Control and Stability of the Time-delay Linear Systems

Transcription

1 ISSN , England, UK Journal of Inforation and Couting Science Vol., No. 4, 206, Control and Stability of the Tie-delay Linear Systes Negras Tahasbi *, Hojjat Ahsani Tehrani Deartent of Matheatics, School of Matheatical Sciences, Shahrood University of Technology, Shahrood, Iran E-ail: (Received Jun 04, 206, acceted August 8, 206) Abstract. This aer resents a ethod for control discrete-tie syste with tie-delay. The ain idea is a convert the discrete-tie delay linear controllable syste into the linear systes without delay. Then by using siilarity transforations a state feedback atrix was obtained, so that tie-delay has no effect on the syste. The roosed technique is illustrated by eans of nuerical exales. Keywords: stability, Discrete-tie, Tie-delay, State feedback atrix, Pole Assignent.. Introduction The roble of investigation of tie delay systes has been exloited over any years. Tie-delay is very often encountered in various technical systes, such as electric, neuatic and hydraulic networks, cheical rocesses, long transission lines, etc. The existence of ure tie-delay, regardless if it is resent in the control or/and the state, ay cause undesirable syste transient resonse, or even instability. During the last four decades, the roble of stability analysis of tie delay systes has received considerable attention and any aers dealing with this roble have aeared []. In the literature, various stability analysis techniques have been utilized to derive stability criteria for asytotic stability of the tie delay systes by any researchers [2]-[6]. The develoed stability criteria are classified often into two categories according to their deendence on the size of the delay: delay-deendent and delayindeendent stability criteria [7]. It has been shown that delay deendent stability conditions that take into account the size of delays, are generally less conservative than delay-indeendent ones which do not include any inforation on the size of delays. Further, the delay-deendent stability conditions can be classified into two classes: frequency-doain (which are suitable for systes with a sall nuber of heterogeneous delays) and tie-doain aroaches (for systes with a any heterogeneous delays). In the first aroach, we can include the two or several variable olynoials [8],[9] or the sall gain theore based aroach [0]. In the second aroach, we have the coarison rincile based techniques [] for functional differential equations [2]-[4] and resectively the Lyaunov stability aroach with the Krasovskii and Razuikhin based ethods [],[5] The stability roble is thus reduced to one of finding solutions to Lyaunov [5] or Riccati equations [2], solving linear atrix inequalities (LMIs) [6],[7] or analyzing eigenvalue distribution of aroriate finite-diensional atrices [8] or atrix encils [0]. For further rearks on the ethods see also the guided tours roosed by [3], [9]-[22]. In this aer we used of results [23],[24] for discrete-tie delay systes. This is ainly due to the fact that such systes can be transfored into augented high diensional systes (equivalent systes) without delay. The reainder of this aer is organized as follows. In Section 2, the roble stateent and soe necessary reliinaries are given. In Section 3, we roosed ethod for stability this systes. Nuerical siulations are rovided in Section 4. Finally, soe concluding rearks are given in Section Proble stateent Consider a controllable linear tie-invariant syste with tie-delay defined by the state equation x(k +) = i0 A i x(k - i) + q i=0 B i u(k - i), () Published by World Acadeic Press, World Acadeic Union

2 292 Narges Tahasbi et al.: Control and Stability of the Tie-delay Linear Systes where is the state vector, u R is the control inut and the atrices constant atrices of diensions and, resectively, with rank( Bi ). By definition state vector such as n x R n n n The syste () with delays in state and syste X( k ) AX ( k) Bu( k) (3) Where x( k ) x( k) x ( k ( )) X (k+)= uk ( ) uk ( ) u ( k ( q )) A0 A A A B B 2 Bq Bq B 0 I n 0 0 I n 0 0 I n A =,B= I I I I 0 0 A i and B i are real q delays in inut vector can be rewritten as a standard n and X R, u R, n n( ) q. We define control law as u ( k ) FX ( k ), (5) Where F is a feedback gain. Therefore, the syste () changes to a standard closed-loo syste X( k ) ( A BF) X( k). (6) In this aer we deterined the state feedback atrix F such that the eigenvalues of the closed-loo syste A BF lie in the self-conjugate eigenvalue sectru {, 2, n }. Karbassi and Bell [25]-[26] have introduced an algorith obtaining an exlicit araetric controller atrix F by erforing three successive transforations T,S and R which transfors the controllable air (A,B) into standard echelon for, riary vector coanion for and araetric vector coanion for, resectively. Let F reresent the riary feedback atrix which assigns the desired set of eigenvalues to the closed-loo syste. 3. Main results Consider the state transforation X ( k) TX ( k), (7) where T can be obtained by eleentary siilarity oerations as described in [26]. By relace (7) in equation (3) we have X ( k ) T AT X ( k) T Bu ( ), (8) In this way, for: k A T AT and B T B are in a coact canonical for known as vector coanion I n G, 0 n,, B B A (9) n,. (2) (4) JIC eail for contribution: editor@jic.org.uk

3 Journal of Inforation and Couting Science, Vol. (206) No. 4, G 0 n B 0 uer triangular atrix. Note that if the Here is a atrix and is an Kroneckerinvariants of the air A, B are regular, then A and B are always in the above for [25]. In the case of irregular Kronecker invariants, soe rows of in A are dislaced [26]. It ay also be concluded I n that if the vector coanion for of A obtained fro siilarity oerations has the above structure, then the A, B Kronecker invariants associated with the air The transfored closed-loo atrix are regular [26]. A BF assues a coact Jordan for with zero eigenvalues 0n, (0) I n, 0n, It is fro this for that the state feedback atrix, which assigns a set of arbitrary eigenvalues to the syste and also the location of araeters, can be obtained. The controller atrix is then odified by adding a diagonal atrix D diag, 2,..., n for an arbitrary set of self-conjugate eigenvalues to as defined in [8]. Then the closed-loo syste atrix V A BF () D A BF becoes Sile eleentary siilarity oerations can be used to obtain the atrix G G A fro F V such that A (2) I n, 0n, where is the first n sub-atrix of. Obviously, ossesses the desired set of eigenvalues and is in the sae canonical for as A. Thus, the riary feedback atrix which gives rise to the assignent of eigenvalues becoes F F B0 G B0 ( G0 G ) (3) Clearly, disturbance is rejected and the eigenvalues of the closed-loo syste A BF are in the sectru, then eigenvalues of the closed-loo A BF are in the sectru, where F FT. 4. Nuerical Exales In this section, we give three exales to show the success of the roosed ethod. Exale. Consider a discrete-tie linear syste with delay in state vector A A x k 0 x ( k ) 2 3 x ( k ) 0 u ( k ) (4) We obtion A, B (5) JIC eail for subscrition: ublishing@wau.org.uk

4 294 Narges Tahasbi et al.: Control and Stability of the Tie-delay Linear Systes The syste (4) is ractically unstable (eigenvalue ofa is {3.98, 2.94,.36, i, i, }, then, syste oen-loo is unstable). stabilization of this syste is our next ai, we find feedback atrix $ F$ which the eigenvalues of the closed-loo syste assign in sectru { 0., 0.3,0,0.,0.3,0.5}. Since, we obtain F (6) With nor Also, The syste resonse to a unit ste inut is shown in Fig.. As you can see in Fig. 2, syste of this exale fro initial value X (0) [,5,3,, 0.5, 3] T is ractically stable. Fig. : State resonse in exale Fig. : Inut resonse in exale Exale 2. Consider a discrete-tie linear syste with delay in state vector [27] x k x ( k ) u ( k ) u ( k ) (7) We find feedback atrix F which the eigenvalues of the closed-loo syste assign in sectru { 0.,0., 0.5,0.5}. therefore, we obtain F (8) As you can see in Fig. 3, syste of this exale fro initial value x(0) =[-,,5,-3 ] T is racticlly stable. Fig. 4 shown Control inut. JIC eail for contribution: editor@jic.org.uk

5 Journal of Inforation and Couting Science, Vol. (206) No. 4, Fig. 3: State resonse in exale2 Fig. 4: Inut resonse in exale2 Exale 3. Consider a discrete-tie linear syste with delay in state and inut vector x k x ( k ) x ( k ) u ( k ) u ( k ) u ( k 2) (9) We find feedback atrix F which the eigenvalues of the closed-loo syste assign in sectru { 0., 0.2, 0.3, 0.4,0.,0.2,0.3,0.4}. therefore, we obtain A, B this syste unstable. For stability syste, state feedback atrix which locates all the eigenvalues of the closed-loo syste is found to be: F (2) Also, The syste resonse to a unit ste inut is shown in Fig. 5. As you can see in Fig. 6, syste of this exale fro initial value x(0) =[-2,2,-,,3,-3,4,-4 ] T is ractically stable (20) JIC eail for subscrition: ublishing@wau.org.uk

6 296 Narges Tahasbi et al.: Control and Stability of the Tie-delay Linear Systes Fig. 5: State resonse in exale3 5. Conclusion Fig. 6: Inut resonse in exale3 The control roble for linear discrete systes with tie-delay is investigated in this aer. We used of augented vector that syste with tie-delay convert to syste without delay. Then by using siilarity transforations, couted otial controller. Also, the stability analysis of tiediscrete systes with tie varying delay and fractional-order syste aear to be an interesting roble and oen for future investigation. 6. References [] J. K. Hale, E. F. Infante, F. S. P. Tsen, Stability in linear delay equations, Journal on Matheatical Analysis and Alications, 05 (985), [2] T. N. Lee, S. Diant, Stability of tie-delay systes, IEEE Transactions on Autoatic Control, 26(4) (98), [3] H. Gorecki, S. Fuksa, P. Grabowski, A. Korytowski, Analysis and Synthesis of Tie Delay Systes, New York : John Wiley and Sons, 989. [4] J. J. Yan, Robust stability analysis of uncertain tie-delay systes with delay deendence,electronics Letters, 37 (200), [5] J. H. Su, Further results on the robust stability of linear systes with a single delay, Systesand Control Letters, 23 (994), [6] H. Ki, Delay and its tie-derivative deendent robust stability of tie delayed linear systes with uncertainty, IEEE Transactions on Autoatic Control 46 (200), [7] S. Huang, H. Shao, Z. Zhang, Stability analysis of large-scale syste with delays, Syste and Control Letters, 25 (995), [8] E. W. Kaen, Linear systes with coensurate tie delays : Stability and stabilizationindeendent of delay, IEEE Transactions on Autoatic Control, 28 (983), [9] D. Hertz, E. I. Jury and E. Zeheb, Stability indeendent and deendent of delay for delay differential systes. Journal of Franklin Institute, 38 (984), [0] J. Chen, H. A. Latchan, Asytotic stability indeendent of delays : sile necessary and sufficient conditions, Proceedings of Aerican Control Conference, Baltiore, USA, (994), JIC eail for contribution: editor@jic.org.uk

7 Journal of Inforation and Couting Science, Vol. (206) No. 4, [] V. Lakshikanta, S. Leela, Differential and integral inequalities, New York:Acadeic Press 969. [2] S. I. Niculescu, A. Trofino-Neto, J. M. Dion, L. Dugard, Delay-deendent stability of linea systes with delayed state : An LMI aroach, Proceedings of 34th IEEE Conference on Decision and Control, New Orleans, Louisiana, USA, (995b), [3] J. P. Richard, Tie-delay systes : an overview of soe recent advances and oen robles, Autoatica, 39 (2003), [4] A. Goubet-Bartholoeus, M. Dabrine, J. P. Richard, Stability of erturbed systes with tie-varying delays, Systes and Control Letters, 3 (997), [5] V. Kolanovskii, J. P. Richard, Stability of soe linear systes with delays, IEEE Transactions on Autoatic Control, 44 (999), [6] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear atrix inequalities in syste and control theory, SIAM Studies in Alied Matheatics, 5 (Philadelhia, USA) 994. [7] X. Li, C. de Souza, LMI aroach to delay deendent robust stability and stabilization of uncertain linear delay systes, Proceedings of 34th IEEE Conference on Decision and Control, New Orleans, Louisiana, USA, (995), [8] J. H. Su, The asytotic stability of linear autonoous systes with coensurate delays, IEEE Transactions on Autoatic Control, 40 (995), [9] S. I. Niculescu, J. M. Dion, L. Dugard, H. Li, Stability of linear systes with delayed state : An LMI aroach. JESA, secial issue on Analysis and control of tiedelay systes, 3 (997a), [20] S. I. Niculescu, J. P. Richard, Analysis and design of delay and roagation systes, IMA Journal of Matheatical Control and Inforation, 9(-2) (2002), [2] S. B. Stojanovic, D. Lj. Debeljkovic, Coents on Stability of Tie-Delay Systes, IEEE Transactions on Autoatic Control (2006). [22] E. Fridan, U. Shaked, An iroved stabilization ethod for linear tie-delay systes, IEEE Transactions on Autoatic Control, 47 (2002b), [23] M. Malek-Zavarei, M. Jashidi, Tie-Delay Systes, Analysis, Otiization and Alications, Systes and Control Series, 9 (987). [24] H. Gorecki, S. Fuksa, P. Grabovski, A. Korytowski, Analysis and synthesis of tie delay systes, John Wiley and Sons, Warszawa (989). [25] S. M. Karbassi, D. J. Bell, Paraetric tie-otial control of linear discrete-tie systes by state Feedback- Part 2 : Irregular Kronecker invariants.international journal of control, 57 (993), [26] S. M. Karbassi, D.J. Bell, New ethod of araetric eigenvalue assignent in state feedback control, IEE Process, 4 (994), [27] S. M. Karbassi, Tie-Otial Control of Disturbance-Rejection Tracking Systes for Discrete-Tie Tie- Delayed Systes by State Feedback, Couters and Matheatics with Alications 50 (2005), JIC eail for subscrition: ublishing@wau.org.uk