Quantitative Robust Optimal Design of Uncertain Multivariable Feedback System

Copyright © IFAC Youth Automation, Beijing, PRC, 1995

QUANTITATIVE ROBUST OPI'IMAL DESIGN OF UNCERTAIN MULTIVARlABLE n :EDBAICK SYSTEM

Ze-RODg Sbi

Chib-MiD LiD

Yuan-Ze Institute ojTechno.'ogy, Chung-Li, 32026, Taiwan.

Abstract: The design algorithm of quantitative perfonnance robustness optimal control for uncertc~n multivariable systems is developed. The frequency-dependent weighting matrices are shaped and two-d~gree-of-freedom controllers are designed to achieve the quantitltive robust optimal control with feedback loop satisfying the return difference equality. So that the designed system achieves quantitative robust optimal control and simultmeously the fE:edback loop possesses the stability margins as in time-domain LQR uptimal system

keyword: Robust performance, Optimal control, Quantitative control.

1. INTRO:I>UCfION

matrix n(s) is called para-Hermitian when n(s) = n"(s) ; n-(s)

In Tsai and Wang (1986,1987), the QFT (quantitative feedback theory) is employild for robust optimal system design. And the considered systems are just for the single-inputsingle-output system. Moreover sinCf~ the Wiener-Hopf optimal system designs ( YClula et al. 1976, Youla and BongJ.omo 1985, PaJk and Bongiomo 1990, Tsai and Wang ,1986,1987) do not satisfy the return difference equality, so that these systems do not possess the stmility margins as in time-domain LQR optimal systems.

trace of n(s);

notations

used

standardized: nOes) is

ill tht~

this

paper

l :

tr 11(s) is the

And a transfer m.atriJc I\(s)

fracted as A(s) =[A(s)]. + [A(s)1- + [1\(s)L

(1.1)

where [A(s)]., [A(s)l and [A(s)l denote the part of the partial fraction expansion associated

with all poles of A(s) in

~

< Re(s) ~ 0,

o< Re(s)< 00, and at s=oo, respectively. 2. WIENER-HOPF LINEAR QUADRATIC OPI'IMAL CONTROL

In this paper, QFT and Wiener-Hopf optimal control techniques are applied for deriving the design algorithm. For the plant with minimunphase zero, the frequency-dependent weighting matrices are shaped and two-degree-of-frE!t~dom controllers are designed to achieve the quantitative performance robustness optimal control with feedback loop satisfying the return difference equality. The

=(n"(s)r

Consider the system shown in Figure 1, where E

C .... , C(s)

E

C;-" , and M(s)

c-

are the nominal plant, controller, and reference model, respectively, and 1'(s), des) and pes)

E

are the reference signal, disturbance, and measurement noise. respectively, and assume that 1'(s), des) and n(s) are mutually uncorrelated: and n(s) E (""'I

are

complex conjugate

T(s) .. P(s)C(sXI. + P(s)C(s)r'

transpose of the transfer m.atrix 11( - .\') . 1\ real

241

(2. 1)

Let ,"(s) , cilaa(s)

,

and

'nn(s) , 'eeCs)

'uuCs) be the power ~ densities of r(s) , des) , n(s) , e(s) and u(s), respectively, and
where ~(s) absorbs all zeros of ales) in Re(s)~O ..

Perform the spectral factorizations .1~(s).1,(s)

=~(sXXs)~(s)+ A,-(s)R(s)A,(s) (2.10)

(2.2) (2. 11)

Consider the linear quadratic cost function

where .1,(s) and .1:(s) are free of poles in

s=jw where the weighting matrices Q(s)

(2.3) E

C"'" and

C-- are para-Hennitian nonnegative definite for all imaginary s (i.e., s=jro ). Assume all the terms in the integration in (2.3) R(s)

~

E

0 (S-2) and are analytic and nonsingular on

the finite part of the s = jw axis (Yoular et al.

Re(s) > 0 and free of zeros in Re(s) ~ 0, respectively. Theorem 1 : In Figure 1, the closed-loop

transfer matrix t(s) which minimizes J of (2.3) is given by t(s) ~ ~(sXU,(s) - K(s)A(s»

where

~(s) = a,,~s) {.1~'(sX!(s)-[a,,(s).17(s)B;(s)Q(s).

1976, Youla and Bongiomo 1985, Park and Bongiomo 1990). Let pes) = A-'(s)B(s) = B,(s)~-' (s)

where the pairs A(s) B,(s) EC""'",

left/right

E C""",

A,(s) EC"x..

coprime,

proper,

(2 .12)

(M(s)' .. (s) + '",(s»A· (s).1;·(s)1.).1;'(s)}

(2 .13)

(2.4)

where I(s) is a real polynomial matrix of

B(s) E e x., and

degree less than or equal to p - 1, by the

constitute

requirement

stable,

any

of

analyticity

of

K(s)

In

,

rational

decompositions of P(s), respectively. We now select proper, stable, rational matrices V;(s),

Re(s) ~ 0, and detCV;(s) + B,(s) K(s»:;eO

Proof : Refer to (Youla et al. 1976, Youla and Bongiomo 1985, Park and Bongiomo 1990)

C,(s), V;(s) and [":(s), such that

B,(s) [ .~(s)

V;(s) IU,(S) -U:(s) A(s)

U,(S) [ A(s)

V;(s) IB,(S) -B(s) A,(s)

V;(S)]_[l. -B(s) - 0

V;(s) -U:(s)

0J

1.

3. (2.5)

J=[1.0 1.0J (26). (2.7)

where a1(s) is the least common denominator

of

adjoint

equality, where F(s) Ec-" and G(s) EC""" . For satisfying the return difference equality, the

a,(s);: a,,(s)a:/(s)

(2.8)

plant pes) has to be with minimum-phase zero(Anderson and Moore 1989).

a,,(s) = O(S')

(2.9)

In Figure 2, t(s) can also be expressed as

A,-'(s),

and

ad} A.(s)

is

an

In general, the system of Figure 1 with the optimal closed-loop transfer matrix t(s) obtain in (2. 13) does not satisfy the return difference equality. In this section, the equivalent twodegree-of-freedom compensation scheme shown in Figure 2 is considered so that the feedback loop can also satisfy the return diffemce

Let A-'(s) = adjA,(s) , a,(s)

THE EQUIVALENT TWO-DEGREEOF-FREEDOM COMPENSATION SYSTEM

polynomial matrix . Define

and

T(s)

242

= p(sXl. + G(s)p(s)r' F(s). T.(s)F(s)

(3.1)

The return difference equality of the feedback loop of Figure 2 is P'(s)Q(s)P(s) + R(s)

1

= (s-IXs+ 2) •

Jl(s)

=(1.. + G(s)P(s»" R(s) (1.. + G(s)P(s»

.

J(5.1)

(s+ 22583Xs+ 9.7417)

18(s+2111) (s+ 18211Xs+ 13.179)

~s+l7T78)

(3 .2)

[

(3.3)

Condition 2:

In this case the prefilter will be F(s)

= (1.. + G(s)P(S»Pf(S)

res)

where P t ( s ) is the pseudo inverse of full

P- s -

1 • 2( )- (s-8Xs+l6)

normal rank P(s). [

4, QUANTITATIVE PERFORMANCE ROBUSTNESS OPTIMAL SYSTEM DESIGN

(S + 1&066)(s+ Tl.fJM)

7l1..s + 14.222)

144(s+16888)

J (5 .2)

(s + 145@Xs+I05.432)

Condition 3:

R s -

1 • 3()- (s-O.l2SXs+025)

Following Horowitz's loop shaping algorithm (Horowitz 1979) for the derivation of bounds on loop transfer matrix L(s) in the Nichols chart. a suitable loop transfer matrix L(s) is chosen. For satisfying the return difference equality, the

[

225(s+Q264) ] (S+0282XS+ 1218) 1125(s+0222) (s+0228Xs+ 1674)

Step 1. The specification of frequency responses bounds for channel 1 and channel 2 are chosen as illustrated in Figure 3. Following the QFT design procedures, the suitable loop transfer

forbidden region of L(s) is shown in Figure 4. As L(s) is determined., the feedback controller is obtain

(4.1)

From the above analysis, we obtain the follOwing design algorithm. step 1: For the uncertain plant and specified closed-loop performance bound, by QFT approach, we can determine the

functions

G(s) - [g,(S) 0

and L'1J)(s) which satisfies the

0 ] g2(S)

(5.6)

where

g,

(s) = 0.4142(s+ 39.396)(s+ 15247 + j0.3286) .s(s+ 16422Xs+ 24.358) . (s+ 15247 - j03286)

G(s) in (4.1).

step 2: From (3.2), we obtain the corresponding weighting matrices Q(s)

(5.7)

(s) = 0.4142(s+51063Xs+ 13673+ jO.0873)

gz

andR(s).

.s(s+ L6422Xs+ 24.358)

From (2.12), we obtain the optimal

system t(s) . step 4: From (3.3),

~o(s)

return difference equality and the robust boundaries. And the feedback controller will be

desired loop transfer matrix L(s) . And then we obtain the feedback controller

step 3:

(5.3)

. (s+ 1.3673- jO.0873)

(5.8)

Step 2. Weighting matrices shaping we

obtain

the

-~

As choosing R(s) = 1- ~ J:, from (3.2), we

corresponding prefiiter F(s).

solve Q(s)

5. EXAMPLE

Q( ) _

1

[qll(S)

S - (l-s' )(269-s')(593.31-s' ) q,,(s)

Consider a system shown in Fig.2 with plant varying within three typical conditions. Condition I:

qll (S)] qll(S)

(5.9)

243

q,.(s) =173(s-0.5IXs-170Xs+203Xs-1325I) q,,(s) = -3.13(s+ 0.51Xs+.170Xs -203XS+ 13251)

d!(s) = S(S +0.89 ±j010Xs+126 ±jO.42Xs+ 1.44 ±j022)

(s+ 161±jo.71Xs + 170±j105)(s + 190+ j0.62) (s + 050Xs + 0.86)(s + 113Xs + 228Xs + 3.31)

q7is) = (056- s2X334 - s·X843.44 - S2)

fl1(s) = 8.837(s + o.40± jO.25)(s + 0.57 ± j0.50)

Step 3. Determine the optimal system

(s + 152 ± jl104Xs + 179 ± j108Xs + 192 + j145)

For the external signals

(s + 238 ± j052Xs + o.34Xs + o.80Xs + 164)

(5.10)

(s+ 172Xs+ 197) fI2(s) = 3.27s(s + 0.30± j0.32Xs + 050± j0.61)

-I

~",(S)=2 12

s

~ .. (s)

= 0.U

2

(5 .11)

(s+ 151±j107Xs + 188±j104Xs + 271 ±jO.41)

(5.12)

(s + 0.22Xs + 164Xs + 2.0XS+ 211Xs + 2. 79Xs - 0.67)

Reference model is chosen as

f21(s) = 4.215(s + 0.34± jO.l2Xs + 0.45± j0.38) (s + 0.64 ± jo.61Xs + 159 ± j105)(s + 178 ± j105)

0] [,'+3:+4 s:+3~+4

(s + 237±jO.60Xs + 164Xs + 177)

(5.13)

(s + 2.01Xs + 2.65) f22(s)

Following (2.5) to (2.15)

=5.97(s + 0.35±j029Xs + 0.51± j0.57)

(s + 150± j107Xs + 266± j021)

1 [tll(S) T(s) = dt(s) t21(s) A

dt(s)

tI2(s)J t22(s)

(5.14)

(s+ 182± j195)(s+ 190± j103XS+ 028XS+ 0.69) (s + 164Xs + 188Xs + 2.00) By applying these optimal controllers to the plant with three different conditions, the responses are shown in Figure 3 and Figure 5. It can be seen the desired quantitative robust linear quadratic optimal control is achieved.

=(s+ 0.89±jo.IOXs + 144±jo.22)

(s + 170± j105)(s + 190± jO.62Xs + 050Xs + 0.86) (s + 228Xs + 3.3IXs + I 161Xs + 30.30) tll(s) = 9.17(s + 0.39 ± j0.27Xs + 054 ± j0.50) (s + 147 ± j100Xs + 237 ± jO.49Xs + 0.35) (s + 2.51 ± j0.17Xs + 24.l2Xs + 28.11)

6

tI2(s) = 0.84s(s + 0.91± jo.07Xs+ 118±jo.08)

The multivariable QFT robust design method and the Wiener-Hopf optimal control technique are employed to propose the quantitative perfonnance robustness optimal control system design algorithm. For the plant with minimumphase zero, two frequency-dependent weighting matrices are shaped and two-degree-of-freedom compensators are designed to achieve the robust optimal control with feedback loop satisfying the return difference equality.

(s + 202±j024Xs + 166± j101Xs + 0.89) (s + 148Xs + 24.42Xs + 46.42) t21(s)

CONCLUSION

=119s(s + 0.89± jo.05)(s + 151±jo.I2)

(s+ 198±jo.2IXs+0.70Xs +s+ 116) (s + 332Xs + 24.36Xs + 24.42Xs + 33.95) t22(s) = lo.89(s + 0.34 ± jO.29Xs + 050 ± j057) (s + 149 ± j108Xs + 248 ± jO.55)(s + 0.28) (s + 263± jO.20Xs + 23.l4Xs + 2524)

REFERENCES

Step 4: Determine the prefilter From (3.3), the prefilter can be obtained I [fII(S) F(s) = df(s) f2I(s)

fl2(s)J f22(s)

Anderson, B.D.O. and Moore, lB. (1989). Linear optimal control, (Englewood Cliffs, N.J.: Prentice-Hall). Horowitz, 1. M. (1979), Quantitative synthesis of uncertain multiple input-output feedback system, 1nt. J. Control, 30, 81.

(5 . 15)

244

Park, K. and Bongiomo, J.J.Jr. (1990), WienerHopf design of servo-regulator-type multivariable control system including feedforward compensator, Int. J. Control, 52,1189. Tsai, T.P. and Wang, T .S. (1986), Optimal design control system with large plant uncertainty, Int. J. Control, 43,1015. Tsai, T.P. and Wang, T.S. (1987), Optimal design of nonminimum-phasee control system with large plant uncertainty, Int. J. Control, 45,2147. Youla, D.C. Bongiomo, J.J.Jr. and Jabr H.A. (1976), Modem Wiener-Hopf-design of optimal controllers; Part2: the multivariable case, IEEE Tram. A utom. Control 21,319. Youla, D.C. and Bongiomo, J.J.Jr. (1985), A feedback of two-degree-of-freedom optimal Wiener-Hopf Design. IEEE Tram. Autom.Control, 30.652.

.~

. .~ -2

-, '"

IYn(jw)l : Bound

Figure 3(b). Spread in

on IYn (jw)1 due to performance specification (solid line), Condition 1 (dashed line), Condition 2 (dashdot line), Condition 3 (dotted line).

' -0.2

1+ ~~~

.;~. ~

():~

'.;..0.6 •••• p •• •

:

I "·~+i ~l +

'0- '

' 0-'

r;;:0~~d-

' ".,

,'1-~',

"/ -..

' \ ... -.

3

'

o.r.< " "1 ' ~" ? "" ~ ""

. ' "it· .,

_.U

;

i

~<~~ .-"

. :'-', " .

" ~.:1'

"~'?0~~;~~:~ ~:,

~

. ~~~ .,.. "1

I

1"(0)

..--r

- 1 ; '

Figure 1. Model reference multivariable feedback control system

- <.""~.,.,.:,,,,~-:: _ )()()~'-::_,:7",-

•

.'7.,,,.,=---.7.,"';:---;;;;;-----:-;;;----;

<0

p,.,- ( o.-q )

Figure 4(a).

J -o,

~

,~"

" "" ,

1

: I;i>t~~,:~:,

":1)

Figure 2. Equivalent two-degree-of-freedom compensation system

-- ....

' ~~~

Channel 1: QFf design

~.,.,.;--"-------",'--,~/---.-~~.~'I.!!!~n-__;.--. . --~---,.,.,t '0

'~-~J"'~---:;_~--~2"'~--'=".,~~., ~"'~~-,~".,--~-"'~~ _ _ COPoI )

~- .....

"

Figure4(b).

, "

,

,, ,

,

,

:1

~

-2

Channel 2: QFf design

=~----,--.,..",-...----~

0', '

-2

0, 6

-~~~'---'cr""",-----".. =-----::,,,:--~--7n , ,,

0 ,<

..... / '11

Figure 3(a). Spread in

IYlIuw)1

0, '

Figure 5(a).

Channel I: Unit-step

response of system output Y 1(t)

245

..,

.-.

~~~~--~~.--~~~~~.o

Figure 5(b).

Channel 2: Unit-step

response of system output Y:2 (t) , Condition l(solid line), Condition 2 (dashed line), Condition 3 (dotted line).

246