On the Zeros of Asymptotically Stable Serially Connected Structures

Transcription

1 43d IEEE Confeence on Decision and Contol Decembe 4-7, 24 Atlantis, Paadise Island, Bahamas WeC5 On the Zeos of Asymptotically Stable Seially Connected Stuctues Jaganath Chandasea, Jesse B Hoagg, and Dennis S Benstein Abstact A seially inteconnected N-mass system is consideed Specifically, we conside the single-input singleoutput (SISO) compliance fom the foce applied on any mass to the position of any mass he main esult of the pape is a esult showing that all SISO compliances of seially connected stuctues ae stictly minimum phase Lastly, we pesent a dynamic compensato that is constucted using genealized oot locus pinciples and the Fibonacci seies he compensato equies only limited nowledge of the plant and povides infinite upwad gain magin I INRODUCION It is well nown that one of the main impediments to achievable pefomance in linea time-invaiant contol systems is the pesence of nonminimum phase zeos, 2 Open ight half plane zeos cause peaing in the sensitivity function, and they limit gain magins fo obust stability Fo noise and vibation contol applications, stability obustness benefits fom senso/actuato colocation, although achievable pefomance can be impoved by sepaating the contol input fom the measuement signal 3 Fo colocated hadwae, it is well nown that the tansfe function is minimum phase; in fact, foce-to-velocity tansfe functions ae positive eal Howeve, fo the case of a noncolocated aangement of contol hadwae it is of inteest to now whethe the esulting tansfe function is minimum o nonminimum phase he ole of nonminimum phase zeos in limiting both achievable pefomance and obust stability suggests the impotance of undestanding the mechanisms that give ise to such zeos in flexible stuctues his issue is discussed in 4, whee it is shown that nonminimum phase zeos aise in noncolocated tansfe functions fo beam models when multiple mechanisms ae involved fo enegy tansfe, fo example, bending and tosion Futhemoe, it is shown in 5 that nonminimum phase zeos aise in noncolocated tansfe functions fo beam models when the dynamics ae dispesive, as occus in bending In the pesent pape we conside the existence of nonminimum phase zeos within the context of lumped stuctues Specifically, we conside mass-sping-dashpot his eseach was suppoted by the Ai Foce Office of Scientific Reseach unde gant F he authos ae with the Depatment of Aeospace Engineeing, he Univesity of Michigan, Ann Abo, MI , (734) , (734) (FAX), dsbaeo@umichedu u u 2 u N 2 N N+ m c c 2 m 2 c N m N c N+ q q 2 Fig N-Mass System systems with seial connections, that is, in the fom of a sting of masses inteconnected by spings and dashpots his stuctual configuation povides an appoximation fo a beam in compession, and is also useful in modeling the dynamics of a sting of vehicles with paiwise contol loops Fo this class of systems, we show that evey foceto-motion tansfe function between evey pai of masses is minimum phase, while the elative degee of each such tansfe function is a simple function of the numbe of intevening masses Since the elative degee of the noncolocated tansfe functions can be as lage as the numbe of masses, oot locus analysis shows that high-gain feedbac fo impoving stuctual esponse can be destabilizing We theefoe apply a high-gain contolle that equies only a bound on the elative degee his contolle, developed in 6, is an extension of the appoach of 7 to the case in which the elative degee is bounded but othewise unnown he contents of the pape ae as follows In Section 2, we pesent the dynamics fo a seially connected N-mass stuctue Section 3 pesents a simple fomula fo the elative degee of evey foce-to-position compliance tansfe function In Section 4, the zeos of the compliance ae shown to be stictly minimum phase A high-gain contolle that impoves stuctual esponse is pesented in Section 5 Ou conclusions ae given in Section 7 II N -MASS SYSEM Conside the N-mass system shown in Figue with foce inputs u,,u N Letq,,q N, denote the position of the mass m,,m N, espectively Fo all i,,n,themassm i and m i+ ae connected by a sping q n /4/$2 24 IEEE 2638

2 with stiffness i+ and a dampe with damping coefficient c i+ he equations of motion ae M q + C q + Kq u, (2) whee M R N N, C R N N, K R N N, q R N, and u R N ae given by K C m M m N , (22) N N + N+ c + c 2 c 2 c 2 c 2 + c 3 c 3 c 3 c 3 + c 4 c N c N + c N+,(23), (24) q q q N, u u u N (25) Futhemoe, (2) - (25) can be witten in the fist ode fom as ẋ Ax + Bu, (26) y C pos x, (27) whee A R 2N 2N, B R 2N N, C pos R N 2N,and x R 2N ae defined by A N I N N M K M, B C M, C pos I N N, (28) x q q N q q N We assume that M, K, andc ae positive definite Hence it can be shown that (26) is asymptotically stable (see 8) he compliance G comp (s) has the ealization A B G comp (s), (29) C pos with SISO enties G adm (s) G comp, (s) G comp,n (s) G compn, (s) G compn,n (s),(2) whee G comp (s) is the compliance fom the foce on mass m j to the position of mass m i Futhemoe, G comp (s) can be expessed as G comp (s) C pos,i (si A) B j, (2) whee C pos,i is the ith ow of C pos and B j is the jth column of B Poposition 2 Fo all i,,n, and j,,n, G comp (s) G compj,i (s) Poof Note that G comp (s) can be expessed as G comp (s) C pos (si A) B ( si + M C + ) s s M K M (22) ( Ms 2 + Cs + K ) It follows fom (29) that G comp(s) has the ealization A G Cpos comp(s) (23) B and hence G comp(s) B (si A ) Cpos C pos (si A) B (Ms 2 + Cs + K ) (24) It follows fom (22)-(24) that M, C, andk ae symmetic, and hence, it follows fom (22) and (24) that G comp (s) G comp(s), which implies that G comp (s) G compj,i (s) III RELAIVE DEGREE OF HE COMPLIANCE In this section, we analyze the elative degee of the compliance Without loss of geneality, we conside G compij (s) fo all i j Patitioning A n into fou N N matices yields whee A n A n A n A n c A n 2 A n 22, (3) RN N and A n R N N have enties A n A n c c,,n N, N,N c, c,n c N, c N,N, (32) Note that A I N and A c N Futhemoe, it follows fom (28) that A N, A 2 M K (33) A c I N, A 2 c M C he following esult is by obsevation Poposition 3 Fo all n 2,,N, andfoall j,,n n, j+n j+ c p, i j + n, m j+ m pj+2 p (34), i > j + n, 2639

3 and j+n c j+ m j+ pj+2 c p, i j + n, m p, i > j + n (35) Poposition 32 Fo all i,,n and j,,i, the elative degee of the compliance G comp (s) is given by i j +2 (36) Poof Let C pos,i A n B j fo all n < and C pos,i A B j, then the elative degee of G comp (s) is given by + It follows fom (28) and (32) that C pos,i A n B j an (37) m j Poposition 3 implies that C pos,i A n B j fo all n< i j + and C pos,i A n B j fo n i j + hus the elative degee is n +i j +2 Note that the elative degee min 2is minimum, when the senso and actuato ae colocated, that is, i j IV ZEROS OF HE COMPLIANCE Next, fo all G comp (s) with i j, define G comp (s) by G comp (s) G comp (s)(s + δ), (4) whee δ> It follows fom (29) that Gcomp (s) has the ealization A B j G comp (s), (42) C pos,i (A + δi) whee C pos,i A B j Poposition 32 implies that > Hence, Gcomp (s) is invetible with the ealization à inv G β comp (s) B j,(43) C pos,i (A + δi) whee Ãinv R 2N 2N is given by à inv A B j C pos,i (A + δi) (44) It follows fom (4)-(43) that ) ( ) spec (Ãinv zeos G compij (s) δ} (45) Since δ>, G compij (s) is stictly minimum phase if and only if Re (λ) < fo all λ spec(ãinv) Patitioning (A+δI) into fou N N matices yields (A + δi) A δ A δ 2 A A δ 22 (46) Expanding (A + δi) into a binomial seies yields (A + δi) h (,)A + h (,)δa (47) + h (, )δ A + h (, )δ I, whee fo all n,,, h (,n)! n!(n )! > Substituting (3) and (46) into (47) yields A δ h (,n)δ n A n, (48) Let A n δ n A h (,n)δ A n c (49) n and A n have enties A n δ A δc δ, δ,n δ N, δ N,N δc, δc,n δc N, δc N,N n, (4) Substituting (4) into (48) yields a δ p,q h (,n)δ n a n p,q, (4) a p,q h (,n)δ n a n c p,q, (42) n fo all p,,n,andq,,n Poposition 4 Fo all i,,n, and j,,n, G comp (s) is stictly minimum phase Poof Note that i j +2 and hence substituting p i and q j into (4) and using the fact that h (, ) h (,) yields a δ a + h (,)δa (43) + h (, )δ a + δ a, and a a + h (,)δa c (44) + h (, )δ a + δ a Case Let 2, which implies that i j Substituting i j into (43) and using (33) yields a 2 δ i,i a 2 i,i +2δa i,i + δ 2 a i,i a 2 i,i + δ 2, (45) a 2 i,i a 2 +2δa + δ 2 a a 2 +2δ 264

4 Hence, if δ> chosen sufficiently lage, then a 2 δ i,i > and a 2 i,i > Case 2 Let > 2, which implies that i > j In that case, it follows fom (34) and (35) (Poposition 3), that thee exists n i j + such that >, n n, >, n n, c, n < n, (46), n < n Substituting (46) into (43) yields a δ a a a + h (,)δa + h (, n )δ n a n, + h (,)δa c + h (, n )δ n a n (47) (48) Hence, if δ is chosen sufficiently lage, then a δ > and a > It follows fom Case and Case 2 that fo all i,,n and j,,i,ifδ> is chosen sufficiently lage, then a δ >, a > (49) Next, we analyze B j C pos,i (A+δI) It follows fom (28) that fo all p,,2n, andq,,2n, the (p, q)th enty of B j C pos,i R 2N 2N is given by, (p, q) (N + j, i) mj (B j C pos,i ) p,q, (p, q) (N + j, i) (42) Hence, it follows fom (42) and (46) that B j C pos,i (A + δi), (42) m j ˆK m j Ĉ whee fo all p,,n, ow p ( ˆK) ow i (A δ ), p j, (422), p j, ow p (Ĉ) ow i (A ), p j, (423), p j, which implies that the jth ow of ˆK R N N and Ĉ R N N is the only non-zeo ow Since, M in (22) is diagonal, B j C pos,i (A + δi) in (42) can also be expessed as B j C pos,i (A + δi) M ˆK M Ĉ (424) Hence, (4) and (422) imply that spec( ˆK) a δ } }, spec(ĉ) a } }(425) Substituting (424) into (44) yields I Ã inv M K M, (426) C whee K R N N and C R N N ae defined by K K + ˆK, C C + Ĉ (427) It follows fom (49), (422) and (425) that ˆK and Ĉ can be made positive semi-definite by choosing a lage δ > Futhemoe, since K and C ae positive definite and >, (427) implies that fo a sufficiently lage δ>, K and C ae positive definite Hence, it follows fom 8 that fo all λ spec(ãinv), Re(λ) <, and hence (45) implies that fo all i,,n,andj,,i, G comp (s) is stictly minimum phase V HIGH-GAIN DYNAMIC COMPENSAION In this section, we conside a high-gain stable dynamic compensato fo the single-input single-output compliance y i (t) G comp (s)u j (t), (5) whee y i (t) is the position of the ith mass and u j (t) is the foce on the jth mass Futhemoe, let δ s ± be the sign of the high-fequency gain, and let β be the magnitude of the high-fequency gain he esults of 8 and Poposition 4 implies that (5) is asymptotically stable and stictly minimum phase Nevetheless, active contol is fequently used on asymptotically stable stuctues to add damping and/o stiffness Unde this motivation, we conside a high-gain stable dynamic compensato that is constucted using genealized oot locus pinciples and the Fibonacci seies 6 he novel aspect of this constuction is that the compensato equies limited infomation of the tansfe function G comp (s) to yield the closed-loop system highgain stable We assume the following infomation (i) he magnitude of the high-fequency gain satisfies β b,wheeb is nown (ii) he sign of the high-fequency gain is nown (iii) he elative degee satisfies < ρ, wheeρ is nown Fo all j let F j be the jth Fibonacci numbe, whee F, F, F 2,F 3 2,F 4 3,F 5 5,F 6 8,F 7 3,F 8 2,, and define f ρ,h Fρ+2 F h+, (52) whee h satisfies h ρ Conside the input u j v u c with the feedbac u c Ĝ(s, )y i, (53) 264

5 and the stictly pope contolle δ s F ρ+2 ẑ(s) Ĝ(s, ) s ρ + fρ,ρ b ρs ρ + f, (54) ρ,ρ b ρ s ρ 2 + f ρ, b whee >, b,,b ρ ae eal numbes, and ẑ(s) is a degee ρ monic polynomial he closed-loop system is G(s, ) +Ĝ(s, )G comp (s) (55) he following two esults ae specializations of esults pesented in 6 heoem 5: Conside the closed-loop system (55) Assume that the polynomials ẑ(s), B ρ 2 (s) s 3 + b ρ s 2 + b ρ s + b, (56) and B i (s) b i+3 s 3 + b i+2 s 2 + b i+ s + b, (57) fo all i,,,ρ 3 ae Huwitz hen G(s, ) is high-gain stable fo all β (,b Futhemoe, as, 2N + ρ poles of the closed-loop system convege to the union of the oots of ẑ(s) and open-loop zeos he eal pats of the emaining + oots appoach Fo implementation puposes, it is desiable that the contolle Ĝ(s, ) be stable Poposition 5 Conside the contolle given by (54), and assume that ˆB(s) s ρ + b ρ s ρ + b ρ s ρ 2 + b 2 s + b (58) is Huwitz hen the contolle (54) is stable fo all > VI EXAMPLE Conside a seially connected 3-mass system as shown in Figue when N 3 Let G comp (s) be the single-input single-output compliance fom the foce of the ith mass to the position of the jth mass We assume that the magnitude of the high-fequency gain is positive and the uppe bound on the high fequency gain is b 4 Poposition 32 implies that the elative degee of G comp (s) must satisfy ρ 4 Now, we use the esults of heoem 5 to design a dynamic compensato fo G comp (s) that will yield the closed-loop high-gain stable Conside the dynamic compensato Ĝ(s, ) 8 ẑ(s) s b 4 s b 3 s b 2 s + 7, (6) b whee > and ẑ(s) is a degee 3 monic Huwitz Bode Diagam Fig 2 Bode plot of the open-loop system fo i and j Bode Diagam Fig 3 Bode plot of the closed-loop system fo i and j,and the gain 5 polynomial he contolle paametes ae chosen to be ẑ(s) (s +5)(s +4)(s +3), (62) b 5, b 2, b 3, b 4 5, (63) to satisfy the assumptions of heoem 5 and Poposition 5 Now, assume that the 3-mass system is given by I ẋ 3 M K M x + C M u, (64) y I 3 x, (65) whee M, C and K ae defined by (22)-(25) and x is defined by (28) with N 3 he masses ae m g, m 2 2g, and m 3 3g he stiffness coefficients ae 5N/m, 2 7N/m, and 3 6N/m he stuctue is lightly damped with the damping coefficients given by c 3Ns/m, c 2 6Ns/m, and c 3 2Ns/m Assume that G comp (s) is the compliance fom the foce of the fist mass to the position of the fist mass Hence, the elative degee 2 he Bode plot fo this tansfe function is shown in Figue 2 o impove pefomance, we implement the high-gain contolle (6)- (63) with the gain set at 5 he Bode plot of the closed-loop system is shown in Figue 3 Note that the addition of high-gain feedbac has attenuated the esonant peas Now, we assume that G comp (s) is the compli- 2642

6 Bode Diagam Fig 4 Bode plot of the open-loop system fo i 2and j Fig 7 Bode plot of the closed-loop system fo i 3and j,and the gain 5 Bode Diagam VII CONCLUSIONS Fig 5 Bode plot of the closed-loop system fo i 2and j,and the gain 6 In this pape, we examined a seially connected N-mass stuctue he elative degee of all SISO foceto-position tansfe functions was shown to be a simple function of the numbe of masses between the senso and actuato Futhemoe, we showed that all SISO foce-toposition tansfe functions ae stictly minimum phase Lastly, we apply a specially constucted contolle that povides infinite upwad gain magin hus high-gain feedbac can be used to impove stuctual esponse REFERENCES ance fom the foce of the second mass to the position of the fist mass so that the elative degee 3he Bode plot of the open-loop system is shown in Figue 4 As seen in Figue 5 the high-gain contol with 6 impoves pefomance Lastly, we assume that G comp (s) is the compliance fom the foce of the thid mass to the position of the fist mass so that the elative degee 4 he Bode plot of the open-loop system and the closed-loop system ae given in Figue 6 and Figue 7, espectively he gain 5 J C Doyle, B A Fancis, and A R annenbaum, Feedbac Contol heoy, Macmillan, New Yo, D S Benstein, What Maes Some Contol Poblems Had?, IEEE Cont Sys Mag, Vol 22, pp 8-9, 22 3 J Hong and D S Benstein, Bode Integal Constaints, Colocation, and Spillove in Active Noise and Vibation Contol, IEEE ans Cont Sys ech, vol 6, pp -2, E H Maslen, Positive Real Zeos in Flexible Beams, Shoc and Vibation, vol 2, pp , D K Miu, Mechatonics, Spinge-Velag, New Yo, J B Hoagg and D S Benstein, Discete-ime Adaptive Feedbac Distubance Rejection Using a Retospective Pefomance Measue, Poc ACIVE 4, Williamsbug, VA, Septembe 24 7 I Maeels, A simple selftuning contolle fo stably invetible systems, Syst and Cont Lettes, vol 4, pp 5-6, D S Benstein and S P Bhat, Lyapunov Stability, Semistability and Asymptotic Stability of Matix Second Ode Systems, ans of ASME, 5th Annivesay Design Issue, pp 45-53, Fig 6 Bode plot of the open-loop system fo i 3and j 2643