Stabilization of MIMO Time Delay System Using Hybrid Controller of Internal Model Control and Feedforward Control: University of the Ryukyus Repository

Title

Stabilization of MIMO Time Delay System Using Hybrid

_{Controller of Internal Model Control and Feedforward Control}

Author(s)

Faramarz Asharif; Tamaki, Shiro; Nagado, Tsutomu; Nagata,

_{Tomokazu; Mohammad Reza Alsharif}

Citation

琉球大学工学部紀要(71)

Issue Date

2010

URL

http://hdl.handle.net/20.500.12000/18511

t c i w H t s K .

1

I b a m s t l I s t c d c s e l d e t r M o a d r r s c w l T e c

Faram

In this paper, w the system whi control object a in distance, the will be an unst HIMAT is unm time-delay elem system using fe Key Words: .

1. Introdu

In this researc by using H∞ and FFC (fee many scheme systems whic the classical w [6]. However, large time-del Integration) [2 scheme and time-delay. H complicated t design the o combination o stabilize the element. Tim long distance distance comm engineering. W the transmitte received sign Moreover, the order to comp also be delaye delay is to re receive the reference sign system usually consider the s we suggest th loop shaping Though contr element. Ther combination o

Stabi

marz Asharif,

we aim to stabi ich is unstable and time-delay e transmitted ref table system du manned aircraft ment. Our aim eed forward con

Time-Delay

uction

ch, we propose Controller, IM edforward Co s to consider a h including t way is PID ( , this scheme, lay. On the ot 2], [10], [11], it warranties However, for o solve the R optimum cont of IMC and closed loop e-delay will h communicati munication is When we hav er’s signal w nal at the con

e feedback si pare the output ed. In this case each the contr

feedback sig nal. Moreover, y is an unstab stabilization o he loop shapin method, we c rol plant will refore, after sta of IMC and FF

ilization

of Int

Shiro Tamak

ilize the unstabl due to time-de el. Time-delay ference signal w ue to time-delay which is requir is to stabilize t ntrol. y System, R e control of th MC (Internal ontroller). Unt

about the desi ime-delay ele do not warra ther hand, LQ [12] method s the stability r MIMO sy Ricatti equation troller. There FFC method p system inc happen durin ion. The appl an important ve a control pl will be delay ntrol plant wi ignal to trans t signal and re e, we have a r rol plant and gnal for com , generally the ble system. Th of the control

ng method wit can make the c

l be unstable abilizing the c FC method to

of MIM

ternal M

ki, Tsutomu N

le system by loo elay element by will occur durin will be delayed.

y elements. In t red be controlle the closed loop

Robust Cont

e time-delay s Model Cont til now there igning control ement. For ex Integral-Deriv anty the stabil QI (Linear Qua is modern c y even for a ystem, it is n [14], [15], [ efore, we co [18], [19], [ cluding time g utilization lication of the t issue in aero lant in the dis yed. Therefor ill also be de smitter locatio eference signa round trip dela another delay mparison wit e control plant herefore, we h plant. In this th H∞ _controll control plant due to time control plant, w

make the syst

O Time

Model Con

Nagado, Tom

Abstract

op shaping meth y Internal Mode ng the long dist . For this reason

this paper we c ed from distance system with In trol, Interna system troller) e were ller for xample vative) ity for adratic control a long very [16] to onsider 20] to e-delay of the e long ospace stance, re the elayed. on (in al) will ay, one y is to th the of the have to paper, ler. By stable. e-delay we use tem stab goo con app shap the robu coe norm bett this unc

2.

In whi and orde bloc con

Delay Sy

ntrol and

mokazu Naga

hod. Moreover el Controller w tance communic n even though w consider an uns e by communic nternal Model al Model Con

ble which was od performanc ntrol plant and proximated by ping method m same as de ustness and pe fficient of ex m and FFC ha ter as much as s design, pr certainty have

The Theor

ground of

this research ich means one d another delay er to compare ck diagram o ntroller.

ystem Us

d Feedfo

ata and Moha

after stabilizing with feed forwar cation. Therefo we stabilize the stable control p cation system th controller and r ntrol, Feedf s unstable du ce. Also, we time-delay ele y using Pade makes the ope esirable loop erformance. IM xternal disturb as role of mak s possible by t roblems of been overcom

ry of Time D

f Research

we consider a e delay eleme y element to with referenc of a round tri Fig. 1: Time

sing Hyb

orward C

ammad Reza

g the system w rd considering ore, when contro e control plant b plant such as H hat means in th realize the bett

forward Con ue to time-dela considered th ement which i e approximat n loop singula shape consi MC method is bance of outpu king the system tuning the filte

the system me.

Delay System

a round trip tim ent to reach t feedback the ce signal. Fig ip time-delay e-delay system

brid Con

Control

a Alsharif

we need to mod the uncertainty ol object is loca by loop-shaping HIMAT. Genera is system we h ter performance ntrol ay element w he uncertainty it estimated an tion. Here lo ar value to ma idering stabil s minimizing ut signal by m’s performan er. Therefore, instability a

m and Back

me-delay syst the control pl output signal gure.1 shows system witho m

ntroller

dify y of ated g, it ally, ave e of with of nd oop ke lity the H nce by and

k

tem ant l in the out

H s a i s S T c c e m a t B s a c e a p w h c m u t

3

f t F f f m C t s F Here “Y”,”H” system, time-d and error sign it is clear t sensitivity (T) S I PKH T I PKH Generally, in controller suc control plant, error. In clas modern contro as LQI metho time-delay is But for Mode system stable and MIMO controller bec equation. In th and contains propose the I with H∞ _contr hybrid system controller is make it stable uncertainty of to stabilize the

3. Loop Sh

For the Proc first, it is requ this case we d For loop sh feedback con frequency resp match as clo Controller has that it matche shape as close

Fig.2: The singula

”,”R”,”P” and delay element nal of the syste that the sens ) are obtained H H PKH n feedback c ch as “K” whi we can make ssical control ol integrator o od are used. A large, system ern scheme L e without erro system we c cause of com his research, t with time-d IMC, FFC m roller which th m controllers. designed co e. IMC and F f time-delay e e closed loop s

haping Met

cess of stabiliz uired to stabi design H∞ _con aping metho ntroller to op ponse of a M osely as poss s the property s the frequenc ely as possible ar value specifica

d “E” are the o t, reference si em, respectivel sitivity (S) a as follows: control system ich is designe the system st usually PID operation and As a result, fo m could not pr LQI method w or. However, could not de mplexity of s the system is delay elemen method and loo he whole syst At first, loo orresponding FFC is designe lement and co system.

thod with H

zing the close lize control o ntroller by loo d, let us de ptimally shap MIMO feedbac sible to a d y that it shape cy response of e.

ations on open loo loop output signal ignal, control ly. Through fi and complem m, by adding ed correspond table or decrea D controller a optimum gain or PID contro reserve the sta we could mak for high dime esign the opt solving the R already an un nts. Therefore op shaping m tem is structur op shaping wi to control pl ed correspond ontrol plant in

H

∞

Controlle

ed loop system object. Therefo op shaping m esign a stab pe the open ck control syst desired loop es the open lo f target desire

op, sensitivity, and

of the object igure 1 mentary (1) (2) some ding to ase the and in n such oller if ability. ke the ension timum Ricatti nstable e, we method red by ith H∞ ant to ding to n order

er

m is at ore, in method. ilizing n loop tem to shape. oop so d loop d closed Thr low low loop bett it is com hav and show Sinc betw y u SM Sinc mus con “C” mi

4.

Int min con con perf time sign IMC time unst IMC by I reas con time dist and Mu com time IMC whi exte “K1 rough figure 2 w frequencies w gain. This i p’s singular v ter performanc s robust to di mplementary s ve low gain in d uncertainty. wn as follows ce stability m ween v_{v } Fig .3: Stan I P C I P I PC C I P ce for good pe st be large nu ntroller such th ” is derived as n I PC C I PC

Combina

Delay Syst

ternal Model nimizes the e nsiders the unc ntroller is an c formance bet e-delay is occ nal is poor. Th C. Also in th e-delay eleme table system d C and FFC m IMC and com son that we nsiderations o e-delay eleme turbance to co d it can be co lti Output) a mpensate the s e-delay eleme C and FFC. F ich “du” is in ernal disturban 1” and “K2” 2, it shows th has high gain is because of values have hi ce and if it ha isturbance, no sensitivity sing high frequenc Derivation s: margin (SM) i  y_{u of infin}

ndard feedback int

C I P

PC I

C I P

PC I P

erformance an umber, for this hat maximize t following con I PC C I PC

ation of IM

tem

Controller is effect of distu certainty of co controller whi tter as much curred in syste herefore, FFC his research w ents. Hence m due to time-de method to mod mpensate the pe suggest the of the uncert

ents. The effe ontrol plant an

orrespondence and high dim stability and p ents, we have Figure 3 show nternal disturb nce to output ” is Internal hat the desired n and in high f in low freq igh gain the o s low gain in oise and uncer

gular values i cies in order to

of loop-shap s inverse of t nity norm that

terconnection PC P PC v v PC P PC _∞ nd robustness , s reason “C” the stability m ndition that we C P C _∞

MC and FF

an optimum urbance to ou ontrol plant an ich has role o h as possible em the perfor is suggested we consider t most of system elay elements dify the stabili erformance by e IMC meth tainty of con fects of intern nd output signa e to MIMO ( mension system performance o designed the h ws the IMC a ance to contro signal, ”n” is Model Cont d loop shape, frequencies h quencies if op output signal h high frequenc rtainty. Also it is desirable o robust to no pe controller transfer functi t is: , stability marg must stabilizi margin. Therefo e have: 3

FC for Tim

controller wh utput signal a nd Feed forwa f make system because wh mance of outp to combine w the existence ms would be . We suggest ity of the syst y FFC. The m hod is due ntrol plant a nal and extern

al are minimiz Multi Input a ms. In order of system due hybrid system and FFC syst ol plant, “dy” s feedback noi troller and fe in has pen has cies for e to oise is ion gin ing fore 3

me

ich and ard m’s hen put with of an the tem main to and rnal zed and to e to m of tem ” is ise, eed

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

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

Through figure 5 the block diagram of IMC and FFC method, we obtained relation between signals and through equations which has shown below, we obtained the complementary sensitivity (4) of figure 4.

y ∆Pdu ∆PCHu y ∆PHu z Hy Hy u K r K e e Q r z n y ∆ PCHM K K Q r n ∆P I CHM K H∆P du I ∆PCHM KH dy 4 where, ∆ I PC , ∆ I PC , G H∆PCH, G H∆PCH, ∆G G G and M I K∆G

Since, in the above equation (4) “dy” multiplied I ∆PCHM KH , IMC method minimizes the effect of disturbance. As a result we consider the minimization of H2

norm of this coefficient. Equation (5), below, shows how to derive “K ” which is Internal Model Controller:

min I ∆PCHM KH 5 that is: ∂ ∂K 1 2π tr I ∆PCHM KH ∗ I ∆PCHM KH dω ∞ ∞ 0

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

K C P I PC (6) where, “K ” is the inverse system of stabled control plant

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

5. Uncertainty of Control plant and Time

Delay Element

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

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 the predicted time-delay element. Therefore, we used the Pade approximation for “ L ” approximated time-delay i.e. time-delay element” H” shown as follows:

H Pade L, n e ∑ _∑ (7) Where, c ! !

! ! ! k 0,1,2, … , n

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















H H H H

D

C

B

A

H

~ ~ ~ ~

~

(8)

But, remember that the dimension “n” of approximated system 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. Because of approximated time-delay element is considered as a system transfer matrix.

6. Simulation and Results

For evaluating the hybrid system of IMC and FFC method, we have simulated for an unstable control plant with time-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 the closed loop system and compare the performance and stability of IMC and IMC+FFC method with their step

response and singular values. Also for IMC+FFC method we have simulated the disturbance response of system and all these processes are simulated for nominal case and non-nominal case.

Simulation conditions

The transfer Matrix of actual system

 

































    0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 30 0 30 0 0 0 0 0 0 30 0 30 0 0 0 0 0 0 0 0 0 1 0 0 0 0 22.3960 31.6040 -0.0009 2.6316 -11.7200 0.0123 0 0 0.0050 -0.1708 -0.0007 - 0.9831 1.8997 -0.0001 0 0 0.7626 -3.2509 32.09 -18.8970 -36.6170 -0.0226 -D C B A P

Here we assume that the reference signal has 1 second time-delay to reach the control plant and 1 second to feedback the output signal to reach controller. Therefore, the actual time-delay L=1 and its transfer function is shown as fallows.

H e I (9) The transfer matrix of model system

 

































    0 0 0 0 5 . 2 0 0 0 0 0 0 0 0 0 5 . 2 0 15 0 120 0 0 0 0 0 0 15 0 120 0 0 0 0 0 0 0 0 0 4 0 0 0 0 89.5840 126.4160 -0.0035 10.5264 -46.8800 0.0494 0 0 0.0199 -0.6832 -0.0029 -3.9325 7.5988 -0.0004 0 0 3.0503 -13.0036 128.3600 -75.5880 -146.4680 -0.0903 -~ ~ ~ ~ ~ D C B A P

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

H Pade L, 6 I (10) IMC and FFC proper for nominal case and non-nominal case filters are set up with following conditions:

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

(2) Initial values of IMC and FFC proper filters are equal to unit matrix. Q 0 I , Q 0 I

(3) IMC and FFC proper filters required be contain with stable poles.

If system is nominal, in other words P P and L L

Q 0 0 _.

_



_











nom IMC nom IMC nom IMC nom IMC Q Q Q Q

D

C

B

A

Q . 0 0 .

















nom nom nom nom Q Q Q Q

D

C

B

A

Q

I

Q

Q

















nom FFC nom FFC nom FFC nom FFC Q Q Q Q

D

C

B

A

Else (non-nominal case)

Q

. ∆ ∆ . ∆ . ∆

_



_











non IMC non IMC non IMC non IMC Q Q Q Q

D

C

B

A

Q ∆ ∆ . ∆ . ∆ . ∆

















non non non non Q Q Q Q

D

C

B

A

Q

I

Q

Q

















non FFC non FFC non FFC non FFC Q Q Q Q

D

C

B

A

Here K and K are IMC and FFC controller respectively which are shown as follows:

K C P I PC Q (11) K C P I PC Q C P I PC I

Q Q (12)

Fig. 5 step response without IMC method

Nominal case

0 1 2 3 4 5 6 7 8 9 10 -20 -15 -10 -5 0 5 10 15 20 25 30 Step Response Time (sec) A m pl itude output1 output2

Fig. 6 step response of IMC and IMC+FFC

Fig. 7 disturbance response of IMC+FFC

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

Fig. 9 Singular values of uncertainty system

Non-nominal case

Fig. 10 step response of IMC and IMC+FFC

Fig. 11 disturbance response of IMC+FFC

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

Fig. 13 Singular values of uncertainty system Results 0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time[sec] Am p litu d e IMC+FFC1 IMC+FFC2 IMC1 IMC2 0 5 10 15 20 25 30 35 40 45 50 -1.5 -1 -0.5 0 0.5 1 1.5 Time[sec] Am pl it ude IMC+FFC y1 IMC+FFC y2 10-2 10-1 100 101 102 103 104 -500 -400 -300 -200 -100 0 100 Singular V alues Frequency (rad/sec) S ing ul ar V a lu e s ( d B ) FF+IMC IMC P 10-2 10-1 100 101 102 103 10 -300 -250 -200 -150 -100 -50 0 50 Singular Values Frequency (rad/sec) S ingul ar V a lues ( d B ) G 0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time[sec] Am pl it ude IMC+FFC1 IMC+FFC2 IMC1 IMC2 0 5 10 15 20 25 30 35 40 45 50 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time[sec] A m pl it ude IMC+FFC y1 IMC+FFC y2 10-2 10-1 100 101 102 103 104 -400 -350 -300 -250 -200 -150 -100 -50 0 50 Singular Values Frequency (rad/sec) S ing ul ar V a lues ( d B ) FF+IMC IMC P 10-2 10-1 100 101 102 103 104 -140 -120 -100 -80 -60 -40 -20 0 20 Singular Values Frequency (rad/sec) S ingu la r V a lu es ( d B ) G

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 and Fig. 10 we can confirm the performance of IMC and IMC+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 in non-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 peek and gain in middle frequencies compare to Fig. 9 the singular values of uncertainty system in nominal case. However the important thing is both in nominal case and non-nominal case in Fig. 8 and Fig. 12 the open loop of controller for IMC and IMC +FFC is shown and it is clear that for both case IMC+FFC method has small gain in high frequencies comparing to IMC. This means that IMC+FFC method is robust to disturbance, noise and uncertainty of control plant and time-delay and it can preserve the stability.

7. Conclusion

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

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