Bull. Faculty of Engineering, Univ. of the Ryukyus No.72, 2011

Evaluation of Linear-Quadratic-Integration and Internal Model Control on Application toWind Turbine Considering Round Trip Delay in Feedback Control SystemFaramarz Asharif*, Shiro Tamaki**, Tsutomu Nagado**, Tomokazu Nagata** andMohammad Reza Alsharif***University of the Ryukyus Graduate School of Science and Engineering**University of the Ryukyus Faculty of Engineering
faramarz_asharif@yahoo.com
Abstract
In this paper, we aim to analysis the system which contains time-delay elements including closed loop system anduncertainty. Time-delay will happen during the long distance communication. Therefore, when we have a control object ina long distance, the transmitted reference signal will be delayed certainly. For this reason even though we assume thecontrol object system is a stable, but it will be an unstable system due to existence of time-delay elements. So, we consider
the Internal Model Controller method which is one of the robust controllers. Internal Model Controller method iscomposed of optimum controller and uncertainty model of control object and time-delay elements. The optimum controller
is designed by minimizing the coefficients of external disturbance of output signal by H2 norm in order to stabilize theclosed loop system considering the uncertainty of control object and predicted time-delay. Also, in order to compare theperformance of the Internal Model Controller, we have compared with Linear Quadratic Integration method. As results theperformance of Internal Model Controller was much better than the Linear Quadratic Integration method.
Keywords: Internal Model Controller, Tele-Operation, H2 Norm, MIMO (Multi Input Multi Output) System.

1. Introduction
In this research, we propose control of the time-delaysystem by using IMC (Internal Model Controller) [1-5]. There
were many schemes and suggestions to consider the designing
controller for systems which includes time-delay elements. For
example the classical way is PID controller [6-8] (Proportional,Integral Derivative). However, this scheme is not suitable forlarge time-delay. On the other hand, LQI (Linear QuadraticIntegration) [9] method which is modern way and it warranties
the stability even for a large time-delay. However, for MIMO(Multi Input Multi Output) system, it is very difficult to solve
the Ricatti equation to design the optimum controller. Therefore,
we consider IMC method to control the system including
time-delay elements. Time-delay will happen during utilizationof the long distance communication. The application of the long
distance communication is an important issue in aerospaceengineering. When we have a control object in the long
distance, the transmitter’s signal will be delayed. Therefore thereceived signal at the control object will also be delayed.
Moreover, the feedback signal to transmitter location will also
be delayed. So, in this case, we have a round trip delay, one
delay is to reach the control object and another delay is toreceive the feedback signal for comparison with the reference
signal. This comparison makes the error signal of the control
system. This error must be decreased for a better performance.
Moreover, control object will be unstable due to time-delay
elements. Therefore, in this paper we consider the stability ofcontrol object and its uncertainty. Also, not only the uncertainty
of control object, but also the uncertainty of time-delay
elements which is estimated and approximated by using Padeapproximation [10] has been considered. IMC method isminimizing the coefficient of external disturbance of output
signal by H2 norm. Therefore, by this design problems of thesystem instability, uncertainty and external disturbance havebeen overcomes. Then, it can be solved by one of the robust
controller such as IMC method.

2. Back Ground of Research and Internal Model
Controller
As we express in introduction in this research we have around trip time-delay system which means one delay element toreach the control object and another delay element to feedback
the output signal in order to compare with reference signal.
Figure 1 shows the block diagram of a round trip time-delaysystem with controller.

Fig. 1: Time-delay system

Here “Y”,”H”,”R”,”P” and “E” are the output signal of thesystem, time-delay element, reference signal, control object anderror signal of the system, respectively. It is clear that thesensitivity (S) and transfer function (T) are:2S  ( I  PKH )

(1)

2 1
T  ( I  PKH ) PKH(2)

Bull. Faculty of Engineering, Univ. of the Ryukyus No.72, 2011

Generally, in feedback control system, by inserting a controller
such as “K” which is designed corresponding to control object,
minimize the error signal. Therefore, in classical control usually
PID controller and in modern control integrator operation andoptimum gain such as LQI method are used. But, minimizing
the error signal of the system is not enough. Also, we have tomake the system stable if it is an unstable. For example, in thecase of PID controller, if time-delay is large, system could notpreserve the stability. But for Modern scheme LQI method we
could make the system stable without error. However, for highdimension and MIMO system we couldn’t design the optimum
controller because of complexity of solving the Ricatti equation.
Therefore, in this research we propose the IMC method which
is one of the robust controllers. Internal Model Controller is an
optimum controller which minimizes the effect of disturbanceto output signal and considers the uncertainty of control object.
Also in this research we consider the existence of time-delay
elements. Hence most of systems would be an unstable system
due to time-delay elements. We suggest the IMC method tomodify the stability of the system and compensate the output
signal. The main reason that we suggest the IMC method isdue to consideration of the uncertainty of control object andtime-delay elements. The effects of disturbance to output signal
are minimized and it can be correspondence to MIMO and highdimension systems. In order to modify the unstable system due
to time-delay elements, we have designed the internal modelcontroller. Figure 2 shows the IMC system which “D” isexternal disturbance to output signal, ”K” is Internal Model~~Controller, ”P”, ”H” and “ P ”, “ H ” are actual system, actual
time-delay element and model system, approximated andpredicted time-delay element, respectively. Follwing figure
shows the structures of IMC.

Since, in the above equation (3) external disturbance multipliedby “ I  PHMKH ” Which IMC method minimizes the effect ofdisturbance. As a result we consider the minimization of H 2
norm of this coefficient. Equation (4), below, shows how toderive “K” which is Internal Model Controller:
min I  PHMKH 2KTheoretically, the optimum Controller K in equation (5) iscalculated when:~ 1(5)K PWhere, “K” is the inverse system of control object’s model
system. Also, if there is no uncertainty exists in system thenequation (3) can be rewritten as follows:y  PHKr  ( I  PHKH )d(6)As it is clear, the effect of time-delay elements is cancelled inclosed-loop system and it exists only in input delay. However,generally, the uncertainty effects cannot be cancelled if actualsystem is unknown. Next Chapter, we discuss more about
uncertainty.

3. Uncertainty of Control Object and Time-Delay
Element
In this case of IMC, we use the model of control object andthe actual control object which is an unknown system. So, as it
obtained through equation (4) when “K” is the inverse systemof model, it is optimum case and minimizes the effects ofexternal disturbances. However, if this model system does notcompletely match with the actual system, it will be feedback
the error between the model and the actual system. So, the ideal
case of this system is when the sensitivity function of IMC
equal to unitary matrix, that is when G  0  M  I ,system is called nominal which is an ideal case. Although in thecase of internal model controller except the model system, itrequires to realize the predicted time-delay element. Therefore,~
it has used the Pade approximation for “ L ” which is anapproximated time-delay i.e. time-delay elements that it can be~
indicate as” H ” has shown as follows:n~k kk ( 1) ck L s~~~sLk
0H e Pade ( L , n) n~k k ck L sk 0

Fig. 2: Structure of IMC with time-delay and disturbanceHere through figure 2 the block diagram of IMC method, we( 2n  k )! n!
obtained relation between output and reference signal as Where, c k2n! k! ( n  k )!
follows.

y  PHMKr  ( I  PHMKH )d(4)(3)

Where, G is Uncertainty of plant and time-delay element:~~ ~G  HPH  HPH and M is the sensitivity function of IMC
1is M  ( I  GK ) .

for(7)(k  0.,1,2,...,n)Here we consider this approximated time-delay element as atransfer matrix.
~ A ~H  H
C H~B~
H

D~
H

(8)

Bull. Faculty of Engineering, Univ. of the Ryukyus No.72, 2011
The step response and phase characteristics of actual time-delay
and Pade approximated time-delay are shown.
~sActual time-delay element: H  e~
Approximated time-delay element (Pade): H  Pade(1,4)~ ~~~~ A B
where, A  A  A , B  B  BandP ~C 0 ~C  C  C . Then Uncertainty system matrix can be shown asfollows:Here we assume that actual system contains with L=10 [sec]time delay to reach the reference signal and L=10 [sec] tofeedback the output signal to controller. However, the predicted~time-delay is considered as L  1 [sec]. Relation between
actual time-delay and predicted time-delay is:~(9)L  L  LAmplitude

1.5
10.5

Which “ L ” is Uncertainty of time-delay. Actual and Padeapproximated time-delay element are as follows.
 sL
Actual: H  eI2 2~~
Pade: H  Pade( L , Dimenison ( A)) I2 2
The IMC proper filter set as follws:

0-0.5

00.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Time (secs)
Phase response2

If system is nominal:
1

0

2(s1)
FNom  1
02s  2.1s  1
Else for non-nominal case:
1


0
 2

F  s  2.1Ls 1
Non  nom 10
s 2 1.9Ls 1
0Phase (deg.) A B C 0  .

padeactual-500

-1000

0

10

1

10
Frequency (rad/s)

210

By these conditions, we have simulated for nominal case andnon-nominal case with Internal Model Controller’s proper filter.
Also we have shown the step response and the phase of 4
dimensional Pade approximated time-delay and actual
time-delay element for 1 second time-delay.

5Fig. 3: Pade approximation and actual time-delay

LQI Output1
LQI Output2

4. Numerical Analysis

Actual system:
0.04150.00730 
 0.0558  0.9968 0.0802
 0.5980  0.1150  0.0318  0.4750  0.4750 0.0077

0.1530 0.1430
 A B   3.0500 0.3880  0.4650 0.1530
P



10000 C 0   0 010000 001000 Predicted Model system has additional uncertainty which has
shown as follows:-5Amplitude

For evaluating the IMC method we have simulated windturbine system. The process of simulation is first to simulate thestep response of LQI method and then with IMC including
nominal case and non-nominal case. The simulation condition
shown as follows.

0-10-15-20-25-30

0

100

200

300

400
Time[s]500

Fig. 4: Step response of LQI600

700

800

Bull. Faculty of Engineering, Univ. of the Ryukyus No.72, 2011

From figure 4, it is clear that system with LQI its performanceis very poor and the output does not converge to reference
signal. Therefore, Internal Model Controller is suggested inorder to realize a good performance and reduce the effects oftime-delay in closed-loop. However, comparing the figure 6and 7, non-nominal case has also very late settle time and big
overshoots comparing to nominal case. This is because of theuncertainty system which exits in sensitivity function of IMC in
the case of non-nominal.

5Amplitude

0-5with controller output1-10-155.
0100200300400time[s]500600700

80015Amplitudewith controller output21050-50100200300400time[s]500600700Conclusion

In this research Internal Model Controller stabilize the closed
loop system including round trip delay containing with muchless error comparing to LQI method. However, for proposed
controller it considered the uncertainty of control object andapproximated time-delay element. Therefore, even thoughsystem is non-nominal, Internal Model Controller can becorresponded. For the system’s performance it can be adjustedwith proper filter and filter itself automatically adjustable byΔL which is uncertainty of time-delay. However the demerit
point of IMC is that realization of controller is quite hard due toexistence of Pade filter, especially when control object includes
high order and in consequence the controller becomes highorder. So, the reduction of controller’s dimension would be oneof the important future works.

800

References[1] R.C.Dorf, R. H .Bishop, 2002 “Modern d ControlSystem”,Prentice Hall

Fig. 5: Step response with controller non-nominal case

Amplitude

1.5
10.5
with controller output10-0.50100200300400time[s]500600700

800

Amplitude1.510.5with controller output20-0.50100200300400time[s]500600700Fig. 6: Step response with controller nominal case800[2] Witold Pedrycz, 2007 “Robust Control Design an Optimal control Approach”
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