Adaptive Repetitive Control Systems for Fast Rejection of Periodic Disturbances with Uncertain Multiple Periods

Periodic Control Systems — PSYCO 2010 Antalya, Turkey, August 26-28, 2010

Adaptive Repetitive Control Systems for Fast Rejection of Periodic Disturbances with Uncertain Multiple Periods Manabu Yamada*, Tomoyasu Yabuki** and Naoki Mizuno*** * Research Center for Nano Device and System, Nagoya Institute of Technology, Showa, Nagoya, 466-8555, JAPAN (e-mail: [email protected] ). ** Brother Industries, LTD., 15-1, Naeshiro-cho, Mizuho-ku,Nagoya,467-8561, JAPAN. *** Department of Scientific and Engineering Simulation, Graduate School of Engineering, Nagoya Institute of Technology,Showa,Nagoya, 466-8555, JAPAN (e-mail: [email protected]) Abstract: In this paper, a discrete-time repetitive control problem is considered for the cases where the disturbance consists of periodic signals with several different periods but the periods are uncertain. A new adaptive repetitive control system for such cases is proposed and consists of three elements; a stabilizing controller, a period estimator and an adaptive reduced order periodic signal generator. The controller not only assures the stability of the overall adaptive repetitive control system including the time-varying generator but also assigns all poles of the closed-loop system on the disk with a given radius. The design problem is reduced to a simple and feasible l1 norm minimization problem. The estimator estimates multiple periods of disturbance from the input/output data of the plant. The algorithm consists of the well-known recursive least squares method. In order to reject periodic disturbances with uncertain multiple periods, the adaptive generator is adjusted on-line by using the period estimated. The repetitive controller with the generator is implemented with much less memory elements than the previous ones. As a result, the degree of the controller is significantly reduced and much faster convergence of the controlled error to zero is provided. The effectiveness is demonstrated by simulations. Keywords: adaptive control, time-varying systems, estimation parameters, disturbance rejection, stabilization, pole assignment 1. INTRODUCTION The repetitive control system is a servo system that achieves zero steady-state tracking error for any periodic desired outputs and any periodic disturbance inputs with given periods. The repetitive control scheme was first proposed by Inoue et al. (1981) and then has been developed by many researchers, for example, Hara et al (1988). This paper is concerned with a design problem of a single-input singleoutput (SISO) discrete-time repetitive control system. From the internal model principle, it has been widely accepted that a repetitive control system requires a generator of periodic signals with given periods and a stabilizing controller, which stabilizes the closed loop system. Tomizuka et al. (1989) has proposed a simple stabilizing controller based on stable pole/zero cancellation and zero phase error compensation. The controller is obtained in an explicit form and the design method involves no equation to be solved. In Chew and Tomizuka (1990), Tsao and Tomizuka (1994) and Yamada et al. (1999), the controller of Tomizuka et al. (1989) has been extended from the viewpoints of stochastic performance, robust one and pole placement, respectively. In Yamada et al. (2000), the method of Yamada et al. (1999) has been extended to rejection of multiple-period periodic signals. However, these previous methods are restricted to the case where the periods of the periodic signals to be regulated are exactly known. Hu (1992), Tsao et al. (1992, 2000) and

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Yamada and Tomizuka (2004) have considered the case where the period of periodic signals is uncertain, and have proposed adaptive repetitive control systems with recursive identification of the period to reject periodic signals with uncertain period. In Yamada and Tomizuka (2004), an adaptive repetitive control system has been presented to guarantee the robust stability of the overall closed-loop system including a time-varying periodic signal generator against plant uncertainties. However, the number of periods of the periodic generators used in these methods is restricted to be single. As a result, if the periodic signals to be rejected consist of several different periods, then the generator is required to have the dead-time length equal to the least common multiple (l.c.m) of all the periods. This is undesirable from the practical viewpoint, since the size of memory for implementation of the dead-time elements may be large and the degree of the controller may be excessively high. In Yamada et al. (2004, 2009), adaptive control systems for multiple-period periodic disturbances have been presented, but the relationship between the design parameters and the convergence rate of the controlled error is not made clear. This paper deals with a repetitive control problem for the cases where the disturbance input consists of periodic signals with several different periods and the periods are uncertain but are within known lower and upper bounds. A new adaptive repetitive control system for such cases is proposed and consists of three elements; a stabilizing controller for the

1

overall adaptive system, a period estimator and an adaptive reduced order generator. The details are as follows. Firstly, the controller assures the stability of the overall adaptive repetitive control system including the adaptive generator, even if the estimated values of the period estimator do not converge to constant ones and the generator remains timevarying element in the steady state. Moreover, the controller assigns all poles of the closed-loop system on the disk with a given radius to adjust fast disturbance rejection. The design problem is reduced to a simple and feasible l1 norm minimization problem. Secondly, the period estimator estimates multiple periods of disturbance from the input/output data of the plant. The parametric model to be estimated consists of a simple FIR model and the algorithm is based on the well-known recursive least squares method. Finally, in order to reject periodic disturbances with uncertain multiple periods, the adaptive generator is adjusted on-line by using the period estimated. The dead-time length is reduced to the sum of all the periods and the repetitive controller with the generator is implemented with much less memory elements than the previous ones. As a result, the degree of the controller is significantly reduced and much faster convergence of the controlled error to zero is provided. The effectiveness is demonstrated by simulations.

Assumption (A1) The periods Li ∈ I + are unknown, but the upper bounds and the lower ones denoted by L i and L i , i = 1, " , h , respectively are given. Then it follows that

The following notations are used throughout this paper. R , C and I + denote the set of real numbers, the set of complex numbers and the set of nonnegative integers, respectively. -1 RH∞ and R[ z ] denote the set of stable and proper real rational functions and the set of real proper polynomials in z −1 , respectively. For given nonnegative values r , r ∈R , define

In this section, we consider the problem above. Figure 1 shows the control system proposed in this paper. The control system consists of two elements. The one is a period estimator to satisfy the condition (C1). The periods estimated are denoted by Lˆi [k ] ∈ I + , i = 1 " h , which satisfy that

D( r , r ] = { z ∈ C | r < z ≤ r } .

L i ≤ Li ≤ Li , i = 1 " h .

In this paper, the following problem is considered. Adaptive Multi-Period Repetitive Control Problem Find a repetitive control system satisfying the following conditions (C1), (C2) and (C3) under the assumption (A1). (C1) To estimate the periods Li ∈ I + , i = 1 " h by using u[k] and y[k]. (C2) To stabilize the overall closed-loop system in dependent of the periods estimated. (C3) To reject the periodic disturbance asymptotically, i.e., lim e[k ] = 0, ∀di ( z ) ∈ R[ z −1 ], i = 1" h .

3. ADAPTIVE REPETITIVE CONTROL SYSTEMS FOR FAST DISTURBANCE REJECTION

L i ≤ Lˆi [ k ] ≤ Li , i = 1 " h .

(1)

Consider a SISO discrete-time system described by y[k ] = P( z )(u[k ] + d [ k ]) ,

(2)

e[k ] = − y[k ] ,

(3)

h

i

i =1

di ( z ) =

The estimation method is shown in Section 4. The other is an adaptive repetitive controller Cˆ ( z , k ) consisting of a periodic signal generator and a stabilizing controller to satisfy both the conditions (C2) and (C3). Let G( z) =

Lˆθi [[k k]

(4)

d i ( z) 1 − z − Li

,

P( z ) . 1 + P( z ) K ( z )

(9)

e[k ] and uˆ[ k ] are the input and the output, respectively.

The z-transform of d[k] and the periodic signals di [ k ] with the period Li ∈ I + are represented as

∑ d ( z),

(8)

K (z ) is a linear time-invariant stabilizing controller for P(z ) , and hence G (z ) is asymptotically stable ( G(z) ∈ RH∞ ). The following repetitive controller Cˆ ( z , k ) is proposed, in which

where P (z ) is a linear time-invariant and proper plant to be controlled. y[k], d[k], u[k] and e[k] denote the controlled output, disturbance input, control input and controlled error at time step k, respectively. d[k] consists of periodic signals with multiple periods, denoted by Li ∈ I + , i = 1 " h , where,

d ( z) =

(7)

k →∞

2. PROBLEM FORMULATION

L1 > L2 > " > Lh .

(6)

e[k ]

(5)

where d i ( z ) ∈ R[ z −1 ] is the z-transform of the finite sequence in the first period component of the periodic signals. The following assumption applies throughout this paper.

Period Estimator

dˆ [ k ] Disturbance Observer

d [k ] u [k ]

Cˆ ( z, k )

P(z)

uˆ [ k ]

y [k ]

K (z )

Fig.1 Proposed adaptive repetitive control system for uncertain multiple periods. 2

Cˆ ( z, k ) ξ1

z

η1

− ( Lˆ1 − L1 )

C1( z)

Fig.1 is BIBO stable for any time varying Lˆ i [k ] ∈ I + satisfying (8), where 1 denotes the l ∞ induced norm.

uˆ1

Proof: The proof is straightforward from Yamada et al. (2004, 2009), and is omitted. Q.E.D.

z − L1 ξ2

e[k ]

ˆ

z −( L2 −L2 )

η2

G (z ) C2 ( z )

uˆ 2

Remark : This theorem means that the design problem of the controller that stabilizes the multiple repetitive control system is reduced to that of C i (z ) satisfying (14), and each C i ( z ), i = 1 " h , can be designed independently. Moreover, this theorem assures the controllers of C i (z ) stabilize the adaptive control system even if the estimated values of the periods do not converge to constant ones and the delay elements of (11) remains time-varying in the steady state.

uˆ [ k ]

z − L2 ξ3

z

−( Lˆ3 − L3 )

η3

G(z ) C3 ( z )

uˆ3

z − L3

Moreover, the controller provides the following attractive properties on the disturbance rejection.

Fig. 2 Structure of multiple repetitive controller.

Theorem 2 h −L Assume that the numerator of G (z ) and ∏i =1 (1 − z i ) are coprime. Consider the controllers (10), (11) and (12). If

ξ i [k ] = η i [k − L i ] +

i −1

∑ G( z )C j ( z )η j [k ] + e[k ] .

(10)

j =1

η i [k ] = ξ i [k − Lˆ i [k ] + L i ] , i = 1 " h . uˆ[k ] =

(11)

h

∑ C j ( z )η j [k ] .

(12)

j =1

h

∑

i =1

Ci ( z) z −( Li − Li ) 1 − z − Li

i −1

⎛

C j ( z) z

⎝

1− z

∏ ⎜⎜1+ G(z) j =1

−( L j − L j ) −Lj

⎞ ⎟ ⎟ ⎠

h

e[k ] = − ∏ S i ( z ) d [k ] , i =1

< 1,

H i ( z ) = z − Li − G ( z ) Ci ( z ),

(i = 1" h) ,

Si ( z ) =

1− z

−L i

1 − Hi ( z) z

−( L i − L i )

(16)

From (14), the numerator and the denominator of Si (z ) are Q.E.D coprime and Si (z) ∈ RH∞ . From (5), we obtain (7).

. (13)

Remark : As shown in Fig.2, the dead-time length of the proposed repetitive controller is reduced to the sum of all the periods and the controller is implemented with much less memory elements than the previous ones, in which the repetitive controllers are required with the dead-time length equal to the least common multiple (l.c.m) of all the periods.

Define

μi = H i

1

,

i = 1" h .

(17)

The following theorem clarifies a relationship between the value of μ i and the poles of the closed loop system. The theorem on the l ∞ induced norm of (17) is an extension to Yamada et al. (1999) on H ∞ norm.

Theorem 1 Consider the controllers (10), (11) and (12). If Ci ( z) ∈ RH∞ (i = 1," , h) satisfy that 1

(15)

k →∞

Proof: For Lˆ i [k ] = L i ( i = 1 " h) , the relation between the disturbance input and the controlled error is obtained in a compact form as follows

This controller is a simple parallel connection of controllers −( L i − L i ) −L Ci ( z ) z (1 − z i ) with the periodic generator of the period of Li ∈ I + adding the pass consisting of proper G(z) ∈ RH∞ , and includes the internal model of periodic signals with the multiple periods Li ∈ I + . This controller provides the following attractive properties on the stability of the closed-loop system including time varying elements of Lˆ i [ k ] ∈ I + in Fig.1.

Hi

lim Lˆ i [ k ] = L i , i = 1 " h ,

then for any periodic disturbances with the multiple periods, we obtain (7).

where Ci ( z) ∈ RH∞ , i = 1 " h are proper and stable controllers to be designed. (11) involves a time-varying delay elements adjusted by the period estimated Lˆi [k ] ∈ I + , i = 1 " h . Figure 2 shows the structure of the proposed controller Cˆ ( z , k ) for the case of h = 3 , where the delay elements of − ( Lˆ i [ k ]− L i ) imply (11). Note that for any estimated periods z ˆ Li [k ] ∈ I + satisfying (8), this controller Cˆ ( z , k ) is proper. If the estimated periods are equal to the true ones, i.e., Lˆ i [k ] = L i ( i = 1 " h) , then the controller is expressed as Cˆ ( z, k ) =

Ci ( z) ∈ RH∞ (i = 1, ", h) satisfy (14) and

(14)

then the proposed multiple-period repetitive control system in

3

Theorem 3 Assume that H i ( z ) ∈ R[ z −1 ] (i = 1,", h) satisfy (14). Then for any periods Li ∈ I + , i = 1 " h satisfying (6), all poles of the

output of the disturbance observer in Fig.1 is given by

closed-loop system in (16) are on the region of D[ 0, ρ ] , i.e., the disk with the radius of ρ and the center at the origin, where ρ = max ρi , i =1" h

ρi = μ

1/( L i − L i + m i )

,

dˆ[k ] = D ( z ) y[k ] − N ( z ) uˆ[k ] .

(18)

Then it follows that dˆ[k ] = N ( z ) d [k ] .

−1

and m i denotes the degree of H i ( z ) ∈ R[ z ] . Proof: From Theorem 1, it is obvious that no pole of in (16) lies outside the open unit disk. Hence we will that no pole of S i (z ) lies on the region of D( ρi ,1) . m the definition of m i , z i H i ( z ) is a polynomial Accordingly, it follows that max

z ∈ D ( ρi , 1)

≤

H i ( z) z

max

z ∈ D ( ρi , 1)

m i

× max

z ∈ D ( ρi , 1)

<

Hi

× ρi 1

− ( L i − L i + m i )

<

Hi

× ρi

− ( L i − L i + m i )

1

z

Lˆ 1[ k ] = arg max θ i [ k ] , i ∈ [ L, L ]

Lˆ 2 [ k ] = arg max θ i [ k ] .

where θi [k ] ∈ \, (i = L," , L ) is given by the following recursive formula. θ [k ] = θ [k − 1] +

(19)

Γ[k ] = Γ[k − 1] +

Consequently, − ( Li − Li )

≠ 0, ∀ z ∈ D( ρi , 1) .

The proof is completed.

(20) Q.E.D

1 + ϕ [k ] Γ[k − 1] ϕ[k ]

ε [k ] .

Γ[ k − 1]ϕ[k ] ϕ T [k ] Γ[k − 1] 1 + ϕ T [k ] Γ[k − 1] ϕ[k ]

.

γ

( L i − L i + m i )

, i = 1" h ,

(27)

where θ [k ] = [ θi [k ], i = L " L , 2 L" 2 L ]T ∈ R3( L − L ) + 2 ,

Remark : This theorem means that for a given γ such that 0 < γ < 1 , if the controllers C i (z ) , i = 1," , h satisfy that 1<

Γ[ k − 1]ϕ[k ] T

ε [k ] = dˆ [k ] − θ T [k ]ϕ[k ] .

1 − Hi ( z) z

Hi

(26)

i ∈ [ L, L ] i ≠ Lˆ1[ k ]

− ( L i − L i + m i )

<1.

(25)

The periods estimated L i [ k ], i = 1, 2 are determined by

S i (z ) prove From in z.

−( L i − L i )

Hi ( z) z

(24)

ϕ[k ] = [dˆ [k − i], i = L" L , 2 L " 2 L ]T ∈ R3( L − L ) + 2 . (28)

(21)

Then the following lemma is obtained. Lemma 4 If d [k ] is a persistently exciting signal, then, the parameter vector θ [k] calculated by the recursive formula (27) converges to the following value.

then not only the adaptive repetitive control system is stable but also all poles of the closed-loop system are assigned on the disk with the given radius γ even if (15) is not satisfied, in other word, the periods estimated is not equal to the true ones.

⎧ 1, i = L1 , L2 ⎪ lim θi [k ] = ⎨ −1, i = L1 + L2 , k →∞ ⎪0, otherwise. ⎩

We will consider the period estimation method to satisfy (15) in Section 4, and the design method of C i (z ) to satisfy (21) for disturbance rejection and pole placement in Section 5.

lim Lˆi [k ] = L i , i = 1, 2 .

(29)

k →∞

4. PERIOD ESTIMATION FOR MULTIPLE PERIODS

Proof: See Yamada et al. (2004, 2009)

In this section, we present a ‘Period Estimator’ in Fig.1 to satisfy (15). For simplicity of description, we consider the case where h=2 and L = L 1 = L 2 , L = L1 = L 2 .

5. CONTROLLER DESIGN

(22)

As shown in Yamada et al. (2004, 2009), the estimation method is presented as follows. Let G ( z ) = N ( z ) D( z ) ,

Q.E.D.

(23)

In this section, we consider a design method of the stable and proper controllers Ci ( z) ∈ RH∞ , i = 1 " h , satisfying (14) and (21). N ( z ) ∈ R[ z −1 ] in (23) is factored as follows. N ( z ) = z − d N s ( z ) Nu ( z ) ,

(30)

where N s ( z ) ∈ R[ z −1 ] contains all asymptotically stable zeros, Nu ( z ) ∈ R[ z −1 ] contains the remaining zeros, e.g., unstable

where N ( z ) ∈ R[ z −1 ] and D( z ) ∈ R[ z −1 ] are coprime. The

4

θˆi [0] = 0.5, i = 40" 44 . Figure 6 shows the controlled output

zeros and d ∈ I + . For simplicity of description, we consider the case where G (z ) has one unstable zero. Then Nu ( z ) ∈ R[ z −1 ] is represented as Nu ( z ) = 1 − n1 z −1 ,

(31)

where n1 > 1 . The following corollary presents stabilizing and pole placement controllers in an explicit form. Corollary 5 For a given mˆ i ∈ I + satisfying mˆ i ≤ Li − d , i = 1" h ,

(32)

consider the following proper controller mˆ i −1

∑

− ( mˆ i − j ) − j

n1

z

∈ RH ∞ .

(33)

j =0

Then we obtain that m i = L i and Hi

1

= n1

− mˆ i

.

< 1, ∀mˆ i ≥ 1 .

Therefore, for any mˆ i ≥ 1 , the controller (33) stabilizes the adaptive repetitive control system for any time varying ˆ i ≥ 1 such that Lˆ i [ k ] ∈ I + . Moreover, for any m mˆ i > L i

log γ log (| n1 |−1 )

4

(34)

,

(35)

Disturbence input d[k]

D( z ) Ns ( z)

ˆ

Ci ( z ) = − z − ( Li − d − mi )

error. These figures demonstrate that, by using the proposed control system, the estimated parameters converges quickly to the true periods of { L1 , L2 } = {43, 41} and moreover the steady-state controlled output error becomes zero. Secondly, in order to investigate the relationship between the value of μ i and the convergence rate of the controlled output error, we set mˆ i = 2, i = 1, 2 . Then we obtained μi = 0.09, i = 1, 2 . For mˆ i = 20, i = 1, 2 , we obtained μi = 3.5 × 10−11 , i = 1, 2 . Figures 7 and 8 show the controlled output responses. Comparing with Fig.6, it is confirmed that as the values of μ i make smaller, the controlled output error converges to zero more rapidly. Table 1 shows ρ of (18) and the dominate poles of the S ( z ) of (16) for some mˆ i ’s. It is confirmed that as increasing mˆ i , the values of ρ become smaller and the dominate poles are closer to the origin

2 0 -2 -4 0.0

the absolute values of all the poles of the closed-loop system are not greater than the given value γ > 0 .

0.2

( n1 )

.

(36)

Therefore, we obtain (34) and (35).

Q.E.D.

Remark : For the general case where G ( z ) has more than one unstable zeros, see Yamada et al. (2004, 2009).

6. SIMULATIONS Consider the following plant. (1 − 0.8 z −1 )(1 − 0.9 z −1 )

1.0

1.5

θ

^

θ

^

40

θ

[k]

^ 42

[k]

[k] 41

1.0 0.5 0.0 -0.5 0.0

0.2

0.4

0.6

Tim e [s]

0.8

1.0

Fig.4 Time response of estimated parameters.

.

(37) Estimated parameters on periods

G( z) =

0.0187 z −1 (1 + 3.35 z −1 )

Estimated parameters on periods

Hi = z

− mˆ i

0.8

Fig.3 Periodic disturbance.

Proof: By substituting (51) and (53), it follows that − ( Li − mˆ i )

0.4 0.6 T im e [s]

The sampling period is given by T = 1 [ms] . Figure 3 shows a periodic disturbance input consisting of periodic signals with the periods { L1 , L2 } = {43, 41} . The upper and lower bound of these periods are given by { Li , Li } = {40, 44}, i = 1, 2 . Firstly, we set mˆ i = 1, i = 1, 2 and designed controllers C i ( z ), i = 1 , 2 such that μi = H i 1 = 0.3, i = 1, 2 by using the proposed design method. Figures 4 and 5 show the time responses of the parameters of θˆi [k ], i = 40" 44 in the period estimator, where the initial parameters are given by

1.5

θ 43 [k]

θ 44 [k]

1.0 0.5 0.0 -0.5 0.0

0.2

0.4 0.6 Tim e [s]

0.8

1.0

Fig.5 Time response of estimated parameters.

5

Controlled output error y[k]

4

REFERENCES Chew, K.K., and Tomizuka, M. (1990). Steady-State and Stochastic Performance of a Modified Discrete-Time Prototype Repetitive Controller. Trans. of the ASME, Journal of Dynamic Systems, Measurement, and Control, Vol.112, No.1, 35-41. Dahleh, M.A., and Diaz-Bobillo, I.J. (1995). Control of Uncertain System – A Linear Programming Approach –. Prentice Hall. Francis, B.A., and Wonham, W.M. (1975). The Internal Model Principles of Linear Multivariable Regulators. Appl. Math. And Opt., Vol. 2, 170-194. Inoue, T., Nakano, M., Kubo, T., Matsumoto, S., and Baba, H. (1981). High Accuracy Control of a Proton Synchrotron Magnet Power Supply. Proc. of the IFAC World Congress, 3137-3142. Hara, S., Yamamoto, Y., Omata, T., and Nakano, M. (1988). Repetitive Control System: A New Type Servo System for Periodic Exogenous Signals. IEEE Trans. on Automatic Control, Vol.33, No.7, 659-668. Hu, J.S. (1992). Variable Structure Digital Repetitive Controller. Proc. of the 1992 American Control Conference, 2686-2690. Tsao, T.C., and Nemani, M. (1992). Asymptotic Rejection of Periodic Disturbances with Uncertain Period. Proc. of the 1992 American Control Conference, 2696-2699. Tsao, T.C., Qian, X.Y., and Nemani, M. (2000). Repetitive Control for Asymptotic Tracking of Periodic Signals with an Uncertain Period, Trans. of the ASME, Journal of Dynamic Systems, Measurement, and Control, Vol.122, 364-368. Tsao, T.C., and Tomizuka, M. (1994). Robust Adaptive and Repetitive Digital Tracking Control and Application to a Hydraulic Servo for Noncircular Machining. Trans. of the ASME, Journal of Dynamic Systems, Measurement, and Control, Vol.116, 24-32. Tomizuka, M. (1987). Zero Phase Error Tracking Algorithm for Digital Control. Trans. of the ASME, Journal of Dynamic Systems, Measurement, and Control., Vol.109 No.1, 65-68. Tomizuka, M., Tsao, T.C., and Chew, K.K. (1989). Analysis and Synthesis of Discrete-Time Repetitive Controller. Trans. of the ASME, Journal of Dynamic Systems, Measurement, and Control, Vol.111, 353-358. Yamada, M., Riadh, Z., and Funahashi, Y. (1999). Design of Discrete-Time Repetitive Control System for Pole Placement and Application. IEEE/ASME Trans. on Mechatronics, Vol.4, No.2, 110-118. Yamada, M., Riadh, Z., and Funahashi, Y. (2000). Design of Robust Repetitive Control System for Multiple Periods. Proc. of the 39th IEEE Conference on Decision and Control, 3739-3744. Yamada, M., and Tomizuka, M. (2004). Robust Repetitive Control System with On-line Identification of the Period of Periodic Disturbances", Proc. of the 2004 ASME International Mechanical Engineering Congress and Exposition, Paper No.IMECE2004-61964. Yamada, M., Yabuki, Y., Funahashi, Y., and Mizuno, N. (2004). Adaptive Repetitive Control System for Asymptotic Rejection of Periodic Disturbances with Unknown Multiple Period”, Proc. of IFAC Workshop on Adaptation and Learning in Control and Signal Processing, 469-474. Yamada, M., Yabuki, and Mizuno, N. (2009). Adaptive Repetitive Control with On-line Estimation of Multiple Periods, Transactions of the Society of Instruments and Control Engineerings, Vol.45, No.5, 243-250. Wellstead, P.E., and Zarrop, M.B. (1991). Self-Tuning Systems Control and Signal Processing -, Wiley. V.Chvatal, (1983), Linear Programming, W.H.Freeman and Comp., New York.

2 0 -2 -4 0.0

0.2

0.4 0.6 T im e [s]

0.8

1.0

Fig.6 Time response of the controlled output error for the case of mˆ i = 1, μ i = 0.3, i = 1, 2 . Controlled output error y[k]

4 2 0 -2 -4 0.0

0.2

0.4 0.6 T im e [s]

0.8

1.0

Fig.7 Time response of the controlled output error for the case of mˆ i = 2, μ i = 0.09, i = 1, 2 . Controlled output error y[k]

4 2 0 -2 -4 0.0

0.2

0.4 0.6 Tim e [s]

0.8

1.0

Fig.8 Time response of the controlled output error for the case of mˆ i = 20, μ i = 3.5×10 −11 , i = 1, 2 .

VALUES OF

ρ

OF

TABLE 1 (18) AND THE DOMINATE POLES OF THE CLOSEDLOOP SYSTEM FOR SOME mˆ i ’S.

(mˆ 1 , mˆ 2 )

ρ

The dominate poles of the closed-loop system

(1,1)

0.973

- 0.972

( 2, 2 )

0.947

± 0.947

( 20, 20 )

0.578

± 0.578

7. CONCLUSION This paper has presented a new and useful adaptive repetitive control system based on pole-placement method to reject quickly any periodic signals with uncertain multiple periods. The effectiveness has been demonstrated by simulations

6