Prototype Angle Domain Repetitive Control - Affine Parameterization Approach

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1 Prototpe Angle Domain Repetitive Control - Affine Parameterization Approach Perr Y. Li Professor Department of Mechanical Engineering Universit of Minnesota Minneapolis, Minnesota U.S.A. lixxx99@mn.ed Angle-domain repetitive distrbances refer to distrbances that are periodic in a generic angle variable which is monotonicall increasing with time bt not niforml. This paper extends the classical prototpe repetitive control methodolog for time periodic distrbances to this sitation. Instead of sing an internal model approach to derive the control, an affine parameterization approach is adopted which redces the control design methodolog to one of estimating and canceling the distrbance. While the reslting control architectres are similar to the classical time-domain periodic case, the stabilit conditions are different and depend on the choice of signal norm. This necessitates an alternate compensator design approach for the non-minimm phase terms. Robst stabilit is also considered in the L 2 setting and an affine Q- filter concept is introdced to achieve robst stabilit. 1 Introdction Repetitive control is a poplar method for compensating for periodic distrbances or for tracking periodic trajectories b learning from previos ccles. In recent ears, a new class of repetitive control problems has arisen in which the distrbances or desired trajectories are periodic in an alternate domain that is not time. A tpical example is a sstem with a rotational element with its own dnamics. If distrbances or desired trajectories are fnctions of the anglar position of the rotational element, the are periodic in the angle of rotation. However, onl if the rotational speed is constant, the distrbances and the desired trajectories are periodic in time. Example applications inclde control of valve timing for engines and for hdralic pmp/motors, track keeping in disk drive dring transient operation, control of banding in printers de to non-niformit of a print drm, etc.. In all these examples, the speed of the rotational element need not be constant all the time. Historicall, repetitive control has been developed from an internal model perspective [1] b embedding the characteristic polnomial of the distrbance in the denominator of the controller and b ensring that the closed loop sstem is stable. A difficlt with repetitive control relative to other internal model controls is that the internal model can have a high order since the time dela can be large compared to the sampling time. Ths, designing and implementing gains for stabilizing the sstem b means sch as pole-placement or LQ will be comptationall prohibitive. Prototpe repetitive control avoids this difficlt b assming that the plant has been sfficientl compensated b an inner loop controller, so that most of its dnamics can be cancelled ot. This allows the closed loop to be stabilized sing onl a scalar gain in conjnction with the internal model. This greatl simplifies the control design and implementation. With angle-domain repetitive distrbances, both timeand angle- domain dnamics are involved. Adaptive control is sed in [4] to render the sstem as an angle-domain invariant sstem so that an internal model based repetitive control can be applied in the angle-domain. Recentl, these sstems have been treated generall as linear time varing (LTV) sstems. For example, in [5, 6], the control design ses a time varing polnomial description and solves a Diophantinelike eqation that can be qite comptationall intensive. In [7], a robst controller is obtained for LTV sstems sing an iterative robst control design methodolog. Interestingl, in all these works, the angle domain periodic distrbances are not treated directl. Rather, onl the first few modes are sed in the design. In this paper, the affine parameterization or innovation feedback approach [2] is applied to the angle domain repetitive distrbance rejection problem to obtain a straightforward design methodolog. The relationship between this approach and time-domain repetitive control is briefl discssed in [8]. In this framework, the controller design is redced to one of estimating and canceling the angle domain periodic distrbance. For this reason, it is sometimes called distrbance observer. Estimators for invertible and non-minimm phase plants are proposed that reslt in con- DS

2 trol strctres similar to the classical time-domain repetitive control. Stabilit conditions are derived for varios signal norms. Becase these conditions are more stringent than the classical time-domain repetitive control case, alternate compensator design methodologies are needed. Moreover, a shaping filter (affine Q-filter) can be incorporated to affinel shape the complementar sensitivit to improve sstem robstness. This serves a similar role as the classical Q-filter in [3] bt has the added advantages of affineness and nconditional nominal sstem stabilit. The rest of this paper is organized as follow. Section 2 defines the angle domain repetitive control problem. In section 3, the affine parameterization framework for repetitive control design is introdced. Section 4 analzes the angle dela operator in the time domain, needed for stabilit analsis. Angle-domain prototpe repetitive controllers are presented in section 5. Robst stabilit is considered in section 6. Section 7 discsses how the compensators are designed for different cases. Section 8 contains simlations to illstrate the control design and reslts. Conclding remarks are given in section 9. Notation: For a sstem G : x( ) 7! ( ) that is a map between time (or angle) signals, we se the notation (t)=(g[x])(t). A sbscript q (e.g. x q (q)) is sed to denote an angle domain signal, a signal with no sbscript (e.g. x(t)) is sed to denote a time domain signal. 2 Angle domain periodic distrbance rejection problem Consider a non-time domain variable q that we genericall refer to as the (nwond) angle. We assme that q(t) is monotonicall increasing bt not necessaril niforml with time sch that q(t)=w(t) > ; q()=q (1) q(t) and w(t) are assmed known or measred. To ensre that the periodicit of the distrbance is captred, we assme that w(t) > is pper and lower bonded: 9 w, w, sch that w w(t) > w > (2) Since q(t) is monotone increasing, it is invertible. i.e. for ever angle e, we can niqel find t s.t. e = q(t). Therefore a signal can be represented in terms of t or q. A sbscript q is sed to denote the angle domain representation of a time domain signal. Hence, for a time domain signal x and an angle e = q(t), x(t)=x q (e)=x(q 1 (e)) = x q (q(t)) (3) We consider a linear time invariant plant pertrbed b a sandwiched angle domain periodic distrbance (Fig. 1): = G o [d + G i []] (4) Fig. 1. Angle-domain repetitive control problem. where G o and G i are linear time invariant (time domain) operators, and d is a angle domain G-periodic signal satisfing: d q (e)=d q (e G) or expressed in time domain d(t)=d(q 1 (q(t) G)). The control objective: Design a controller to generate control inpt so that! in (4) despite the angle domain periodic distrbance d(t). Note that trajector tracking problems can also be addressed in this framework b taking G o = 1 and the desired trajector to be d(t). Remark 1 If d is a time domain periodic distrbance, the sandwiched distrbance sstem can be transformed into a simpler inpt distrbance sstem = G o G {z } i [x + ] G o,new with x, sch that G i [x]=d, being the inpt distrbance which is also time periodic. This transformation however is not meaningfl if d is angle domain periodic since the reslting x is generall neither angle- nor time- periodic. G i and G o mst therefore be treated independentl. 3 Affine Parameterization Controller Fig. 2 shows the affine parameterized control for the plant (4) in which the plant model is sed to obtain a prediction of the otpt. Here, the feedback signal is the innovation e which is the difference between the actal otpt and the predicted otpt: e := (G o G i )[]=G o [d] (5) DS

3 Q a G i d(t) G o e kc o e sto + e s qg e st i C i (1 k B o ) G i G o + Fig. 4. Angle domain repetitive control. Fig. 2. Affine parameterized control strctre with innovation feedback. e + Q a e The control (9) is different from conventional distrbance observer in that it estimates the distrbance of the previos period instead of the crrent instant. This is advantageos since Gi 1 and Go 1 are normall not casal. The repetitive controller in (9) will be generalized in sections 5 and 6 to the strctre in Fig. 4 where G o G i C o G o = e st o Q o (1) G i C i = e st i B i Q i (11) Fig. 3. Relationship between otpt feedback and innovation feedback. Becase e is directl related to the distrbance d and can be sed to recover it, affine parameterization approach is sometimes referred to as distrbance observer. Instead of designing an otpt feedback controller directl, we design the controller Q a which acts on the innovation e: = Q a [e] (6) Here, C o and C i are casal compensators (LTI transfer fnctions), B o is the residal from canceling G o, k is a scalar, e st o and e st i are time dela operators, and Q o and Q i are shaping filters for improving robstness. 4 Angle Dela Operator e s qg Let e sqg be the angle-invariant, angle domain operator that represents a G (angle) dela: e s qg [x q ](q) := x q (q G) (12) If we denote then, d :=(G i Q a )[e]=(g i Q a G o )[d] where s q is the Laplace variable for angle-domain signals. The corresponding time dela (from time t) to achieve a G angle dela is: T (t) := t q 1 (q(t) G) (13) =(G o (1 G i Q a G o ))[d]=g o [d d] (7) Since G i and G o are stable, the closed loop sstem is stable if and onl if the affine parameterization controller Q a is stable [2]. Moreover, from (7), in order for (t)!, we mst have d(t)! d(t). Hence Q a shold estimate d(t) from its inpt e = G o [d] and to cancel it. When G o and G i are minimm phase, a straightforward soltion to the repetitive control problem is to set: which is accomplished with d q (q(t)) = d q (q(t) G) (8) Q a = G 1 i e s qg G 1 o (9) where e s qg is the G angle dela, angle-domain operator that will be defined and analzed in detail in the next section. We define the time-domain operator corresponding to e s qg, the time-varing dela operator e st ( ) as: e st ( ) [x](t) := x(t T (t)) (14) where T (t) is given b (13). Hence, the angle domain periodicit of a distrbance can be expressed in either the angle domain or in the time domain as: (1 e s qg )[d q ]= 1 e st ( ) [d]= Note that since e st ( ) is time varing, it does not generall commte with other LTI operators. The following theorem derives the indced norms of e st ( ) which are sefl for stabilit and robstness analsis. DS

4 Theorem 1 Sppose that the angle q satisfies (1)-(2). The angle-domain dela operator has the following properties in the time-domain: for all time-domain signal x( ), for p = 1,2,..., w w kxkp p appleke st ( ) [x]k p p apple w w kxkp p (15) ke st ( ) [x]k = kxk (16) where kxk p :=( R xp (t)dt) 1/p and kxk := sp t x(t) are the L p norm and the L norm of the time domain signal x( ). Moreover, the bonds are tight in that there exist an anglar speed trajector w( ) satisfing (2) and a time domain signal x( ) sch that the bonds are achieved. Proof: Since (t)=e st ( ) [x](t)=x(t sp ( ) = sp x( ) so that kk = kxk which is (16). Consider now p = 1,2,..., Z Z kk p p = x p (t T (t)) dt = 1 T (t)), x p (t) dt(t) dt dt (17) where t := t T (t). From (13), we have q(t T (t)) = q(t) G. Differentiating this w.r.t. t, we get 1 dt dt (t) = w(t) w(t T ) (15) follows b sbstitting this into (17) and appling the bonds: w w apple w(t) w(t T ) apple w w To show the bonds are tight, let x(t) be the time-domain signal sch that its angle domain representation is: x q (q)= ( 1 if apple q apple q 1 otherwise (18) where q 1 << 2p is a small nmber. To achieve the pper bond in (15), define w(t) sch that the dration of the plse in x(t) is the shortest while the dration of plse in e st ( ) [x] is the longest (see Fig. 5): 8 >< w if apple q apple q 1 w(t = q 1 (q )) = w G apple q >: apple q 1 + G Don t care otherwise Fig. 5. Illstration of a time domain signal that achieves maximm gain after going throgh an angle dela operator. Note that x(t)=1 for t 2 [,q 1 / w]. Then, = e st ( ) [x] is: (t)= ( 1 if t 2 q 1 (G)+[, q 1 /w] otherwise so that kxk p p = q 1 / w and kk p p = q 1 /w. Ths, kk p p = w w kxkp p. To achieve the lower bond in (15), define w(t) with the roles of w and w swapped, sch that 8 >< w if apple q apple q 1 w(t = q 1 (q )) = w Gapple q >: apple q 1 + G Don t care otherwise Over an interval of q 1 /w, x(t) =1, and over an interval of q 1 / w, (t) =e st ( ) [x](t) =1. The desired lower bond is obtained b integration of x p (t) and p (t). Remark 2 1. Theorem 1 shows that in continos time, for all p =[, 1,..., ] the indced p norm ke st ( ) k i,p = w 1/p 1. w In contrast, a constant time-dela operator e st has indced p norm being 1 for all p 2 [,1,..., ]. This difference will have significance when establishing stabilit condition. 2. Using similar proof concepts, the p-norms of a signal represented in angle domain and in time domain can be shown to be bonded as: wkxk p p applekx q k p p apple wkxk p p 3. If the signal is in discrete time, the delaed signal x(t k T (t k )) ma not correspond to an past vales. In this DS

5 e m kc o + P(t i,t ) C i kc o e sto + e s qg e st i C i (1 k B o )e s(t i+t o) Lin. Angle Invariant G o G i -Q a m kc o + P(t i,t ) C i Fig. 6. Affine parameterized form of the angle domain repetitive controller in Fig. 4 after look aheads are absorbed. case, a linear interpolation can be sed. For example, if t j apple t k T (t k ) apple t j+1, e s(t i+t o) Fig. 7. Internal model implementation of the angle domain repetitive controller in Fig. 6 with compensators satisfing (23)-(24). Top: before absorbing the time-advance terms; Bottom: after absorbing the time-advance terms. x(t k T (t k )) = e st ( ) [x](t k ) = x(t j )l + x(t j+1 )(1 l) where l =(t k T (t k ) t j )/T s and T s is the sampling time. 4. We will have for all discrete time seqence x(t k ), sp k e st ( ) [x](t k ) applesp x(t k ) k A seqence can easil be constrcted so that the above is an eqalit. Hence, the indced l norm of e st ( ) as a mapping between discrete time seqences will also be Moreover, if the sampling time T s is small compared to the variation of w(t), then the indced L p norm for the continos time signals can be sed as a close approximation of the indced l p norm for the discrete time seqence. 5 Prototpe Repetitive Control In this section, we analze the nominal design of the repetitive controller in Fig. 4 with relationships Eqs.(1)- (11) where Q o = Q i = 1. With G i assmed invertible, cases of G o being invertible and not invertible are investigated. The case of G i not being invertible will be investigated in Section 7 where an approximate inverse is proposed. The time leads in Fig. 4 can be absorbed into the feedback loop as in Fig. 6 where P(t 1,t 2 ) : x( ) 7! ( ) is the mixed domain-dela time-domain operator, h i P(t 1,t 2 ) := e st 1 e st ( ) e st 2 It can be npacked sing (12) as: (19) (t)=p(t 1,t 2 )[x](t)=x(q 1 (q(t + t 1 ) G)+t 2 ) (2) or sing T (t) := t q 1 (q(t) G) from (13) in (14) as (t)=p(t 1,t 2 )[x](t)=x(t + t 1 + t 2 T (t + t 1 )) (21) Notice that P(t 1,t 2 ) reqires knowledge of q p to t 1 ahead of time. In the nominal case where Q o = Q i = 1 in (1)-(11), sing the relation e = (G o G i )[], and (23)-(24), the innovation feedback control can be converted into the otpt feedback form in Fig. 7. This is exactl the internal model based prototpe repetitive control in [3] except that the constant time dela operator is replaced b the constant angle dela operator. In time domain, the controller in innovation feedback (Fig. 6) and otpt feedback form (Fig. 7) are given below: Innovation feedback: m(t)= (1 k B o ) P( t o,t o ) [m](t)+kc o [e](t) =(1 k B o )[m 1 ](t)+kc o [e](t) m 1 (t)=m(t T (t t o )) + kc o [e](t) 1 (t)=p(t i,t o )[m](t)=m(t + t i + t o T (t + t i )) (t)= (C i P(t i,t o ))[m](t)= C i [ 1 ](t) Otpt feedback: m(t)=p( t o,t o )[m](t)+kc o [](t) = m(t T (t t o )) + kc o [](t) 1 (t)=p(t i,t o )[m](t)=m(t + t i + t o T (t + t i )) (t)= (C i P(t i,t o ))[m](t)= C i [ 1 ](t) Becase the amont of the time dela T ( ) associated with the G angle dela varies with time, care mst be taken to absorb the time advance/dela properl. 5.1 G o and G i invertible When G o is invertible (stable and minimm phase), d can be reconstrcted via d = Go 1 [e], bt tpicall noncasall. However, as T (t) is sall large compared to the DS

6 acasalit of Go 1, so that d q (q(t) G)=d(t T (t)) is often available. The repetitive controller in (9) can be generalized to a first order filtered version of d q (q G) in the angle domain: d q (q) := l d q (q G)+(1 l) d q (q G) (22) B o [d] can be sed as inpt to the estimator. To proceed, we note that (22) for the invertible G o,g i case can be written in an error feedback form: d q (q) := d q (q G)+k[d q d q ](q G) (27) This reslts in the estimation error d q := the dnamics: d q d q satisfing which is a G-angle periodic integrator with a G-delaed error as inpt. Mimicking this and sing B o [d d] instead of d d, the distrbance estimator is defined as: d q (q)=l d q (q G) Ths, for l 2 ( 1,1), the first order (angle-domain) filter is exponentiall stable. Since G o is stable, we also have, as t!. (t)=g o [d d](t)! The distrbance estimator/controller in (22) corresponds to the innovation feedback form in Fig. 4 with k =(1 l) 2 ( 1,1), B o = 1. C i and C o are the casal inverses of G i and G o p to time delas as below: G i C i = e st i (23) C o G o = e st o (24) Theorem 2 The affine parameterization controller given in Fig. 4 or Fig. 6 with B o = 1, and the corresponding closed loop sstem with (4) is stable if and onl if k = 1 l 2 (,2). Frthermore if G i and G o are invertible, and the compensators C i and C o satisf (23)-(24), then the estimate of the distrbance d q (q(t))! d q (q(t)) and the otpt (t)!. The convergence is exponential in time and in angle. Proof: Since the repetitive controller implements the angle invariant distrbance estimator (22), and = G o (d d), we need onl show that (22) provides asmptotic convergence of d! d. Since (22) is a linear angle-invariant sstem, we analze it in the angle domain. The feedback loop is stable iff l =2 ( 1,1) or k 2 (,2) as specified. In angle domain, the exponential convergence rate is ln(l)/g. Since w(t) w, convergence rate in time will be greater than wln(l)/g. Note that as long as k 2 (,2), the stabilit is independent of w, w, the bonds on q. 5.2 G i invertible, G o not invertible When G i can be perfectl cancelled ot bt G o ma not be, let C i and C o be given b G i C i = e st i (25) C o G o = B o e st o (26) with B o 6= 1 containing the non-minimm phase terms. The isse is that we cannot recover d directl from e and onl d(t) := d(t T (t)) + k B o [d d](t T (t)) (28) = (1 k B o )[ d]+k B o [d] (t T (t)) (29) which involves both time- and angle-domain operators bt ses B o [d] (which is available) as inpt as intended. From (28) and d(t T (t)) = d(t), the distrbance estimation error d := d d satisfies: d(t)=(1 k B o ))[ d](t T (t)) (3) The distrbance estimator in (28) or (29) are represented in Figs. 4, 6 and 7. Note that the innovation feedback controller in Fig. 6 depends on B o, bt the otpt feedback controller in Fig. 7 does not. Theorem 3 Consider the mixed domain repetitive controller in Figs. 4, 6 or 7 and Eqs.(25)-(26) which implements the distrbance estimator (29). The distrbance estimate d(t) converges to d(t) and (t) converges to in the L p sense with p 2 [1,2,..., ] if k1 k B o k i,p < (w/ w) 1/p (31) norm of the time do- where kzk i,p denotes the indced p main operator Z. Proof: From (3), the distrbance estimation error d = d satisfies: 1 e st ( ) (1 k B o ) [ d]= whose stabilit can be viewed as a feedback loop between e st ( ) and (1 k B o ) (see the feedback loop in Fig. 4). From small gain theorem, the feedback loop is stable if the loop gain, as time domain operator, with the respective indced norm, is less than 1: ke st ( ) (1 k B o )k i < 1 From Theorem 1, ke st ( ) k i,p apple ( w/w) 1/p ; and ke st ( ) k i, = 1. Therefore, conditions in (31) ensres that loop gains are less than 1 in the p-norm with p =[1,2,..., ]. d DS

7 kc o e st o + e s qg e st i C i 1 k B o (1 Q shape ) m kc o + P(t i,t ) C i e s(t i+t o ) (1 k B o (1 Q shape )) Fig. 8. Otpt feedback form of the repetitive controller with affine Q-Filter. Top: before absorbing the time-advance terms; Bottom: after absorbing the time-advance terms. Remark 3 1. Recall that kzk i,2 is the maximm freqenc response gain of Z and kzk i, is the L 1 norm of the implse response of Z. 2. The indced- norm condition in Theorem 3 generalizes Theorem 2 in that k 2 (,2) is eqivalent to 1 k < 1 which is the norm condition in Theorem The p norm stabilit condition for the angle domain distrbance case is stricter than that of the time-periodic distrbance case which mst have k1 k B o k i,p < 1. However, as ( w w)! the time domain distrbance case is recovered. 4. For p 6=, the p norm condition is stricter than the case in Theorem 2 where G o is invertible and B = 1. Moreover, except with k = 1, even as B o! 1, the stabilit condition does not converge to that in Theorem 2. This discrepanc is becase Theorem 3 ses a mixed angle-time domain analsis, whereas Theorem 2 ses the angle-domain analsis alone. 5. If kc o is designed sch that k B o 1, the stabilit condition will be satisfied for a wide range of anglar velocities. As k B o! 1, the stabilit condition is satisfied b arbitrar w/ w. 6. G i being invertible is necessar for asmptotic convergence with the crrent architectre. This is not the case with time-domain repetitive control since the sandwiched sstem can be converted into a sstem with periodic inpt distrbance as discssed in Remark 1. Nonminimm phase G i can be dealt with b sing an approximate inverse (see Section 7.) 6 Robst Stabilit We consider now the isse of robst stabilit. A ke advantage of affine parameterization concept is that the complementar sensitivit fnction can be shaped affinel. This featre is sed to design affine Q-filters to ensre robst stabilit (at the expense of convergence). We shall see that the affine Q-filter will be eqivalent to the conventional Q-filter in [3] when k B = 1. For robst stabilit, the plant ncertaint is assmed to be given in terms of mltiplicative ncertaint in the freqenc domain, i.e. G actal ( jw)=g( jw)(1 + D( jw)w ( jw)) where D is an LTI operator sch that kdk = sp w D( jw) < 1 and W is the ncertaint weighting. Since the maximm freqenc gain is the indced 2-norm of the operator, sch a description imposes that stabilit robstness mst be analzed sing 2-norm of the time signals. Let the compensators in the repetitive controller in Figs. 4 or 6 satisf: C i G i = e st i Q si (32) C o G o = e st o B o Q so (33) instead of (25)-(26). These choices define an affine Q-filter: Q shape = Q si Q so (34) Note that from the perspective of robstness, Q shape can be implemented either in C i or C o. Besides the freqenc shaping fnction, Q o and Q i can also contain residals from canceling G o and G i, particlarl those with non-minimm phase zeros. Theorem 4 For a given mltiplicative plant ncertaint D( jw)w ( jw), the repetitive controller in Fig. 6 with compensators defined sing (32)-(33) will be robstl stable in the L 2 sense if Q shape is stable and: apple w 1/2 kq shape W k B o k < k1 k B o k (35) w where Q shape is given in (34). DS

8 Fig x est ( ) 1 k B o Q shape k B o W D Interaction of ncertaint with repetitive controller. Proof: The feedback loop containing the model ncertaint is shown in Fig.9. The otpt and inpt x to the angle dela operator e st ( ) are related b: x =(1 k B o )[]+(Q shape k B o W D)[] Since D( jw) can have arbitrar phase, kxk 2 apple k(1 k B o )k i,2 + k(q shape k B o W D)k i,2 kk 2 = k(1 k B o )k + k(q shape k B o W k kk 2 Since from Theorem 1, ke st ( ) k i,2 apple ww 1/2, from small gain theorem, the feedback loop in Fig. 9 is stable if: w 1/2 k(1 k B o )k i,2 + k(q shapek B o W D)k i,2 < 1 w The desired relation is a rearrangement of the above. Remark 4 1. Becase Q shape enters affinel in (35), Q shape can be compted directl to achieve robst stabilit and distribted in (32)-(33). 2. How Q shape is distribted between (32) and (33) has no effect on robst stabilit bt can reslt in different performances becase angle and time domain operators do not commte. 3. B setting W = in (35), we see that the L 2 stabilit condition in Theorem 3 for the nominal sstem is a necessar condition for robst stabilit. 4. It also implies that a sstem that is stable according to Theorem 2 ma not be robst to an size of plant ncertaint. An example is if k 2 (,2) bt 1 k > (w/ w) 1/2. 5. The right hand side of (35) is maximized when k B o = 1. Ths, from the perspective of robst stabilit, the compensator kc o shold aim to approximate Go 1 (p to a dela). The affine Q-filter has the advantage that the nominal sstem remains stable as long as the Q-filter is also stable. The affine Q-filter does not change the form of the innovation feedback controller as in Fig. 4 or Fig. 6. Onl C o and C i are changed. In contrast, the strctre of the otpt feedback controller changes from Fig. 7 to Fig. 8. Direct implementation in the innovation feedback form ma have some advantage in its simplicit in the presence of affine Q-filter. From Fig.8, the otpt feedback form of the controller with the affine Q-filter is seen to redce to the conventional Q-filter in [3] onl if k B o = 1. Conversel, nless k B o = 1, the conventional Q-filter wold shape the complementar sensitivit fnction in a non-affine wa. With the affine Q-filter, performance degradation is expected. The degree of performance degradation for the case when Q so = 1 can be obtained from the following reslt: Proposition 1 If the repetitive controller in Fig. 4 is designed sing Eqs.(32)-(33) with Q so = 1, then the otpt error in response to the angle-domain periodic distrbance d, is Proof: This can be seen from (7): = G o (1 Q si )[d] (36) = G o [d G i Q a G o ]d = G o [d G i C i e st i[ d]] {z } Q si In the absence of Q si, the estimator (22) or (28) is designed sch that d(t)! d(t G) =d(t). When Q si is present, the otpt of the controller is Q si [d]. Therefore, we have = G o [d Q si [d]] = G o (1 Q si )[d] Ths, (1 Q si ) plas the role of a performance filter. However, the freqenc content of d(t) depends both on the freqenc content of d q (q) and on the anglar velocit w(t). The formla for the case when Q o 6= 1 is more complicated becase we cannot se the convergence propert of the distrbance estimator since the inpt to the filter will nolonger be angle-periodic. However, it is clear that the error wold be affine in k(1 Q so )[d]k and in k(1 Q si )[d]k. 7 Compensator Design via Approximate Inverse The compensators C i and C o are designed according to (32)-(34) and (35) in order to satisf Theorem 4. The wold nominall be inverses of G i and G o. When G o or G i is not invertible, the non-minimm phase zeros need to be compensated appropriatel. In classical discrete time repetitive control, zero phase compensators are sggested [3]. For both stabilit (see e.g. Theorem 3) and performance (see e.g. (36) with Q si incorporating the inpt residal), it is desirable for the residal to be close to 1. Unfortnatel, zero-phase compensator can onl ensre, DS

9 e kc o e st o e s qg e st i C i Fig. 1. Angle domain repetitive controller with k B o = 1. b picking k small enogh, that k1 k B o k < 1 bt not necessaril smaller than ( w/w) 1/p or to minimize performance degradation. Here, the forward series expansion approach in [9] is proposed instead. Case 1: Non-minimm phase G o via Theorem 3 Consider the discrete time sstem example, G o (q)= q n o (1 bq 1 ) A(q 1 ) where b > 1. Or objective is to define compensator C i sch that k1 k B o k is small enogh to satisf robst stabilit. Here, the inverse of the non-minimm phase zero term is approximated as in: Then define: q (1 bq 1 ) b + q2 b 2 + q3 b qn b {z n } (1 bq 1 ) 1 = 1 + qn b n q kc o = Q so A(q 1 ) b + q2 b 2 + q3 b qn b n q n (37) where the q n term is to keep C o casal. Hence, (33) becomes: kc o G o = Q so 1 + qn b n q (n o+n) with k B o =(1 + qn b n ) and t o = T s (n o + n). This wa, b sing larger n, we can make k1 k B o k = qn b n = 1 b n arbitraril small to satisf the nominal stabilit in Theorem 3. Q shape can then be chosen to frther satisf the robst stabilit condition in Theorem 4. Comptationall, the forward series expansion can be obtained as a deconvoltion. This approach is onl one wa of generating an acasal FIR approximation to the inverse of the non-minimm phase terms. The FIR filter design can be combined with the affine Q f ilter design to redce filter length. Other acasal filter design methodologies, sch as those based on optimization, can also be applied. Case 2: Non-minimm phase G o /G i via Theorems 2 & 4 Even if G i and/or G o is non-minimm phase, we can still take k B o = 1 in Eqs.(32)-(33), bt consider the residals for inverting G i and G o as part of the affine Q-filter. The forward series expansion approach described above can be sed. If the zeros are important onl at high freqencies where the plant ncertaint W is also large, the non-minimm phase zero ma not even need to be compensated. An advantage of taking k B o = 1 and sing Theorem 2 to ensre nominal stabilit is that the nominal sstem will be nconditionall stable for ALL anglar speeds w(t). Also, the feedback controller becomes Fig. 1 which will be simpler to implement as it does not involve a feedback loop. As an example, sppose that G i and G o are discrete time LTI sstems with sfficientl fast sampling. If G i is invertible and G o (q)= q n o (1 zq 1 ) A(q 1 ) where A(q 1 ) is a monic polnomial of q 1 with stable roots and z >, we can design C i = Gi 1 e st i C o = Q so1 A(q 1 ) and se Q shape = Q so1 R with R being the residal in compensating for the non-minimm phase zero term in (35) to ensre robst stabilit. A similar approach can be taken if G i is not invertible. 8 Simlations To illstrate the control design, we assme that the anglar freqenc w(t)= q(t) is shown in Fig. 11 with w/w = 4. The distrbance in angle domain, d q (q), and in time domain, d(t), are shown in Fig. 12. The sandwich plant is given b two LTI discrete time sstems with sampling time of.2s: G i (q 1 )= q ( q 1 ) (1.948q 1 ) 2 (38) G o (q 1 )= q (1 + 2q 1 ) (1.948q 1 ) 2 (39) so that G i is invertible bt G o is non-minimm phase. We define the compensators as: C i (q 1 (1.948q 1 ) 2 )=.4679( q 1 ) kc o (q 1 )= (1.948q 1 ) q 1 4 q 2 2 DS

10 Anglar speed: rad/s Nominal controller d estimate d actal Time s Fig. 11. Anglar speed of nwond angle w = q Distrbance Distrbance Fig Angle rad Time s of time (bottom)..3.2 Angle domain distrbance as a fnction of angle (top) and Error with no Affine Q filter Fig Time nominal controller. Magnitde (db) Actal and estimated distrbance (at otpt of G i ) with Normalized Freqenc ( π rad/sample) Fig. 15. Magnitde Response (db) Freqenc response of the 21 tap FIR affine Q-filter. Error.1.3 Error with Affine Q filter Time s Error.1 Fig. 13. Otpt error with nominal controller. with the inpt and otpt delas t i = T s and t o =(1 + 3)T s. Becase 1 k B o = q 3 /8 (its implse response is a single implse), the indced p norm for all p 2 [1,2,..., ] satisf: Time Fig. 16. Otpt error with controller and affine Q-filter. k1 k B o k i,p = 1 8 < w w 1/p.2.18 With Affine Q filter d estimate d actal.16 so that the sstem is nominall L p and L stable according to Theorem 3. The controller is implemented in the innovation feedback form (Fig. 6). The performance of this nominal control are shown in Figs. 13 and 14 which show that error converges to and the estimate of d converges after 1-2 ccles. This nominal control is then robstified b incorporating a 21 tap linear-phase FIR low pass affine Q-filter into C o as in (33). The freqenc response of the affine Q-filter is shown in Fig. 15. The phase of the filter is acconted for b increasing the time dela to t o =( )T s. Figs. 16 and 17 show that the robstified controller Fig Time Actal and estimated distrbance (at otpt of G i ) with controller and affine Q-filter. DS

11 Error Fig. 18. Error with Q filter (no feedback) n=6 term forward series Time Otpt error with simple controller in Fig. 1, n = 6 term series expansion, and affine Q-filter. is also effective in estimating and canceling ot the distrbance. Compared with Figs. 13 and 14, the error is onl slightl degraded. Finall, we consider the controller in which we compensate for the non-minimm phase zero sing a 6-term forward series expansion, bt take k B o = 1 as in Fig. 1. The residal is lmped into the affine Q-filter in Fig. 15. Fig. 18 shows that the error decreases in 1 ccle and the residal error magnitde is smaller than the case where k B o 6= 1 in Fig. 16. If a 5-term series expansion is sed, the error wold be slightl worse than that in Fig. 16. [5] Sn, Z., Zhang, Z, and Tsao, T.-C., 29, Trajector tracking and distrbance rejection for linaer timevaring sstems: inpt/otpt representation, Sstems & Control Letters, Vol. 58, pp [6] Gillella, P.K., Song, X., and Sn, Z., 214, Time- Varing Internal Model-Based Control of a Camless Engine Valve Actation Sstem, IEEE Transactions on Control Sstems Technolog, Vol. 22(4), pp [7] Chen, C., and Chi, G., 28, Spatiall Periodic Distrbance Rejection with Spatiall Sampled Robst Repetitive Control, J Dn Sst-T ASME, Vol. 13(2), #212. [8] Chen, X., and Tomizka, M., 214, New Repetitive Control With Improved Stead-State Performance and Accelerated Transient, IEEE Transactions on Control Sstems Technolog, Vol. 22(2), pp doi: 1.119/TCST [9] Gross, E., Tomizka, M., and Messner, W., 1994, Cancellation of discrete time nstable zeros b feedforward control, J Dn Sst-T ASME, Vol. 116, pp Conclsions In this paper, the classical prototpe repetitive control scheme is extended to the angle-domain repetitive distrbance scenario. This is achieved b sing an affine parameterization perspective instead of an internal model perspective to derive the controller. The design methodolog is comptationall straight forward. While the reslting controller strctre is similar to the time-periodic case, the stabilit conditions are more stringent. This reqires a different approach to compensating the non-minimm phase zeros than sing a zero-phase error compensator. The forward series expansion is shown to be a sefl means for doing so. The affine parameterization perspective also allows for the direct comptation of an affine Q-filter for improving robstness. References [1] Francis, B. A., and Wonham, W. M., 1976, The internal model principle of control theor, Atomatica, Vol. 12(5), pp [2] Goodwin, G. C., Graebe, S. F., and Salgado, M. E., 21, Control Sstem Design, Prentice Hall, Upper Saddle River, N.J., USA. [3] Tomizka, M., Tsao, T. C., and Chew, K.-K., 1989, Analsis and Snthesis of Discrete-Time Repetitive Controllers, J Dn Sst-T ASME, Vol. 111, pp [4] Wang, J., and Tsao, T.C., 214, Repetitive control of linear time varing sstems with application to electronic cam motion control, Proceedings of the American Control Conference, Vol. 4, pp DS