Nonconservative and Noniterative Solution to the Robust Stabilization Problem for Classes of Systems with Real Parametric Uncertainties

Copyrighl @ 1996 IFAC. 13th Triennial World Congress, San Francisco, USA

2d-06 6

NONCONSERVATIVE AND NONITERATIVE SOLCTION TO THE ROBUST STABILIZATION PROBLEM F'OR CLASSES OF' SYSTEMS WITH REAL PARAMETRIC UNCERTAINTIES M . Abrisbamchian K. N. Toosi , University of Technology Departm e 1~t of Electrical Engineering P. O. Box 16315-1355

Tehran, Jran EmaiJ: [email protected]

Abstract: This paper considers the problem of ro bustly stabilizing a famil:; of linear time-invariant. systems with a li near time- invariant cont roller. More specifically, systems with structured real parametric uncertainties are considere d. By taking advant.age of t he sped ...) un cert ainty struct ure du ~ to certain impo rt.ant system !s configurations aud modeling, a nonconservat ive and noniterative solution to t.he posed rob ust sta.hilization problem in terms of a standard HOQ optimization problem is given. Keywords: Robust control, H-infinity control , Robust litaLili ty, Robust stab ilizability, Uncerta in linear systems, Stabilizing controllers.

1.

INTRODUCTION

of robust stabilization in a non·:onservative and noniterative way for systems wi t h real parametric uncertainties, it is important and necessary t.) look at the way the uncertainties enter int.o the syst.~ ms; some examples are given next.

In t his paper, the problem of the robust stabilization for a family of linear time invariant single-input singleou t put. tlytlteJIlB is considered. The control systt!IrI ill this paper consist.s of an uncertain linear time-invariant plant. with a specified bound on t.he uncertainty and a controll er whidl is connected in a feedback configur ati on. More specificallYl a class of systems having real parametric uncertainties entering into the t.ransfer function coefficients is considered and furthermore l common uncertainty structure enters int.o t.h e numerator a nd denominat.or of the transfer fun ctiou representiug the pla nt . The un certai nty structure can be linea r o r noulinear rea l parametric but sho uld satisfy certain rea li zab ility and bala ncin g co ndit ions. The kind of uncertain ty st ru ctu re co nsidered in t.his paper is uu avoida.ble for interest ing classes of importanl. systems. Furt.hermore, the goal is t o motivate t.hat. to solve t.h e problem Funding for this research was. provided by 1( . N. Toosi, Uni-

Example 1.1 (Input-Outpu t Blocks Configuration): Consider a syst.em wit.h inpu t a nd output blocks as in Figure L [t is assumed t.hat th e uhcp.rt.a int.y ent.ers in

~~ Input

~

Output

Figure 1: ]nput-Olltput Blocks the output block, Zout I for ex;tmple du e to the change

versity of Technology under Grant No. MP/1710.

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in the load applied to the system. The resulting transfer function from the input to the output is written as _ Z~",{s) + t:>.Z,"t{S, r) P{ 8,1' ) _ Z;n(S) + zg",(S) + t:>.Z,",(.<, r) where Z~ut(s) and ~Zout(s) represent the nominal and the uncertain portion of the out.put. block respectively. Hence, it can be seen that the transfer function pes: r) has common uncertainty structure in the numerator and denominator. Example 1.2 (lnner~Outer Configuration): In this example, a feedback system with inner and outer networks is considered as in Figure 2; applications involving this type of configuration are discussed in Truxal (1955). Furthermore, it is assumed that the compensator C 1(s) is a PD controller described by

C,(.)

= Kp + KD8

~

I~~I

No{s) = Np(s)DH(S)(Kp + KDS); N, Cs) Np{s)DH(S); Do(s) = DH{s)Dp{s) + Np{s)NH(s)(Kp DI(s) Np{s)Nfl(s); t:>.F(s,r) rls+ro.

= =

+ [(D8);

It can easily be verified that the uncertain plant P(s, r) has common uncertain POlYIlO~Ilial entering into the numerator and denominator. Example 1.3 (Simplified Velsion of an Auto SuspenThe transfer function for a simplified sion System): version of the sll~pel1sion system, assuming the input of the system to be the motion l"n at a specified point P and the output to be the vertin.! motion Xout of the auto body, as given by Ogota (199C), is

X,",(s) X;n{S)

i,s + k ms' + bs + k'

Let) band k to be the uncertain parameters. It is easy to see that common uncertain·;y structure enters in the numerator and the denominatcr of the transfer function.

Figure 2: Inner-Outer Configuration

Further motivation for the app roach in this paper drives from the fact that the robust stabilization problem for t.he case of rf!;).1 paraInetric uncertainties is not com~ pletely solved. Examples of work in this area are systems with a single uncertain Jarameter corresponding to the gain or a single pole (,r zero) for example) see Khargonekar and Tannenbauu, (1985) and the more recent paper by Olbrot and Nikodem (1992), for t.he case of stable uncertainty enterinE: only in either the numerator or the denominator as in Abrishamchian and Barmish (1996) and the case~: for which the iterative methods lead to a solution of classes of robust stabilization problems) e.g., see the D-K iteration scheme in Doyle (1985) and the convex programming approac.h of Rantzer and Magretski (1992)

and Co(s) and H( ..,) can be any linear time-invariant controller. In many applicat.ions, the goal is to design Cots), C, (s) and H(s) to meet, cert.ain performance criterion; furthermore the choices of these controllers should not jeopardize the stability of overall closed loop system subject to uncertainty in the two gains [{p and K D . It i~ iniere8t.ing to 8ee that the configuration shown above gives rise to a special class of uncertain plants. This class has simultaneous uncertainty in the numerator and denominator and moreover, robust stabilization can be studied in a nonconservative manner using H oo theory. Indeed, since the gains f{p and [{D include uncertainty, the controller C\(s) can be rewritten as

Cl (s, r) = (I{ P + fO) + {Kn + rl)s with a bounding set R={r:lril:'Or; for i=O,l}. Now, it is straightforward to see that the inner loop can be replaced with the following transfer function given

In this paper) systems with more than one real uncertain parameters are considereol and the uncertainty is allowed to enter simultaneously in the numerator and denominator of the transfer f _lDction representing the system. Furthermore, the robllst stabilization problem is solved in a nonconservative way and in a noniterative way by taking advantage of the special structure of the uncertainty and reducini~ the robust stabilization problem to a standard 11= problem.

by

Writing

2. NOTATION AND STABILIZATION CONCEPTS Throughout this paper, a unit~r feedback system is considered and it consists of an uncertain plant connected in cascade with a proper controller C(s); see Figure 3. The theory to follow also applies when C( s) is in the

it is obvious that pes c) _ Nu(s)

,

with real uncertainties r E U and appropriate choice of Ni(s) and D;(s) for i = 0, 1. Here,

+ t:>.F(s, r)NI(s) - Do(s) + t:>.F(s, r)DI{s)

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Definition 2,3 (Balanced U lcertain Family): The family of uncertain polynomials LlF(s, r) is said to be balanced if, for any positive real number 0',

C(s)

maxLlF(a,r) = -minLlF{a,r). '"eR

rER

Assumption 2.4 (Realizability and Balanced CondiThroughout this papl~r, the uncertain polynotion): mial !:::.F(s, r) is (',onsidered to be balanced and the maximal uncert.ainty magnitude rf:alizable by a fixed polynomial with no root. in the open right. half plane.

Figure 3: Cascaded Unity Feedback Configuration feedback path in~tead. In this paper, it is assumed that certain common structured uncert.ainty ent.ers into the numerator and the denominator of the uncertain plant. Thai is) t.he uncert.ain plant.

( ). No(s) Ps, r = Do(s)

+ LlF(s, r)N,{s) + LlF{s, r)D,{s)

Assnnlption 2.5 (One Right Half Plane Root of The Difference Polynomial): It is assumed that the difference polynomial

where No(s) and Do{s) are fixed polynomials of order m and n respectively with no common roots in the dosed right half plane. The uncertainty is represented by the polynomials LlF{s, r)N,(s) wit.h order of m or less and LlF(s,r)D,{s) wit.h order of n - I or less and each uncertain parameter r'i il:i real and the bounding set R== {r: Ir,l S r, for ;=0, 1,2, ... ,£,.}. for r; Le., it is required that r E R. Note that NI(s) and DI(S) are fixed polynomials. Definitions that are needed in this paper are now stated (for more detailed discussion, see Abrishamchian and Barmish (1996)).

£(8)

D,{s)No(s)

has exactly one zero in the closed right half plane. The notat.ion Pc is used to represent a family satisfying the t.WO cv:;slImpt.ions above.

Remark 2.6: With regard tJ the Assumption 2.5, although it is assumed that the difference polynomial L( s) has one root in the closed right half plane, but the numerator and denominator can have several zero~ in the closed right half plane. Furthermore, there are rich classes of polynomial families satisfying the Assumption 2.4; e.g., see Abrishamchian and Barmish (1996). The uncertainty considered here can be a linear or nonlinear function of the uncertain parameters. Two examples are now given.

Definition 2.1 (f\.·faximal Uncertaint.y Magnitudes): Consider a family of uncertain polynomials !:::.:F described by

, LlF(s, r) == L a,{r)s'

ExaIIlple 2.7 (Uncertain Parameters Entering Nonlinearly): Consider the uncertain polynomial

with each coefficient a'i (r) depending continuously on r and uncertainty hounding set R which is assumed compact. Then, associat.ed with this family is the frequency dependent quantity

.6.F(s, r) where Iril

= rrr2s2 + rOrlT2s + TrT2.

.s 1 for i = 0,1,2. It is easy to see that .6.Fma l,.(s)

Llfmax{w) == maxILlF(jw,r)1 ,ER

= s:l + s + 1

serves as a polynomial realization which is stable. Furthermore, the family is balanced.

which is called the maximal uncertainty magnitude.

Definition 2,2 (Polynomial Realizability): family of uncertain polynomials

= N1(8)Do{s) -

Given a

ExalIlple 2.8 (Uncert.ain Parameters Entering Linearly): Consider an affine inear uncertainty which is expressed as

LlY = {LlFe. r) : r ER}, the maximal uncert.ainty magnitude !:::.fmax(w) is said to be realizable by a fixed polynomial if there exists ~olIle r* E R such t.hat

LlF(s, r) =

:L'. r,F,(s) i=J

=

where each F,(s) is a fixed polynomial of degree n - 1 or less, each uncerta.in paramfter ri if-! rea.l and r E n. Without loss of generality, let

Llfmox{w) ILlF;"ax(jw, r")1 for all w E R That is, the maximal uncertainty magnitude can be represent.ed by the fixed polynomial !:::.F(s, .,.*) evaluated at s = jw. Henceforth, the fixed polynomial

deg F,{s) ;0: deg F,{s) ~: ... ;0: deg Fds)

and assume that for

!:::.Fmax(s) == !:::'F(s, r*)

i2

M

is called a polynomial realization for the maximum uncertainty magnitude !:::.fmax(w).

> i 1 > 0, each of the quotients . ~ £,,(8)

;1,'2 -

Fi1(S)

is positive real; i.e., Re lvfi1 ,i 2(jW)

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2: 0 for all w E R.

Abrishamchian (1994) shows that for this case i,

LlFmax(s) =

2:: ri Fi(S) i:::::O

and furthermore, ~Ffnax("') is a polynomial with no roots in the open right half plane. It is also obvious that the uncertain polynomial LlF(s, r) is balanced. Definition 2.9 (Robust Stabilizers and the Set C): Consider the family of plants Pc 1:\.ud a controller

C(s)

=N,(s) D,(s)

where Nr;{s) and Dc(s) are fixed polynomials. The controller C(s) is called a robust. stabilizer if the resulting closed loop polynomial

W(B, ,.)

=%(8) + Llw(s, 1')

0.22s + 2.45 + 10(r,s + rD) + 0.2420s' + 2.6958' + lls2(r,s + 1'0) real uncertain parameters Irll S 0_018 84

where the and Irol ~ 0.155. Next, by adding to the denominator of the uncertain plant E(S + 1) where f is a sufficient small number such that. t.he poles at jw axis of the uncertain plant described above is harmlessly avoided. By choos-0.001, the reformula~ed nominal denominator ing f is given by

=

=

and Llw(s,r) LlF(s, r)N, (s)N,(s) + LlF(s, riD, (s)D,(s) is stable for all r E R; i.e., it does not have any roots in the closed right half plane for all r E R. Henceforth, C denotes the set of all stabilizing c:ontrollers for the nominal plant.

=

0.18s+ L55 WT(S) = 0.22s+ 2.45' it results in

3. MAIN RESULT In this section, the main result of this paper is given.

'r" = CEe inf 11 WsSo + WrTa 1100'" 0.6357.

Theorem 3.1: (see Section 5 for proof) For the plant family Pc, a robust stabilizer exists if and only if

inf

_ P( s,r ) _

Do(s) = 8 4 + 0.24208" + 2.t.958' - O.OOls - 0.001. Note that the difference polynomial above has exactly one root in the do~;ed right. half plane, at s 0.1856. Now, applying the main result in this paper with the weighting functions 0.1988" + 1.7058' Ws(s) S4 + 0.2420,,3 + 2.1>95s' - 0.0018 - 0.001 '

where

GEe

where 0(8) and Tc(s) are the :~aplace transforms of the angle and t.he control torque r'~spectiveIy. Furthermore, the parameters k associated with spring constant and d associated with viscous damp lng are the ullcertain parameters; e.g.) it is assumed ;hat 0.09 S k :::; 0.4 and 0.004 S d S 0.04. Using the notation of previous section, the uncertain plant is giyen by

Therefore, a robust stabilizer exits. Remark 3.3: Note that t.le Theorem 3.1 gIVes us a necessary and sufficient condition for existence of a robust stabilizer for the case 'Jf one zero in the closed right half plane for the difference polynomial. For other cases when the difference polyuomial has more than one zero, the result becomes sufficient.

11 WsSo + WTTo 1100< I

where So(s) and To(s) afe the nominal sensitivity function and complementary sensit.ivity function respect.ively, and LlFmax(s)D, (s) Ws(s) Do(s)

4.

=

D.Fmax(s)N, (s) Wr(s) Nu(s) serve as nonconservat.ive weighting functions.

MACHINERY NEEDED IN THE PROOF OF THEOREM 3.1

In this section, the plant family Pt is considered and two lemmas that are needed in the proof of Theorem 3.1 are discussed. For the first lemma, the a.
Note that the theorem above gives both necessary and sufficient conditions for existence of a robust stabilizer for families with real parametric structured uncertainties in a nonconservative way. An example is now given to show the application of the theorem above.

Lemma 4.1: Consider the jar'lily 'Pc with the difference

Example 3.2 (Satellite Attitude Control): In this ex· ample, the problem of controlling the attitude of a satellite in orbit as given by Franklin, Powell and EmamiNaeini (1986) is considered. The transfer [unction for the attitude control is given by O(s) 10ds + IOk Tc(s) s'(s' + llds + llk)

polynomial L( s) having k real ;:eros Ctt, 0'2, ... , O'k in the closed r(qht half plane. If a ,·obust stabilizer exists, it

follows th.at

maxLlF("i,r)D,(a;j < IDo("i)1 'ER

for i

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= 1, 2, ... , k.

Proof: Without loss of generality, this t.heorem is proven for s -== at; i.e., it must be shown that if a robust st.abilizer exists, thf'TI

functions and taking K(s) to Je a stable polynomial of order n, define

maxt>F("I,r)DI(atl < IDo(atll·

. 1I'i(S) Ni(s) = ]"(s) ,

Indeed, if a robust stahilizer exit.s, then the resulting dosed loop polynomial \If(s, r) is stable for all T E Ri i.e., it does not have any roots in the closed right half plane for all r' E R. Then, it follows that

VieS) '" gi(S) F(s) for i = 0, 1, and t.ake Xo(s) E Q and Yo(s) E Q satisfy-

'ER

\I1(''1,r) for allr' E R; equivalently:

iD

Xo(s)Alc,(s)

+ Yo( .• )Vo(s) '"

1.

Now] using the fact. that

No(adN,(atl + t>F(al, r)N I (cr,)N,(aJ) +Do(adD,(o-d + Ll.F(a" r)D,(aIlD,(aIl i 0 for all r E R. Next., the both sides of the inequality above are multiplied by Do(adNo(aI) (Note that Do(a,) and No(ad are not equal to zew), then to the re.."iulting inequality, t.he term No(a,)N,(adNo(aJ)t>F(a" r)D,(atl is added to and subtracted from respect.ively. gwuping them, it results in

ing the Bezout identity

Ws(s)So(s) = Ws(s)Vo(s:(Yo(s) - No(s)Q(s»; WT(s)1o(s) = WT(s)No(sl(Xo(s) + Vo(s)Q(s)) and furthermore] defining the rat.ional [unctions

Tl(s) '" Ws(s)Vo(s)Yo(s) + WT(s)No(s)Xo(s) = t>Fmax(s)(V, (s)Yo(s) +NI(S)XO(s» and

After appropriately re-

=No(s)Vo(s)(Ws (') -

T,(s)

oi

WT(s)) Vo(s)N, (s»,

= t>Fm"x(s)(No(s)l), (s) -

t>F(a"r)No(O'I)N,(ad[N,(adDo(a,) -No(aIlD,(oIll + (Ll.F(O'" r)D,(aIl Do(aJ)No( "', )[D,(",,)Du(ad + N,(a,)No(aIlJ, for all r E R. Since, the first term of the right side of

it. is st.raight.forward to verify 1hat

the inequality is equal to zero and

In view of t.his reformulation, it. suffices to show that.

D,(a,)Do(a,) + N,(adNo(a,)

i

0,

inf

GEe

11 WsSo + WTTO 11== QEQ iuf 11 T, - T,Q 11= inf

11 T, - T,Q Ilx= !:I.Fmax(cr)D,(Q)!

D o(") To this end] two cases are considered: Case 1: If ilFmax has no zeros on t.he imaginary axis, using the fact that the numera.tor of T2(S) is the polynomial ~Fmaz(s)L(s) which has only one zero in the open right half plane (at s = a), it. follows that the only closed right. half plane zero of [2(S) is at s a. Hence, by existing results,

it follows that.

QEQ

Do(aIl i t>F(a"r)D,(ad· Now, noting that t>F(s,O) = 0 and t.hat. the coefficients of LiF(s, r) depends continuously on r: it follows that

t>F(""r)D,(aIl = [-kmax>kma
=

where

km" = rnaxLl.F("" r)D,(aI). 'ER

iuf

Combining this fact. with t.he condit.ion,

QEQ

Do«>Il!/c t>F(<>"r)D,("Il

11 T, - T2Q

II~=

11 1(")1.

In addition, sinr:e

the desired result is obtained.

Vo(<»Yo(a)

+ No(cr)Xo(a)

=1 ,

after multiplying the equality above by V,(a) and then adding and subtracting N,(a)Vo(a)X(Q) to and from t.he resulting equalit.y, it follows that

Lemma 4.2: Conside.r thr. plant family Pc satisfying Assumptwns 2.4 and 2.5 and Id s = (.1:" denote the location of the right half plane zero of the difference polynom'iai L(s). Taking the "Weighting function as in Theorem 3.1, 'it f()llow,~ that

Xo( ,,)(Vo(a)N, (a) - No("')V, (a» +Vo(crj(V, (a)Yo(a) + N, :,,)Xo(a» = V,(a). Note that since the first term of the left side of the

. ~ Ll.Fm,,(ao)D,(o:) mf 11 Ws'<'o + WTTO 1100= DOD () . CK

equality is equal to zero, by substituting

V,(a)Yo(a) +N,(o:)Xo(a) =

'Do v,«Q» (.1:"

Proof:

The problem of finding the intimal value

into Tl(O), it. is easy t.o see

" =11 WsSo + WI'To 1100

IT ' (<> )1

is reformulated as a model matching problem; see Vidyasagar (1985) for a review of existing results about the standard model matching problem. Indeed, letting Q represent the set of all proper stable rat.ional

=

thc~t

It>Fm "

(a)V, (a)! Do(a)'

Case 2: If ilFmaAs) has at le:l.st one zero on the imaginary axis~ it follows that T 2 (;;) has zeros both on the

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imaginary axis and in the open right half plane. Since the zeros on the imaginary axis of T2 (s) are also zeros ofT,(s), by existing lemmas (see Vidyasagar (1985»,

J~~ liT,

-T2FII== IT,(a)1

5.

= I~F=;:~~~I(a)1

where C is the set of all the stabilizing controllers for the nominal plant. Therefore, it is obvious that a controller exists such t.hat inf

GEe

11 WsSo + ItTTo 1100< 1.

Now, it i::; obvious that whene'rer a robust stabilizer exists, it is obtained by solving the standard H OO problem described in the theorem. .

PROOF OF THEOREM 3.1

rn this section, the proof of Theorem 3.1 is given. 6.

Proof: To establish sufficiency, it is a..."Isnmed that a controller C( s) exists which stabilizes the nominal plant and satisfies the inequality

11 WsSu + WTTO 1100< 1 or equivalently,

I

~F='"DID'+~F="NINcll < NoNe + DoDe eo

I.

Jt. must be shown that C(s) is a robURt. st.abili:;:er. To prove that C(s) is a robust stabilizer, it sufficies to show that for all w ~ 0, the Zero Exclusion Condition (for example, see Barmish (1994) for morc detail) is satisfied by the closed loop polynomial

"'(jw,r) = "'o(jw)+~"'(jw,r) where

Wo{iw) = No(jw)N,(jw)

+ Do(jw)D,(jw)

and ~"'Uw, r) = ~FUw, r)(DI(jw)D,(jw)+NI(jw)NcUw)) That. is, considering the value set

W(jw, R)

= {"'(jw, r);

rE R},

it. t-:iufficies to t-:ihow that

o rt "'(jw, R). Note that it is obvioUIi t.o see that

1 >

1 ~Fma"(DID, + NINe) 1 NoNe + DoDe eo

>

sup sup

-

wER,ER

1MU~, r) I. IJIo(Jw)

Thus, I~"'(jw, r)1

< l"'o(jw)1

for all w E R and all 'T' E R, and hence, the Zero Exclusion Condition is satisfied. To est.ablish necessity, it is assumed that C(s) IS a robust st.abilizer and must prove that inf

CEe

11 WsSo + WT10 1100< I.

Indeed, note that Lemma 4.1 guarantees that

~Fma"(,,)DI(") 1 < I. 1

Do(a)

and furthermore, by Lemma 4.2, we obtain . f III Gee

11 W 8S0 + WTOM ~ 11 = 1 ~Fma"(")DI(") 1 D() Du

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CONCLUSION

rn this pa.per, the robURt st.ai::ilization problem for the case of plants with structured real parametric uncertainties was considered. It wa~l shown that certain configurat.ions and modeling of mlcert.ain systems result in plants with special uncertainty structure. By taking advantage of this special structur~, the robust stabilization problem was reduced to a standard HX problem. This reduction was accomplished in a nonconservative and in a noniterative way.

REFERENCES M. Abrisharnchian (1994), "Rf:duction of Classes of Robust Stabilization Problems to Classical Reo Problems," Ph.D. Dissertation, Department of Electrical and Computer Engineering, Universit.y of Wisconsin-Madison. M. Abrishamchian and B. R. Barmish (1996), "Reduction of R.obust Stabilization P:'oblems to Standard HOC! Problems for Classes of Systems with Structured Uncertainty", A utomatica, to appear'. B. R. Barmish (1994), Ne-.v Tools for Robustness oJ LineaT Systems, Macmillc.n Publishing Company, New York. J. C. Dayle (1985), "Structured Uncertainty in Control System Design," Proceedings o/the IEEE Conference on Decision and Control, Fort Lauderdale. G. F. Franklin, .1. D. Y)well and A. EmamiNaeini (1986), Fer.dback Control of Dynamics Systems l Addison-Wesley, New York. P. P. Khargonekar and A. Tannenbaum (1985), "NonEuclidean Metrics and the Robust Stabilization of Systems with Parameter Uncertainty," Ib'EH transactions on Automatic Control, vol. AC-30, pp. 1005-1013. K. Ogata (1990), Mode,., Control Engineering, Prentice-Hall, New Jersey. A. W. Olbrot and M. Nikodern (1992), "Robust Stabilization of SISO Systems with Linear-Fractional Parametric Uncertainties,)' ProCf!~dings of the American Control Conference, Chicago, A. R.antzer and A. Magretski (1992), "A Convex Paramcterization of Robustly Stabilizing Controllers," in Rubustness of Dynamic Systems with Parameter Uncertainties, 1'. 1. Mansour, S. Bal·~mi and W. Truo), eds., Birkhauser, Basel. J. G. Truxal (1955), Automa,ic Feedback Control Sys. tern Synthesis, McGraw-Hill, New York . M. Vidyasagar (1985), Cont1'01 System Synthesis: A Factorization Approach, MIT Press, Cambridge,