THE PARAMETERIZATION OF ALL ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS FOR MIMO TD PLANTS WITH THE SPECIFIED INPUT-OUTPUT CHARACTERISTIC

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1 International Journal of Innovative Computing, Information Control ICIC International c 218 ISSN Volume 14, Number 2, April 218 pp THE PARAMETERIZATION OF ALL ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS FOR MIMO TD PLANTS WITH THE SPECIFIED INPUT-OUTPUT CHARACTERISTIC Yun Zhao 1, Yinglu Lyu 1, Takaaki Suzuki 2 Kou Yamada 2 1 Graduate School of Science Technology 2 Division of Mechanical Science Technology Gunma University Tenjincho, Kiryu , Japan { t ; t171b86; suzuki.taka; yamada }@gunma-u.ac.jp Received July 217; revised November 217 Abstract. In this paper, we investigate the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output time-delay plants with specified input-output characteristic. The multi-period repetitive control system is a type of servomechanism for a periodic reference input. When multi-period repetitive control design methods are applied to real systems, the influence of uncertainties in the plant must be considered. In some cases, uncertainties in the plant make the multi-period repetitive control system unstable, even though the controller was designed to stabilize the nominal plant. The stability problem with uncertainty is known as the robust stability problem. Recently, the parameterization of all robust stabilizing multi-period repetitive controllers for time-delay plants was obtained by Chen et al. In addition, Sakanushi et al. proposed that for multiple-input/multiple-output time-delay plants. However, using their method, it is difficult to specify the low-pass filter in the internal model for the periodic reference input of which the role is to specify the input-output characteristic, because the low-pass filter is related to four free parameters in the parameterization. To specify the input-output characteristic easily, this paper proposes the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output time-delay plants with specified input-output characteristic such that the input-output characteristic can be specified beforeh. Keywords: Repetitive control, Multi-period repetitive controller, Parameterization, Robust stability, Multiple-input/multiple-output systems 1. Introduction. A modified repetitive control system is a type of servomechanism for a periodic reference input, 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 [1, 2, 3, 4, 5]. However, the modified repetitive control system has a bad effect on the disturbance attenuation characteristic [6], in that at certain frequencies, the sensitivity to disturbances of a control system with a conventional repetitive controller becomes twice as worse as that of a control system without a repetitive controller. Gotou et al. overcame this problem by proposing a multi-period repetitive control system [6]. However, the phase angle of the low-pass filter in a multi-period repetitive controller has a bad effect on the disturbance attenuation characteristic [7, 8]. Yamada et al. overcame this problem proposed a design method for multi-period repetitive controllers to attenuate disturbances effectively [9, 1] using the time advance compensation described in [7, 8, 11]. Using this multiperiod repetitive control structure, Steinbuch proposed a design method for repetitive control systems with uncertain period time [12]. 387

2 388 Y. ZHAO, Y. LYU, T. SUZUKI AND K. YAMADA As we know, the multi-period repetitive controller design procedure is usually complicated by the fact that only those controllers are allowed for which the closed-loop system is stable. References [13, 14, 15, 16, 17, 18, 19, 2] showed that it was possible to parameterize all stabilizing controllers for a particular system in a very effective manner. When designing a controller with specific properties, one can simply without loss of generality search over the space of all stable transfer functions. The parameterization, which is based on the coprime factorization of the plant, has been widely used for designing control systems, provides an elegant efficient way towards solving the stabilizing design problem, with which all stabilizing controllers are characterized thus a constrained design procedure can be replaced by an unconstrained optimization. The parameterization of all stabilizing multi-period repetitive controllers was solved in [21, 22]. When multi-period repetitive control design methods are applied to real systems, the influence of uncertainties in the plant must be considered. In some cases, uncertainties in the plant make the multi-period repetitive control system unstable, even though the controller was designed to stabilize the nominal plant. The stability problem with uncertainty is known as the robust stability problem [23]. However, multi-period repetitive controllers in [21, 22] cannot guarantee the stability of control system for plants with uncertainties. Satoh et al. proposed the parameterization of all robust stabilizing multi-period repetitive controllers for plants with uncertainties [24]. However, the method in [24] cannot guarantee the stability of control system for time-delay plants with uncertainties. To solve this problem, Chen et al. proposed the parameterization of all robust stabilizing multi-period repetitive controllers for time-delay plants [25]. Sakanushi et al. exped the result in [25] proposed the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output time-delay plants [26]. However, using the method in [26], it is complex to specify the low-pass filter in the internal model for the periodic reference input of which the role is to specify the inputoutput characteristic, because the low-pass filter is related to four kinds of free parameters in the parameterization proposed by Sakanushi et al. When we design a robust stabilizing multi-period repetitive controller, if the low-pass filter is set beforeh, we can specify the input-output characteristic more easily than in the method employed in [26]. This is achieved by parameterizing all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output time-delay plants with specified input-output characteristic, which is the parameterization when the low-pass filter is set beforeh. However, no paper has considered the problem of obtaining the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output time-delay plants with specified input-output characteristic. In addition, the parameterization is useful to design stabilizing controllers [13, 14, 15, 16, 17]. Therefore, the problem of obtaining the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output time-delay plants with specified input-output characteristic is important to solve. In this paper, we propose the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output time-delay plants with specified input-output characteristic such that the low-pass filter in the internal model for the periodic reference input is set beforeh. A parameterization is derived on the basis of the definition of the internal stability. This paper is organized as follows. In Section 2, the plant under consideration is formally defined some important background is introduced. In Section 3, the parameterization is derived for MIMO plants with time-delays. In Section 4, some control characteristics are explained. In Section 5 some applications of the parameterization are discussed. Finally, conclusions are given in Section 6.

3 ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS 389 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 rational functions. H the set of stable causal functions. D orthogonal complement of D, i.e., [ D D ] [ ] D or D is unitary. A T transpose of A. A pseudo inverse of A. ρ{ }) spectral radius of { }. σ{ }) the largest singular value of { }. { } H norm of { }. [ ] A B represents the state space description CsI A) C D B + D. L{ } the Laplace transformation of { }. L 1 { } the inverse Laplace transformation of { }. 2. Problem Formulation. Consider the unity feedback control system in { y = Gs)e sl u + d, 1) u = Cs)r y) where Gs)e sl is the multiple-input/multiple-output time-delay plant, L > is the time-delay, Gs) R m p s) is assumed to be stabilizable detectable. Cs) is the multi-period repetitive controller with the m-th input p-th output defined later, u R p is the control input, d R m is the disturbance, y R m is the output r R m is the periodic reference input with period T > satisfying rt + T ) = rt) t ). 2) It is assumed that m p rank Gs) = m. The nominal plant of Gs)e sl is denoted by G m s)e sl m, where G m s) R m p s). Both Gs) G m s) are assumed to have no zero or pole on the imaginary axis. In addition, it is assumed that the number of poles of Gs) in the closed right half plane is equal to that of G m s). The relation between the plant Gs)e sl the nominal plant G m s)e slm is written as Gs)e sl = e slm I + s) ) G m s), 3) where s) is an uncertainty. The set of s) is all functions satisfying σ { jω)} < W T jω) ω R + ), 4) where W T s) Rs) is a stable rational function. The robust stability condition for the plant Gs) with uncertainty s) satisfying 4) is given by where T s) is given by T s)w T s) < 1, 5) T s) = I + G m s)e sl m Cs) ) 1 Gm s)cs). 6) According to [6, 9, 1, 21, 22], the general form of multi-period repetitive controller Cs) which makes the output y follow the periodic reference input r with period T in 1)

4 39 Y. ZHAO, Y. LYU, T. SUZUKI AND K. YAMADA with small steady state error, is written by N Cs) = C s) + C i s)c ri s), 7) where N is an arbitrary positive integer, T i > R i = 1,..., N), C s) R p m s), C i s) R p m s) i = 1,..., N) satisfying rank C i s) = m i = 1,..., N), C ri s) i = 1,..., N) is the internal model for the periodic signal with period T written as C ri s) = q i s)e st i I N 1 q i s)e i) st, 8) where q i s) R m m s) i = 1,..., N) is low-pass filter satisfying N q i) = I rank q i s) = m i = 1,..., N). In order to compare the difference between the characteristic of the internal model of modified repetitive controller that of the internal model of multi-period repetitive controller, we show Bode plots of 1 qs)e T s) N q is)e i) st in Figure 1 Figure 2, respectively, where qs) = 1/.1s + 1), T = 1, N = 5, q 1 s) = qs), q i s) = q 1 s) {.1s/.1s+1)} i 1 i = 2,..., N) T i = T i i = 1,..., N). Figures 1 2 show that, the gains of multi-period structure are higher than those of singleperiod structure. Therefore, in order to obtain good tracking precision or disturbance attenuation performance, the multi-period structure is employed. In addition, from the same reason, we find that even if the period T has some uncertainty, using the multiperiod repetitive controller, we have good tracking precision or disturbance attenuation performance. The general form of the multi-period repetitive controller Cs) is shown in Figure 3. Gotou et al. [6] proposed the design method for multi-period repetitive controller as T i = T i i = 1,..., N). 9) On the other h, Yamada et al. [1] proposed the design method for multi-period repetitive controller such that T i i = 1,..., N) do not necessarily satisfy 9). Therefore, Gain [db] ω [rad/s] Figure 1. Bode plots of 1 qs)e T s) 1

5 ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS Gain [db] Figure 2. Bode plots of ω [rad/s] 1 ) 1 N q is)e st i P N Ci s) q i s) Cs) C s) e àst i + P N qi s)e àst i Figure 3. Structure of a multi-period repetitive controller in this paper, we attach importance to the generality assume that T i i = 1,..., N) do not necessarily satisfy 9). According to [6, 9, 1, 21, 22], if the low-pass filters q i s) i = 1,..., N) satisfy { } N σ I q i jω k ) k =, 1,..., n), 1) where ω k i =, 1,..., n) is the frequency component of the periodic reference input r written by ω k = 2π T k k =, 1,..., n) 11) ω n is the maximum frequency component of the periodic reference input r, then the output y in 1) follows the periodic reference input r with small steady state error. Using the result in [26], in order for q i s) i = 1,..., N) to satisfy 1) in wide frequency range, we must design q i s) i = 1,..., N) to be stable of minimum phase. If we obtain the parameterization of all robust stabilizing multi-period repetitive controllers such that

6 392 Y. ZHAO, Y. LYU, T. SUZUKI AND K. YAMADA q i s) i = 1,..., N) in 7) is settled beforeh, we can design the robust stabilizing multi-period repetitive controller satisfying 1) more easily than the method in [26]. The problem considered in this paper is to propose the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output timedelay plants with specified input-output characteristic. That is, when q i s) RH m m i = 1,..., N) are set beforeh, we obtain the parameterization of all controllers Cs) in 7) satisfying 5) for multiple-input/multiple-output time-delay plants Gs)e sl in 3) with any uncertainty s) satisfying 4). 3. The Parameterization. In this section, we clarify the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output timedelay plants with specified input-output characteristic. w u Ps) z y Cs) Figure 4. Block diagram of H control problem In order to obtain the parameterization of all robust stabilizing multi-period repetitive controllers for time-delay plants with specified input-output characteristic, we must see that controllers Cs) satisfy 5). The problem of obtaining the controller Cs), which is not necessarily a multi-period repetitive controller, satisfying 5) is equivalent to the following H control problem. In order to obtain the controller Cs) satisfying 5), we consider the control system shown in Figure 4. P s) is selected such that the transfer function from w to z in Figure 4 is equal to T s)w T s). The state space description of P s) is, in general, ẋt) = Axt) + B 1 wt) + B 2 ut L m ) zt) = C 1 xt) + D 12 ut) yt) = C 2 xt) + D 21 wt), 12) where A R n n, B 1 R n m, B 2 R n p, C 1 R m n, C 2 R m n, D 12 R m p, D 21 R m m, xt) R n, wt) R m, zt) R m, ut) R p yt) R m. P s) is called the generalized plant. P s) is assumed to satisfy following assumptions: 1) C 2, A) is detectable, A, B 2 ) is stabilizable. 2) D 12 has [ full column rank, ] D 21 has full row rank. A jωi B2 3) rank = n + p ω R C 1 D + ), [ 12 ] A jωi B1 rank = n + m ω R C 2 D + ). 21 4) C 1 A i B 2 = i =, 1, 2,...). Under these assumptions, from [27], the following lemma holds true.

7 ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS 393 Lemma 3.1. There exists an H controller Cs) for the generalized plant P s) in 12) if only if there exists an H controller Cs) for the generalized plant P s) written by qt) = Aqt) + B 1 wt) + B 2 ut) zt) = C 1 qt) + D 12 ut) ỹt) = C 2 qt) + D 21 wt), 13) where B 2 = e AL m B 2. When us) = Cs)ỹs) is an H control input for the generalized plant P s) in 13), ut) = L 1 {Cs)ỹs)} 14) is an H control input for the generalized plant P s) in 12), where } ỹs) = L {yt) + C 2 e Aτ+Lm) B 2 ut + τ)dτ. 15) L m From Lemma 3.1 [23], the following lemma holds true. Lemma 3.2. If controllers satisfying 5) exist, both X A B 2 D 12C ) 1 + A B 2 D 12C ) T { 1 X + X B 1 B1 T B 2 D T 12 D 12 ) 1 BT 2 } X + D 12C 1 ) T D 12 C 1 = 16) Y A B 1 D 21C ) T 2 + A B 1 D 21C ) { 2 + Y C1 T C 1 } C2 T D21 D21) T 1 C2 + B 1 D21 B1 D21) T = 17) have solutions X Y such that both A B 2 D 12C 1 + ρ XY ) < 1 18) {B 1 B T1 B 2 D T12 D 12 ) 1 BT2 } X 19) A B 1 D 21C 2 + Y { C T 1 C 1 C T 2 D21 D T 21) 1 C2 } 2) have no eigenvalue in the closed right half plane. Using X Y, the parameterization of all controllers satisfying 5) is given by where Cs) = C 11 s) + C 12 s)qs)i C 22 s)qs)) 1 C 21 s), 21) [ ] C11 s) C 12 s) = A c B c1 B c2 C C 21 s) C 22 s) c1 D c11 D c12, 22) C c2 D c21 D c22 A c = A + B 1 B1 T X B 2 D 12C 1 + E12 1 B ) 2 T X ) I Y X) 1 C2 B 1 D 21 + Y C2 T E D 21 B1 T X ), ) B c1 = I Y X) 1 B 1 D 21 + Y C2 T E21 1,

8 394 Y. ZHAO, Y. LYU, T. SUZUKI AND K. YAMADA B c2 = I Y X) 1 B2 + Y C T 1 D 12 ) E 1/2 12, C c1 = D 12C 1 E 1 12 B T 2 X, C c2 = E 1/2 21 C2 + D 21 B1 T X ), D c11 =, D c12 = E 1/2 12, D c21 = E 1/2 21, D c22 =, E 12 = D T 12D 12, E 21 = D 21 D T 21 Qs) H p m is any function satisfying Qs) < 1. Remark 3.1. Cs) in 21) is written using Linear Fractional Transformation LFT). Using homogeneous transformation, 21) is rewritten by Cs) = Z 11 s)qs) + Z 12 s)) Z 21 s)qs) + Z 22 s)) 1 = Qs) Z 21 s) + Z ) 1 22 s) Qs) Z 11 s) + Z ) 12 s), 23) where Z ij s) i = 1, 2; j = 1, 2) Z ij s) i = 1, 2; j = 1, 2) are defined by [ ] [ Z11 s) Z 12 s) C12 s) C = 11 s)c21 1 s)c 22 s) C 11 s)c21 1 s) Z 21 s) Z 22 s) C21 1 s)c 22 s) C21 1 satisfying [ Z11 s) Z12 s) Z 21 s) Z22 s) ] = [ C21 s) C 22 s)c 1 12 s)c 11 s) C 1 [ Z22 s) Z12 s) 12 s)c 11 s) 12 s) C 22 s)c 1 12 s) C 1 ] [ Z11 s) Z 12 s) ] ] ] 24) 25) Z 21 s) Z11 s) Z 21 s) Z 22 s) [ ] [ Z11 s) Z = I = 12 s) Z22 s) Z12 s) Z 21 s) Z 22 s) Z 21 s) Z11 s) ]. 26) Using Lemma 3.1, Lemma 3.2 Remark 3.1, the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output time-delay plants with specified input-output characteristic is given by following theorem. Theorem 3.1. If multi-period repetitive controllers satisfying 5) exist, both 16) 17) have solutions X Y such that 18) both 19) 2) have no eigenvalue in the closed right half plane. Using X Y, the parameterization of all robust stabilizing multi-period repetitive control laws with specified input-output characteristic satisfying 5) is given by where ut) = L 1 {Cs)ỹs)}, 27) } ỹs) = L {yt) + C 2 e Aτ+Lm) B 2 ut + τ)dτ L m 28) Cs) = Z 11 s)qs) + Z 12 s)) Z 21 s)qs) + Z 22 s)) 1 = Qs) Z 21 s) + Z ) 1 22 s) Qs) Z 11 s) + Z ) 12 s), 29)

9 ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS 395 where Z ij s) i = 1, 2; j = 1, 2) Z ij s) i = 1, 2; j = 1, 2) are defined by 24) 25) satisfying 26), C ij s) i = 1, 2; j = 1, 2) are given by 22) Qs) H p m is any function satisfying Qs) < 1 written by ) 1 N N Qs) = Q n s) + Q ni s)q i s)e st i Q d s) + Q di s)q i s)e i) st, 3) Q n s) RH p m RH m m, Q d s) RH m m, Q ni s) RH p m i = 1,..., N) Q di s) i = 1,..., N) are any functions satisfying rank {Z 11 s) Q n s) + Q ni s)) + Z 12 s) Q d s) + Q di s))} = m i = 1,..., N) 31) Proof: First, the necessity is shown. That is, we show that if the multi-period repetitive controller Cs) in 7) stabilizes the control system in 1) robustly q i s) i = 1,..., N) are set beforeh, then Cs) is written by 29) 3), respectively. From Lemma 3.2 Remark 3.1, the parameterization of all robust stabilizing controllers Cs) for Gs)e sl is written by 29), where Qs) < 1. In order to prove the necessity, we will show that if Cs) written by 7) stabilizes the control system in 1) robustly q i s) i = 1,..., N) are set beforeh, then Qs) in 29) is written by 3). Substituting Cs) in 7) for 29), we have 3), where Here, N n s) RH p m RH m m Q n s) = N n s)n d s), 32) Q ni s) = N in s) + N n s)n d s) i = 1,..., N), 33) Q d s) = D n s)d d s)n d s)n d s) 34) Q di s) = D in s) D n s)d d s)) N d s)n d s) i = 1,..., N). 35), N in s) RH p m, D n s) RH m m, D in s) RH m m are coprime factors satisfying D d s) RH m m i = 1,..., N), N d s) RH m m, N d s) i = 1,..., N), D d s) RH m m Z 21 s)c s) Z 11 s) = D n s)d 1 d s), 36) Z 21 s)c i s)d d s) = D in s)d 1 d s) i = 1,..., N), 37) Z22 s)c s) Z 12 s)) D d s)d d s) = N n s)n 1 d s) 38) Z 22 s)c i s)d d s)d d s)n d s) = N in s)n 1 d s) i = 1,..., N). 39) From 32)-35), all of Q n s), Q ni s) i = 1,..., N), Q d s) Q di s) i = 1,..., N) are included in RH. Thus, we have shown that if Cs) written by 7) stabilizes the control system in 1) robustly q i s) i = 1,..., N) are set beforeh, Qs) in 29) is written by 3). From the assumption of rank C i s) = m i = 1,..., N) from 37) 39), rank D in s) = m i = 1,..., N) 4) rank N in s) = m i = 1,..., N) 41) hold true. From 4), 41), 33) 35), 31) is satisfied. Thus, the necessity has been shown. Next, the sufficiency is shown. That is, it is shown that if Cs) Qs) H p m are settled by 29) 3), respectively, then the controller Cs) is written by the form in

10 396 Y. ZHAO, Y. LYU, T. SUZUKI AND K. YAMADA 7) rank C i s) = m i = 1,..., N) hold true. Substituting 3) into 29), we have 7), where C s) C i s) i = 1,..., N) are denoted by C s) = Z 11 s)q n s) + Z 12 s)q d s)) Z 21 s)q n s) + Z 22 s)q d s)) 1, 42) C i s) = {Z 11 s) Q n s) + Q ni s)) + Z 12 s) Q d s) + Q di s))} Z 21 s)q n s) + Z 22 s)q d s)) 1 i = 1,..., N). 43) We find that if Cs) Qs) are settled by 29) 3), respectively, then the controller Cs) is written by the form in 7). From 31) 43), holds true. Thus, the sufficiency has been shown. We have thus proved Theorem 3.1. rank C i s) = m i = 1,..., N) 44) 4. Control Characteristics. In this section, we explain control characteristics of the control system in 1) using the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output plants. In addition, roles of Q n s), Q ni s), Q d s) Q di s) in 3) are clarified. From Theorem 3.1, Qs) in 3) must be included in H. Since Q ni s) RH in 3), if Q d s) + 1 N Q dis)q i s)e i) st H, then Qs) satisfies Qs) H. That is, the role of Q d s) Q di s) is to assure Qs) H, the role of Q n1 s) Q n2 s) is to guarantee Qs) < 1. Next, the input-output characteristic of the control system in 1) is shown. The transfer function Ss) from the periodic reference input rs) to the error es) = rs) ys) of the control system in 1) is written by where Sns) = { I S d s) = Z 21 s)q n s) + Ss) = S n s)s 1 d s), 45) } N q i s)e st i C21 1 s) C 22 Q n s) + Q d s)) 46) N Z 21 s)q ni s) + Z 22 s)q di s)) q i s)e st i + Z 22 s)q d s) + Gs) + { Z 11 s)q n s) + Z 12 s)q d s) } N Z 11 s)q n2 s) + Z 12 s)q d2 s)) q i s)e st i. 47) According to 46), if Q n s), Q d s), Q ni s) Q di s) are selected satisfying 1), then σ { S n jω i ) } σ {I qjω i )} σ { C 1 21 jω i ) } σ { C 22 Q n1 jω i ) + Q d1 jω i )) }, 48) the output ys) follows the periodic reference input rs) with frequency components without steady state error. ω i = 2π i i =, 1,..., ) 49) T

11 ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS 397 Next, the disturbance attenuation characteristic of the control system in 1) is shown. The transfer function from the disturbance ds) to the output ys) of the control system in 1) is written by 45). From 45), for ω i i =, 1,..., ) in 1) of the frequency component of the disturbance d 1 s) that is the same as that of the periodic reference input rs), if 48) holds, then the disturbance ds) is attenuated effectively. This implies that the disturbance with the same frequency component ω i i =, 1,..., ) of the periodic reference input rs) is attenuated effectively. That is, the role of Q n2 s) Q d2 s) is to specify the disturbance attenuation characteristic for the disturbance with the same frequency component ω i i =, 1,..., ) of the periodic reference input rs). When the frequency components of disturbance ds), ω k k =, 1,..., h), are not equal to ω i i =, 1,..., ), even if the disturbance ds) cannot be attenuated, because σ {I qjω k )}, 5) e j ω kt 1 51) σ { I q j ω k ) e j ω kt }. 52) In order to attenuate the frequency components ω k k =, 1,..., h) of the disturbance ds), we need to satisfy σ { C 22 j ω k ) Q n j ω k ) + Q d j ω k )}. 53) This implies that the disturbance ds) with frequency components ω k ω i i =, 1,...,, k =, 1,..., h) can be attenuated effectively. That is, the role of Q n1 s) Q d1 s) is to specify disturbance attenuation characteristics for disturbance of frequency ω d ω i i =, 1,..., ). From above discussion, the role of Q ni s) Q di s) is to specify the input-output characteristic for the periodic reference input rs) to specify for the disturbance ds) of which the frequency components are equivalent to that of the periodic reference input rs). The role of Q n s) Q d s) is to specify for the disturbance ds) of which the frequency components are different from that of the periodic reference input rs). 5. Numerical Example. In this section, numerical examples are made to illustrate the validity of the proposed approach. Consider the problem to obtain the parameterization of all robust stabilizing multi-period repetitive controllers for the set of plants Gs) written by 3), where G m s) = s + 3 s 2)s + 9) s + 3 s 2)s + 9) 2 s 2)s + 9) s + 4 s 2)s + 9) 54) W T s) = s ) The period T of the periodic reference input r is given by T = 1 [sec], the time-delay L m is given by L m = 5 [sec]. We settle N = 3 the low-pass filters of multiple-period repetitive controller in 8) as 1 q 1 s) = I, 56).2s + 1

12 398 Y. ZHAO, Y. LYU, T. SUZUKI AND K. YAMADA q 2 s) = q 1 s).2s q 3 s) = q 2 s).2s + 1 ).2s I 57).2s + 1 ) I. 58) Solving the robust stability problem using Riccati equation based H control as Theorem 3.1, the parameterization of all robust stabilizing controllers Cs) is obtained. In addition, we find that C 22 s) is of minimum phase as s s C 22 s) =. 59) s s Since C 22 s) is of minimum phase, we set Q n s), Q ni s), Q d s) Q di s) in 3) as Q d s) = I RH, 6) Q n s) = C 1 22 s) q d s) RH, 61) Q ni s) = RH 62) Q di s) = I q d s)) RH, 63) where q d s) is written by 1 q d s) = I. 64).2 s + 1 In order to verify that Qs) in 3) belongs to H satisfy Qs) < 1, we show the Nyquist plot of det Q d s) + ) N Q dis)q i s)e st i the largest singular value plot of Qs) are shown in Figure 5 Figure 6, respectively. Since the Nyquist plot of Im Re Figure 5. The Nyquist plot of det Q d s) + ) N Q dis)q i s)e st i

13 ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS Largest singular value Frequency[red/sec] Figure 6. The largest singular value plot of Qs) detq d1 s)+q d2 s)e st ) does not encircle the origin, we find that Qs) in 3) is included in H. Figure 6 illustrates σ{qjω)}.9432 < 1 ω R), i.e., Qs).9432 < 1. Using above-mentioned parameters, we have a robust stabilizing multi-period repetitive controller. When s) is given by s 1 1 s) = s s + 5 s + 5 s 1 s ) the designed robust stabilizing multi-period repetitive controller Cs) is used, the tracking error et) = rt) yt) in 1) for the periodic reference inputs ) 2π [ ] r1 t) sin rt) = = T t r 2 t) ) 2π 2 sin T t 66) is shown in Figure 7. Here, the largest steady state peak to peak relative error is.2%. These tracking errors are small can be further reduced by modifying the cutoff frequency of the low-pass filters qs) q d s). Next, the disturbance attenuation characteristic is shown. The response of the output yt) for the disturbance dt) of which the frequency components are equivalent to that of the periodic reference input rt) ) 2π ] sin dt) = [ d1 t) d 2 t) = 2 sin T t ) 2π T t 67)

14 4 Y. ZHAO, Y. LYU, T. SUZUKI AND K. YAMADA is shown in Figure 8. Figure 8 shows that the disturbance dt) is attenuated effectively. Next we show that the response of the output yt) for the disturbance dt) of which the frequency components are different from that of the periodic reference input rt). The response of the output yt) for the disturbance dt) of which the frequency components Figure 7. Response of the error et) for the reference input rt) in 66) Disturbances d1t), d2t) d 1 t) d 2 t) Time[s] Output y1t), y2t) y 1 t) y 2 t) Time[s] Figure 8. Response of the output yt) for the disturbance dt) in 67)

15 ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS 41 Disturbance d1t), d2t) d 1 t) d 2 t) Time[s] Output y1t), y2t) y 1 t) y 2 t) Time[s] Figure 9. Response of the output yt) for the disturbance dt) in 68) are different from that of the periodic reference input rt) ) 2π [ ] d1 t) sin dt) = = 1.7T t d 2 t) ) 2π 2 sin 1.7T t 68) is shown in Figure 9. Figure 9 shows that even if the frequency components of the disturbance ds) are different from that of the periodic reference input, the disturbance dt) is attenuated effectively. The results shown in Figure 7, Figure 8 Figure 9 demonstrate that the design method presented here provided not only good robustness, but also satisfactory tracking performance for the reference inputs attenuation performance for the disturbance. In some cases, the period T of the reference input has uncertainty. Next, we show the responce of the tracking error when the period T of the reference input has uncertainty. The tracking error et) = rt) yt) in 1) for the periodic reference inputs r ) 2π [ ] r1 t) sin rt) = = T + T t r 2 t) ) 2π 2 sin T + T t 69) is shown in Figure 1, where T is an uncertainty of the period of the reference input. Here, the largest steady state peak to peak relative errors are.1%,.2%,.1%.5% for T =.5T, T =, T =.3T T =.7T, respectively. These tracking errors are small can be further reduced by modifying the cutoff frequency of the lowpass filters qs) q d s). The results shown in Figure 1 demonstrate that the design

16 42 Y. ZHAO, Y. LYU, T. SUZUKI AND K. YAMADA Figure 1. Response of the error et) for the reference input rt) in 69) method presented here provided satisfactory tracking performance for the reference inputs with uncertain period. 6. Conclusions. In this paper, we proposed the parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output time-delay plants with specified input-output characteristic such that low-pass filters in the internal model for the periodic reference input are settled beforeh. Advantages of the multi-period repetitive control system using the proposed parameterization are that its input-output characteristic is easily specified the system to guarantee the robust stability is easy to design. This control system is expected to have practical applications in, for example, engines, electrical motors generators, converters, other machines that perform cyclic tasks. Acknowledgment. The authors would like to express their gratitude to the anonymous reviewers Dr. Z. X. Chen of Hunan Normal University for their valuable comments suggestions, which helped to improve the paper greatly. REFERENCES [1] S. Hara, T. Omata M. Nakano, Stability condition synthesis methods for repetitive control systems, Trans. of Society of Instrument Control Engineers, vol.22, no.1, pp.36-42, [2] Y. Yamamoto S. Hara, The internal model principle stabilizability of repetitive control systems, Trans. of Society of Instrument Control Engineers, vol.22, no.8, pp , [3] S. Hara Y. Yamamoto, Stability of multivariable repetitive control systems Stability condition class of stabilizing controllers, Trans. of Society of Instrument Control Engineers, vol.22, no.12, pp , [4] S. Hara, Y. Yamamoto, T. Omata M. Nakano, Repetitive control system: A new type of servo system for periodic exogenous signals, IEEE Trans. Automatic Control, vol.ac-33, no.7, pp , 1988.

17 ROBUST STABILIZING MULTI-PERIOD REPETITIVE CONTROLLERS 43 [5] G. Weiss, Repetitive control systems: Old new ideas, Systems Control in the Twenty-First Century, pp , [6] M. Gotou, S. Matsubayashi, F. Miyazaki, S. Kawamura S. Arimoto, A robust system with an iterative learning compensator a proposal of multi-period learning compensator, J. Soc. Instrument Control Engineers, vol.31, no.5, pp , [7] H. Sugimoto K. Washida, A proposition of modified repetitive control with corrected dead time, Trans. of Society of Instrument Control Engineers, vol.34, no.6, pp , [8] H. Sugimoto K. Washida, A design method for modified repetitive control with corrected dead time, Trans. of Society of Instrument Control Engineers, vol.34, no.7, pp , [9] T. Okuyama, K. Yamada K. Satoh, A design method for repetitive control systems with a multi-period repetitive compensator, Theoretical Applied Mechanics Japan, vol.51, pp , 22. [1] K. Yamada, K. Satoh, T. Arakawa T. Okuyama, A design method for repetitive control systems with multi-period repetitive compensator, Transactions of the Japan Society of Mechanical Engineers Series C, vol.69, no.686, pp , 23 in Japanese). [11] H. L. Broberg R. G. Molyet, A new approach to phase cancellation in repetitive control, Proc. of the 29th IEEE IAS, pp , [12] M. Steinbuch, Repetitive control for systems with uncertain period-time, Automatica, vol.38, pp , 22. [13] D. C. Youla, J. J. Bongiorno H. Jabr, Modern Wiener-Hopf design of optimal controllers, Part I: The single-input-output case, IEEE Trans. Automatic Control, vol.ac-21, pp.3-13, [14] V. Kucera, Discrete Linear System: The Polynomial Equation Approach, Wiley, [15] C. A. Desoer, R. W. Liu, J. Murray R. Saeks, Feedback system design: The fractional representation approach to analysis synthesis, IEEE Trans. Automatic Control, vol.ac-25, pp , 198. [16] J. J. Glaria G. C. Goodwin, A parameterization for the class of all stabilizing controllers for linear minimum phase system, IEEE Trans. Automatic Control, vol.ac-39, pp , [17] M. Vidyasagar, Control System Synthesis A Factorization Approach, MIT Press, [18] K. Yamada T. Moki, The parameterization of all internally stabilizing controllers for linear minimum phase systems their characteristic, Transactions of the Japan Society of Mechanical Engineers Series C, vol.68, pp , 22 in Japanese). [19] K. Yamada, K. Satoh, Y. Mei, T. Hagiwara, I. Murakami Y. Ando, The parametrization for the class of all proper internally stabilizing controllers for multiple-input/multiple-output minimum phase systems, ICIC Express Letters, vol.3, no.1, pp.67-72, 29. [2] Y. Zhao, T. Sakanushi, K. Yamada, N. Smitthimedhin H. Yamaguchi, The parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers for multiple-input/multipleoutput plants with specified frequency characteristic, Proc. of the 54th Annual Conference of the Society of Instrument Control Engineers of Japan SICE), pp , 215. [21] K. Yamada, K. Satoh T. Arakawa, The parameterization of all stabilizing multiperiod repetitive controllers, Int. Conf. Cybernetics Information Technologies, System Applications, vol.2, pp , 24. [22] K. Yamada, K. Satoh T. Arakawa, A design method for multiperiod repetitive controllers design method using the parameterization of all multiperiod repetitive controllers), Transactions of the Japan Society of Mechanical Engineers Series C, vol.71, no.71, pp , 25. [23] J. C. Doyle, K. Glover, P. P. Khargonekar B. A. Francis, State-space solution to stard control problems, IEEE Trans. Automatic Control, vol.ac-34, pp , [24] K. Satoh, K. Yamada Y. Mei, The parametrization of all robust stabilizing multi-period repetitive controllers, Theoretical Applied Mechanics, vol.55, pp , 26. [25] Z. Chen, T. Sakanushi, K. Yamada, S. Tohnai, N. L. T. Nguyen Y. Zhao, The parameterization of all robust stabilizing multi-period repetitive controllers for multiple-input/multiple-output plants, ICIC Express Letters, vol.6, no.1, pp.41-46, 212. [26] T. Sakanushi, Y. Zhao, K. Yamada N. Smitthimedhin, Robust stabilizing multi-period repetitive controllersfor multiple-input/multiple-output time-delay plants, Proc. of the 11th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications Information Technology, 214. [27] N. Abe A. Kojima, Control in Time-Delay Distributed Parameter Systems, Corona Publishing, 27.