Controller synthesis technique for systems with affine linear uncertainty

CONTROLLER SYNTHESIS TECHNIQUE FOR SYSTEMS WITH AF...

14th World Congress of IFAC

G-2e-08-5

Copyrigbt (Q 1999 IFA.C

14th Triennial \Vorld

Congrcss~

Beijing, P.R. China

CONTROLLER SYNTHESIS TECHNIQUE FOR SYSTEMS WITH AFFINE LINEAR UNCERTAINTY Nusret Tan and Derek P .. Atherton

School of Engineering, Un2:ve r sify of Sussex~ Fa lrn er, Br'ighton B.l·{j 9QT ilK. d.p. [email protected]. Q.C. 71k; n. t(l,'n,@~"'i1tS.r.;ex. o.c. uk

Abstract: The paper deals "\\rith the design of comp€nsators that provide a guaranteed level of performance, in the sense of gain and phase rnargins~ for systems with affine linear uncertainty. To achieve this goal, po\verful procedures for COIllputing the Bode, Nyquist and Nichols envelopes of a transfer function, \vhose. numerator and denominator POlY110111ials arc polytopic polynomials, arc first developed by using the convex par-polygonal value set of polynornials '\Jvith affine linear uncertainty. lJsing these frequency envelopes, a controller design strategy is then presented v.rith two ilIustrative examples. Copyright @1999 IPAC

Keywords: Uncertain linear systcnlS; _~ffiIle; Frequency responses; Bode diagrarns; l\Tyquist tliagrarns; Nichols charts; Envelopes; R.obust control

1. INTRODUCTION In rer.ent years, a

SB hstantial

arnount of research

concerning robustness analysis of control systems affected by real parametric uncertainty has been done. The rnain rnethods used in the research hav~ been the \vell-kno"\\rn I(haritonov theorem on interval polynomials and the Edge theorem for affine polynomials. lVlotivated by the results obtained in the paranletric area~ several pa.pers (see e.g. Bailey and Hui, 1989; FUr 1990; Bartlett~ 1993; HoUot and Tempo, 1994; Keel and Bbattacharyya~ 1994; Tan and Athcrton~ 1998 and references t.herein) have studied the cornplltation of the frequency response of control systems under parametric uncertainty. It is also necessary to mention that a large part of the literature in tlle field of robust parauletrie control ha~ be~n devoted to t.he robust stabilit:r ana.lysis paradigm, rather than the robust performance paradiglu. This is not because the robust performance problelns of systerlls with pararnctric uncertainty have been soJved, but silnply because the research ha~ had little success in this field so far. The paper by I{eel and Bhattacharyya (1994) addresses ra bust versions of the

po\vcrful [2;raphica.I tools such as the ~yquist plot, Bode plot, and the Nichols chart for uncertain systems defined by an interval plant structure and then the problem of a controlJer design using extensions of a classical approach. However, the interval plant strncture is too re~trictive for real physica,l systems. In this paper ~ the nUIllcrator and tIle denOluillator polynomials of the traIl.sf~r function of a given system are assumed to be a poly topic polynomial family of the form

pes, q)

== ao(q)

+ al (q)s +

+ an(q)sn

(1)

"\vhose coefficients ai(q) depend linearly on q = ..•• , pq] T and the uncertainty box is Q == {q : PiE[Pi,PiJ,i == 1~2, .... ,q} \vhcrc Pi and Pi are specified lo,ver and npper bounds of the ith perturbation Pi ~ respectively. In other words, the syste:r:n's t,ransfer function is assumed to be

(PI : P2:

3331

Copyright 1999 IFAC

ISBN: 0 08 043248 4

14th World Congress ofIFAC

CONTROLLER SYNTHESIS TECHNIQUE FOR SYSTEMS WITH AF...

bo (r)

+ b 1 ( 1~ ) oS +

(lD ((]) + al (q)S +

+ b"'l ( r ) Sur. + an (q)sn

Cl ( Si

(2)

vvhere q ~ [r)l~I)Z~ .... ,pfJJT and l' == [rl 1 ~ d J1 •. ~.~ dr]T == {q : Pi E {Pi, Pi L 'i ~ 1 ~ 2, .... , q} , R = {,. ; d i E[d d i ], i =: 1,2, .... , r}. t7sing the result given in Shaw and Jayasuriya (1996) where a simplification algorithnl was given for testing the stability of a polytopic polynornial farnily of the fortIl of Eq.( 1) by constructing it.s 2q-convex parpolygons (for each s == jw, the 2q-convex parpolygon is defined as the outer edges of the iInage of exposed edges ((2 fJ - 1 )q edges) of the Q-box), the a.mplitude and phase extremums of Eq.(l) at s :::::: jw* (W*E [0, c:>o)) arc obtained. The anlplitude and phaoe extrernurns of P(s; q) Illnltiplied v/jth a fixed polynomial, 6(8) J are also discussed. Then, the procedures for constructing the Bode, Nyquist and Nichols envelopes of a control systenl defined by Eq.(2) a.re presented. The distinguishing feature of the present paper is the efficient procedure obtained for constructing the 2q-c.onvex parpolygon of a poly topic polynornial farnily and u:-;ing the fact t.hat one does not need to identify the edges of a 2q-collvex parpolygon for each frequency. Therefore, fi·orIl a eornputational point of view, the results of this pa.per can be more helpful than the existing results for computing the boundary of Bode, Nyqllist and NichoIs envelopes of a sy~tem v/ith affine 1in~a.r uncertainty of the form of Eq.(2). Finally, using the Bode) Nyquist and Nichols envclopes~ a controller design strategy is given. Prcliulinary versions of the results, on cOI~struct.ingthe Bode envelopes, presented in this pa.per were given in Tan and .A.therton (1998).

q) ~ f 0 ( s ) + f 1 ( S ) PI + f 2 ( S ) P2 +

C2( s, q) ==

an cl Q

C2q

+ f q ( S ) I Jq fo (.'-i) + f1 (S)Pl + 12(8 )P2 + + fq(s)pq

(s~ q) = 10(s) + /1 (S)Pl

+ f2(S)P2 + ... + fq(s)p q

1:,

The autlinp, of the paper is a~ follo\vs: In Section 2 the construction of the 2q-convex parpolygon is given. rrhe magnitude and phase extrcmUIllS of EQ.(l) a.re obtained in Section 3 and the construction of the Bode, Nyquist and ~~iehols envelopes for a transfer function of the form of Eq.(2) are discussed in Section 4. The extension of classical controller design t~chIliqlles for an uncertain system using these frequency envelopes is presented in Section 5 via two illustrative examples.

2. CONS'T'RUCTION OF CON\'EX

PARPOLYGON The corresponding polytope of a farnily of polynomials of Eq.(l) in the coefficient space has 2 Q vertices and (2 lf - 1 )q exposed edges and it can be re\vritten as

(4) 1~he value set of Eq.(l) can be obtained by mapping the (2 q - 1 )q exposed edges in the cornplex pla.ne for each .s = ju.) and the outer edges of the value set define a 2q-convex parpolygon. The (2 q - J )q edges in the cOlllplex plane can be divided into q groups ,vhere each group has the same nunlber of edges (Shaw and Jayasuriya, 1996) . .t\ll edges in group i(i ~ 1,2, ... , q) arc parallel to each other \vith the same slope. Thus, kno'A.ring one edge from each group if: sufficient to construct the 2q-convex parpolygon. For example, let cC Ci, Cj) denote the edge ,vith end points Ci and Cj and for clarity of presentation consider Fig.la which is the image of the exposed edges of a polytope \vith q :=::: 3 parameters. It can be easily shO~Tn that. the edges e(el' C2)~ e(c3~ (4), e(c5~ Ctd and e(c7' c~) are parallel to each other as shown in Fig.la. Two of thenl which have the luaximum and minimum intersections VIi~ith the irnaginary axis identify tV\,o edges of a 2q-convex pa.rpolygon as shown in Fig~l. Similarl3'\ t.he other edges needed for construction of the 2q-coHvex parpolygoIl can be identified. If there are vertical edges ,;tlhich have no intersection with the imaginary axis, in this case from the maximum and the minimum intersections with the real axis, the two required edges can be found. The general forIllulas (Shaw and Jayasuriya, 1996) fot the intersection point of the edge line yvith the imaginary and real axes are

(1

L)Ok E , - O,Ek)(Pk - Pk), k-/;i

yi = ; ,;,

,

x'

=

1

(5)

k~l fj,

.

' .

0; ~)OiEk - OkE;)(Pk - Pk), ki=2

(6)

k=l

where i = 1) 2, , q, Pk takes either Pk or Pk depending Oll with 1\Thich edge it is associated and E i and Oi are the even a.nd odd parts of fi(S). Further informa.tion about the value set of the uncertain polynomials and the construction of the 2q-convcx par'polygon can be found in Barluish (1994) and Sha,-v and Jayasuriya (1996).

For different values of frequency, t.he edges of a 2qconvex parpolygon may be different. The following theol'eln is given in order to divide the frequency axis> wE[O, (0), into a finite number of intervals ~rhere in each interval the edges of the 2q-convex parpolygon arc the same. 'l~he proof of the follov.. ing theoreln can be found in Shaw and J ayasl1riya (1996) and Tan and AthertoIl (1998). r

-

The 2 9 vertex polynolllials of thp, polytope of P(.r.;, q) can be written in the following pattern

3332

Copyright 1999 IFAC

ISBN: 0 08 043248 4

CONTROLLER SYNTHESIS TECHNIQUE FOR SYSTEMS WITH AF...

Theorem 1: For i, j l)ositive real roots of

==

1, 2, ... , q and if:. j

~

the

divide the frequency axis into finite intervals \vhere in each interval the 2q edges of the (2 q - 1 )fj exposed edges \vhich c.onstitute the boundary of

the convex parpolygon ren1ain unchanged. The frequencies v~rherc the edges \vhich constitute the bOllndary of the convex parpolygon may change will be referred to as transition frequencies.

14th World Congress of IFAC

pa...-.;s through the points (vp;r" Vpi) and (vpy ~ Vpi). Let rph 1 and rfh2 be the angles of the lines h 1 and h 2 ,vhich are draV\ln frOIn the origin and are perpendicular to the lines II and l2, respectively. ~4..ssume also that the 2q-convex parpolygon does not include the origin(if the value set includes the origin for s ~ .iw* t.hen t.he miniIIlUIIl rnagnitude is equal to zero). Then, rnJnjP(jw*, q)/ is

\vhere 01

(a)

(b)

CT2

== [(2Iv p1, !Pl)2

-

== [(2fV pl I(2)2 -

2 (IV p i 1 -

~'Vp.T, 12

+ pi )2]1/2

(1'L'pi 1

~V]JY 1

+ p~)2]l/2

2

-

2

Pi ==

Fig. 1. a)Irnage of exposed edges b)2q-convex parpolygon

((R,efvpa:J - R,e[vpi)Y~ + (Im[vpT,] - Irnlvpi.J)2Jl/2 fJ2

==

[CRe[vpy ] - Re[Upi]) 2 + (Im('v py ] - Im[v piJ)2]1/2 (9)

3. IVIAGNITUDE ~t\.);D PH.A.SE· EXTRE,I\1Ul\lS OF' U~CERTAIN POI;~{NOlVTIALS In this section, the magnitude and phase extrcllIUlllS of a linear polytopic polynolnial falnily: pes, q), at s == jw* are first investigated. 'Then, the magnitude and phase extrerDums of P(s, q) n1ultiplied \vith a fixed polynomia.l, des) ~ 60 + 81 S + .... + r)'nsn, are obtained. The details and the proofs nf an ~he results given in this section can be found in Tan and Atherton (1998).

3.1 A-lagnitude and phase cxtr-elftUrns of

Ho\vever, the application of this theorem may give

SOllle difficul tics. Because for each frequency, one needs to find the lin~s 11 112, h l and h 2 and the phases c/Jpi, 1>px ~ tP py , q,hl and 4>h2· Therefore, an equivalent result "\vhich is easily applied is given by the follo\ving lemlna, Lemma 1: Define the segments SI == (1 - A)Vpi + '\v px and 52 == (1 - A)Vpi + AVpy where AErO, 1J. Take a value for A~ say .\ *, \vhich is very close to zero such as 0 < ). * ~ 10- 6 and write k] == (1 ..\*)Vpi+"\*V px and k z == (l~A*)Vpi+A*Vpy. Then, the rnininlulll lnagnitude of P (s 7 q) at s == .i w* is

pes, q)

Let the 2q vertices of the 2q-convex parpolygon of P(s,q) at s::::::;; jw* bevp l,v p 2, .... ,vp 2q(seeFig.lb). Then the Jnaxinlum magnitude and the phase extreIllUIIlS of pes, q) at s == ju.. * are l

rnaxIP(jw* ,q)l==max(lvpll,

minarg[P(ju/", q)l maxarg[P(jw*, q)J

IVp 2L

"'j

IVp 2ql)

\vhere 0"1,0"2, Pi and P2 are given in Eq.(9) and IV]ri ~ is define<} by Eq.(8).

= min(arg[vplL ... , arg[v p 2Q]) = max(arg[v p l], "'} argfv p 2qJ)

To find the minimurll rnagnitude, the following theorem is given: Theorem 2: Define

where i == 1,2,3, , 2q. Let e(vp-l' vp:t.) and F; (v pl ~ 'V py ) be t\~lO edges of the 2q-convex parpolygon ","'hieh have the common vertex 1J p i. Let rPpi, 1'px and rPpy be the angles of Vpi; v px and v py respectively. Dra\'\' the lines [1 and l2 \vhich

3.2 Magnitude and phase extremums of8(s)P(s, q) Gconletrically: the affect of multiplying P(s, q) ,vith a fixed polynomial <5(8) is to rotate and st:ale the value set of P(s, q) at s == jw* ~ but not to distort its shape. Therefore, the value set of b(ju.J* )P(jw*, q) is still a 2q-convcx parpolygon. So~ for each frequency: t.he Inagnitutle and phase extremllnl~ of 5(s)P(,,,, q) can be computed by multiplying the magnitude and phase extremums of P( s ~ q) ,vith the value of b( s).

3333

Copyright 1999 IFAC

ISBN: 0 08 043248 4

CONTROLLER SYNTHESIS TECHNIQUE FOR SYSTEMS WITH AF...

14th World Congress of IFAC

4. BODE, NY'"Q1JIST i\ND NICHOLS E·N\TELOPES

4.2 .lVyquist and .iVichols envelopes

lJ sing the 2q-coIlvex parpolygonal value set of a In t.his section, the construct.ion of the Bode, Nyquist and Nichols envelopes of a given uncertain system of the forlu of Eq. (2) is discllssed. Throughout this section~ it is assluned that Of$D(s, q). If this assumption fails to hold one can exclude those values of frequency ,vhich violate the assumption. Consider the transfer function given in Eq.(2) and let Vnl, ... , 'tI n 2r and Vd! ~ ... , 'Vd2q be the vertices of the 21' and 2q-convex parpolygons of lvT(s, '1-) and D(s, q) at 8 == ju... * (see Fig.l.b), respectively. T'hen define the sets S.Nv and SA~E \vhich contain the vertices and the edges of the 2r'-convex parpolygon of ]\l(s, T) at s == jw* as l

poly topic polynomial family of the form of Eq. (1), the follo,ving theorem is given for construction of the boundary of a Nyquist or a Nichols telnplate Theorem 3: -<..:\t s = jw*,

' * ,q,r ) C (SA'\' SNE) - - ' U-~ DG( Ju... SUE SD v 1

where [) denotes the boundary and S N v,SNE' S Dv and ---'iDE are defined in Eq.(lO) and Eq.(11) . Proof: Let _4 1 and -<4 2 be the t"v·o complex plane polygons ~rith vertex sets 1l Al and VA2' and edge sets EAI and EA2' respectively. Then~ from the complex plane geoInetry, it is known (Bhattacharyya, et al' i 1995) that A E V,. 8(_1 )C(-?U~) A2 1/A2 EA 2

similarly define

SDv

and

SDp

(12)

(13)

for the denominator

&5

Since the value set of the numerator N(s, r) and the denominator D(s, q) of Eq.(2) at s ~ jw* are independent 21" and 2q-convex parpolygons, "ye

where en! ~ ... , e n 2r and edl, ... , e2q are the edges of 2r and 2q-convex parpolygons~ respectively.

and EA 2

can Vv~rite l'~4I

==

::::=

SD~.

Sl\r,v,

\l~12 ::::::

The rnagnitude and phase envelopes of G(.f;J: ql r) can be computed by the following procedure

1) R.e\vrite N(s, 1') and Eq.(3).

D(s~ q)

in the [arID of

2) Solve Eq.(7) both for }\l(.s~ r) and D(s~ q) and find the tr ansition frequencies of JV (s r) and D(8~ q). 1

3) Obtain the frequency intervals. within these intervals the edges of the 2r and 2q-convex parpolygans of 1\-7(s, r) and D(s, q) rClnain unchanged, as (0, Wnl) ~ (Wnl ~ W n 2), 0_, (W'ru:l ~ CXJ) for J.V (,c;~ To) and (0; ~dl); (Wdl' Wd2), ... , (WdcZ' (0) for D(s, q) \vhere Wnl < W n 2 < ... < WHEl and Wdl < Wd2 < ... < Wdt''2 are the transition frequencies of ]\l(s~r) and D(s ~ q), respectively.

E Al

:::::::: SiVE

.' S~N SN BG(Jw*,q,r)c(S VU~) DE

4.1 Bode envelopes

SJ)y" ,

Thus, from Eq.(13)

i:JDv

The advantage of theorem 3 is that it is not necessary to consider all the exposed edges and vertices of t.he corresponding poly tapes of the numerator and denominator polynomials. For example, in

order to find a Nyquist template of a transfer fUIlctioIl of Eq. (2) ,vi th 'f' == 3 and q == 4 uncertain parameters then using the known result (Bhattacharyya, et al., 1995) one needs to find the iInage of 7(2°) ~ 448 edges, however, from theorenl 3, one needs to find the ilnage of only 96 edges. l~sing the procedure which is given in Section 4.1 for construction of the Bode envelopes and theorem 3 7 the Nyquist and Nichols envelopes of G(s, q, r) can be constructed.

o.

4) Choose a.n arbitrary value of frequency l,~,;ithin each interval found in .1 and by nsing E,tIR(5-6), identify the 27· and 2q-convex parpolygons edges and obtain the vertex and edge sets(SNv, S/v E ~ SL)v and SD B ) of the numerator and the denominator polynoulilat> for each interval. 5) For eaeh s === jw, End the magnitude and phase extremums of f\l(s~ r) and D(s, q) using the results obtained in Section 3 and obtain the Bode envelopes of G(S1 (j, r).

5. CONTROLLER Sy"NTHESIS TECHNIQUE In this Section, using the tools(Bode, Nyquist and Nichols envelopes) developed in the previous section, classical controller design methods are used to design robust controllers for aysterns "i.vith affine linea.r uncertainty. Classical controller design techniques are basically based on t,vo approaches. One is the root-locus approach and the other is the frequency-response approach. Here, the frequency-response approach is used in ord~r to design a controller which compensates a system of the form of Eq.(2). The design procedure is given by the follo\-ving t,\.~o examplcs~ The first

3334

Copyright 1999 IFAC

ISBN: 0 08 043248 4

CONTROLLER SYNTHESIS TECHNIQUE FOR SYSTEMS WITH AF...

14th World Congress of IFAC

eXaIllple deals v,"ith a Lead controller design. The objective of the second eXaIl1ple is to design a. robust PI controller for a given uIlcertain syst.euL

(P4

+ PI )S4 + (P4 + P3 + P2)s3 + (P2 + Pl)s2 + P3 S

vlhere R ~ {r == [d 1 d 1 d 3 1

[0.01, o.08L dJ

Example 1 Consider

: d 1 E[O.4, 0.81, d 2 ::::: [0.2,0.6J) and Q {q:=

[PI P2 ])3 P4] : PI E[O.4~ O.8],P2 E [O.2, O.6],P3E[O.02~ O.08Lp4E[O.OOl,O.005]}. The aim is to design a PI controller of the {orIn C~(s) == (Kps + Ki)/s \vhich guarantees that the eIltire family" }.l3S a phase nlargin of at least lp :;::= 45 0 .

~ N (8, r) ~ G( s;q~r) - D(s,q) -

d1 \vhere R

==

{r

=:

[d1 l

: d 1 E[5: 7]}

and Q ~ {q

==

[PI P'2 P:3 P4] ;PI E[O.OS, O.25],P2E[O.4,O.5],p;1E[O.5, O.7],P4E(O.09,O.11]}. The objective is to design a Lead controller of the form C (s ) == Kc (T s + 1) I(Ctl~s + 1) \vherc 0 < 0' < 1 \tvhich guarantees that the Hystern has a phase rnargin of at least tp :::: 45° and a gain margin of not less than 1£ =:;;; 12db. It is desired that the band\vidth of the closed loop systern be equal to 01' greater' than O.5radlsec.

Since IV(s~T) == r·l, for allwE(O,oo)~ S1VF == {5~ 7} and S]V E == {5 + 2,\} vlhere AE[O, 1J. It is easy to show from theorem 1 that D(s, q) has no transition frequency~ Thus, one single value of frequency within wE(OJ (X)) is sufficient to identify the edges of the 2q-convex parpolygons of D (s, q). It. is found that the €dgcs~ e(c7, C8)~ e(cg, CIO), e (Co, C8 ), e ( C9 , ell), e ( (.;2 , C(; ), e ( ell, Cl f) ) , c( CL. , Cl;J) and e(ci,c15)(here, C2; Ca, C7, ,.' are the vertex polynomials of the polytope of D(s, q) \vhich con~ stitute the edges of a 2q-convex parpolygoll and they ean be obtained by using E,q.(4)), constitute the boundary of the 2q-convex parpolygons of D(s, q). Thus for all L4,.IE(O,. 00), the vertex and edge sets SDv and ,-9DE can be obtained frOIl} Eq.(11). Using the magnitude and phase envelopes of the uncompensated system~ G(s, q, r), '~lhich arc sho,vn in Figure 2, the Lead controller C( s) 0.3(1.28 + 1) 1(0.128 + 1) is obtained. The phase =::;;

margin of the compensated system is 52.4 c and the gain margin is 14~ldb. From the magnitude envelope of the closed loop tra.nsfer function of the eOlnpensated systeII1~ it \yas found that the bandv.."idth of every system in the family lies between O.64r·adJ sec and 1.28rad! scc. Therefore, the designed Lead corrt.TolIer provides the design specification robustly. Bode, Nyquist and Nichols envelopes of the uncompensated and compensated systems arc shown in Figures 2, 3 and 4~ respectively.

E.'l:arnple 2 Consider

G(s, q, 1')

=

~~;: ~?

=

There is no transition frequency for I\l (" , r) and the edges, e(c:1~c4), e(c5,c6), e(c6,c8), e(cl,cs), e( C4 ~ cs); e( Cl! C5J; constitute the boundary of the 2r-convex parpolygon for all frequencies. However, for D(::;, q), one transit i 011 frequency Vlas found at lradjsec. So, the edges of the 2qconvex parpolygon of D(s, q) for wE(O,l) a.nd for wE(l, (0) can be identified by using Eqs(5-6). FroIll Eqs(lO-11), t.he vertex and edge sets (Sj¥V' Sr-lE' SD v and SjJ E ) can be calculated. Using

the Bode envelopes of the uncompesated system shown in Figure [) ~ the P I controller C ( s) === (1.29 X 10- 4 S + 1.55 x 10- 7 ) / s is designed for \vhich the phase margin of the compensated system is greater than 56.7°. Figures 5, G and 7 show the Bode, Nyquist a,nd Nichols envelopes of the uncompensated and compensated systems. 6. CONCLUSION In this paper, the design of controllers for systems \vith affine linear uncertainty has been studied. Effective procedures have been proposed for COlllputing the Bode, Nyquist and Xichols envelopes of uncertain systems defined by Eq.(2). The given procedures are based on efficiently constructing a 2q-conv€x parpolygon of a polynonlial fanIily of the form of Eq.{l) and making use of theorem 1 v.,rhich enables one to divide the frequency axis into finite int.ervals \vhere in each interval the edges of the 2q-convex parpolygon are unchanged. TheYefor~, the results of this paper give a big reduction in comput.ation cOll1.pared with the previously presented approaches. Using this falnily of plots, classical control rip-sign techniques have been used to design robust control systems. 7. R,EFER,E!\lCES Bailey, F~ N. and C. H. Hlli (1989). A fast algorithm for computing parametric rational functions. IEEE TT-ans. A u.toma.t. Contr., 34, 1209-1212. Barmish~ B. R. (1994) . .1\'{e'w Tools for J-lobustness of Lincar~ S.'t.jsterns . :;\1a.ClnillaIl, Nel-v )'Tork. Bartlett, A. C. (1993). COlllput.ution of frequency response of systelns "\vith uncertain parameters: a sjmplification. Int. J. Contr. ~ 57, 1293-1309.

3335

Copyright 1999 IFAC

ISBN: 0 08 043248 4

CONTROLLER SYNTHESIS TECHNIQUE FOR SYSTEMS WITH AF...

14th World Congress of IFAC

i~~f~~:~ 10° f:"roqU0rtcy(rad,'secI

10- J

10'

1 o-~

10--

10'

10

10 J

lO" F ~QUO=Il"\CV( ... d/$8CJ

fQ"

r~r==-::-~~~>_ , ' - .,.~

&::1

", _.::~

',',

~300

'O-~

ld~

10- 1

10~

10 1

1&

FrequBliCY (mdlsec)

~requency(rad!sac)

Fig. 2. Bode envelopes( ... uncolnp., -colnp.)

Fig. 5. Bode enve.lopes( ... uncornp., -comp.)

_6l-~

---L._ _--J.--_ _...L-_ _.l.....-.-.--_-----'

-4

-5

-2

-3

-1

_

Q

Real

Real

Fig. 6.

Fig. 3. Nyquist envelopes

~yquist

envelopes

50r 40r 30/ 20r ::§ .:1: c ~

10

ot---~~F---~~~~:""""--..,..-~""':-~_----:"_~

--30 -40'"-----'-_--1..-~--\...-..-..l...--...------.:_---.l-_-"---_....l-...-_l...-...-_--

-:220

-.2.00

-180

-160

-140

-120 -100 Pha.se in. deg

-80

-60

-40

-:J2D

-20

-3DO

-280

-260

-24D

-200 -180 Phase In deg

-220

-HiD

-140

-120

Fig. 4. Nichols envelopes

Fig- 7. !\Tichols envelopes

Bhattacharyya, S. P.) II. Chapellat and L_ H. Keel (19!J5). Robust Control: 1'hc Parametric Approach. Prentice HalL Fu, ~1. (1990). Computing the frequency response of linear systems with parametric perturbatiOIlS. Syst. Contr. Lctt.~ 15. 45-52. Bollot, C. \T. and R" Ternpo (1994). On the Nyquist envelope of an interval plant family. IEEE Trans. Automat. GYontr., 39, 391-396. Keel, L. H. and s~ P. Bhattacharyya (1994). Control system design for paran1etrir: uncertainty. Int. J. Robust and Nonlinear Contr.~ 4, 87100.

Sha,v, J. and S. Jayasuriya (1996). A new algorit.hm for testing the stability of a polytope: a geometric approach for simplification. ~l. of Dynamic, Sys., }.{eas- and Contr., 118, 611614. Tan~ N. an(l D. P. .A.. therton (1998). 11agniturle and phase ~nvelopes of systems with affine linear uncertainty. International Conf. on Control'98, (]KAC~ 2, 1039-1044. J

3336

Copyright 1999 IFAC

ISBN: 0 08 043248 4