lee, “Delaydependent robust stabilization of uncertain state-delayedsystems,” International Journal of Control, vol. 74, no. 14, pp.1447–1455, 2001.[18] L. Xie, “Output feedback control of systems with parameteruncertainty,” International Journal of Control, vol. 63, no. 4,pp. 741–750, 1996.[19] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, LinearMatrix Inequalities in System and Control Theory, SIAM,Philadelphia, Pa, USA, 1994.[20] M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall,Upper Saddle River, NJ, USA, 1993.[21] W. Gao, Y. Wang, and A. Homaifa, “Discrete-time variablestructure control systems,” IEEE Transactions on IndustrialElectronics, vol. 42, no. 2, pp. 117–122, 1995.[22] J. K. Hale and S. M. V. Lunel, Introduction to FunctionalDifferential Equations, Springer, New York, NY, USA, 1993.Journal of Control Science and Engineering

International Journal ofRotatingMachineryEngineeringJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014The ScientificWorld JournalHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014International Journal ofDistributedSensor NetworksJournal ofSensorsHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Journal ofControl Scienceand EngineeringAdvances inCivil EngineeringHindawi Publishing Corporationhttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Volume 2014Submit your manuscripts athttp://www.hindawi.comJournal ofJournal ofElectrical and ComputerEngineeringRoboticsHindawi Publishing Corporationhttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Volume 2014VLSI DesignAdvances inOptoElectronicsInternational Journal ofNavigation andObservationHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Hindawi Publishing Corporationhttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.comChemical EngineeringHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Volume 2014Active and PassiveElectronic ComponentsAntennas andPropagationHindawi Publishing ingHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2010Volume 2014International Journal ofInternational Journal ofInternational Journal ofModelling &Simulationin EngineeringVolume 2014Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Shock and VibrationHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014Advances inAcoustics and VibrationHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014

the design of SMC for uncertain systems with time delay. One approach based on lumped sliding surface has been applied to design an SMC law for state-delay systems in [10]. In [11], the SMC method is proposed for systems with mismatched parametric uncertainties and time delay, and a delay-independent sufficient condition for the existence of linear