A design method for two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems

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1 A design method for two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems Zhongxiang Chen Kou Yamada Tatsuya Sakanushi Iwanori Murakami Yoshinori Ando Nhan Luong Thanh Nguyen Shun Yamamoto Gunma University, Tenjincho, Kiryu, Japan t ; yamada; t ; murakami; o; t ; Abstract: Multi-period repetitive controller was proposed by Gotou et al., in order to improve the disturbance attenuation characteristic of modified repetitive control systems that follows the periodic reference input with small steady state error. Recently, the parameterization of all stabilizing multi-period repetitive controllers was studied. However, when the parameterization of all stabilizing multi-period repetitive controllers is used, the input-output characteristic the feedback characteristic cannot be specified separately. From the practical point of view, it is desirable to specify the input-output characteristic the feedback characteristic separately. In addition, the parameterization is useful to design stabilizing controllers. Yamada et al. solved this problem by obtaining the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers. However, the method by Yamada et al. cannot be applied to multipleinput/multiple-output plants. Because, the method by Yamada et al. uses the characteristic of single-input/single-output system. In this paper, we propose the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers for multiple-input/multipleoutput systems. Keywords: multi-period repetitive control; modified repetitive controller; parameterization; low-pass filter; two-degree-of-freedom control; periodic signal. 1. INTRODUCTION A repetitive control system is a type of servomechanism for periodic reference signals, i.e., it follows a periodic reference input without steady state error, even when there exists a periodic disturbance or an uncertainty of a plant Inoue et al., 1980, 1981; Hara et al., 1986; Yamamoto Hara, 1987; Hara Yamamoto, 1986; Hara et al., 1988; Omata et al., 1987; Watanabe Yamatari, 1986; Ikeda Takano, 1990; Katoh Funahashi, Because a repetitive control system that follows any periodic reference input without steady state error is a neutraltype of time-delay control system, it is difficult to design stabilizing repetitive controllers for plants Watanabe Yamatari, The reason is we must stabilize infinite numbers of poles on the imaginary axis those are included in the repetitive controller. To design a repetitive control system that follows any periodic reference input without steady state error, the plant needs to be biproper Hara et al., 1986; Yamamoto Hara, 1987; Hara Yamamoto, 1986; Hara et al., 1988; Omata et al., 1987; Watanabe Yamatari, However, almost all plants are strictly proper. To design stable repetitive control systems for strictly proper plants, repetitive controllers with low-pass filters have been considered Hara et al., 1986; Yamamoto Hara, 1987; Hara Yamamoto, 1986; Hara et al., 1988; Omata et al., 1987; Watanabe Yamatari, Because a repetitive control system with a low-pass filter has a simple structure is easily designed, this design method for repetitive control systems, called the modified repetitive control system, has been applied to many applications Inoue et al., 1980, 1981; Omata et al., However, the modified repetitive control system has a bad effect on the disturbance attenuation characteristic Gotou et al., 1987, in that at certain frequencies, the sensitivity to disturbances of a control system with a modified repetitive controller becomes twice as bad as that of a control system without a modified repetitive controller. Gotou et al overcame this problem by proposing a multi-period repetitive control system. However, the phase angle of the low-pass filter in a modified repetitive controller that of a multi-period repetitive controller have a bad effect on the disturbance attenuation characteristic Sugimoto Washida, 1998a,b. Okuyama et al. 2002; Yamada et al overcame this problem proposed a design method for multi-period repetitive controllers to attenuate disturbances effectively using the time advance compensation described in Sugimoto Washida 1998a,b. Using this multi-period repetitive control structure, Steinbuch 2002 proposed a design method for repetitive control systems with uncertain period time. On the other h, there exists an important control problem to find all stabilizing controllers named the pa- Copyright by the International Federation of Automatic Control IFAC 5753

2 rameterization problem Vidyasagar, Yamada et al. 2004, 2005 gave the parameterization of all stabilizing multi-period repetitive controllers. However, when the parameterization of all stabilizing multi-period repetitive controllers in Yamada et al. 2004, 2005 is used, the input-output characteristic the feedback characteristic cannot be specified separately. From the practical point of view, it is desirable to specify the input-output characteristic the feedback characteristic separately. In addition, the parameterization is useful to design stabilizing controllers. Therefore, the problem of obtaining the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers those can specify the input-output characteristic the disturbance attenuation characteristic separately is important to solve. Yamada et al solved this problem by obtaining the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers Yamada et al., However, the method by Yamada et al cannot be applied to multiple-input/multiple-output plants. Because, the method by Yamada et al uses the characteristic of single-input/single-output system. In this paper, we propose the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems those can specify the input-output characteristic the feedback characteristic separately. Control characteristics are also presented. Notation R the set of real numbers. R + R. Rs the set of real rational functions with s. RH the set of stable proper real coefficient rational functions. H the set of stable causal functions. A pseudo inverse of A σ the maximum singular value of. 2. TWO-DEGREE-OF-FREEDOM MULTI-PERIOD REPETITIVE CONTROLLERS FOR MULTIPLE-INPUT/MULTIPLE-OUTPUT SYSTEM AND PROBLEM FORMULATION Consider the two-degree-of-freedom control system shown in Fig. 1 that can specify the input-output characteristic the feedback characteristic separately. Here, Gs Cs d 1 s + us + Gs zs Fig. 1. Two-degree-of-freedom control system R m p s is the plant satisfying + + ys d 2 s rank Gs m, 1 Cs is the controller written by Cs[C 1 s C 2 s ], 2 us R p s is the control input written by uscs [ ] zs [C 1 s C 2 s ] [ ], 3 zs ys R m s is the output, d 1 s R p s d 2 s R m s are disturbances, R m s is the periodic reference input with period T>0 satisfying rt + T rt t 0 4 zs ys+d 2 s. In the following, we call C 1 s the feed-forward controller C 2 s the feedback controller. From the definition of internal stability Vidyasagar, 1985, when all transfer functions V i si 1,...,6 written by [ ] us ys [ ] [ ] V1 s V 2 s V 3 s d V 4 s V 5 s V 6 s 1 s d 2 s are stable, the two-degree-of-freedom control system in Fig. 1 is stable. According to Gotou et al. 1987; Okuyama et al. 2002; Yamada et al. 2003, 2004, 2005, when the plant Gs has a periodic disturbance d 1 s with period T an uncertainty, in order for the output ys to follow the periodic reference input with period T with a small steady-state error, the feedback controller C 2 s must be written by N C 2 sc 20 s+ C 2i se sti i1 5 1 I q i se sti, 6 i1 where N is an arbitrary positive integer, C 20 s R p m s, C 2i s R p m si 1,...,N q i s R m m si 1,...,N are low-pass filters satisfying q i 0 I, 7 i1 T i > 0 Ri 1,...,N. Without loss of generality, we assume that rank C 2i s mi 1,...,N rank q i s mi 1,...,N. The feedback controller C 2 s written by 6 is called the multi-period repetitive controller Gotou et al., 1987; Okuyama et al., 2002; Yamada et al., 2003, 2004, Gotou et al proposed the design method for multi-period repetitive controller as T i T i i 1,...,N. 8 On the other h, Yamada et al proposed the design method for multi-period repetitive controller such that T i i 1,...,N do not necessarily satisfy 8. Therefore, in this paper, we attach importance to the generality assume that T i i 1,...,N do not necessarily satisfy

3 The problem in this paper is to clarify the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems Cs in 2 defined as follows: Definition 1. stabilizing two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems We call the controller Cs in 2 a stabilizing two-degreeof-freedom multi-period repetitive controller for multipleinput/multiple-output system, if following expressions hold true: 1 The feedback controller C 2 s in 2 works as a multiperiod repetitive controller. That is, the feedback controller C 2 s is written by 6, where C 20 s R p m s, C 2i s R p m si 1,...,N satisfies rank C 2i s mi 1,...,N q i s R m m si 1,...,N satisfies 7 rank q i s mi 1,...,N. 2 The two-degree-of-freedom control system in Fig. 1 is stable. That is, all transfer functions V i si 1,...,6 in Fig. 1 are stable. 3 The transfer function V er s from the periodic reference input to the error es ys in Fig. 1 satisfies σ V er jω k 0 k 0,...,n, 9 where ω k k 0,...,n is a frequency component of the periodic reference input given by ω k 2π k k 0,...,n 10 T ω n is the maximum frequency component of the periodic reference input. Q 2 s Q 2n0 s+ Q 2ni se sti i1 1 Q 2d0 s+ Q 2di se sti Here, Ns RH m p Ds RH m m satisfying i1 H p m. 14, Ds RH p p, Ñs RH m p are coprime factors of Gs onrh Gs NsD 1 1 s D sñs. 15 Xs RH p m, Y s RH p p, Xs RH p m Ỹ s RH m m are functions satisfying [ Y s ][ ] Xs Ds Xs Ñs Ds Ns Ỹ s I [ ][ ] Ds Xs Y s Xs. 16 Ns Ỹ s Ñs Ds Q 1 s H p m is any function to satisfy σ I Njω k Q 1 jω k 0 k 0,...,n. 17 Q 2n0 s RH p m, Q 2nis RH p m i 1,...,N, Q 2d0 s RH m m Q 2di s RH m m i 1,...,N are any functions satisfying Ỹ 0Q2di 0 N0Q 2ni 0 i1 3. THE PARAMETERIZATION In this section, we clarify the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems defined in Definition 1. The parameterization of all stabilizing two-degree-offreedom multi-period repetitive controllers for multipleinput/multiple-output systems is summarized in the following theorem. Theorem 1. The controller Cs is a stabilizing twodegree-of-freedom multi-period repetitive controller for multiple-input/multiple-output system if only if where Cs[C 1 s C 2 s ], 11 C 1 s Y s Q 2 sñs 1 Q1 s, 12 1 C 2 s Xs+DsQ2 sỹ s NsQ2 s 1 Y s Q 2 sñs Xs+Q 2 s Ds 13 1 Ỹ 0Q2d0 0 N0Q 2n0 0 I, 18 rank Q 2d0 s m 19 rank Q 2ni s Q 2n0 sq 1 2d0 sq 2dis m i 1,...,N. 20 Proof of this theorem requires following lemma. Lemma 1. Unity feedback control system in y Gsu 21 u Csr y is stable if only if Cs is written by 1 Cs Xs+DsQs Ỹ s NsQs 1 Y s QsÑs Xs+Qs Ds, 22 where Ns RH m p, Ds RH p p, Ñs RH m p Ds RH m m are coprime factors of Gs on RH satisfying 15. Xs RH p m, Y s RH p p, Xs RH p m Ỹ s RHm m satisfy 16 Qs H p m is any function Vidyasagar,

4 Using Lemma 1, we shall show the proof of Theorem 1. Proof. First, the necessity is shown. That is, we show that if the controller Cs in 11 is a stabilizing two-degreeof-freedom multi-period repetitive controller for multipleinput/multiple-output system, then the controller Cs takes the form 11, 12, From simple manipulation, we have V 1 s I + C 2 sgs 1 C1 s, 23 V 2 s I + C 2 sgs 1 C2 sgs, 24 V 3 s I + C 2 sgs 1 C2 s, 25 V 4 s I + GsC 2 s 1 GsC1 s, 26 V 5 s I + GsC 2 s 1 Gs 27 V 6 s I + GsC 2 s 1 GsC2 s. 28 From the assumption that all transfer functions in 23, 24, 25, 26, are stable, the feedback controller C 2 s is a stabilizing controller for the plant Gs. From Lemma 1, the feedback controller C 2 s must take the form 1 C 2 s Xs+DsQ2 sỹ s NsQ2 s 1 Y s Q 2 sñs Xs+Q 2 s Ds, 29 where Q 2 s H p m. Substituting the feedback controller C 2 s in 6 for 29, we have Q 2 s as 14, where Q 2n0 sxsq d sc 20d sc 2d s Y sc 20n sc 2d s RH p m, 30 Q 2ni s Xsq in sc 20d sc 2d s +Y sc 2in s RH p m i 1,...,N, 31 Q 2d0 s ÑsC 20nsC 2d s Dsq d sc 20d sc 2d s RH m m 32 Q di s Dsq in sc 20d sc 2d s +ÑsC 2ins RH m m i 1,...,N. 33 Here, q in s RH m m i 1,...,N q d s RH m m are coprime factors of q i s i 1,...,Non RH satisfying q i sq in sq 1 d s i 1,...,N. 34 C 20n s RH p m C 20ds RH m m are coprime factors of C 20 sq d s onrh satisfying C 20 sq d sc 20n sc 1 20d s. 35 C 2in s RH p m i 1,...,N C 2d s RH m m are coprime factors of C 20 sq in s C 2i sq d sc 20d s i 1,...,NonRH satisfying C 20 sq in s C 2i sq d s C 20d s C 2in sc 1 2d s i 1,...,N. 36 From 30 36, Q 2n0 s RH p m, Q 2ni s RH p m i 1,...,N, Q 2d0 s RH m m Q 2di s RH m mi 1,...,N hold true. Thus, it is shown that the feedback controller C 2 s is written by 13 Q 2 s H p m is written by 14, where Q 2n0 s RH p m, Q 2nis RH p m i 1,...,N, Q 2d0 s RH m m Q 2di s RH m m i 1,...,N. From the assumption that transfer functions in written by V 1 sds Y s Q 2 sñs C 1 s 37 V 4 sns Y s Q 2 sñs C 1 s 38 are stable, unstable poles of C 1 s are included in unstable zeros of Y s Q 2 sñs. Therefore, the feed-forward controller C 1 s is written by C 1 s Y s Q 2 sñs 1 Q1 s, 39 where Q 1 s H p m. Next, we show that 17, are satisfied. From 38 12, the transfer function V er s from the periodic reference input to the error es ys is written as V er s I NsQ 1 s. 40 From the definition of stabilizing two-degree-of-freedom multi-period repetitive controllers for multiple-input/multipleoutput systems in Definition 1 40, 17 holds true. From 7, 30, 31, 32 33, 18 is satisfied. From the assumption that rank C 2i s mi 1,...,N, 20 holds true. Thus, the necessity has been shown. Next, the sufficiency is shown. That is, it is shown that if the feed-forward controller C 1 s the feedback controller C 2 s in 11 takes the form 12 13, then the controller Cs makes the two-degree-of-freedom control system in Fig. 1 stable, the feedback controller C 2 s is written by the form in 6 proper low-pass filters q i si 1,...,N in 6 satisfy N i1 q i0 I. After simple manipulation, we have V 1 s DsQ 1 s, 41 V 2 s Xs+DsQ2 s Ñs, 42 V 3 s Ds Xs+Q 2 s Ds, 43 V 4 s NsQ 1 s, 44 V 5 s Ỹ s NsQ 2 s Ñs

5 V 6 s Ns Xs+Q 2 s Ds. 46 Since Ns RH m p, Ds RH p p, Ñs RH m p, Ds RH m m, Xs RHp m, Y s RHp p, Xs RH p m, Ỹ s RH m m, Q 1 s H p m Q 2 s H p m, transfer functions in are stable. From 44 17, the transfer function V er s from the periodic reference input to the error es ys written by V er s I NsQ 1 s 47 satisfies 9. Next, we show that the feedback controller C 2 s in 13 works as a multi-period repetitive controller. The feedback controller C 2 s in 13 is rewritten by the form in 6, where C 20 s XsQ2d0 s+dsq 2n0 s 1 Ỹ sq2d0 s NsQ 2n0 s, 48 C 2i s Y s Q 2n0 sq 1 2d0 sñs 1 Q2ni s Q 2n0 sq 1 2d0 sq 2dis Ỹ sq2d0 s NsQ 2n0 s 1 i 1,...,N 49 q i s Ỹ sq2di s NsQ 2ni s 1 Ỹ sq2d0 s NsQ 2n0 s i 1,...,N. 50 From the assumption of 20, rank C 2i s mi 1,...,N hold true. In addition, from 50 the assumption in 18, N i1 q i0 I is satisfied. These expressions imply that the feedback controller C 2 s in 13 works as a multi-period repetitive controller. Thus, the sufficiency has been shown. We have thus proved Theorem CONTROL CHARACTERISTICS In this section, we describe control characteristics of twodegree-of-freedom control system in Fig. 1 using the stabilizing two-degree-of-freedom multi-period repetitive controller Cs for multiple-input/multiple-output system in 11 with the feed-forward controller C 1 s in 12 the feedback controller C 2 s in 13. First, we mention the input-output characteristic. The transfer function from the periodic reference input to the error es ys is written by V er si NsQ 1 s. 51 From 51 17, for ω k k 0,...,n in 10, which are frequency components of the periodic reference input, the output ys follows the periodic reference input with small steady state error. Next, we mention the disturbance attenuation characteristic. The transfer function Ss from the disturbance d 1 s to the output ys is written by Ss I + Ỹ sq2di s NsQ 2ni s i1 Ỹ sq2do s NsQ 2n0 s 1 e st i Ỹ sq2d0 s NsQ 2n0 s 1 Q 2d0 s+ Q 2di se sti Ñs. 52 i1 From 52, for ω k k 0,...,n in 10 of frequency components of the disturbance d 1 s those are same to those of the periodic reference input, if σ I + Ỹ jωk Q 2di jω k Njω k Q 2ni jω k i1 1 Ỹ jωk Q 2do jω k NsQ 2n0 jω k 0 k 0,...,n, 53 then the disturbance d 1 s is attenuated effectively. In order to satisfy 53, for example, Q 2ni s is settled by Q 2ni sn o sỹ sq 2dis N o s q dis Ỹ sq2d0 s NsQ 2n0 s i 1,...,N, 54 where N o s RH m p is an outer function of Ns satisfying Ns N i sn o s. N i s RH m m is an inner function satisfying N i 0 I σn i jω 1 ω R +. q di s i 0,...,N are proper low-pass filters satisfying σi N i jω d N i1 q dijω d 0 to make N o s q dis proper. For ω d of the frequency component of the disturbance d 1 s that is different from that of the periodic reference input, that is ω d ω k k 0,...,n, even if σ I + Ỹ jωd Q 2di jω d Njω d Q 2ni jω d i1 Ỹ jωd Q 2do jω d Njω d Q 2n0 jω d 1 0, 55 the disturbance d 1 s cannot be attenuated, because e jω dti 1 i 1,...,N 56 σ I + Ỹ jωd Q 2di jω d Njω d Q 2ni jω d i1 5757

6 Ỹ jωd Q 2do jω d Njω d Q 2n0 jω d 1 e jω d T i In order to attenuate this frequency component, we need to settle Q 2d0 s Q 2n0 s ofq 2 s in 14 satisfying σ Ỹ jωd Q 2d0 jω d Njω d Q 2n0 jω d In order to satisfy 58, Q 2n0 s is settled by Q 2n0 s N o s q d0sỹ sq 2d0s, 59 where q d0 s is proper low-pass filter satisfying σi N i jω d q d0 jω d 0 to make N o s q d0s proper. From above discussion, the role of Q 1 s in 12 is different from that of Q 2 s in The role of Q 1 s is to specify the input-output characteristic for the periodic reference input. The role of Q 2 s is to specify the disturbance attenuation characteristic. Especially, the role of Q 2ni si 1,...,N Q 2di si 1,...,Nisto specify for the disturbance d 1 s of which the frequency component is equivalent to that of the periodic reference input. The role of Q 2n0 s Q 2d0 s is to specify for the disturbance d 1 s of which the frequency component is different from that of the periodic reference input. 5. CONCLUSION In this paper, we proposed the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers for multiple-input/multiple-output systems. Control characteristics of the control system using the proposed parameterization were presented. Roles of free parameters were clarified. A more detailed design method for control system which satisfies using the proposed parameterization a numerical example are described in another article. REFERENCES T. Inoue et al. High accuracy control magnet power supply of proton synchrotron in recurrent operation. The Trans. of The Institute of Electrical Engineers of Japan, volume C100-7, pages , T. Inoue, S. Iwai, M. Nakano. High accuracy control of play-back servo system. The Trans. of The Institute of Electrical Engineers of Japan, volume C101-4, pages 89 96, S. Hara, T. Omata, M. Nakano. Stability condition synthesis methods for repetitive control system. 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