Simple Stability Conditions of Linear Discrete Time Systems with Multiple Delay

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1 SERBIA JOURAL OF ELECRICAL EGIEERIG Vol. 7, o., May, UDK: : Siple Stability Conditions of Linear Discrete ie Systes with Multiple Delay Sreten B. Stoanovic, Dragutin LJ. Debelovic Abstract: In this paper we have established a new Lyapunov-Krasovsii ethod for linear discrete tie systes with ultiple tie delay. Based on this ethod, two sufficient conditions for delay-independent asyptotic stability of the linear discrete tie systes with ultiple delays are derived in the shape of Lyapunov inequality. uerical exaples are presented to deonstrate the applicability of the present approach. Keywords: Discrete tie-delay systes, Lyapunov-Krasovsii ethod, Delayindependent stability, Liner atrix inequality. Inotrduction During the last three decades, the proble of stability analysis of tie delay systes has received considerable attention and any papers dealing with this proble have appeared. In the literature, various stability techniques have been utilized to derive stability criteria for tie delay systes by any researchers. he techniques can be grossly classified into two categories: frequency doain approach (which are suitable for systes with a sall nuber of heterogeneous delays) and tie-doain approach (for systes with a any heterogeneous delays). he second approach is based on the coparison principle based techniques for functional differential equations [, ] or the Lyapunov stability approach with the Krasovsii and Razuihin ethods [3, 4]. In the past few years stability probles are thus reduced to one of finding solutions to Lyapunov [5] or Riccati equations [6] solving linear atrix inequalities (LMIs) [7] or analyzing eigenvalue distribution of appropriate finite-diensional atrices [8]. It is well-nown that the choice of an appropriate Lyapunov Krasovsii functional is crucial for deriving stability conditions [9]. he general for of this functional leads to a coplicated syste of partial differential equations Faculty of echnology, University of is, Lesovac, Serbia, e-ail: ssreten@ptt.rs Faculty of Mechanical Engineering, University of Belgrade, Belgrade, Serbia; E-ail: ddebelovic@as.bg.ac.rs 69

2 S.B. Stoanovic, D.LJ. Debelovic []. Special fors of Lyapunov Krasovsii functional lead to sipler delayindependent [, 7, 9] and less conservative delay-dependent conditions [9,, ]. In the past few years, there have been various approaches to reduce the conservatis of delay-dependent conditions by using new bounding for cross ters (Par s and siilarly inequalities) [, 3] or choosing new Lyapunov Krasovsii functional and odel transforation. However, the odel transforation ay introduce additional dynaics [4]. In [5] it is shown that descriptor transforation leads to a syste which is equivalent to the original one, does not depend on additional assuptions for stability of the transfored syste and requires bounding of fewer cross-ters. Since ost physical systes evolve in continuous tie, it is natural that theories for stability analysis are ainly developed for continuous-tie. However, it is ore reasonable that one should use a discrete-tie approach for that purpose because the controller is usually ipleented digitally. Despite this significance entioned, less attention has been paid to discrete-tie systes with delays [6-3, 5, 6]. It is ainly due to the fact that the delaydifference equations with nown delays can be converted into a higher-order delay less syste by augentation approach. However, for systes with large nown delay aounts, this schee will lead to large-diensional systes. Furtherore, for systes with unnown delay the augentation schee is not applicable. In this paper, new delay-independent asyptotic stability conditions are derived for discrete state-delayed systes with ultiple delays. hese conditions are derived using Lyapunov-Krasovsii ethod for discrete tiedelay systes which is presented in [6]. hroughout this paper we use the following notation. R denote real vector space or the set of real nubers, Z+ denotes the set of all non-negative integers. he superscript denotes transposition. For real atrix F the notation F > eans that the atrix F is positive definite. I and represent identity atrix and zero atrix. In syetric bloc atrices or long atrix expressions, we use an asteris (*) to represent a ter that is induced by syetry. Blocdiag {.} stands for a bloc-diagonal atrix. Matrices, if their diensions are not explicitly stated, are assued to be copatible for algebraic operations. v and denote nor of vector v and atrix 7 F, while v n v i= i = and F = λ ax ( F F) denote their Euclid nors. λ ( F ) is the eigenvalue of atrix F and ( F) σ denotes singular value of atrix F ( σ ax ( F) = F ).

3 Siple Stability Conditions of Linear Discrete ie Systes with Multiple Delay Model Description and Preliinaries A linear, autonoous, ultivariable discrete tie-delay syste can be represented by the difference equation x( + ) = Ax( t) + A x( h), < h < h, () with an associated function of initial state n x ( ) R, ( ) ( ), {,,..., } x θ =ψ θ θ h h + Δ, () n n A R is a constant atrix of appropriate diension and h Z + is unnown tie delay in general case. Let x ( ) ( ) ( ) x x x h, is state vector, (, n D Δ R ) - space of continuous functions apping the discrete interval Δ into R n and φ = sup φ ( θ ), ( ): n φ θ Δ R - the nor of an eleent φ in D. Further, D θ Δ { :, } D γ = φ D φ <γ γ R D. For initial state, the next condition is D assued ψ D. (3) Evidentially, x : θ x ( θ ) x( +θ) D and x ( ) = x (, ψ ). D Definition. he equilibriu state x = of () is asyptotically stable if any function of initial state ψ( θ ) which satisfies holds ψθ ( ) D, (4) li x (, ψ). (5) Lea. [5] If there exist positive nubers α and β and continuous functional V : D R such that < V( x ) α x, x, V() =, (6) D ΔV( x ) V( x ) V( x ) β x( ), (7) + x D satisfying (), then the equilibriu state x = of () is global asyptotically stable. 7

4 S.B. Stoanovic, D.LJ. Debelovic Definition. Discrete syste with tie delay () is asyptotically stable if and only if it s the equilibriu state x = is asyptotically stable. Lea. For any two atrices F and G of diension n and for any square atrix P= P > of diension n, the following stateent is true ( ) ( ) ( ) ( ) F + G P F + G +ε F PF + +ε G PG, (8) where ε is soe positive constant. Lea 3. hebyshev s inequality holds for any real vector v i vi vi vi vi i= i= i=. (9) 3 Main Results heore. he linear discrete tie-delay syste () with A is asyptotically stable if there exists real syetric atrix P > such that ( + ε ) + ε A PA + A PA P<. ε ( ) ε = A A. () Proof. Let the Lyapunov functional be ( ) = ( ) ( ) + ( ) ( ) V x x Px x l S x l l= P= P >, S = S, h, () where x = x( +θ ), θ { h, h,..., } +. () he forward difference along the solutions of syste () is 7

5 Siple Stability Conditions of Linear Discrete ie Systes with Multiple Delay Δ V x = Ax + Ax h P Ax + Ax h ( ) ( ) ( ) ( ) ( ) x Px h x l S x l l= ( ) ( ) ( ) ( ) h l= ( ) ( ) x l S x l. Applying Lea on (3), one can get ( ) ( ) ( ) ( ) ΔV x +ε x A PA x + ( ) x ( h ) A P Ax( h) ( ) ( ) ( ) ( ) ( h) Sx( h) + +ε x Px x S x + x Based on Lea 3 follows ΔV( x) x ( ) ( +ε ) A PA + S Px( ) + ( ) x ( h) A PAx( h) + +ε ( h) Sx( h) x ΔV( x) x ( ) ( +ε ) A PA + S Px( ) + ( h ) ( ) A PA S x( h) + x +ε.., (3) (4) (5) (6) then If one adopt ( ) S = +ε A PA, (7) 73

6 S.B. Stoanovic, D.LJ. Debelovic Δ V ( x ) x ( ) ( +ε ) A PA + ( +ε ) A PA P x ( ). (8) Let us define the following function f ( ε, x( ) ) = x ( ) ( +ε ) A PA + ( +ε ) A PA x ( ). (9) Since atrices A PA and A PA, =,,, are syetric and positive seidefinite then, based on Rayleigh and Air-Moez inequalities [3, 4] f ε, x ( ( )) x +ε λ + +ε λ = g ε λ P ( ) ( ) ax ( A PA ) ( ) ax ( A PA) x( ) ax, ( ) ( ) x ( ) () where ( ε ) = ( +ε) σ ax ( A ) + ( +ε ) σax ( A ) g. () Miniu of the scalar function g( ε ) is obtained fro condition d g() ε = σ ( A ) σ ( A ) =, () ε ε ax ax d ε=ε = σax ( A) σ ax ( A) = A A, (3) where fro follows Δ V x f ε, x f ε, x. (4) ( ) ( ( )) ( ( )) Putting ε instead of ε into (8) we obtain ΔV( x) x ( ) ( +ε) A PA P+ ( +ε ) A PA x ( ) (5) If the condition () is satisfied then ( ) { } { } ΔV x λ Q x( ) = β x( ) <, β λ Q >. (6) in 74 in

7 Siple Stability Conditions of Linear Discrete ie Systes with Multiple Delay Liewise, for x holds ax ( x ) < V h ax x x + +ε x x l= ( ) P ( ) ( ) ( l) A PA ( l) { } ( ) ax { } D λ P x( ) + +ε h λ A PA x( ) λ ax { P} + ( +ε ) h λax { A PA } x( ) =α x ( ), D D (7) where ax { P} ( ) h ax { A PA} α λ + +ε λ >. (8) So, based on Lea, syste () is asyptotically stable. Corollary. he linear discrete tie-delay syste () is asyptotically stable if there exist real syetric atrix P > and scalar ε > such that ( +ε) P (*) (*) (*) ( +ε) PA P (*) (*) ( +ε) PA εp (*) >. (9) ( +ε) PA εp Proof. Fro (), for ( ) Aˆ () ε = A +ε, Aˆ () ε = A +ε / ε (3) follows ( ) ˆ ˆ ˆ ˆ,. (3) P A PA A P A > P> Using Schur copleents [7] it is easy to see that the condition (3) is equivalent to 75

8 S.B. Stoanovic, D.LJ. Debelovic ˆ ˆ ˆ P A PA A > P > ˆ A P,. (3) Siilarly, the condition (3) is equivalent to ˆ ˆ ˆ ˆ P A PA A A ( ) ˆ P ˆ A A P > ˆ ˆ ˆ ˆ P A PA A A (33) ˆ A P >. ˆ A P Finally, the condition (3) is equivalent to P Aˆ Aˆ Aˆ > ˆ A P ˆ A P ˆ A P. (34) Pre and post ultiply (34) with blocdiag{ I, P,, P} we obtain ˆ ˆ ˆ P A P A P A P PAˆ P PAˆ P >. (35) PAˆ P Using (3) and pre and post ultiply (35) with blocdiag{i, I /( +ε ), I ε / ( +ε ),, I ε / ( +ε )} we obtain 76

9 Siple Stability Conditions of Linear Discrete ie Systes with Multiple Delay P A P A P A P PA P +ε ε PA P >. (36) ( +ε) ε PA P ( +ε) With P / ( ( + ε )) replaced by P we obtain (9). 4 uerical Exaple Exaple. Let us consider a discrete delay syste described by ( + ) = A ( ) + A ( h ) + A ( h ) x x x x, A..3 =. α,.3 A =γ.., A..5 =.3., where γ is adustable paraeter and syste scalar paraeter α taes the following values:.5 and.5. o deterined the largest paraeter γ for various values of ε by Corollary, the feasibility of equation (36) with γ as a variable can be cast into a generalized eigenvalue proble in P> (*) (*) (*) (*) (*) α, γ= / α, ( +ε) P (*) (*) (*) ( +ε) PA P (*) (*). <α ( +ε ) PA (*) εp (*) ( + ε) PA εp he delay-independent asyptotic stability conditions are characterized by eans of range of paraeter γ and are suarized in able. 77

10 S.B. Stoanovic, D.LJ. Debelovic able Stability Conditions. Paraeter α Conditions heore γ <.37 ε = γ<. ε = heore γ <.468 ε =ε =.66 γ<.3 ε =ε =.886 Corollary γ <.469 ε =ε =.44 γ<.5 ε =ε =.767 op op 5 Conclusion In this paper we have established a new Lyapunov-Krasovsii ethod for linear discrete tie systes with ultiple tie delay. Based on this ethod, two sufficient conditions for delay-independent asyptotic stability of the linear tie systes with ultiple delays are derived. hese conditions stabilities have been expressed in the shape of Lyapunov inequality. uerical exaples are presented to deonstrate the applicability of the present approach. 6 References [] S.I. iculescu, C.E. de Souza, L. Dugard, J.M. Dion: Robust Exponential Stability of Uncertain Linear Systes with ie-varying Delays, IEEE ransaction on Autoatic Control, Vol. 43, o. 5, May 998, pp [] L. Dugard, E.I. Verriest: Stability and Control of ie-delay Systes, Springer Berlin/Heidelberg, 998. [3] J.K. Hale, S.M.V. Lunel: Introduction to Functional Differential Equations, Springer- Verlag, ew Yor, 993. [4] V.B. Kolanovsii, V.R. osov: Stability of Functional Differential Equations, Acadeic Press, Orlando, Florida, 986. [5] J.H. Su: Further Results on the Robust Stability of Linear Systes with a Single Delay, Systes and Control Letters, Vol. 3, o. 5, ov. 994, pp [6] S.I. iculescu, C.E. de Souza, J.M. Dion, L. Dugard: Robust Stability and Stabilization for Uncertain Linear Systes with State Delay: Single Delay Case (I). IFAC Syposiu on Robust Control Design, Rio de Janeiro, Brazil, 994, pp [7] S. Boyd, L. El Ghaoui, E. Feron, V. Balarishnan: Linear atrix inequalities in syste and control theory, SIAM Studies in Applied Matheatics, Philadelphia, USA, 994. [8] J.H. Su: he Asyptotic Stability of Linear Autonoous Systes with Coensurate ie Delays, IEEE ransactions on Autoatic Control, Vol. 4, o. 6, June 995, pp

11 Siple Stability Conditions of Linear Discrete ie Systes with Multiple Delay [9] V. Kolanovsii, J.P. Richard: Stability of Soe Linear Systes with Delays, IEEE ransactions on Autoatic Control, Vol. 44, o. 5, May 999, pp [] M. Male-Zavarei, M. Jashidi: ie-delay Systes, Analysis, Optiization and Applications, Elsevier Science Inc., ew Yor, USA, 987. [] P. Par: A Delay-dependent Stability Criterion for Systes with Uncertain ie-invariant Delays, IEEE ransactions on Autoatic Control, Vol. 44, o. 4, Apr. 999, pp [] S.I. iculescu: On Delay-dependent Stability under M odel ransforations of Soe eutral Linear Systes, International Journal of Control, Vol. 74, o. 6, Apr., pp [3] J. Chen, G. Gu, C.. ett: A ew Method for Coputing Delay Margins for Stability of Linear Delay Systes, 33rd IEEE Conference on Decision and Control, Lae Buena Vista, Florida, USA, Vol., Dec. 994, pp [4] P. Par, Y.S. Moon, W.H. Kwon: A Delay-dependent Robust Stability Criterion for Uncertain ie-delay Systes, Aerican control conference, Philadelphia, Pennsylvania, June 998, pp [5] E. Fridan, U. Shaed: A Descriptor Syste Approach to H Control of Linear iedelay Systes, IEEE ransactions on Autoatic Control, Vol. 47, o., Feb., pp [6] E. Fridan, U. Shaed: An LMI Approach to Stability of Discrete Delay Systes, European Control Conference, Cabridge, United Kingdo, Sept. 3. [7] E. Fridan, U. Shaed: Delay-dependent H Control of Uncertain Discrete Delay Systes, European Journal of Control, Vol., o., 5, pp [8] E. Fridan, U. Shaed: Stability and Guaranteed Cost Control of Uncertain Discrete Delay Systes, International Journal of Control, Vol. 78, o. 4, March 5, pp [9] V. Kapila, W.M. Haddad: Meoryless H Controllers for Discrete-tie Systes with ie Delay, Autoatica, Vol. 34, o. 9, Sept. 998, pp [] Y.S. Lee, W.H. Kwon: Delay-dependent Robust Stabilization of Uncertain Discrete-tie State-delayed Systes, 5th riennial World Congress, Barcelona, Spain,. [] M.S. Mahoud: Robust H Control of Discrete Systes with Uncertain Paraeters and Unnown Delays, Autoatica, Vol. 36, o. 4, April, pp [] M.S. Mahoud: Linear Paraeter-varying Discrete ie-delay Systes: Stability and l- gain Controllers, International Journal of Control, Vol. 73, o. 6, April, pp [3] C.D. Meyer: Matrix analysis and applied linear algebra, SIAM, Philadelphia,. [4] A.R. Air-Moez: Extree Properties of Eigenvalues of a Heritian ransforations and Singular Values of Su and Product of Linear ransforations, Due Matheatical Journal, Vol. 3, o. 3, Sept. 956, pp [5] S.B. Stoanovic, D.L. Debelovic, I. Mladenovic: Siple Exponential Stability Criteria of Linear Discrete ie Delay Systes, Serbian Journal of Electrical Engineering, Vol. 5, o., ov. 8, pp [6] S.B. Stoanovic, D.L. Debelovic, I. Mladenovic: A Lyapunov-Krasovsii Methodology for Asyptotic Stability of Discrete ie Delay Systes, Serbian Journal of Electrical Engineering, Vol. 4, o., ov. 7, pp