Controllers. Yale University, New Haven, CT Nagoya Works, Japan

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1 Servisor Control of Families of Noise Sressing Controllers Shogo Fjii Jo~ao P. Hesanha z A. Stehen Morse z shogo.fjii@ale.ed joao.hesanha@ale.ed morse@ssc.eng.ale.ed Det. of Electrical Engineering Yale Universit, New Haven, CT z Nion Steel Cororation Nagoa Works, Jaan Abstract This aer describes a high-level \servisor" caable of switching into feedback with a single-int/singleott (SISO), linear time-invariant (LTI) rocess a seqence of linear controllers, so as to sress from the ott of the rocess the eect of a ersistent distrbance int consisting of the sm of a nite nmber of sinsoids of nknown amlitde, hase, and freqenc. Each of the controllers being servised is designed so as to solve the distrbance rejection roblem for a secic set of distrbance freqencies. The stabilit reslts stated in this aer are valid when the freqencies of the distrbance sinsoids are known to belong to a - nite set. 1 Introdction It is not ncommon for the int to a rocess to be corrted b a ersistent signal which cannot be measred. This aer deals with this sitation. Or goal is to asmtoticall sress from the ott of a SISO, LTI rocess the eect of a noise signal or distrbance d which is added to its int. We assme the distrbance is eqal to the sm a nite nmber of sinsoids of nknown amlitde, hase, and freqenc. Control algorithms to solve the above roblem have a wide range of alications in active noise and vibration control in which the distrbance d is ticall de to rotating machiner. Tical alications inclde noise cancellation in atomobile engines [1], trboro aircraft engines [2], and ventilation sstems [3]. In sch alications, the freqenc and amlitde of the distrbance ma var de to changes in the oerating conditions of the sstems that generate it. We are artic- This research was sorted b the Air Force Oce of Scientic Research, the Arm Research Oce, and the National Science Fondation. larl interested in sitations in which ractical considerations revents the addition of sensors caable of accratel measring the distrbance. Were the freqencies of the sinsoids in d known, we cold design a linear time invariant controller based on the internal model rincile to asmtoticall reject the eect of the distrbance d from the ott of the rocess [4]. Since the freqencies are not resmed known, we roose a servisor control sstem which orchestrates the switching among a famil of candidate controllers, each designed for a secic set of distrbance freqencies, so as to reject the eect of d from. Following certaint eqivalence, controller selection is based on the crrent estimate of what the freqencies of d. The algorithm roosed is insired b [5, 6, 7]. The roblem of sressing from the ott of a rocess the eect of a distrbance eqal to the sm a - nite nmber of sinsoids of nknown amlitde, hase, and freqenc, is addressed in [8, 9, 1, 11]. In [8] a model reference adative algorithm is roosed to solve this roblem with an nknown rocess model. To this eect the famil of model reference controllers is over-arameterized to allow for distrbance cancellation. Althogh in [8] the roblem is solved onl for onedimensional rocesses, the aroach seems to be generalizable to higher dimensional sstems which are minimm hase. In [9] an adative control law based on the Yola arameterization [12] is roosed to solve the roblem considered here. The reslting closed loo sstem is shown to be locall exonentiall stable. In [1] two adative algorithms based on exact cancellation of a distrbance consisting of a single sinsoid are considered. These algorithms make se of online estimation of distrbance's amlitde, hase, and freqenc. The analsis given is aroximate and onl valid when the initial estimates are close to their tre vales. The remaining of this aer is organized as follows. The roblem addressed here is formalized in Section 2 In Proc. of the 37th Conf. on Decision and Contr., Dec

2 and Section 3 describes the servisor control sstem roosed. The main reslt of the aer namel, that the roosed control law is caable of asmtoticall sressing from the ott of the rocess the eect of the distrbance is stated and roved in Section 4. Finall Section 5 contains some conclding remarks and direction for ftre research. 2 Problem The roblem of interest is to constrct a control sstem caable of asmtoticall sressing from the ott of a rocess P the eect of a ersistent distrbance int d consisting of the sm of a nite nmber of sinsoids of nknown amlitde, hase, and freqenc. The rocess is resmed to admit a SISO, LTI model with strictl roer transfer fnction P (s) and realization _x P = A P x P b P ( d); = c P x P (1) where denotes the measred ott and the control signal. It is assmed that d consists of the sm of m sinsoids whose freqencies! (1) ;! (2) ; : : :;! (m) all lie within a re-secied closed, bonded sbset on the ositive real line. Withot loss of generalit, we assme that the elements of the freqenc list! = 4 f! (1) ;! (2) ; : : :;! (m) g are ordered so that! (1)! (2)! (m). We write P for the set of all sch lists. Presmed given is an m-arameter famil of \o-theshelf" feedback controller transfer fnctions K 4 = f : 2 Pg, with at least the following roerties: Pole Content Proert: For each element (i) of each list = f (1) ; (2) ; : : : ; (m) g 2 P, s = j (i) is either a transmission zero of P (s) or a ole of (s), bt not both. Stabilit Margin Proert: There is a ositive constant sch that for each 2 P,? is greater than the real arts of all of the closed-loo oles 1 of the feedback interconnection P Figre 1: Feedback Interconnection Also resmed given is an integer n C and a famil of n C -dimensional realizations fa ; b ; f ; g g, one 1 B the closed-loo oles are meant the zeros of the olnomial?, where and are the redced transfer fnctions P and resectivel. for each 2 K. These realizations are reqired to be chosen so that for each 2 P, (c ; I A ) is detectable and (I A ; b ) is stabilizable. As noted in [13], there are man dierent was to constrct sch realizations, once one has in hand an er bond n on the McMillan Degrees of the. Given sch a famil of realizations, the sb-sstem to be servised is ths of the form shown in Figre 2, where C () is the C () d P Figre 2: Servised Sb-Sstem n C -dimensional \state-shared" dnamical sstem _x C = A x C b ; = f x C g ; (2) called a mlti-controller and is a iecewise constant switching signal taking vales in P. As a conseqence of the Pole Content Proert, if we were to set (t) =!; t (3) then this wold case converge to zero at t! 1 [4]. Since the list! is not resmed to be known, imlementation of (3) is not ossible. What we shall do instead is to constrct a rovabl correct \servisor" which is caable of generating so as to achieve 1. global bondedness fof all sstem signalsg and 2. asmtotic reglation fi.e.,! g. 3 Estimator-Based Servision The servisor to be considered can be exlained informall in terms of the \mlti-" architectre shown in Figre 3. Here each is a sitabl dened estimate of which wold be asmtoticall correct if the distrbance d consisted of a sm of m sinsoids whose freqencies were exactl the comonents of the m-vector. For each 2 P, e 4 =? denotes the th ott estimation error and is a \normed" vale of e or a \erformance signal" which is sed b the servisor to assess the otential erformance of controller. H is a switching logic whose fnction is to determine on the basis of the crrent vales of the. The nderling decision making strateg sed b an -based servisor is basicall this: From time to time select for, that candidate control index In Proc. of the 37th Conf. on Decision and Contr., Dec

3 1 2 i 1 2 i e 1 e 2 e i 1 2 i H s 2? (i) 2, where (i) 2 P, (s) = m i=1 is the ith comonent of. The c are sed in the denition of W which will be given in a moment. The c also enable s to dene ott estimates 4 = c x E ; 2 P (6) and the corresonding ott estimation errors e 4 =? ; 2 P (7) m m e m m Since d is the sm of m sinsoids whose freqencies are the comonents of!, one has 2 Figre 3: Estimator-Based Servisor q whose corresonding erformance signal q is the smallest among the ; 2 P. Motivation for this idea is straightforward: the set of distrbance freqencies whose associated erformance signal is the smallest, \best" aroximates what the tre set of freqencies is, and ths the candidate controller designed on the basis of that set of distrbance freqencies oght to be able to do the best job of rejecting d. The secic servisor of interest shown in Figre 4, while int-ott eqivalent to the servisor jst described, admits a slightl dierent realization than that shown in Figre 3. Internall the servisor we want to discss consists of three sbsstems: a mlti- dnamic E, a erformanceweight generator W, and a scale-indeendent hsteresis switching logic H. x E W E W H Figre 4: Estimator-Based Servisor E is a n E -dimensional linear dnamical sstem of the form _x E = AE be x A E (4) E b E 4 where n E = 2(n 2m) and (A E ; b E ) is a arameterindeendent, (n 2m)-dimensional SISO, controllable air with I A E asmtoticall stable. Here n is the McMillan degree of the rocess transfer fnction P (s). In [5] it is exlained how to constrct a continos fnction 7?! c so that for each 2 P, 1? c E =! E ; c E =! E (5) where! E is the characteristic olnomial of A E ; 4 E = (si? AE )?1 b E ; and are corime olnomials with monic sch that P = =; and, for each! d = (8) From this and (5), one can dedce that e! mst go to zero as fast as e?t fcf. Aendixg. While the error signals dened b (7) are not actall generated b the servisor, the la an imortant role in exlaining how the servisor fnctions. The servisor's second sbsstem, W, is a casal dnamical sstem whose ints are x E and and whose state and ott W is a \weighting matrix" which takes vales in a linear sace W. W together with a sitabl dened erformance fnction : P W! R determine a famil of scalar-valed erformance signals of the form 4 = (; W ); 2 P (9) Each is viewed b the servisor as a measre of the exected erformance of controller C (). W and are dened b and _W =?2W xe xe ; W () > (1) (; W ) = c?1 W c?1 (11) resectivel. The denitions of W and are romted b the observation that if are given b (9), then _ =?2 e 2 ; () > ; 2 P (12) becase of (7), (1) and (11). The servisor's third sbsstem, called a scaleindeendent hsteresis switching logic H, is a hbrid dnamical sstem whose int is W and whose state and ott is. To secif H it is necessar to rst ick a ositive nmber h > called a hsteresis constant. H 's internal logic is then dened b the comter diagram shown in Figre 5 where the are de- ned b (9) and, at each time t, q denotes the element In Proc. of the 37th Conf. on Decision and Contr., Dec

4 n Initialize (1 h) q = q Figre 5: Comter Diagram of H. of P that minimizes (q; W ). In interreting this diagram it is to be nderstood that 's vale at each of its switching times t is eqal to its limit from above as t # t. Ths if t i and t i1 are an two sccessive switching times, then is constant on [t i ; t i1 ). The fnctioning of H is roghl as follows. Sose that at some time t, H has jst changed the vale of to. is then held xed at this vale nless and ntil there is a time t 1 > t at which (1h) q for some q 2 P. If this occrs, is set eqal to q and so on. 4 Main Reslt The overall sstem jst described, admits the block diagram in Figre 6. The following theorem is the main W H W C () x E E d P Figre 6: Servisor Control Sstem reslt of this aer. Theorem 1 Assme that the set of admissible freqencies is nite. For an initialization of the closedloo sstem with W () > and an distrbance signal d consisting of the sm of an m sinsoids whose freqencies are the comonents of some m-vector in P, the signals x C ; x E, and W are bonded. Moreover, converges to faster than e?t. Proof of Theorem 1. It is shown in the Aendix that for an iecewise-constant signal s : [; 1)! P 2 For an signal and olnomial we se the notation (s) to denote the action of the dierential oerator olnomial (s)j s=d=dt on. and an initialization of the sstem consisting of (1), (2), (4), and (1) with = s, e! converges to zero faster than e?t. From this and e 2t! (t) = Z t e 2 e 2!d! () fcf. eqation (12)g we frther conclde that e 2t! is bonded on [; 1). It follows that if we dene the scaled erformance signals (t) = e 2t (t); t ; 2 P (13) then (i)! will be bonded on [; 1). Moreover, becase of (12), _ = e 2t e 2 ; () > 2 P ths we also conclde that (ii) each is ositive and has a limit as t! 1. This is becase each is a nondecreasing fnctions of t that starts ositive. In the seqel we make se of the above roerties (i) and (ii). The interconnected sstem dened b (1), (2), (4), and (1) is a dnamical sstem of the form _z = f (z); = g (z); 2 P (14) where z 4 = fx P ; x C ; x E ; W g and, for each 2 P, f and g are locall Lischitz. Becase of the hsteresis constant h, for each initial state fz(); ()g for with W () >, there mst be an interval [; T ) of maximal length on which there is a niqe air fz; g with z continos and iecewise constant, which satises (14) assming is generated b the scale-indeendent hsteresis switching logic [14]. Moreover, on each roer sbinterval [; ) [; T ), can switch at most a nite nmber of times. Sose that T was nite. Since d is bonded and the sstem dened b (1), (2), and (4) is linear for ever iecewise constant, one wold conclde that x P, x C, x E,, and were bonded on [; T ). Moreover, W wold also be bonded, becase of (1). In this case, the soltion to (1), (2), (4), and (1) cold be contined onto at least an oen half interval of the form [T; T 1 ) thereb contradicting the hothesis that [; T ) is the sstem's interval of maximal existence. B contradiction we can therefore conclde that T = 1. The term \scale-indeendence" is romted b the fact that if is an iecewise continos signal which is ositive on [; 1), the state of the switching logic remains nchanged if each erformance signal, 2 P is relaced. Ths, becase of (13), and as far as the signal is concerned, we can think of the switching logic as being driven b the scaled erformance signals 4 = with (t) 4 = e 2t, t. The facts that P is a nite set and that the ossess roerties (i) and (ii) noted above, enable s to exloit the Scale-indeendent Hsteresis Switching Lemma [14, 15] and conseqentl to draw the following conclsion. In Proc. of the 37th Conf. on Decision and Contr., Dec

5 Lemma 1 For an initialization of the close-loo sstem with W () >, let fx P ; x C ; x E ; W; g denote the niqe soltion to (1), (2), (4) and (1) with generated b H. There is a time T < 1 beond which is constant and e 2t (T ) (t) is bonded on [; 1). Let x P ; x C ; x E ; W;, and T be as in Lemma 1 and set q = 4 (T ). For t T, (t) = q and therefore C () is a linear time-invariant sstem with the transfer fnction q. Becase of the stabilit margin roert and the bondedness of d, the signals x P, x C,, and are bonded on [T ; 1). Bondedness of x E and W on [; 1) follows becase of (4), (1), and the asmtotic stabilit of A E. Therefore, z is bonded on [; 1). Becase of (12) and the observation that e 2t (T ) (t) is bonded on [; 1), we frther conclde that Z 1 Z 1 e 2 qd (e e q ) 2 d < e 2T q < 1 Ths, e q 2 L 2. Moreover _e q is bonded on [; 1), becase of (7), (14) and bondedness of z. Therefore, e q converges to as t tends to 1 [16, Lemma 1,. 58]. Becase of eqations (4), (5), and (6), we conclde that! E q = (! E? q ) q (15) Since the transfer fnction of the rocess is, = ( d) (16) becase of (1). Using (16) to eliminate from (15) ields! E ( q? ) =? q d This and eqation (7) frther iml that! E e q =? q d (17) Since d is a sm of m sinsoids, q d is also a sm of m sinsoids. Hence one can write q d as q d = mx? A (i) e j!(i)t B (i) e?j!(i) t (18) i=1 where each A (i) and B (i) is a xed comlex nmber. From eqation (17) and (18), we get! E e q =? mx? A (i) e j!(i)t B (i) e?j!(i) t (19) i=1 Since (19) is a stable dierential eqation forced b a sinsoidal term and its soltion e q converges to zero, we mst have mx? A (i) e j!(i)t B (i) e?j!(i) t = ; t i=1 Becase all the! (i) are distinct, the exonentials in the above eqation are linearl indeendent fnctions of time and therefore we mst have A (i) = B (i) = ; which means that 8i 2 f1; 2; : : :; mg q d = (2) For = q the redced transfer fnction of the mlticontroller is q q, ths q = q From this and (16) we conclde that ( q? q ) = q d (21) Bt, becase of ole content roert, an root of q is also a root of q and therefore, in view of (2), q d =. Ths (21) imlies that ( q? q ) = The stabilit margin roert then garantees that the roots of q? q have real art smaller then? and therefore converges to faster than e?t. 5 Conclsions In this aer we roosed a servisor control aroach to sress from the ott of a SISO, LTI rocess the eect of a ersistent distrbance int consisting of the sm of a nite nmber of sinsoids of nknown amlitde, hase, and freqenc. The aroach roosed in insired b the literatre on servisor control. The main shortcoming of the resent aer is that the stabilit analsis in Section 4 is onl valid when the set of freqencies is nite and when there is no measrement noise and nmodeled dnamics in the rocess model. However, simlation reslts sggest that the overall closed loo sstem ma, in fact, be stable even when has innitel man elements (e.g. is an interval) and ma also be robst with resect to nmodeled dnamics and measrement noise. Acknowledgments The second athor wishes to thank Antonio Pascoal for sefl discssions on the control of atonomos nderwater vehicles close to the srface of a wav sea. These discssions artiall motivated this work. In Proc. of the 37th Conf. on Decision and Contr., Dec

6 Aendix In the seqel we show that for an iecewise-constant signal s : [; 1)! P and an initialization of the sstem consisting of (1), (2), (4), and (1) with = s, e! converges to zero faster than e?t. Since the transfer fnction of the rocess is, = ( d) (22) becase of (1). From eqations (8) and (22), we conclde that! =! (23) Eqations (4), (5), and (6) iml that! E! = (! E?! )! (24) Sbtracting (23) from (24) and sing the fact that e! =!?, ields! E e! = Since! E is the characteristic olnomial of A E and I A E is asmtoticall stable, e! converges to zero faster than e?t. References [1] R. Shoreshi and P. Knrek. Atomotive alications of a hbrid active noise and vibration control. IEEE Control Sst. Mag., 16(6):72{78, Dec [2] U. Emborg and C. F. Ross. Active control in the saab 34. In Proc. 2nd Conf. Recent Adv. Active Control Sond Vibrations, Blacksbrg, VA,. S67{ S73, [3] L. J. Eriksson. A ratical sstem for active attenation in dcts. Sond and Vibration, 22(2):3{34, [4] E. J. Davison. The robst control of a servomechanism roblem for linear time-invariant mltivariable sstems. IEEE Trans. Atomat. Contr., AC-21(1):25{ 34, Feb [5] A. S. Morse. Servisor control of families of linear set-oint controllers art 1: exact matching. IEEE Trans. Atomat. Contr., 41(1):1413{1431, Oct [6] A. S. Morse. Servisor control of families of linear set-oint controllers art 2: robstness. IEEE Trans. Atomat. Contr., 42(11):15{1515, Nov [7] J. Hesanha and A. S. Morse. Certaint eqivalence imlies detectabilit. To aear in Sstems & Control Letters. [8] K. S. Narendra and A. M. Annaswam. Stable Adative Control. Prentice Hall information and sstem sciences series. Prentice-Hall, [9] F. Ben-Amara, P. T. Kabamba, and A. G. Ulso. Robst adative sinsoidal distrbance rejection in linear continos-time sstems. In Proc. of the 36th CDC, vol. 2,. 1878{1883, Dec [1] M. Bodson and S. C. Doglas. Adative algorithms for the rejection of sinsoidal distrbances with nknown freqenc. Atomatica, 33(12), Dec [11] G. Feng. Robst adative control with internal model rincile. Int. J. Sst. Science, 25:1333{1336, [12] D. C. Yola, J. J. Bongiorno, and H. A. Jabr. Modern wiener-hof design of otimal controllers art I. the single int-ott case. IEEE Trans. Atomat. Contr., 21(3):3{13, Mar [13] A. S. Morse. Control sing logic-based switching. In A. Isidori, editor, Trends in Control: A Eroean Persective,. 69{113. Sringer-Verlag, [14] J. P. Hesanha and A. S. Morse. Scaleindeendent hsteresis switching. Technical reort, Lab. for Control Science & Eng., Yale Universit, Ma [15] J. P. Hesanha. Logic-Based Switching Algorithms in Control. PhD thesis, Yale Universit, New Haven, CT, [16] M. A. Aizerman and F. R. Gantmacher. Absolte Stabilit of Reglator Sstems. Holden-Da, In Proc. of the 37th Conf. on Decision and Contr., Dec