Stabiilization of MIMO Time Delay Syystem Ussing Hybbrid Conntroller
MConntrol andd Feedfoorward CControlof Intternal ModelFarammarz Asharif, Shiro Tamakki, Tsutomu Nagado,
NTommokazu Nagaata and Mohaammad Rezaa Alsharif

Abstract
In this paper, w
we aim to stabiilize the unstablle system by looop shaping methhod. Moreover after stabilizingg the system wew need to moddify
tthe system whiich is unstable due to time-deelay element byy Internal Modeel Controller withw feed forwarrd considering the uncertaintyy ofccontrol object aand time-delay eel. Time-delay will occur durinng the long disttance communiccation. Therefoore, when controol object is locaatediin distance, the transmitted refference signal w
will be delayed.. For this reasonn even though wew stabilize thee control plant bby loop-shapingg, itw
will be an unsttable system duue to time-delayy elements. In tthis paper we cconsider an unsstable control plantpsuch as H
HIMAT. Generaally,H
HIMAT is unm
manned aircraft which is requirred be controlleed from distancee by communiccation system thhat means in this system we havettime-delay elemment. Our aim is to stabilize tthe closed loop system with Innternal Model controller and rrealize the bettter performancee ofssystem using feeed forward conntrol.

KKey Words: Time-Delayy System, RobustRConttrol, Interna
al Model Conntrol, Feedfforward Conntrol.

11. Introdu
uctionIIn this researcch, we proposee control of the time-delay ssystem
bby using H∞ Controller, IM
MC (Internal Model Conttroller)
aand FFC (feeedforward Coontroller). Unttil now theree werem
many schemes to consider aabout the desiigning controlller forssystems which including time-delay eleement. For exxample
tthe classical w
way is PID ((Proportional-Integral-Derivvative)[6]. However,, this scheme, do not warraanty the stability forllarge time-dellay. On the otther hand, LQ
QI (Linear QuaadraticIIntegration) [22], [10], [11], [12] method is modern ccontrol
sscheme and it warrantiess the stabilityy even for a long
ttime-delay. HHowever, forr MIMO syystem, it is very
ccomplicated to solve the R
Ricatti equationn [14], [15], [[16] toddesign the optimum
oconttroller. Thereefore, we coonsider
ccombination oof IMC and FFC method [18], [19], [20] tosstabilize the closed loopp system inccluding timee-delayeelement. Time-delay will hhappen during utilization of thellong distance communicatiion. The appllication of thee long
ddistance commmunication is an importantt issue in aeroospaceeengineering. W
When we havve a control pllant in the disstance,tthe transmitteer’s signal w
will be delayyed. Thereforre therreceived signnal at the conntrol plant wiill also be deelayed.
M
Moreover, thee feedback siignal to transsmitter locatioon (inoorder to comppare the outputt signal and reeference signaal) will
aalso be delayeed. In this casee, we have a roundr
trip delaay, one
ddelay is to reeach the contrrol plant and another delayy is torreceive the feedback siggnal for com
mparison witth therreference signnal. Moreover,, generally thee control plant of thessystem usuallyy is an unstabble system. Thherefore, we hhave tocconsider the sstabilization oof the control plant. In this paper,w
we suggest thhe loop shapinng method witth H∞ controlller. By
lloop shaping method, we ccan make the ccontrol plant stable.
TThough contrrol plant willl be unstable due to timee-delayeelement. Therrefore, after staabilizing the ccontrol plant, w
we useccombination oof IMC and FF
FC method to make the systtem

stabble which wass unstable duue to time-delaay element withw
goood performancce. Also, we considered thhe uncertainty ofconntrol plant and time-delay eleement which it
i estimated annd
appproximated byy using Padee approximattion. Here looop
shapping method m
makes the open loop singulaar value to makethe same as deesirable loop shape consiidering stabillity
robuustness and peerformance. IMMC method iss minimizing thecoefficient of exxternal disturbbance of outpuut signal by Hnormm and FFC haas role of makking the system
m’s performannce
bettter as much ass possible by tuningt
the filteer. Therefore, by
thiss design, prroblems of the system instability aandunccertainty have been overcomme.

2. The Theorry of Time Delay
DSystemm and Backk
ground off Research
In this research we consider a round trip tim
me-delay systtem
whiich means onee delay elemeent to reach tthe control plant
andd another delayy element to feedback the output signall inordeer to compare with referencce signal. Figgure.1 shows theblocck diagram oof a round triip time-delay system withooutconntroller.

mFig. 1: Timee-delay system

H
Here “Y”,”H””,”R”,”P” andd “E” are the ooutput signal of thessystem, time-ddelay elementt, reference siignal, control object
aand error signnal of the systeem, respectivelly. Through fiigure 1
iit is clear tthat the senssitivity (S) aand complem
mentaryssensitivity (T)) are obtained as follows:SI

PKH
HTI

PKHH(1)
PKH(2)Generally, inn feedback ccontrol system
m, by adding someccontroller succh as “K” whiich is designeed correspondding toccontrol plant, we can make the system sttable or decreaase theeerror. In classsical control usually PID
D controller aand inm
modern controol integrator ooperation and optimum gainn suchaas LQI methood are used. A
As a result, foor PID controoller if
ttime-delay is large, systemm could not prreserve the staability.
B
But for Modeern scheme LQIL
method w
we could makke thessystem stablee without erroor. However, for high dimeension
aand MIMO system we couldc
not deesign the opttimum
ccontroller beccause of com
mplexity of solving
sthe Ricatti
Reequation. In thhis research, tthe system is already an unnstableaand contains with time-ddelay elemennts. Thereforee, we
ppropose the IIMC, FFC method
mand looop shaping m
methodw
with H∞ contrroller which thhe whole systtem is structurred byhhybrid systemm controllers. At first, looop shaping wiith H∞
ccontroller is designed coorresponding to control plant tom
make it stablee. IMC and F
FFC is designeed correspondding touuncertainty off time-delay element and coontrol plant inn ordertto stabilize thee closed loop ssystem.

33. Loop Sh
haping Metthod with H∞ Controlleer
For the Proccess of stabilizzing the closeed loop system
m is atffirst, it is requuired to stabilize control oobject. Therefoore, intthis case we ddesign H∞ conntroller by looop shaping m
method.
FFor loop shaping method, let us deesign a stabilizing
ffeedback conntroller to opptimally shappe the openn loop
ffrequency respponse of a M
MIMO feedbacck control systtem tommatch as cloosely as posssible to a ddesired loop shape.
CController hass the propertyy that it shapees the open looop sotthat it matches the frequenccy response off target desired loop
sshape as closeely as possiblee.

FFig.2: The singulaar value specificaations on open looop, sensitivity, andd closed
loop

Thrrough figure 22, it shows thhat the desiredd loop shape, inlow
w frequencies has high gainn and in high frequencies hashlow
w gain. This iis because off in low freqquencies if oppen
loopp’s singular vvalues have hiigh gain the outputosignal hashbettter performancce and if it has low gain in high frequenccies
it iss robust to diisturbance, nooise and uncerrtainty. Also forcom
mplementary ssensitivity singgular values it
i is desirablee tohavve low gain in high frequenccies in order too robust to nooise
andd uncertainty. Derivation of loop-shappe controller isshow
wn as followss:
Sincce stability mmargin (SM) is inverse of transfertfunctiionyv
betw
ween v  u of infinnity norm thatt is:
Fig .3: Stanndard feedback intterconnectionyu

I PC
PC
C I P

I PPC
I PCP vv

SMI PCCC I PPCI PPC
I PPCP∞

Sincce for good peerformance annd robustness , stability marggin
musst be large nuumber, for thiss reason “C” must stabiliziing
conntroller such thhat maximize thet stability mmargin. Therefo
fore“C”” is derived as following conndition that wee have:
min4.

I PC
C I PCCCI PC
CI PC

P

3∞

Combinaation of IM
MC and FF
FC for Timme
Delay Systtem

Intternal Model Controller is an optimum controller which
minnimizes the eeffect of distuurbance to ouutput signal aandconnsiders the unccertainty of coontrol plant annd Feed forwaard
conntroller is an ccontroller whiich has role of make systemm’sperfformance bettter as muchh as possible because whhen
timee-delay is occcurred in systeem the performance of outpput
signnal is poor. Thherefore, FFC is suggested to combine withw
IMCC. Also in thhis research w
we consider thet existence oftimee-delay elemeents. Hence m
most of system
ms would be an
unsttable system due
d to time-deelay elements. We suggest theIMCC and FFC method
mto moddify the stabiliity of the systtem
by IMCIand com
mpensate the peerformance byy FFC. The mainmreasson that we suggest thee IMC methhod is due toconnsiderations oof the uncerttainty of conntrol plant aandtimee-delay elemeents. The effe
fects of internnal and extern
rnal
distturbance to coontrol plant annd output signaal are minimizzed
andd it can be coorrespondencee to MIMO (Multi Input aandMulti Output) aand high dim
mension system
ms. In order tocom
mpensate the sstability and pperformance oof system duee totimee-delay elemeents, we have designed the hybrid
hsystemm ofIMCC and FFC. F
Figure 3 show
ws the IMC aand FFC systtem
whiich “du” is innternal disturbance to controol plant, “dy”” isexteernal disturbannce to output signal, ”n” iss feedback noiise,“K11” and “K2”” is Internal Model Conttroller and feeed

forward controller respectively and ”P”, ”H” and“P”,”H”,”C“are actual system, actual time-delay element,
model system, approximated and predicted time-delayelement and model of loop shape controller, respectively.

model system. But note that to realize the controller, if thecase of K is an improper system, it is required to add afilter the same as K ’s dimension to make controller properor strictly proper. The reason of this is for realization ofmodel or controller, system requires to be a proper or strictlyproper.

5.

Fig.4: IMC and FFC including a round trip time-delay elements, internaland external disturbance and feedback noise

Through figure 5 the block diagram of IMC and FFC
method, we obtained relation between signals and throughequations which has shown below, we obtained thecomplementary sensitivity (4) of figure 4.
y∆Pdu ∆PCHuy ∆PHuz Hy Hyu K r K ee Q rz ny ∆ PCHM
KI
CHM K H∆P duwhere, ∆H∆PCH, ∆GK Qr n
∆P I
∆PCHM KH dy

I PC , ∆G G and M

I PC , GI K∆G

4H∆PCH, G∆PCHM KH

5tr IC P

∆PCHM

KH

∗I

∆PCHM

KH dω

0∞

IPC

Pade L, n(6)where, “K ” is the inverse system of stabled control plant

∑e! !
! !

(7)

∑!

k0,1,2, … , n

Here we consider the realization of approximated time-delay
element as below.

~  A~H  H
C H~BH~ DH~ (8)But, remember that the dimension “n” of approximatedsystem transfer matrix must be the same as control plant
dimension. The reason of this is that in the multiplication of
two system transfer matrix systems, both of them are
required to have the same dimension, theoretically. Becauseof approximated time-delay element is considered as asystem transfer matrix.

6.
∞Theoretically, the optimum Controller “K ” in equation (6)is calculated as follows:K

In this case, the system is called nominal system which is
an ideal case. Although in the case of internal model
controller except the model system, we have to realize thepredicted time-delay element. Therefore, we used the Padeapproximation for “ L ” approximated time-delay i.e.
time-delay element” H” shown as follows:

Where, cthat is:
∂ 1
∂K 2π

In this case of IMC, we use the model of control plant andthe actual control plant which it is an unknown system. So,
as we obtained through equation (6) when IMC controller isthe inverse system of model, it is optimum case. But, if thismodel system does not completely match with the actualsystem, it will be feedback the error between the modelsystem and the actual system. So, the ideal case of thissystem is when M equal to unit matrix, that is: when
ΔG 0 therefore, M=I.

HSince, in the above equation (4) “dy” multiplied I∆PCHM KH , IMC method minimizes the effect ofdisturbance. As a result we consider the minimization of H2
norm of this coefficient. Equation (5), below, shows how toderive “K ” which is Internal Model Controller:
min I

Uncertainty of Control plant and Time
Delay Element

Simulation and Results

For evaluating the hybrid system of IMC and FFC method,
we have simulated for an unstable control plant withtime-delay. Control plant is considered to be 6 dimensional
MIMO system which has 2 inputs and outputs.
The process of simulation is to confirm the stability of theclosed loop system and compare the performance andstability of IMC and IMC+FFC method with their step

response and singular values. Also for IMC+FFC method wehave simulated the disturbance response of system and allthese processes are simulated for nominal case andnon-nominal case.

Q0The transfer Matrix of actual system

- 0.0226 - 36.6170 -18.8970 - 32.09 3.2509 - 0.7626 0 0 
 0.0001 -1.8997 0.9831 - 0.0007 - 0.1708 - 0.0050 0 0 
 0.0123 11.7200 - 2.6316 0.0009 - 31.6040 22.3960 0 0 
A B
0010000 0P
C D0000030 0
 30
 0

00000 30
 30
 0
100000 0
 0001000 0 Here we assume that the reference signal has 1 second
time-delay to reach the control plant and 1 second tofeedback the output signal to reach controller. Therefore, theactual time-delay L=1 and its transfer function is shown asfallows.
HeI.

(9)

0Q

I

 AQIMCnom
Cnom
 QIMCBQ nom 
IMC
DQ nom 
IMC 0 AQ nom
C Q nom

BQ nom 
DQ nom .

Simulation conditions
Q0.

Q AQFFCnom
Cnom QFFC

Q

BQ nom 
FFC
DQ nom 
FFC Else (non-nominal case). ∆ AQIMCnonCnon QIMC

∆Q.

∆. ∆∆

 AQ non BQ non C Q non DQ non 

∆

Q.

. ∆

∆

.

BQ non IMC
DQ non IMC ∆

The transfer matrix of model system

- 0.0903 -146.4680 - 75.5880 -128.3600 13.0036 - 3.0503 0 0 
 0.0004 - 7.5988 3.9325 - 0.0029 - 0.6832 - 0.0199 0 0 
 0.0494 46.8800 -10.5264 0.0035 -126.4160 89.5840 0 0 ~ ~A B0040000 0~P ~ ~ C D0000120015 0
 00000
120 0 15 02.500000 0
 0002.5000 0 

For predicted time-delay element we assume that ourestimated time-delay is 0.001 second and to realize thetime-delay element in the model system we used 6
dimensional Pade approximation.
H

Pade L, 6 IQIQ

 AQFFC
nonCnon QFFC

Q

BQ non FFC
DQ non FFC Here K and K are IMC and FFC controller respectively
which are shown as follows:
KC PIPC QKQC PQIPC Q(11)
C PIPC I(12)Step Response

(10)

3025

IMC and FFC proper for nominal case and non-nominal casefilters are set up with following conditions:

output1
output2

2015

10Amplitude(1) In non-nominal case IMC and FFC proper filter’s
parameters are adjustable with uncertainty of time-delay∆LL L and in nominal case with actualtime-delay.

50-5

(2)

Initial values of IMC and FFC proper filters are equal
to unit matrix. Q
0I,Q0I(3) IMC and FFC proper filters required be contain withstable poles.
If system is nominal, in other words P P and L L-10-15-200123456789Time (sec)Fig. 5 step response without IMC method

Nominal case

10

1.4

1.4
IMC+FFC1
IMC+FFC2
IMC1
IMC2

1.2

1

Amplitude

10.8

Amplitude

IMC+FFC1
IMC+FFC2
IMC1
IMC2

1.2

0.6

0.8

0.6

0.4

0.4

0.2
0.2

000

5

10152025
Time[sec]

303540450510152050

25
Time[sec]

3035404550

Fig. 10 step response of IMC and IMC+FFC

Fig. 6 step response of IMC and IMC+FFC21.5
IMC+FFC y1
IMC+FFC y2

1

IMC+FFC y1
IMC+FFC y2

1.5
10.5
Amplitude

Amplitude

0.5

00-0.5-0.5-1-1-1.5-1.50510152025Time[sec]30354045-2500Fig. 7 disturbance response of IMC+FFC

510152025Time[sec]3035404550Fig. 11 disturbance response of IMC+FFC

Singular Values

Singular Values

10050

FF+IMC

FF+IMC

IMC
P0IMC

0P-50-100
Singular Values (dB)Singular Values (dB)

-100-200-300-150-200-250-300-400-350-500-21010-101102103101010-400-2104-110010110210341010Frequency (rad/sec)

Frequency (rad/sec)

Fig. 12 Singular values of open loop system of plant, IMC and IMC+FFC

Fig. 8 Singular values of open loop system of plant, IMC and IMC+FFC

Singular Values

Singular Values

2050GG00-20

Singular Values (dB)Singular Values (dB)-50-100-150-40-60-80-200-100-250-120-300-210-110010110210Frequency (rad/sec)

31010-140-210010110210Frequency (rad/sec)

Fig. 9 Singular values of uncertainty system

Non-nominal case-110Fig. 13 Singular values of uncertainty system

Results310410

Through simulation results in Fig. 5 we can see that for
time-delay system without IMC method it is unstable.
Therefore IMC method is suggested. However in Fig. 6 andFig. 10 we can confirm the performance of IMC andIMC+FFC for both nominal case and non-nominal case that
IMC+FFC has better performance than IMC. For
disturbance response of IMC+FFC method it is clear that innon-nominal case the response of system contains with more
oscillation than nominal case. This is because of in Fig. 13
the singular values of uncertainty system contains with peekand gain in middle frequencies compare to Fig. 9 thesingular values of uncertainty system in nominal case.
However the important thing is both in nominal case andnon-nominal case in Fig. 8 and Fig. 12 the open loop ofcontroller for IMC and IMC +FFC is shown and it is clearthat for both case IMC+FFC method has small gain in highfrequencies comparing to IMC. This means that IMC+FFCmethod is robust to disturbance, noise and uncertainty ofcontrol plant and time-delay and it can preserve the stability.

7.

Conclusion

In this paper we suggest the IMC+FFC method in order tocompensate the performance of the step response andpreserve the stability of closed loop system against
disturbance and uncertainty. Also in non-nominal case theperformance of system is adjustable automatically byuncertainty of time-delay in order to avoid overshoots.
However our future works is to design a lower order ofcontroller and preserve the stability against continuously
external disturbance.

References[1] R.C.Dorf, R. H .Bishop “Modern Control System” Prentice Hall, 2002
[2] Witold Pedrycz “Robust Control Design an Optimal Control Approach”
Wiley 2007
[3] G.F.Franklin,J.D.Powell,M.Workman ”Digital Control of Dynamic System”
Addison – Wesley, 1997
[4] R. Oboe, K. Natori, K. Ohnishi”A Novel Structure of Time Delay Control
System with Communication Disturbance Observe” IEEE’08[5] J.E.Normey-RicoandE.F.Camach “Control of dead-time processes” springer[6] Guillermo J. SilvaAniruddha Datta S.RBhattacharyya“PID ControllerforTime-DelaySystem” Birkhauser[7] ATewari, “Atmospheric andSpace Flight Dynamics”,Birkhauser[8] G.J Balas, J.C Doyle, K Glover, A. Packard, R Smith “RobustControlToolbox TM 3 User’S Guide” The Math Works[9] Sigurd Skogestad, Ian Postlethwaite “Multi Variabla Feedback
Control Analysis and Design” JOHN WIEY & SONS
[10] F .Asharif, S.Tamaki, T.Nagado, T,Nagata, M. Rashid, M.
Asharif “Feedback Control of Linear Quadratic Integration
Including Time-Delay System” ITC-CSCC 2009[11] Dan Simon “Optimal State Estimation Kalman, H infinityand Nonlinear approaches” WILEY, INTERSCIENCE[12] Magdi S.Mahmoud “ROBUST CONTROL AND FILTERINGFOR TIME-DELAY SYSTEMS” MARCEL DEKKER, Inc[13] L.F. SHAMPINE, I. GLADWELL, S. THOMPSON “Solving
ODEs with MATLAB” CAMBRIDGE[14] L. Dugard, E.I. Verriest “Stability and Control of Time-delay
Systmes” Springer[15] Qing-Chang Zhong “Robust Control of Time-delay Systems”
Springer[16] Gary J. Balas, John C. Doyle, Keith Glover, Andy Packard,

Roy Smith “mu-Analysis and Synthesis Toolbox for Use withMATLAB” The MathWorks[17] Richard Y. Chiang, Michael G. Safonov “MATLAB RobustControl ToolBox” The MathWorks[18] Gary Balas, Richard Chiang, Andy Packard, Michale Safonov“Robust Control ToolBox TM 3 Getting Started Guide”[19] Buhai SHI, Xuefeng ZHU “Research on Multivariable Time
Delay System Multi-step Predective Dynamic Modeling” 2007
IEEE International Conference on Control and Automation[20] Pascal. Gahinet, Arkadi. Nemirovski, Alan. J. Laub,Mahmoud. Chilali “LMI Control Toolbox for Use with
MATLAB”[21] KEMIN ZHOU, JOHN C. DOYLE, KEITH GLOVER“”ROBUST AND OPTIMAL CONTROL” PRENTICE HALL,Englewood Cliffs, New Jersey 07632[22] Faramarz Asharif, Shiro Tamaki, Tsutomu Nagado, Nagata
Tomokazu, Mohammad Reza Alsharif “Design of MIMOInternal Model Controller Including Uncertainty ofTime-Delay System” SIP-2009[23] Faramarz Asharif, Shiro Tamaki, Tsutomu Nagado, Nagata
Tomokazu “Design of Robust Optimum Controller for
Unstable System Including Time-Delay Element” Conferenceof Electrical Electronics Computer Branch of Kyushyu 2009