TRANSACTIONS ON ELECTRICAL AND ELECTRONIC ENGINEERINGIEEJ Trans 2006; 1: 345–348Published online in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/tee.20058

Letter

A Design Method of Generalized Minimum Variance Control in aMulti-rate System

Takao Satoa∗Akira Inoue∗∗

Generalized minimum variance control is designed in a multi-rate system where the sampling interval ofa plant output is longer than the holding interval of a control input. Because new design parameters areintroduced, intersample ripples in the multi-rate system are suppressed by weighting the variation in the controlinput. Furthermore, the proposed method is an extension of a conventional design method. Finally, simulationresults are illustrated in order to show the effectiveness of the proposed method. 2006 Institute of ElectricalEngineers of Japan. Published by John Wiley & Sons, Inc.

Received 30 March 2006; Accepted 20 July 2006

1. Introduction

In digital control, a plant output in continuous timeis sampled, and a calculated control input in discretetime is updated with a holder. In a multi-rate systemwhere the sampling interval of a plant output is longerthan the holding interval of a control input, ripples oftenarise between sampling instants [1]. Hence, many designmethods have been proposed for suppressing the ripplesin the multi-rate system [2,3]. However, in designingthese methods the dead-time needs to be known. Hence,this paper proposes a design method of generalizedminimum variance control (GMVC) in the multi-ratesystem. Use of new design parameters makes clear therelation between the standard GMVC [4] in a slow-ratesingle-rate system and the proposed GMVC in the multi-rate system.

2. Problem Statement

A controlled plant is a single-input single-outputcontinuous-time system, and the plant is controlled usinga digital controller. It is assumed that a control input isupdated at intervals of Ts , but a plant output is sampledat intervals of lTs . Because of this constraint, a fast-rate

a Correspondence to: Takao Sato. E-mail: tsato@eng.u-hyogo.ac.jp∗ Division of Mechanical System, Department of Mechanical Engi-

neering, Graduate School of Engineering, University of Hyogo, 2167Shosha, Himeji, Hyogo 671-2201, Japan

∗∗ Division of Industrial Innovation Sciences, Graduate School ofNatural Science and Technology, Okayama University, 3-1-1 Tsushima-naka, Okayama 700-8530, Japan

single-rate system cannot be obtained. It is expected thatthe control performance of a multi-rate system is betterthan that of a slow-rate single-rate system. In this paper,a multi-rate system is transformed into an l-inputs single-output slow-rate single-rate system [2,3], and a controlleris designed on the basis of the following l-inputs single-output model.

A[z−1l ]y[k] = z

−km

l B [z−1l ]T u[k − l] (1)

A[z−1l ] = 1 + a1z

−1l + · · · + anz

−nl (2)

B [z−1l ] = [B1[z−1

l ] · · · Bl[z−1l ]]T (3)

Bj [z−1l ] = bj0 + bj1z

−1l + · · · + bjmz−m

l (4)

u[k] = [u[k] · · · u[k + l − 1]]T (5)

where, y[k] and u[k] are the sampled plant output andthe control input at step k. z−1

1 is the one-step backwardshift operator, and z−1

j = z−j

1 . lkm is the dead-time, andl and km are integer.

3. Multi-rate GMVC

Even though an actual system is a single-input single-output system, because the plant model (1) is an l-inputssingle-output system, a control law is derived in an l-inputs single-output system. A multi-rate system has tobe designed taking into account ripples between samplinginstants. To suppress the ripples, a new generalizedoutput vector, which contains the variation in the controlinput between sampled plant outputs, is proposed. The

2006 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

T. SATO AND A. INOUE

proposed multi-rate GMVC law is derived by minimizingthe following performance function:

J [k] = E[�[k + l(km + 1)]T �[k + l(km + 1)]] (6)

�[k + l(km + 1)] = P[z−1l ]y[k + l(km + 1)]

+Q[z−1l ]u[k] − R[z−1

l ]w [k] (7)

P[z−1l ] =

[P1[z−1

l ] 01×(l−1)

0(l−1)×1 z−(km+1)l P[z−1

l ]

](8)

P[z−1l ] = diag{P2[z−1

l ], · · · , Pl[z−1l ]} (9)

Q[z−1l ] = diag{Q1[z−1

l ], · · · , Ql[z−1l ]} (10)

R[z−1l ] =

R1[z−1l ]

... 0l×(l−1)

Rl[z−1l ]

(11)

y[k] = [y[k] �u[k + 1] · · · �u[k + l − 1]

]T(12)

w [k] = [w[k] 0T

1×(l−1)

]T(13)

w [k] is the reference vector to be followed by y[k].Pj [z−1

l ], Qj [z−1l ] and Rj [z−1

l ] are design polynomials.The control objective is to make y[k] follow the referenceinput w[k] while suppressing the variation in the controlinput. Therefore, w [k] is set as (13).

To derive a control law, the following Diophantineequation is introduced:

P1[z−1l ] = A[z−1

l ]E[z−1l ] + z

−(km+1)l F [z−1

l ] (14)

E[z−1l ] = 1 + e1z

−1l + · · · + ekmz

−km

l (15)

F [z−1l ] = f0 + f1z

−1l + · · · + fnf

z−nf

l (16)

nf = max{n − 1, np1 − km − 1} (17)

where, np1 is the order of the polynomial P1[z−1l ]. Solv-

ing the Diophantine equation, the following prediction isderived.

P [z−1l ]y[k + l(km + 1)]

= F [z−1l ]y[k] + E [z−1

l ]B [z−1l ]u[k] (18)

F [z−1l ] = diag{F [z−1], 0, · · · , 0} (19)

E [z−1l ] = diag{E[z−1], 1, · · · , 1} (20)

B [z−1l ] = [B [z−1

l ]T B̃[z−1l ]T ]T (21)

B̃ [z−1l ] = [0(l−1)×1 P[z−1

l ]]

−[P[z−1l ] 0(l−1)×1] (22)

Substituting the prediction (18) into the generalizedoutput vector � of (7), the multi-rate GMVC law

minimizing the performance function is given by

G[z−1l ]u[k] = R[z−1

l ]w [k] − F [z−1l ]y[k] (23)

G[z−1l ] = E [z−1

l ]B [z−1l ] + Q[z−1

l ]. (24)

In designing the proposed method, the ripples can besuppressed by adjusting P2, · · · , Pl which are introducednewly.

4. Numerical Example

A controlled plant model is expressed by thecontinuous-time transfer function G(s) = (1/s2 + s +1)e−2s . Because of the constraint a control input isupdated at intervals of Ts , but a plant output is sam-pled at intervals of lTs . Simulation results in the case ofl = 2 are conducted. A reference input is given as a unitstep signal.

The simulation result by using the standard slow-rate single-rate GMVC [4] is designed first. Use of thesampling time Ts = 2 transforms the transfer functionG(s) into the following:

(1 + 0.12z−12 + 0.14z−2

2 )y[k]

= z−12 (0.85 + 0.40z−1

2 )u[k − 1] (25)

The design parameters of the slow-rate single-rateGMVC are set as follows:

P [z−12 ] = 1, Q[z−1

2 ] = 0.01, R[z−12 ] = 1.0 (26)

Next, the proposed multi-rate GMVC is designed.Because of l = 2, the following 2-inputs single-outputsystem is given:

(1 + 0.12z−12 + 0.14z−1

2 )y[k] (27)

= z−12 [0.51 + 0.089z−1

2 0.34 + 0.32z−12 ]u[k − 2]

The design parameters of the multi-rate GMVC are setthe same as the slow-rate single-rate GMVC:

P1[z−12 ] = 1, Q1[z−1

2 ] = Q2[z−12 ] = 0.01 (28)

R1[z−12 ] = 1.0, R2[z−1

2 ] = 0.01 (29)

where, the weighting factor of the variation in the controlinput is set as P2[z−1

2 ] = 10−4.The simulation results are shown in Fig. 1. The over-

shoot is smaller than the slow-rate single-rate GMVC,and the plant output in continuous-time converges tothe reference input faster than the slow-rate single-rateGMVC. Furthermore, ripples do not emerge because ofsuppressing the variation in the control input.

346 IEEJ Trans 1: 345–348 (2006)

A DESIGN METHOD OF MULTI-RATE GMVC

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

Pla

nt o

utpu

t

Time [s]

0 2 4 6 8 10 12 14 16 18 200.6

0.8

1

1.2

1.4

Con

trol

inpu

t

Time [s]

proposed methodconventional method

proposed methodconventional method

Fig. 1 Simulation results by using the conventional method and the proposed method

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

Time [s]

Pla

nt o

utpu

t

P2=10-4

P2=10-2

P2=100

0 2 4 6 8 10 12 14 16 18 200.6

0.8

1

1.2

1.4

Time [s]

Con

trol

inpu

t

P2=10-4

P2=10-2

P2=100

Fig. 2 Simulation results by using the proposed method with P2[z−12 ] = 10−4, 10−2, 100

The larger the P2[z−12 ], the smaller the variation in

the control input between sampled plant outputs. Hence,the difference between the slow-rate single-rate and themulti-rate GMVC can be adjusted by designing P2[z−1

2 ].For example, the simulation results for P2[z−1

2 ] is set as

10−4, 10−2 and 100 are shown in Fig. 2. As is shown, theproposed multi-rate GMVC can be identical to the slow-rate single-rate GMVC with large P2[z−1

2 ]. Therefore, itcan be seen that the proposed method is a general designmethod including the slow-rate single-rate GMVC.

347 IEEJ Trans 1: 345–348 (2006)

T. SATO AND A. INOUE

5. Conclusion

In this paper, a design method of GMVC in a multi-rate system has been proposed. Although multi-ratecontrol has interesting advantages, intersample rippleshave to be suppressed, and it is not easy to design multi-input single-output GMVC in a polynomial approach.Hence, in this paper, by weighting the variation in acontrol input, the problems have been overcome. Theproposed multi-rate GMVC can be linked to conventionalslow-rate single-rate GMVC by adjusting the weight.Therefore, it can be seen that the proposed GMVCincludes the slow-rate single-rate GMVC.

References

(1) Tangirala A, Li D, Patwardhan R, Shah S, Chen T. Issues inmulti-rate process control. Proceedings of the American ControlConference, San Diego, California, USA, 1999; 2771–2775.

(2) Lu W, Fisher D, Shah SL. Multi-Rate constrained adaptive con-trol. The International Journal of Control 1990; 51(6):1439–1456.

(3) Ishitobi M, Kawanaka M, Nishi H. Ripple-suppressed multi-rateadaptive control. Proceedings of 15th IFAC World Congress,Barcelona, Spain, 2002; 327–332.

(4) Clarke D. Self-tuning control of nonminimum-phase systems.Automatica 1984; 20(5):501–517.

348 IEEJ Trans 1: 345–348 (2006)