Journal of Robotics, Networking and Artificial Life
Vol. 5(4); March (2019), pp. 273–277
DOI: https://doi.org/10.2991/jrnal.k.190220.015; ISSN 2405-9021 print; 2352-6386 online
https://www.atlantis-press.com/journals/jrnal

Research Article

Quantitative Evaluation in Discrete-time Domain and­Improvement in Intersample Response on Multirate
Optimal Control
Takao Sato*, Natsuki Kawaguchi, Nozomu Araki, Yasuo Konishi
Graduate School of Engineering, University of Hyogo, Himeji, Hyogo, 671-2280, Japan

ARTICLE INFO
Article History
Received 22 October 2018
Accepted 20 November 2018Keywords
Sampled-data control systemdual-rate systemintersample response
discrete-time responsenull-space
redundancy

ABSTRACTIn this study, the dual-rate sampled-data control system is designed, where the sampling interval of a plant output is an integer
multiple of the holding interval of a control input. Because the discrete-time Linear Quadratic Integrator (LQI) is used for
optimizing the discrete-time control performance, intersample output might oscillate. To eliminate the oscillation of theintersample ripple caused by the input oscillation, the designed LQI controller is extended. Finally, the original LQI control law
and the extended control laws are quantitatively compared.

© 2019 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
Most recent control systems are implemented using the discretetime digital computer. On the other hand, the controlled object is
not only a discrete-time system but also a continuous-time system.
When the continuous-time system is controlled using the digitalcomputer, such a control system is called sampled-data control
system [1]. In the single-input/-output system, there are twointervals: the holding interval of a control input and the samplinginterval of a plant output. This study discusses a design method forcontrolling a dual-rate control system, where the sampling interval
is longer than the holding interval.
In such a dual-rate control system, intersample ripples might occur
even if the sampled output converges to the reference input because
the control input can be changed between the sampling intervals[2]. Therefore, the condition for eliminating the intersample ripples
caused by the input oscillation has been proposed [3].
This study proposes two design methods for eliminating the intersample ripples: one is an extension method based on the nullspace [4], and the other is to weight on the input deviation. Whenthe intersample response is improved, the discrete-time sampled
response might be degraded, and hence the performance degradation should be fairly evaluated. The extension method using
the null-space has been proposed [5,6]. The conventional method[5] is derived using the polynomial approach, whereas the proposed method is designed based on the state-space representation.
Corresponding author. Email: tsato@eng.u-hyogo.ac.jp*

As a state-space approach, the conventional method [6] is designed
using the null-space. However, because the integral compensation
is not included, the robust-tracking for the step-type reference
input or the step-type disturbance is not achieved. On the other
hand, a state-space design method using the integral compensation has been proposed [7]. In this method, the intersample ripples
are eliminated. However, an existing discrete-time response is not
maintained because the null-space is not used. Finally, the effectiveness of the proposed method is demonstrated through numericalexamples, and the control performances are quantitatively evaluated based on the numerical results. Furthermore, this study showsthat the discrete-time performance of the state variable is notchanged using the extension method using the null-space.

2. CONTROLLED PLANT MODEL
A controlled plant is an unknown linear continuous-time singleinput and -output time-invariant system given as the following
Equations (1) and (2):		x (t ) = Ac x(t ) + bcu(t ) (1)		

y(t ) = ccT x(t ) (2)where x(t) Î Rn is the state variable, u(t) Î R is the control input,and y(t) Î R is the plant output, respectively. Ac Î Rn×n is a coefficient matrix, and bc Î Rn and cc Î Rn are coefficient vectors. Because
the sampled-data control system is designed, a continuous-time

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plant output is sampled, and a discrete-time control input is held.
Since the sampling interval of the plant output is an integer multiple of the holding interval of the control input, the following lifteddual-rate model is obtained [Equations (3)–(5)].
		x(k + l ) = Al x(k) + Bl u(k) (3)Tu(k) = éëu(k)  u(k + l - 1)ùû (4)		

y(k) = clT x(k) (5)		

where l is a positive integer. The dual-rate model is assumed to be
known, and a sampled-data control system is designed based on
this model. It is assumed that (Al, Bl) is stable.

3. OPTIMAL CONTROLLER DESIGN
The control system is designed using the following state feedback
control law [Equations (6)–(8)]:u(k) = − Fz (k) (6)		
Tz (k) = éë x(k) xi (k)ùû (7)		
x (k + l ) = xi (k) + ( r (k) - y(k ))Ts (8)		 i
where r(k) Î R is the reference input to be followed by the plant
output, and Ts denotes the sampling interval of the plant output.
F Î Rl×(n+1) is the feedback gain, and it is designed using the Linear
Quadratic Integrator (LQI) [Equation (9)] [8]:¥		(k =0T						z (k + l ) = ( Az − Bz F )x (k ) + Er (k ) + Bz B⊥w(k) (18)From the definition of B^, the last-term in the right-hand sideof Equation (18) is 0(n+1)×1. As a result, the closed-loop system is
not affected by w(k). In this study, w(k) is designed so that all theelements in u(k) are equivalent.

4.2. Input-weighting Method
In the other extension method, the deviation of the control input issuppressed in LQI. To this end, the difference between u(k + i) andu(k + i + 1) is penalized as the following Equations (19) and (20):z (k + l ) = ( Az - Bz F )z (k ) + Er (k ) (13)é AAz = ê l T
ë -Tsc0ùé0 ùéB ù, B = l , E = ê n´1 ú (14)1 úû z êë01´l úûë Ts û

4. TWO INTERSAMPLE SUPPRESSIONMETHODS
To eliminate the intersample ripples caused by the input oscillation between the sampling instants, two extension methods areproposed.

d i (u(k + i - 1) - u(k + i))2 =éu(k + i - 1)ù (19)[u(k + i - 1) u(k + i)]Di êúë u(k + i) ûé 1 -1ù
Di = d i êú (20)ë -1 1 û		
where δi Î R is a non-negative weighting factor. To evaluate theweighted input deviation in LQI, Ju in Equation (12) is replacedwith the following criterion (21):¥J u = åu(k)T (R + D)u(k )∞The closed-loop system using control law (6) is obtained as follows:u(k) = − Fx(k) + B⊥w(k) (17)The closed-loop system using the extended control law (11) isobtained as follows:

k =0

k =0Ker (Bl ) = {b^ Î R l | Bl b^ = 0n,1 , b^ ¹ 0} (16)where w(k) Î Rl–1 is a newly introduced vector.

J z = ∑z (k)T Qz (k) (11)J u = ∑u(k)T Ru(k) (12)Im(B^ ) = Ker (Bl ) (15)Using the defined B^, the control law is extended as follows:		

∞		B^ Î Rl×(l–1) is defined as the following Equations (15) and (16):)J = J z + J u (10)		

One extension method is based on the null-space.

J = å z (k) Qz (k) + u(k) Ru(k) (9)Twhere Q Î R(n+1)×(n+1) is a positive definite matrix, and R Î Rl×l is apositive semi-definite matrix. The performance index is rewritten
as the following Equations (10)–(12):		

4.1. Null-space Method

(21)

k =0

where Δ is a matrix which satisfies Equation (22):l -1

åd (u(k + i - 1) - u(k + i)) = u(k) Du(k) (22)i =1i2T

In the proposed weighting method, Ju must be redesigned.
Therefore, the Riccati equation is resolved by replacing Equation(12) by Equation (21).

5. DISCUSSION ON DISCRETETIME AND INTERSAMPLE CONTROLPERFORMANCEIn the dual-rate sampled-data control system, the discrete-time model
is available, and the continuous-time model is not always obtained.

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In such a case, when the discrete-time performance is optimized,
the intersample ripples might be caused by the input deviation
between the sampling instants, although the discrete-time performance is optimized.
In this study, the discrete-time performance is optimized based on
the known lifted dual-rate model, and the designed control law isextended so that the intersample performance is improved. In thediscrete-time performance evaluation, because the discrete-time
performance function consists of Jx and Ju, two indices are quantitatively evaluated.
In the first extension method based on the null-space, the intersample ripples can be eliminated by introducing the new design
parameter. Further, the parameter is designed independent of thediscrete-time closed-loop system. Therefore, Jx is maintained andJu is changed. Hence, the optimization on the control input is notmaintained.
In the second extension method, since the difference of the control input is always penalized, the input deviation is suppressed. Asa result, the intersample ripples caused by the input deviation iseliminated. However, both Jx and Ju are affected by the weighting
factor δi.

Figure 1 | Simulation result using the original state feedback control law(upper: output trajectory, lower: input trajectory).

6. NUMERICAL EXAMPLE
Consider a continuous-time system given as the following
Equation (23):G(s) =13s + 1(23)The sampling interval and the holding interval are 2s and 1s,respectively, and the following lifted dual-rate system is given asEquations (24) and (25):é0.72 ùx(k + 2) = 0.51x(k) + êú u(k) (24)ë 1. 0 û		y(k) = 0.28 x(k ) (25)		

Figure 2 | Simulation result using the extended method is given in
Subsection 4.1.

The feedback gain is obtained by solving the LQI problem with
Q = R = I2. The reference input is a unit step function.

output response is the same as Figure 1 even though the control
input is redesigned.

The control result using the original feedback control law (6) isshown in Figure 1. This figure shows that the sampled output converges to the reference input at the sampling instants. However, thecontrol input oscillates, and the intersample output hence oscillates. In the LQI design, the amount of the control input is evaluated, but the intersample output is not evaluated. Therefore, thedeviation of the control input is not attenuated, and the intersample
ripples occur.

The control results using the extended controller designed inSubsection 4.2 are shown in Figures 3 and 4, where δ1 is set to
100 and 105, respectively. In both figures, the amplitude of the­difference in the control input is attenuated since the control input
deviation between the sampling instants is penalized. The larger δ1,
the smaller the amplitude of intersample ripples.

To eliminate the intersample ripples caused by the input deviation, the extended controllers are applied. The result using
the extended controller designed in Subsection 4.1 is shown inFigure 2. The lower figure shows that the control input is constant
between the sampling instants, and the upper figure shows thatthe i­ ntersample ripples are eliminated. Furthermore, the sampled

For comparing the controllers, the performance indices are quantitatively evaluated in Table 1, where the evaluation length is 50 steps.
Table 1 shows that the index J50 of the original LQI is best, and thatof the extended methods are increased, although the intersample
ripples are suppressed. In the index of state z, J z50 , the original LQI
is best and it is the same as the extended method of Subsection 4.1
because the original closed-loop system is maintained in theextended controller designed in Subsection 4.1.

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7. CONCLUSION
This study has discussed the design methods for controlling a dualrate sampled-data control system. First, an optimal control law isdesigned using LQI.
Second, the designed control law is extended using two methods: one
is based on the null-space, and the other is to weight the input deviation between the sampling instants. At last, the effectiveness of theproposed method is demonstrated through the numerical examples,
and the control performances are quantitatively evaluated.

ACKNOWLEDGMENT
This study was supported by JSPS KAKENHI Grant Number
16K06425.
Figure 3 | Simulation result using the extended method is given in
Subsection 4.2 with d1 = 100.

Figure 4 | Simulation result using the extended method is given in
Subsection 4.2 with d1 = 105.
Table 1 | Index values
Controller

J z50 (´103 )

J u50

J 50

Original
4.1
4.2 (d1 = 100)
4.2 (d1 = 105)

3.029
3.029
3.034
3.037

93.82
96.38
95.49
96.35

3.122
3.125
3.130
3.133

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Authors Introduction
Dr. Takao SatoDr. Nozomu Araki

He received the B. Eng. and M. Eng.
degrees from Okayama University, in1997 and 1999, respectively, and hisD. Eng. degree from Okayama Universityin 2002. He is an associate professor inthe Graduate School of Engineering atUniversity of Hyogo. His research interests
are in the area of PID control, mechanical system and multirate control. Dr. Sato is a member of theSociety of Instrumentation and Control Engineers in Japan(SICE) and the Institute of Systems, Control and Information
Engineers in Japan (ISCIE).
Dr. Natsuki Kawaguchi
He received the D. Eng degree fromOkayama University in 2018. He is anassistant professor in the Graduate Schoolof Engineering at University of Hyogo. Hisresearch interests are in the area of fault
detection and fault tolerant control.

He received his PhD degree fromUniversity of Hyogo, Japan in 2007. He ispresently working as an Assistant Professor
in the Department of Mechanical System,
University of Hyogo, Japan. His researchinterest includes control engineering,
signal processing and medical engineering.

Prof. Yasuo Konishi
He received his PhD in mechanical engineering from Keio University, Tokyo,
Japan, in 1989. He is a professor at theUniversity of Hyogo, and is presently
the head of the Laboratory of Control
Engineering. His research focuses on thegame-theoretical approach to mechanicaldesign, time optimal control by binary
input using MLDS, and positioning control with static andkinetic friction using hybrid controllers.