Design of controllers for time delay systems: Integrating and unstable systems

6 Design o Con olle s o Time Delay Sys ems: In eg a ing and Uns able Sys ems Pe Dos ál, F an išek Gazdoš, and Vladimí Bobál Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín Nad S áněmi 4511, 760 05 Zlín 5, Czech Republic 1. In oduc ion The p esence o a ime delay is a common p ope y o many echnological p ocesses. In addi ion, a pa o ime delay sys ems can be uns able o ha e in eg a ing p ope ies. Typical examples o such p ocesses a e e.g. pumps, liquid s o ing anks, dis illa ion columns o some ypes o chemical eac o s. Plan s wi h a ime delay o en canno be con olled by usual con olle s designed wi hou conside a ion o he dead- ime. The e a e a ious ways o con ol such sys ems. A numbe o me hods u ilise PI o PID con olle s in he classical eedback closed-loop s uc u e, e.g. (Pa k e al., 1998; Zhang and Xu, 1999; Wang and Clue , 1997; Sil a e al., 2005). O he me hods employ ideas o he IMC (Tan e al., 2003) o obus con ol (P okop and Co iou, 1997). Con ol esul s o a good quali y can be achie ed by modi ied Smi h p edic o me hods, e.g. (Ås öm e al., 1994; De Pao , 1985; Liu e al., 2005; Majhi and A he on, 1999; and Ma ausek and Micic, 1996). P inciples o he me hods used in his wo k and design p ocedu es in he 1DOF and 2DOF con ol sys em s uc u es can be ound in pape s o au ho s o his a icle (Dos ál e al., 2001; Dos ál e al., 2002). The con ol sys em s uc u e wi h wo eedback con olle s is conside ed (Dos ál e al., 2007; Dos ál e al., 2008). The p ocedu e o ob aining con olle s is based on he ime delay i s o de Padé app oxima ion and on he polynomial app oach (Kuče a, 1993). Fo uning o he con olle pa ame e s, he pole assignmen me hod exploi ing he LQ con ol echnique is used (Hun e al., 1993). The esul ing p ope and s able con olle s ob ained ia polynomial Diophan ine equa ions and spec al ac o iza ion echniques ensu e asymp o ic acking o s ep e e ences as well as s ep dis u bances a enua ion. S uc u es o de eloped con olle s oge he wi h analy ically de i ed o mulas o compu a ion o hei pa ame e s a e p esen ed o i e ypical plan ypes o in eg a ing and uns able ime delay sys ems: an in eg a ing ime delay sys em (ITDS), an uns able i s o de ime delay sys em (UFOTDS), an uns able second o de ime delay sys em (USOTDS), a s able i s o de plus in eg a ing ime delay sys em (SFOPITDS) and an uns able plus in eg a ing ime delay sys em (UFOPITDS). P esen ed simula ion esul s documen use ulness o he p oposed me hod p o iding s able con ol esponses o a good quali y also o a highe a io be ween he ime delay and uns able ime cons an s o he con olled sys em. www.in echopen.com Time-Delay Sys ems 114 2. App oxima e ans e unc ions The ans e unc ions in he sequence ITDS, UFOTDS, USOTDS, SFOPITDS and UFOPITDS ha e hese o ms: 1() ds K Gs e s τ − = (1) 2() 1ds K Gs e s τ τ − =− (2) 312 () (1)(1) ds K Gs e ss τ ττ − =−+ (3) 4,5() (1) ds K Gs e ss τ τ − =±. (4) Using he i s o de Padé app oxima ion, he ime delay e m in (1) – (4) is app oxima ed by 2 2 dsd d s es τ τ τ −− ≈+. (5) Then, he app oxima e ans e unc ions ake o ms 01 121 (2 ) () (2 ) d A d Ksbbs Gsss sas τ τ −− == ++ (6) whe e 02 d K b τ = , 1 bK = and 12 d a τ = o he ITDS, 01 2210 (2 ) () (1)(2 ) d A d Ks bbs Gs ss sasa τ ττ −− == −+ + + (7) wi h 02 d K b τ τ =,1K b τ = , 02 d a τ τ =− , 12d d a τ τ ττ − = and τd ≠ 2τ o he UFOTDS, 312 (2 ) () (1)(1)(2) d A d Ks Gs ss s τ ττ τ − =−++ 01 32 210 bbs sasasa − = + +− (8) whe e 012 2 d K b τ ττ =, 112 K b τ τ =, 012 2 d a τ ττ =, 12 112 2( ) d d a τ ττ τττ −− =, 12 1 2 212 2dd d a τ τττττ τττ +− = and τd ≠ 2τ1 o he USOTDS, and, 01 4,5 32 21 (2 ) () (1)(2 ) d A d Ks bbs Gs ss s sasas τ ττ −− == ±+ ++ (9) www.in echopen.com Design o Con olle s o Time Delay Sys ems: In eg a ing and Uns able Sys ems 115 whe e 02 d K b τ τ =, 1K b τ = , 12 d a τ τ =± , 22d d a τ τ ττ ± = and τd ≠ 2τ o he SFOPITDS and UFOPTDS, espec i ely. All app oxima e ans e unc ions (6) – (9) a e s ic ly p ope ans e unc ions () () () A bs Gs as = (10) whe e b and a a e cop ime polynomials in s ha ul ill he inequali y de g de g ba<. The polynomial a(s) in hei denomina o s can be exp essed as a p oduc o he s able and uns able pa () () ()as asas +− = (11) so ha o ITDS, UFOTDS, USOTDS and SFOPITDS he equali y de g de g 1aa + = − (12) is ul illed. 3. Con ol sys em desc ip ion The con ol sys em wi h wo eedback con olle s is depic ed in Fig. 1. In he scheme, w is he e e ence, is he load dis u bance, e is he acking e o , u0 is he con olle ou pu , y is he con olled ou pu , u is he con ol inpu and GA ep esen s one o he app oxima e ans e unc ions (6) – (9) in he gene al o m (10). Rema k: He e, he app oxima e ans e unc ion GA is used only o a con olle de i a ion. Fo con ol simula ions, he models G1 – G5 a e u ilized. Bo h w and a e conside ed o be s ep unc ions wi h Laplace ans o ms 0 () w Ws s =, 0 () Vs s =. (13) The ans e unc ions o con olle s a e assumed as () () () q s Qs p s =# #, () () () s Rs p s =# (14) whe e ,andq p ## a e polynomials in s. - - y u u 0 e w R Q G A Fig. 1. The con ol sys em. www.in echopen.com Time-Delay Sys ems 116 4. Applica ion o he polynomial me hod The con olle design desc ibed in his sec ion ollows he polynomial app oach. Gene al equi emen s on he con ol sys em a e o mula ed as i s in e nal p ope ness and s ong s abili y (in addi ion o he con ol sys em s abili y, also he con olle s abili y is equi ed), asymp o ic acking o he e e ence and load dis u bance a enua ion. The p ocedu e o de i e admissible con olle s can be pe o med as ollows: T ans o ms o basic signals in he closed-loop sys em om Fig.1 ake ollowing o ms ( o simpli ica ion, he a gumen s is in some equa ions omi ed) () () () b Ys Ws pVs d =+ ⎡ ⎤ ⎣ ⎦ # (15) 1 () ( ) () ()Es ap bqWs bpVs d =+ − ⎡ ⎤ ⎣ ⎦ ## # (16) () () () a Us Ws pVs d =+ ⎡ ⎤ ⎣ ⎦ # (17) whe e () ()() () () ()ds asps bs s qs=+ + ⎡ ⎤ ⎣ ⎦ ## (18) is he cha ac e is ic polynomial wi h oo s as poles o he closed-loop. Es ablishing he polynomial as () () () s s qs = +# (19) and subs i u ing (19) in o (18), he condi ion o he con ol sys em s abili y is ensu ed when polynomials p # and a e gi en by a solu ion o he polynomial Diophan ine equa ion ()() ()() ()asps bs s ds + = # (20) wi h a s able polynomial d on he igh side. Wi h ega d o ans o ms (13), he asymp o ic acking and load dis u bance a enua ion a e p o ided by di isibili y o bo h e ms ap bq + ## and p # in (16) by s. This condi ion is ul illed o polynomials p #and q #ha ing o ms () () p ssps = #, () ()qs sqs = #. (21) Subsequen ly, he ans e unc ions (14) ake o ms () () () q s Qs p s =, () () () s Rs s p s = (22) and, a s able polynomial p(s) in hei denomina o s ensu es he s abili y o con olle s ( he s ong s abili y o he con ol sys em). The con ol sys em sa is ies he condi ion o in e nal p ope ness when he ans e unc ions o all i s componen s a e p ope . Consequen ly, he deg ees o polynomials q and mus ul il hese inequali ies www.in echopen.com Design o Con olle s o Time Delay Sys ems: In eg a ing and Uns able Sys ems 117 de g de g qp ≤ , de g de g 1 p ≤ +. (23) Now, he polynomial can be ew i en o he o m () () () s s sqs=+ . (24) Taking in o accoun sol abili y o (20) and condi ions (23), he deg ees o polynomials in (19) and (20) can be easily de i ed as de g de g de g a==, de g de g 1qa = −, de g de g 1pa≥−, de g 2de g da≥. (25) Deno ing deg a = n, polynomials , and q ha e o ms 0 () ni i i s s = = ∑ , 0 () ni i i s s = = ∑ , 1 1 () ni i i qs qs − = =∑ (26) and, ela ions among hei coe icien s a e 00 = , iii q + = o 1,... ,in = (27) Since by a solu ion o he polynomial equa ion (20) only coe icien s i can be calcula ed, unknown coe icien s i and qi can be ob ained by a choice o selec able coe icien s 0,1 i γ ∈ such ha iii γ = , (1 ) iii q γ = − o 1,... ,in = . (28) The coe icien s γi di ide a weigh be ween nume a o s o ans e unc ions Q and R. Rema k: I 1 i γ = o all i, he con ol sys em in Fig. 1 educes o he 1DOF con ol con igu a ion (Q = 0). I 0 i γ = o all i, and, bo h e e ence and load dis u bance a e s ep unc ions, he con ol sys em co esponds o he 2DOF con ol con igu a ion. The con olle pa ame e s hen esul om solu ions o he polynomial equa ion (20) and depend upon coe icien s o he polynomial d. The nex p oblem he e is o ind a s able polynomial d ha enables o ob ain accep able s abilizing and s able con olle s. 5. Pole assignmen The polynomial d is conside ed as a p oduc o wo s able polynomials g and m in he o m () () ()ds gsms= (29) whe e he polynomial g is a monic o m o he polynomial g ′ ob ained by he spec al ac o iza ion () () ()() () ()sa s sa s b s b s g s g s ϕ ∗∗∗ ′ ′ += ⎡⎤⎡⎤ ⎣⎦⎣⎦ (30) whe e ϕ > 0 is he weigh ing coe icien . Rema k: In he LQ con ol heo y, he polynomial g ′ esul s om minimiza ion o he quad a ic cos unc ion www.in echopen.com Time-Delay Sys ems 118 {} 22 0 () ()Je u d ϕ ∞ =+ ∫$ (31) whe e ()e is he acking e o and ()u $is he con ol inpu de i a i e. The second polynomial m ensu ing p ope ness o con olle s is gi en as 2 () () d ms a s s τ + ==+ (32) o bo h ITDS and UFOTDS, 2 21 () () d ms a s s s τ τ +⎛⎞ ⎛⎞ ==+ + ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ (33) o he USOTDS, and, 21 () d ms s s τ τ ⎛⎞ ⎛⎞ =+ + ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ . (34) o bo h UFOPITDS and SFOPITDS. The coe icien s o he polynomial d include only a single selec able pa ame e ϕ and all o he coe icien s a e gi en by pa ame e s o polynomials b and a. Consequen ly, he closed loop poles loca ion can be a ec ed by a single selec able pa ame e . As known, he closed loop poles loca ion de e mines bo h s ep e e ence and s ep load dis u bance esponses. Howe e , wi h espec o he ans o m (13), i may be expec ed ha weigh ing coe icien s γ in luence only s ep e e ence esponses. Then, he monic polynomial g and de i ed o mulas o hei pa ame e s ha e o ms 32 210 () g ss g s g s g = +++ (35) o bo h ITDS and UFOTDS, whe e 2 01221 2 21 14 2 4 ,, dd d KK gggKgg τϕ ϕτ ϕ τ ⎛⎞ ==+=+ ⎜⎟ ⎜⎟ ⎝⎠ (36) o he ITDS, and, 22 01 2 22 2 2 21 211 1 1 ,4 1, 11 24 dd dd dd d K ggKgK gg ττ τ τ τττϕ ϕ ϕ ττ τ τ ττ ϕ ⎛⎞ == ++ ⎜⎟ ⎝⎠ =++ (37) o he UFOTDS, and, 432 3210 ()gs s gs g s gs g = ++++ (38) www.in echopen.com Design o Con olle s o Time Delay Sys ems: In eg a ing and Uns able Sys ems 119 o USOTDS, SFOPITDS and UFOPITDS, whe e 2 012 22 222 12 12 12 12 22 2 12 213 3 2 222 2 2 2 12 12 1 2 21 4 1 4 , 4( ) 241 2411 , dd d d ddd KKK ggg K ggg g g τττ τττ ϕ ϕτττττ ττ τ ϕτττϕ τ ττ ϕ τ τ τ ⎛⎞ ==++ ⎜⎟ ⎜⎟ ⎝⎠ ++ =− + =+++ (39) o he USOTDS, and, 2 012 2 213 3 2 22 24 11 ,, 41 1 2 41 2, dd dd d KK K ggg ggg K g g ττ ϕ ττ ϕτ ττ ττ ϕ ϕττ ⎛⎞ ==+ ⎜⎟ ⎜⎟ ⎝⎠ ⎛⎞ =+ − =++ ⎜⎟ ⎜⎟ ⎝⎠ (40) o bo h SFOPITDS ans UFOPITDS. The ans e unc ions o con olle s a e 21 0 () qs q Qs sp + =+, 2 210 0 () () s s Rs ss p + + =+ (41) o bo h ITDS and UFOTDS, and, 232 321 3210 22 10 10 () , () () qs qs q s s s Qs Rs spsp sspsp + ++++ == ++ ++ (42) o he USOTDS, SFOPITDS and UFOPITDS. 6. Con olle pa ame e s Fo he sake o limi ed space, o mulas de i ed om (20) o all conside ed sys ems oge he wi h condi ions o he con olle s’ s abili y a e in oduced in he o m o ables. Pa ame e s i and qi in (41) and (42) can hen be calcula ed om i acco ding o (28). 02 1 0 (2 ) 4 dd p ggg ττ =+ + , 00 1 g K = 110 1() d gg K τ =+ , 210 (2 ) 4 dd gg K ττ =+ p0 > 0 o all τd Table 1. Con olle pa ame e s o he ITDS www.in echopen.com Time-Delay Sys ems 120 210 0 2( )2 2 2 d d d ggg p τ ττ ττ ⎡⎤ + ++ ⎢⎥ ⎣⎦ =− 00 g K τ =, 1 010 1() d pgg K ττ =++ ⎡ ⎤ ⎣ ⎦, 202 1()1 pg K τ = −− ⎡ ⎤ ⎣ ⎦ p0 > 0 o τd < 2τ Table 2. Con olle pa ame e s o he UFOTDS 31 2 1 0 1 01 2 22( ) 2 2 d d d gggg p τ ττ τ ττ ⎡⎤ ++++ ⎢⎥ ⎣⎦ =−, 131 1 pg τ =+ 1 00 g K τ =, () 101120 1() d pg g K τττ ⎡ ⎤ =+++ ⎣ ⎦ 12 2 2120312121 1 14 4 1 1 dd pggg K ττ τ ττ τ ττ τττ ⎡ ⎤ ⎡ ⎤ ⎛⎞⎛⎞ ⎛⎞ ⎢ ⎥ =+−−+++− ⎢ ⎥ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎢ ⎥ ⎢ ⎥ ⎝⎠ ⎝⎠⎝⎠ ⎣ ⎦ ⎣ ⎦ 2 31023 1 1 () pgg K ττ τ ⎡ ⎤ =−−− ⎢ ⎥ ⎣ ⎦ p1 > 0 o all τd , p0 > 0 o τd < 2τ1 Table 3. Con olle pa ame e s o he USOTDS 02 1 0 (2 ) 4 dd pg g g ττ =+ + , 13 1 pg τ = + 00 1 g K =, 11 0 1() d g g K ττ =++ ⎡ ⎤ ⎣ ⎦, 2100 1(2 )(2 ) 2 4ddd gg g K ττ τ ττ =+++ ⎡ ⎤ ⎣ ⎦ 310 2 4 dd gg K ττ τ =+ ⎡ ⎤ ⎣ ⎦ p1, p0> 0 o all τd Table 4. Con olle pa ame e s o he SFOPITDS 2 3210 0 4 4(2 ) 24 2 dd d d gggg p ττ ττ τ ττ ⎛⎞ + ++++ ⎜⎟ ⎜⎟ ⎝⎠ =−, 13 2 pg τ = + 00 1 g K =, 11 0 1() d g g K ττ =++ ⎡ ⎤ ⎣ ⎦, 20321 44 18 8 11 ddd d pggg K ττ τ τ ττ ττ ⎡⎤ ⎛⎞ ⎛⎞ =−−−+−− ⎢⎜⎟ ⎜⎟ ⎥ ⎜⎟ ⎜⎟ ⎢⎝⎠ ⎝⎠ ⎦ ⎣, 3023 12 ()2 pgg K τ τ ⎡ ⎤ =−−− ⎢ ⎥ ⎣ ⎦ p1 > 0 o all τd , p0 > 0 o τd < 2τ Table 5. Con olle pa ame e s o he UFOPITDS www.in echopen.com Design o Con olle s o Time Delay Sys ems: In eg a ing and Uns able Sys ems 121 7. Simula ion esul s The simula ions we e pe o med by MATLAB-Simulink ools. Fo all simula ions, he uni s ep e e ence w was in oduced a he ime = 0 and he s ep load dis u bance a e se ling o he s ep e e ence esponses. 7.1 ITDS In he ans e unc ion (1), le K = 1. The esponses in Fig. 2 o τd = 5 show he e ec o ϕ upon he con ol quali y. An inc easing alue ϕ imp o es con ol s abili y, and, by choosing i s alue highe , ape iodic esponses can be ob ained. Simula ion esul s shown in Fig. 3 demons a e he in luence o pa ame e s γ on he con ol esponses. Thei smalle alues accele a e s ep e e ence esponses bu hey do no a ec load dis u bance esponses. Highe alues o γ can lead o o e shoo s and oscilla ions. The e ec o pa ame e s γ on he con ol 0 20 40 60 80 100 120 140 160 180 0.0 0.2 0.4 0.6 0.8 1.0 1.2 y( ) Time ϕ = 100 ϕ = 400 ϕ = 900 w Fig. 2. ITDS: con olled ou pu esponses (τd = 5, = - 0.1, γ1 = γ2 = 0) 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 y( ) Time γ1 = γ2 = 0 γ1 = γ2 = 0.25 γ1 = γ2 = 0.4 w Fig. 3. ITDS: con olled ou pu esponse (τd = 5, = - 0.1, ϕ = 900). 0 50 100 150 200 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 y( ), u( ) Time γ1 = γ2 = 0 γ1 = γ2 = 0.4 y( ) 5xu( )w Fig. 4. ITDS: Con ol inpu and con olled ou pu esponses (τd = 5, ϕ = 900) www.in echopen.com