SIMPLE LOW-ORDER AND INTEGRAL-ACTION CONTROLLER SYNTHESIS FOR MIMO SYSTEMS WITH TIME DELAYS

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1 Appl. Comput. Math., V.10, N.2, 2011, pp SIMPLE LOW-ORDER AND INTEGRAL-ACTION CONTROLLER SYNTHESIS FOR MIMO SYSTEMS WITH TIME DELAYS A.N. GÜNDEŞ1, A.N. METE 2 Abtact. A imple finite-dimenional contolle ynthei method i developed fo ome clae of linea, time-invaiant, multi-input multi-output ytem that ae ubject to time delay. The popoed ynthei pocedue give low-ode contolle that achieve cloed-loop tability, with imple modification, alo achieve integal-action. Keywod: Stability, Delay Sytem, Integal-Action, Contolle Deign. AMS Subject Claification: 93D Intoduction A wide ange of dynamical phenomena cannot be modeled ufficiently accuately a finitedimenional linea time-invaiant (LTI) ytem due to being ubject to time delay. The effect of thee delay often cannot be ignoed and have to be included in the model [3]. Thi pape peent a tabilizing contolle ynthei method fo ome clae of linea, timeinvaiant (LTI), multi-input multi-output (MIMO) ytem that ae ubject to time delay. The popoed contolle ae imple finite-dimenional LTI contolle that achieve cloed-loop tability of the delayed plant. The contolle deign can be modified eaily o that thee contolle alo povide integal-action in ode to achieve aymptotic tacking of tep-input efeence with zeo teady-tate eo. Fo the plant clae conideed hee, the finite-dimenional pat of the plant may have any (finite) numbe of pole; while the pole in the table egion ae uneticted, thoe in the untable egion (cloed ight-half complex plane) have to be mall, i.e., cloe to the oigin. Thee ae no etiction on the numbe o location of the tanmiion-zeo of the finite-dimenional pat of the delayed plant. The contolle deigned fo thee plant ae low-ode, and in the SISO cae they ae eithe table (if integal-action i not equied in the deign) o have all pole in the open left-half complex plane except a pole at the oigin fo integal-action. The implicity of the deigned contolle explain why the plant clae have etiction on the pole in the egion of intability: thee plant ae in fact tongly tabilizable. Stability of delay ytem of etaded type and of neutal type wa tudied extenively and many delay-independent and delay-dependent tability eult ae available (ee [3], [6]). Mot tuning and intenal model contol technique ued in poce contol ytem apply to delay ytem [8]. The eult on obut contol of infinite-dimenional ytem apply to the ubcla of ytem with delay [1]. The poblem of exitence of popotional-deivative (PD) and popotional-integal-deivative (PID) contolle fo time delay ytem wa alo conideed and computational PID-tabilization method have been extended to cove cala, ingle-delay ytem (ee e.g., [5, 7]). Fo MIMO table plant and fo untable plant whoe finite-dimenional pat have no moe than two pole in the untable egion, PD and PID contolle ynthei method wee developed fo I/O delay, i.e., delay that affect the plant input and output (ee [4]). The ynthei method developed in thi pape allow abitay delay tem to affect 1 Depatment of Electical and Compute Engineeing, Univeity of Califonia, Davi, CA 95616, angunde@ucdavi.edu 2 Depatment of Electical and Electonic Engineeing, Mein Univeity, Mezitli/Mein 33343, Tukey Manucipt eceived 8 Septembe

2 A.N. GÜNDEŞ, A.N. METE: SIMPLE LOW-ORDER AND INTEGRAL-ACTION diffeent entie of the plant tanfe-matix. The popoed contolle deign applie unde cetain aumption on the pole of the finite-dimenional pat of the untable delayed plant: We ue the following tandad notation: Notation: Let C, R, R + denote complex, eal, and poitive eal numbe. The extended cloed ight-half complex plane i U = { C Re() 0} { }; R p denote eal pope ational function (of ); S R p i the table ubet with no pole in U; M(S) i the et of matice with entie in S. The pace H i the et of all bounded analytic function in C +. Fo h H, the nom i defined a h = e up C + h(), whee e up denote the eential upemum. A matix-valued function H i in M(H ) if all it entie ae in H ; in thi cae H = e up C + σ(h()), whee σ denote the maximum ingula value. Fom the induced L 2 gain point of view, a ytem whoe tanfe-matix i H i table iff H M(H ). A quae tanfe-matix H M(H ) i unimodula iff H 1 M(H ). We dop () in tanfematice uch a G() whee thi caue no confuion. Since all nom of inteet hee ae H nom, we dop the nom ubcipt, i.e.,. We ue copime factoization ove S ; i.e., fo G R m m p, G = D 1 N denote a left-copime-factoization (LCF), whee N, D S m m, det D( ) Poblem deciption Conide the feedback ytem Sy(G Λ, C) in Fig. 1, whee C R m m p i the tanfe-function of the contolle and G Λ i the tanfe-function of the plant with time delay. It i aumed that the feedback ytem i well-poed and that the delay-fee pat of the plant (i.e, the plant without the time delay tem) and the contolle have no untable hidden-mode. The finite-dimenional pat of the plant i denoted by G R m m p. v u e w C y G Λ Figue 1. The feedback ytem Sy(G Λ, C). Let G = D 1 N be an LCF of G. Suppoe that each ij-th enty N Λ ij of N Λ contain any abitay delay tem and that the delay ae known. We aume that G Λ can be witten a G Λ = D 1 N Λ, whee N Λ ij = e h ij N ij, i, j = 1,..., m. (1) If the finite-dimenional pat G of the delayed plant G Λ i table, then (1) implie that the entie of G Λ may contain all diffeent abitay known delay tem. If the finite-dimenional pat G of the delayed plant G Λ i not table, then we aume that the delayed plant tanfe-function G Λ ha etiction on it pole in the untable egion. In the ytem Sy(G Λ, C), let u, v, w, y denote the input and output vecto. The cloed-loop tanfe-matix Ĥ fom (u, v) to (w, y) i [ C(I + G Ĥ = Λ C) 1 C(I + G Λ C) 1 G Λ ] G Λ C(I + G Λ C) 1 (I + G Λ C) 1 G Λ. (2) Let the (input-eo) tanfe-function fom u to e be denoted by H eu and let the (input-output) tanfe-function fom u to y be denoted by H yu ; then H eu = (I + G Λ C) 1 = I G Λ C(I + G Λ C) 1 = I H yu. (3) Definition 1. a) The feedback ytem Sy(G Λ, C) hown in Fig. 1 i table if the cloed-loop map Ĥ i in M(H ). b) The contolle C tabilize G Λ if C i pope and Sy(G Λ, C) i table. c) The ytem Sy(G Λ, C) i table and ha integal-action if the cloed-loop tanfe-function fom (u, v) to (w, y) i table, and the (input-eo) tanfe-function H eu ha blocking-zeo at

3 244 APPL. COMPUT. MATH., V.10, N.2, 2011 = 0. d) The contolle C i aid to be an integal-action contolle if C tabilize G Λ and Y (0) = 0 fo any RCF C = XY 1. Let G = D 1 N be an LCF of the finite-dimenional pat of G Λ, and let G Λ = D 1 N Λ ; let C = XY 1 be an RCF, whee D, N, Y, X S m m, det D( ) 0, det Y ( ) 0. Then C tabilize G Λ if and only if M 1 M(H ), whee M := D Y + N Λ X. (4) Suppoe that the ytem Sy(G Λ, C) i table and that tep input efeence ae applied at u(t). Then the teady-tate eo e(t) due to tep input at u(t) goe to zeo a t if and only if H eu (0) = 0. Theefoe, by Definition 1-(c), the table ytem Sy(G Λ, C) achieve aymptotic tacking of contant efeence input with zeo teady-tate eo if and only if it ha integal-action. By (4), wite H eu = (I + G Λ C) 1 = Y M 1 D. Then by Definition 2-(d), Sy(G Λ, C) ha integal-action if C = XY 1 i an integal-action contolle ince Y (0) = 0 implie H eu (0) = (Y M 1 D)(0) = 0. Note that the ytem Sy(G Λ, C) would alo have integalaction if evey enty of the MIMO plant ha pole at = 0 ince D(0) = 0 implie H eu (0) = 0 even if the contolle Y (0) 0. Howeve, fo obut deign, integal-action i achieved with pole duplicating the dynamic tuctue of the exogenou ignal that the egulato ha to poce; thee integal-action contolle obey the well-known intenal model pinciple [2]. We aume thoughout that the finite-dimenional pat G ha no tanmiion-zeo at = 0. In fact, thi condition i a neceay condition fo exitence of integal-action contolle: Let G R m m p, with (nomal) ankg() = m. If G Λ admit an integal-action contolle, then G ha no tanmiion-zeo at = Contolle ynthei In thi ection, we popoe tabilizing contolle fo a cla of (MIMO a well a SISO) plant with time delay. Theoem 1 peent contolle ynthei fo cloed-loop tability. Coollay 1 include integal-action in the tabilizing contolle ynthei. Let G = D 1 N R m m p have full (nomal) ank m. Let G have no tanmiion zeo at = 0, equivalently, ankn(0) = m. Let p j U, j = 1,...,, denote the pole of G with non-negative eal pat (counting multiplicitie). Define := ( p j ). (5) Since the et { p j } include all U-pole of G, fo any α R +, i a laget invaiant- D 1 M(S). facto of the denominato D. Theefoe, G M(S), equivalently, Hence, G Λ M(H ). Theoem 1. (Stabilizing contolle ynthei): Let G Λ = D 1 N Λ be a in (1), with a in (5). Define Suppoe that Chooe α R + uch that and Φ := 1 [ N Λ () N(0) 1 I ]. (6) p j < Φ 1. (7) α p j fo j = 1,...,, (8) α < 1 ( Φ 1 p j ). (9)

4 A.N. GÜNDEŞ, A.N. METE: SIMPLE LOW-ORDER AND INTEGRAL-ACTION Then a contolle that tabilize G Λ i given by C = [ ( + α) ] N(0) 1 D(). (10) Poof of Theoem 1. Since the ode of the polynomial tem d := [ ( + α) ] i 1, the tem X := [ ] N(0) 1 M(S) (it i in fact tictly-pope and table). The tem Y := D 1 M(S) ince contain all U-pole of G. Theefoe, C = XY 1 i an RCF of the popoed contolle in (10). By (4), C tabilize G Λ if and only if M 1 M(H ), whee = M = D Y + N Λ X = ( + α) DD 1 + [ ( + α) ] ( + α) N Λ N(0) 1 ( + α) I + [ ( + α) ] ( + α) N Λ N(0) 1 = I + [ ( + α) ] ( + α) [N Λ N(0) 1 I ] = I + [ ( + α) ] [N Λ N(0) 1 I ] ( + α) = I + [ ( + α) ] ( + α) Φ = I + d ( + α) Φ. A ufficient condition fo M to be unimodula i that α > p j, the nom Since α atifie (9), d i: d ( + α) = unimodula and hence, C tabilize G Λ. We now pove (11): Fo all 1, [ ( + α) ( p j ) ] ( + α) = α + d d Φ Φ = (α + ( [ ] d Φ < 1. Unde the condition p j. (11) p j )Φ < 1. Theefoe, M i ) = = α + p j implie the nom in (11) i geate than o equal to the ight-hand ide. We pove the nom in (11) i le than o equal to the ight-hand ide by iteation: Fo = 1, p 1 R + and (11) hold ince (α+p 1) +α = α + p 1. Fo = 2, [(2α + p 1 + p 2 ) + α 2 p 1 p 2 ] ( + α) 2 = (2α + p 1 + p 2 ) ince α 2 p 1 p 2 and hence, (11) hold. Fo = 3, let p 3 R + ince at leat one of the thee U-pole ha to be eal. Then +α = 1 and 2 ( p j )] 2 [( + α) 3 3 ( p j )] ( + α) 3 = 1 fo α 2 p 1 p 2 imply that [( + α) ( p j )] ( p j )] + α [ ( + α) 2 + α + p 3 ( + α) 2 ] = (2α + p 1 + p 2 + α + p 3 ) and hence, (11) hold. Continuing imilaly, uppoe that (11) hold fo and how that it hold fo ( + 1): Cae (i): if at leat one of the U-pole of G i eal, let p (+1) R +. Then

5 246 APPL. COMPUT. MATH., V.10, N.2, 2011 ( p j )] = 1 fo α p j implie that [( + α) ( p j )] ( + α) +1 [( + α) ( p j )] + α [ ( + α) + α + p (+1) = (α + p j + α + p (+1) ) ( p j )] ( + α) ] 1 ( p j )] and hence, (11) hold. Cae (ii): if none of the U-pole of G i eal, let p (+1) = p be the complex- conjugate pai of the pole p U. Then = 1 and 1 fo α p j imply that [( + α) ( p j )] ( + α) +1 α(2+α) 1 p p ( p j ) [( + α) ( 1) 1 ( p j )] + α [ ( + α) ( + α) 1 α(2 + α)( + α) 1 1 p p ( p j ) ( p j )] + ( + α) + (p + p ) + α ( + α) 1 ] 1 = ( [ 1]α + p j + 2α + p + p (+1) ) = 2α and hence, (11) hold. Coollay 1. (Integal-action contolle ynthei): Unde the aumption of Theoem 1, chooe α R + atifying (8) and α < ( Φ 1 Then an integal-action contolle that tabilize G Λ i given by C i = [ ( + α)+1 ] p j ). (12) N(0) 1 D(). (13) Poof of Coollay 1. With X := [ +1 ] N(0) 1 M(S), and Y := D M(S), C i = XY 1 i an RCF of the popoed contolle in (13). By (4), C i tabilize G Λ if and only if M 1 M(H ), whee M = D Y + N Λ X = ( + α) +1 DD 1 + [ ( + α)+1 ] ( + α) +1 N Λ N(0) 1 = I + [ ( + α)+1 ] [N Λ N(0) 1 I ] ( + α) +1 = I + [ ( + α)+1 ] ( + α) +1 Φ = I + d ( + α) Φ,

6 A.N. GÜNDEŞ, A.N. METE: SIMPLE LOW-ORDER AND INTEGRAL-ACTION whee d := [ ( + α) +1 ]. Unde the condition α > p j, the nom Since α atifie (12), d [ ( + α) +1 ( p j ) ] ( + α) +1 = ( + α) +1 ( + 1)α + d Φ = [( + 1)α + +1 d +1 i: p j. (14) p j ]Φ < 1. Theefoe, M i unimodula and hence, C i tabilize G Λ. Remak 1. a) The contolle (10) popoed in Theoem 1 i tictly-pope ince the polynomial tem d := [( + α) ] i of ode 1. Similaly, the integal-action contolle (13) popoed in Coollay 1 i tictly-pope. b) In the cae that G Λ i SISO, fo any -th ode tictly-huwitz polynomial θ(), the finite-dimenional pat of G Λ can be witten a G = η γ = ( θ ) 1 ( η γ θ ) = D 1 N, (15) whee γ i a tictly-huwitz polynomial of degee δ(γ), whoe oot ae the (finitely many) pole of G in the table egion C \ U. The ode of G i [ + δ(γ)], and the numeato polynomial η i of ode [ + δ(γ)]. Theefoe, fo the SISO cae, the contolle in (10) become C = [ ( + α) ] N(0) 1 D() = γ(0)θ(0)η(0) 1 [ ( + α) ] θ(), (16) which i a table contolle of ode, the ame a the numbe of U-pole of G. Theefoe, C in (16) tongly tabilize SISO plant G Λ. Fo the cae of SISO plant with time delay, the integal-action contolle in (13) become C i = γ(0)θ(0)η(0) 1 [ ( + α)+1 ] θ(). (17) The contolle in (17) i of ode + 1, and ha pole in the table egion C \ U except fo the pole at = 0 poviding integal-action. c) A in the SISO cae, wheneve the finite-dimenional pat G of the MIMO plant G Λ i uch that an LCF G = D 1 N i given by D = θ I, N = θ G, i.e., ank ( θ G) =pj = m fo each U-pole p j of G, then the contolle in (10) become C = [ ( + α) ] N(0) 1 D() = [ ( + α) ] θ() N(0) 1, (18) which i a table contolle, whoe pole ae the oot of the abitaily choen tictly- Huwitz polynomial θ(). Delayed plant G Λ in thi cla ae theefoe tongly tabilized by the contolle in (18). d) Any numbe of the U-pole of G may be at the oigin = 0. In fact, if p j = 0 fo j = 1,...,, then condition (7) i obviouly atified ince 0 < Φ 1. In thi cae, chooe α R + in (9) uch that α < 1 Φ 1. e) In the MIMO cae, the finite-dimenional pat G of G Λ may have coinciding pole and zeo (in the untable egion U a well a the table egion C \ U of the complex plane). The only aumption i that G ha no zeo at = 0. f) If the finite-dimenional pat G of G Λ ha only one U-pole (although it may obviouly have any numbe of pole in the open left-half complex plane), then the contolle (10) and the

7 248 APPL. COMPUT. MATH., V.10, N.2, 2011 integal-action contolle in (13) become C = [ ( + α) ( p 1) ] p 1 N(0) 1 D() = ( α + p 1 ) p 1 N(0) 1 D(), (19) C i = [ (2α + p 1) + α 2 ] ( p 1 ) N(0) 1 D(). (20) Following Remak-(b), if G can be factoized a in (15), i.e., D = ( p 1) θ I, whee θ i a fit-ode tictly-huwitz polynomial, then the contolle (19) and (20) become PD and PID contolle, epectively: C = ( α + p 1 ) θ C i = [ (2α + p 1) + α 2 ] θ 4. Concluion N(0) 1, (21) N(0) 1. (22) The main eult (Theoem 1) peented a imple LTI contolle ynthei method to achieve cloed-loop tability fo ome clae of LTI, MIMO plant that ae ubject to time delay. The pocedue i modified in Coollay 1 to include integal-action in the contolle deign o that aymptotic tacking of tep-input efeence with zeo teady-tate eo i alo achieved in addition to cloed-loop tability. The eult apply to plant whoe finite-dimenional pat ha etiction on the pole in the untable egion of the complex plane. Pefomance implication fo choice within the contolle paamete can alo be exploed fo pecific application of the ynthei method peented hee. Refeence [1] Foia, C.C., Özbay, H., Tannenbaum, A. Robut contol of infinite dimenional ytem, LNCIS 209, Spinge-Velag, London, [2] Fanci, B.A., Wonham, W.A. The intenal model pinciple fo linea multivaiable egulato, Applied Mathematic & Optimization, V.2, N.2, 1975, pp [3] Gu, K., Khaitonov, V.L., Chen, J. Stability of time-delay ytem, Bikhäue, Boton, [4] Gündeş, A.N., Özbay, H., Özgüle, A.B. PID contolle ynthei fo a cla of untable MIMO plant with I/O delay, Automatica, V.43, N.1, 2007, pp [5] Lain, V.B. On tabilization of ytem with delay, Intenational Applied Mechanic, V.44, N.10, 2008, pp [6] Niculecu, S.-I. Delay effect on tability: a obut contol appoach, Spinge-Velag: Heidelbeg, LNCIS, V.269, [7] Silva, G.J., Datta, A., Bhattachayya, S.P. PID contolle fo time-delay ytem, Bikhäue, Boton, [8] Skogetad, S. Simple analytic ule fo model eduction and PID contolle tuning, Jounal of Poce Contol, V.13, 2003, pp

8 A.N. GÜNDEŞ, A.N. METE: SIMPLE LOW-ORDER AND INTEGRAL-ACTION A.N. Gündeş - i a pofeo in the Electical & Compute Engineeing Depatment and a membe of the Gaduate Pogam in Mechanical & Aeopace Engineeing and a membe of the Gaduate Goup in Applied Mathematic at the Univeity of Califonia, Davi. She eceived the BS, MS, PhD (1988) degee in EECS fom the Univeity of Califonia at Bekeley. A.N. Mete - eceived the B.S. degee (2000) in Electical and Electonic Engineeing fom Univeity of Gaziantep, Tukey, the M.S. (2004) and Ph.D. (2009) degee in Electical and Compute Engineeing fom Univeity of Califonia, Davi. Cuently he i a lectue in the Depatment of Electical and Electonic Engineeing at Mein Univeity. Hi eeach inteet include eliable contol and decentalized contol.