A Comparison Between Robust and Adaptive Control of Uncertain Systems

Copvright

©

IFAC .-\d"pt ive Svstellls ill COlltrol

and SiKllal Proct:'ssing. LUIle! . S\\"t'den. I ~IX6

A COMPARISON BETWEEN ROBUST AND ADAPTIVE CONTROL OF UNCERTAIN SYSTEMS K.

J.

Astrom*, L. Neumann** and P. O. Gutman**

*Departlllfllt uf Alltulllatif CUlltml. LUlld IlI stitute uf Tnhllulogy . Bux 11 8. 221 00 Luml. Sll'erlfll **EL-OP. Israel Eleftm-Optirs I IIdllStries. POB 11 65 . Rehovot i 611 0. Israel Abstract. Design of control systems for plants with conslder.ble p.ramet.r unc.rtalnty can be approached by robust control and adaptive control. Robust design m.thods glv.s fixed compensators with s.tlsfactory performance over a specified r.nge of plant par.m.ter variations. Adaptive design methods extract knowledge of the pl.nt p.r.meters on line .nd redesigns the control law. These two appro.ches are compared In the paper.

1. INTRODUCTION function Is known exactly. The sltuatlon Is v.ry different when the transfer function Is only known approxlmatively as Is shown by the following .xampl ••

This paper considers control of systems with unknown or tlmevarylng dynamics. It is attempted to understand the possibilities and limitations of controlling such system with linear constant parameter controllers, and to compare with adaptive control. One goal Is to indicate when one type of control is adv.ntag.ous over the other. The problem of plant uncert.inty inspired Black's invention of the feedback amplifier. It has in different forms been In the mainstream of feedback control research ever since. The problem Is now frequentl y popul.rized as robust control. The problem of parameter variations c.nnot be dealt with in isolation. A proper approach must, however, also include other .spects of control systems d.sign like command following, lo.d disturbances and me.surem.nt nois •• In this p.per the main emphasis is, how.ver, on the par.meter vari.tions.

Example 2.1 Consider a system with the loop tr.nsfer function G(s)

1

= s(s+1)

Th. corr.spondlng closed loop system Is cl.arly stable. Now consider the system with the loop tr.nsf.r function G (s) E

= s-a-E -----a S -

GCs)

The transfer function G .nd G c.n be m.d. arbitrarily cl os. by choosing E sufflci.nti y small. Th. closed loop system which corr.sponds to G Is, howev.r, unstabl ••

Some fund.ment.1 issu.s that ch.racterize the control of uncert.ln systems .re discussed in Section 2. A few illustrative ex.mple. are giv.n. Dlffer.nt w.y. to formul.te design problems which c.n cope with p.r.met.r varlatlons are presented in Section 3. The m.thods of Horowitz, Coyl. and St.ln, and the multi model approach .re included. Conditions for the method. to work are st.ted. In Section 4 the ad.ptiv. .pproach is bri.fly described, as well •• condition. for it to work. Section 5 contains thr_ ex.mples, solved both with the Horowltz method and with an ad.ptive method. The ex.mples ar.: a simpl. ..rvo, a .yst.m with time del.y, and. robotic exampl •• Th. discussion In Section 6 covers a comparison of the various conditions for robust and adaptlv. control, .nd a f.w conclusions .r. drawn from the .xampl.s. Section 7 fln.lly v.rlous sugg.sts ways to combln. ad.ptiv. and robust control.

E

D

The example shows th.t In order to guarant_ st.blllty It Is not enough to know th. simplified Nyqulst curv•• It Is also nec.ssary to know If th.re ar. some unstable right half pl.ne pol.s which .r. clo.. to c.nc.lIation. This Is .lso cl •• r from the Nyqulst st.blllty crlt.rlon which st.t•• th.t th. closed loop syst.m Is st.bl. If the number of .nclrcl.m.nts of • st.nd.rd contour equals the number of singul.rlti •• of th. open loop tr.nsfer function In th. right half pl.n.. It Is thus e ...nti.l to know the unst.bl. proc... pol.s .xplicltly. In the robust de.lgn method introduced In the following section It I. crucial to know th. sign of th. high frequ.ncy g.ln. Th. following r •• ult illustr.t.s why this I. nec....ry. Exampl.2.2

2. FUNDAMENTAL ISSUES An Integr.tor who.. sign Is not known c.nnot be controlled using high g.ln robu.t control.

Som. fundam.ntal limitations th.t unc.rt.lntl.s in proc.ss dyn.mlcs Impos.s on the perform.nc. of a control syst.m will be discussed In this section. A clos.ly related problem is how much Information is required about a control syst.m In ord.r to m.ke • control d.slgn. Sinc. the robu.t design m.thods ar. b.sed on the Nyqulst curve .t the pl.nt it Is e ..entlal to know If knowl.dg. .bout the Nyquist curv. .lon. suffices for d.slgn. Slnc. the transf.r function Is .n.lytic .p.rt from slngul.rltl.s It Is In principle sufflci.nt to know th. transf.r function In .n • rbltr.rily sm.ll region. The full tr.nsf.r function can th.n In princlpl. be recovered by analytic continu.tlon. This argument requir.s th.t th. tr.nsf.r

D

Uncertainties In th. tIme d.lay po... anoth.r ....ntl.l constraint. This I. illustr.ted by the following example. Ex.mple 2.3 - Time d.l.ys Consider a lIn.ar plant where the major unc.rt.lnty Is due to variations In the tlm. del.y. A.sum. th.t the time d.lay v.rl.s betw_n T .nd T . Furth.rmor• .ssume that It Is required toDtliap th.lI{,'\.l}lations In the ph.se m.rgln 1... th.n 20°. It th.n follows th.t the

43

K. J . ..\Sl ro m *. L. ~ e um a nn a nd P. O. G Ulm a n

44

cro••-ov.r frequ.ncy I.) c must satisfy < I.)c -

T

0.3:5

m.x

- T

min

Th. unc.rt.inty in th. tlm. d.lay thus induc.. .n upper bound to the achi.vabl. cross-ov.r frequ.ncy. o

3. APPROACHES TO ROBUST DESIGN Diff.r.nt approach.s to design of robust control .y.t.ms wUI be summarized in this chapter. Although w. are .mph.sizing the .ffect of p.r.m.ter v.riatlons it .hould be k.pt in mind that the control should .1.0 be c.p.ble of d.al1ng with comm.nd following, load d1&turbanc••• nd m••sur.m.nt noise.

Th. Horowltz Design Method Thi. m.thod is a d i rect descendent of Bodes work on feedback ampl1fiers (Bode, 1945), wh.r. the k.y probl.m was to d ••ign a f. .dback .ystem th.t couid cope with the changing characteristics of vacuum tube .mplifi.rs, se. (Horowitz, 1963). Th. k.y id.as of the Horowitz design are the following: A•• uming that the output and the reference are both av.U.ble for m.asur.ment, a two degree-of-freedom structure, con.isting of a feedback compensator, .nd a prefUt.r i. propos.d. The purpose of the feedback loop is to st.b1l1z. the plant (if neces.ary), to reduce the ••nsitivity to plant variations, and to r.ject di.turbances. The prefUter is d.signed to shape the nominal tran.mission from refer.nce to output. One con.equence of feedback i. th.t m••• urement noise is led b.ck to the input chann.l, pos.ibly I••ding to • aturation. Ther.fore it i. • ...ntial to keep the b.ndwidth and the gain of the f.edb.ck loop a •• m.ll as pos.ible whU. .atisfying the .pecification. on di.turbanc. rejection and sen.itivity reduction. A k.y id.a in the Horowitz d •• ign m.thod 1& d •• ign to .pec1fications. Horowitz call. hi. m.thod "quantitativ." in contr•• t to mo.t oth.r m.thod. (Horowitz, 1975a). Origin.lly the m.thod w.. d.v.loped for minimum ph... SISO pl.nts (Horowitz, 1972). It h.. been ext.nded in v.riou. degr. . . to non-minimum pha.. plants, (Horowitz, 1984), plant. with saturation., c . .c.ded pl.nt., tim. varying pl.nt., non-lln•• r pl.nt., multi-v.ri.bl. pl.nt., .tc. Th1& m.thod i • • till v.ry much .ubject of furth.r r . . .arch .nd .xt.n.lon. to n.w types of .y.t.m., see •• g. Y.niv <1986.). Th. m.jor dr.wb.ck of the m.thod i. that .xc.pt for c.rt.in cl ..... of .y.t.m. it i. impo•• ibl. to know. priori if the de.ired closed loop .peciflc.tion. .r. attain.bl.. In such c.... the d •• ign becom.. • tri.l-and-.rror procedur., .nd the d •• ign.r n.v.r know. if • more .kiiful d •• ign.r would h.v. been .ble to .ati.fy th. .pecification.. Th. .dv.nt.g. i., how.v.r, th.t • .y.t.m.tic d •• ign procedure i • • v.U.bl. which .llow. the d ••ign.r to g.in insight into the tr.d.-off. between differ.nt .peciflc.tlon., .uch •• closed loop •• nsitlvity .nd b.ndwidth, compl.xity of regul.tor, m••• ur.m.nt noi . . .mplific.tlon, .tc.

(1n d.cllog or dB) .re called TOLERANCES. V.rious suggestions have been made how to tran.l.te tlm. domain specifications into specifications on the closed loop gain (Horowitz, 1972). There .xi.ts no .y.t.m.tlc procedure to do so. Notic. In p.rtlcul.r th.t if the uncertain plant 1& such that the specific.tions are satisfied for all pl.nt cases, then an open loop control could be considered. Step 2: Determine the plant unc.rtainty. For ••ch frequency the plant unc.rtalnty glv.. ri.. to • TEMPLATE, I.e. a set In the compl.x plan•. Th. plant may be given In parametric form. Th. unc.rtainty I. then specified as bounds for ••ch par.m.t.r. Alternatively, a number of experlm.ntal tr.n.f.r function may be acquired. The t.mpl.t •• are th.n r ••d directly for each frequency of Int.r.st. ~: GI ven the toleranc.s and th. t.mpl.t••, calculate constraints on the open loop Including th. feedback compensator. In the compl.x pl.n. the constraint for each fr.quency will tak. th. form of • border between an allowed and forbldd.n region for th. nominal compensated open loop syst.m. We c.ll th. borders HOROWITZ BOUNDS.

Step 4: Design a feedback compensator so th.t th. compensated open loop transmi ..lon satl.fi" the bounds. This Is convenlentl y done in • Nlchols chart. Minimum phase and stable I1nks .r. Included, d.l.ted and changed untU the loop transf.r function for •• ch frequency is on the corr.ct side of the bound, or untU the task Is considered Impossibl • • ~: Design a pr.m t.r so th.t the tot.1 nomln.l or average transfer function from r.f.r.nc. to output I. the desired one. This design is conv.nl.ntly done in • Bode chart.

Notice that phase tolerances could be included in Step 1. Dlsturbanc. r.jection .peclfic.tion. .r. normally ent.red for high frequencl •• In St.p 3 in order to produc. high frequency bound.; .uch specifications can al.o be introduced for low.r frequ.ncl.s . It has be.n shown in (Horowltz 1978, 1984) th.t th. d.slgn procedure wUI alway. give • solution if th. plant Is minimum-ph •••, if the .ign of th. high frequ.ncy g.in is known .nd if the p.r.m.ter uncert.lnties .re .uch that for .v.ry gain, 0 S .rg G(il.) S 2n-,,(I.), wh.r. " Is posl ti v., for frequencl.s around the crossov.r frequ.ncy I.) c ' Th.re .re s.v.r.1 Int.r.ctiv. comput.r progr.ms for the Horowltz d.sign m.thod. On. of th.m is HORPAC (Gutman, 1985). Oth.r D!llgn M.thods Th.re .r. oth.r d.sign m.thod. which c.n give robust control laws. On. technlqu. h•• been proposed by Ooyl. .nd St.ln. This m.thod Is b.sed on the LOG methodology. Anoth.r appro.ch i. to d.scrlbe the sy.t.m by a ..t of mod.l. (M ) .nd it i • •tt.mpted to k find a const.nt g.in regul.tor which wUI glv. s.ti.f.ctory performance for .11 mod.l. in the cl .... Th. d •• lgn m.thods used In thl. case .r. frequently based on multiobjectiv. optimization.

4 . ADAPTIVE CONTROL Design Procedure Th. b •• ic d •• ign procedure m.y be .umm.rized in five .t.p.: ~:

Det.rmin. the closed loop .pecific.tlon. •• upper and low.r llmit. on the g.in of the closed loop tr.n.fer function. Th. diff.r.nc•• betw. .n the llmit.

When comparing robu.t .nd .d.ptiv. control it I • ....nti.l that similar control probl.ms .r. discussed. For thi. r ...on w. wUI discus• •n .d.ptiv. regul.tor for a det.rmlnlstic control probl.m. It I •••• umed th.t the proc ••s to be controlled can be mod.led by A(q)y(t) = B(q)u(t)

(4 . 1>

Lnce nain System s

45

where u Is the control variable y the controlled output and A and B are polynomials In the torward shift operator. Assume that It Is desired to find a control law such that the relation betw. .n the command signal and the output la gl ven by A.(q)y(t)

=

O+-----~~------------10

(4.2)

Bm(q)u(t)

-20

Furthermore let the observer polynomial be Ao' When the parameters are known and certain technical conditions are satisfied the design procedure can be expressed as tollows.

CD ~ -30 c c l')

Design Procedure

10

Factor the polynomial B as B+B-, where B+ Is monlc stable and well damped. Determine polynomials Rand S which satlsty the equation

100 w

Fig. 1

Closed loop gain tolerance speciflcalions, and envelope of final transfer tunctlons from reference to output for Example 1.

(4.3) The control law Is then Ru

= Tc

(4.4)

- Sy

+ where R=R B and T=AoBm/B • 1

20

c 10

The equation (4.3) has many solutions. Under the compatibility conditions a solution which Is realizable can always be found. In the case when there are known disturbances characterized by the polynomial A.d a solution where A divides R can be tound. This corresponds to the1nternal model principle.

0

-10

An Adaptl ve control law can now be obtained as tollows.

-20

ALGORITHM Repeat the tollowlng steps at each sampling period:

-30

Step 1. Update the estimates ot the parameters ot the model (4.1) by some recursive estimation method.

_40 Step 2. Apply the design procedure to obtain the polynomials R, Sand T of the control law (4.4) and lbe control signal u. c In the examples presented In Section 5 we will use recursive least squares as an estimation method. The signals u and Y will be high pass filtered before they are entered Into the estimator. It will be assumed that the polynomial Ad which characterizes the disturbances Is A = q-1. This ensures that the regulator will have ~ntegral action. There are of course many other choices.

-50 -270 Fig. 2

-180

-90

0

Nlchols chart CdB vs. degrees) displaying bounds and the compensated nominal open loop for Example 1.

function

GCB) =

7 4·10 CB+0.25)(8+1.5) 8(B+30)C8+5008+250000)

5. EXAMPLES In this Section we will give examples ot robust and adaptive designs.

An integrator Is included to satisfy the bounds more easily and to ensure rejection of low frequency disturbances. A suitable pretilter is

=

2.89

Example 1: A simple servo. The servo model is taken trom (Astram, 1979):

FCB)

PCS)=k/C1+TB)2,k E[1,4],

The system Is thus of second order with variations In gain and time constants. The aim of the control system Is to make the servo follow step commands well In face ot the above specified parameter variations, and constant load disturbances.

Fig. 1 shows that this prefilter gives a closed loop system which satisfies the tolerance specifications. The time domain properties of the compensated system are Illustrated by simulation. Fig. 3 shows the response when lbe gain k is changed trom 1 to 4 at time t-15s and Fig. 4 shows the response when the time constant T is changed from 1 to 0.5 at time t=15s.

Robust design: The closed loop gain tolerances are shown in Fig. 1. Step 3 In Horowltz design procedure gives the bounds shown In Fig. 2. It. suitable compensator can now be designed by trial and error. Fig. 2. shows one compensator which satisfies the bounds. The corresponding controller has the transfer

Adaptive control: An explicit 2nd order pole placement Self Tuning Regulator is suggested in (Astram, 1979). The procedure described In Section 4 Is used. A second order model is estimated. The desired response is specified to be ot second order with the bandwidth 1.5 and the relative damping 0.707. The sampling interval

T E [0.5,2]

(5.1>

K.J. :\SIl"Om *. L. \/ e umann a nd P. O . G Ullllan

46

J 0.

[/ 10.

20 .

30 .

l

..

,

so Fig. 3

Simulation of the robust control system. The gain is changed from 1 to 4 at time t=15s. The time constant is 1.

Fig. £'

Simulation of the adaptive control system. The time constant T changes from 1 to 0.5 at time t=15s. The gain is k=1.

Reference

"roT' •

i=-

t

t

Fig. 4

Simulation of the robust control system. The time constant T changes from 1 to 0.5 at time t=15s . The gain is k=1.

Fig. 7.

Control system configuration for the integrator with time delay.

instantaneously. The responses of the robust systems are not perfect although they are within the specifications. See e.g. Fig. 4 where there Is a dlstorslon in the step response. The responses of the adaptive systems are batter when they have adapted to the changed condl tions.

~,

Fig. 5

Simulation of the osdaptive control system. The gain Is changed from 1 to 4 at time t=15s. The time constant is 1.

Is 0.3s. The performance of the adaptive system when

the gain and time constants are changed are shown in Fig. 5 and Fig. 6. These figures are the same experiments as were shown in Fig. 3 and Fig. 4 for the robust regulator. Comparison: A comparison of the Fig. 3 with Fig. 5 and Fig. 4 with Fig. 6 gives some of the characteristics of the different approaches. It is clear that the robust control responds much faster and cleaner to parameter variations than the adaptive system. In Fig. 5 and Fig. 6 It takes thr_ transients after a parameter change before the adaptive systems have adjusted to the changed parameters. The robust system responds

Example 2: An integrator with variable time delay. The plant is a simple non-minimum phase system. It Is also of Interest because it is a typical part of many industrial processes. pea) =

!

a

exp(-aT)

'

T E [0,1]

(5.2)

The system is assumed to be subject to step disturbance inputs at the plant input. The aim of the control system is to follow step commands as fast as posSible, and to eliminate the effect of the load disturbances as fast as possible. Robust design: The system configuration is shown in Fig. 7. The unknown time delay causes a phase uncertainty that will violate the maximum pha. . condition of Section 3 for high frequencies. The attainable bandwidth is therefore limited. The maximally attainable crossover frequency Is about 3 rad/s. It s_ms Impossible, however, to find a rational, proper, stable and minimum-phase compensator that achieves this, since the phase advance demands and the gain limitation demands s_m incompatible (cf. Bode's relations, Bode (1945». Instead a simple lead-lag network Is suggested that gives a crossover frequency of 0.7 rad/s (s_ Fig. 8): 0.6(1 + (Ha)

• B

(1

a)

1.3

47

L' nce rt a in Syste m s Adaptive control: The model

20

= y(t)

y(t+h)

+ b1u(t) + b u(t-h) 2

was used In the adaptive regulator. The sampling period was chosen as 11 which Is adequate to cover tbe variations In the time delay with the model. A shorter sampling period will requlre a more complicated model . The polynomials Am and Ao were chosen as

10

o

A (z)

m

= z - a and A (z) = z 0

-10

The procedure described In Section 4 was then applied. The results obtained are show In Fig. 11 and Fig. 12. - 20 -360

-1 80

-270

o

-90

Nlchols chart (dB vs. degrees) displaying bounds and the compensated nominal open loop for the Integrator with delay.

Fig. 8

It may be possible to get a slightly higher crossover frequency without too great an effort. To Increase the bandwidth of the transfer function from reference to output, the following prefllter was tried: =

F(s)

ComparllOn: Since tbe system In this example Is non-minimum phase Horowltz design procedure Is not guaranteed to give a solution. Fig. 9 and Fig. 10 also shows that altbough the frequency domain specifications are satisfied there Is qulte a variation In the step responses wben tbe time delay Is cbanglng. The responses of tbe adaptive system are much better. Apart from the time delay there Is practically no difference between the responses shown In Fig. Hand Fig. 12. Also notice the differences In the control signals. The high gain nature of tbe robust regulator Is clearly noticeable.

(1+s)

[1

+

5)

In the slmulatlons, step command following and step disturbance rejection was tested for various values of T, with and without the prefllter. The results are shown In Fig. 9 and Fig. 10.

t 1\ r I~r o

20

60

40

tl[L[ o

Fig. 9.

20

40

Fig. 11. Step response and step disturbance rejection for control with the adaptive regulator when Td =0.1 .

Step response and step disturbance rejection for control with the robust regulator when T d =0.1.

6'0

.b:Y= 20

40

o

o

20

40

6'0

.1

u

6

r\ I~ IT I~I(

v

60

Fig. 10 Step response and step disturbance rejection for control with the robust regulator when T d =1.

f 0

, 40

,

60

Fig. 12 Step response and step disturbance rejection for control with tbe adaptive regulator when Td =1.

K. J. As tro m *. L. ~ e um a nn and P. O . (;utma n Discussion Based on the examples shown In this paper and other that we have solved a few observations can be made about robust and adaptive control. Robust design will In general give systems that respond faster to variations In process parameters. They will, however, only respond well to variations Inside the design specifications. The adaptive systems respond more slowly to parameter changes. The adaptive systems give better responses to set point changes and load variations when the parameter estimates have converged. This Is particularly noticeable for non-minimum phase systems. The robust systems will In general have higher loop gains which makes them more sensitive to noise. Also notice that a few examples are not sufficient for a comparison between two design methods. More work Is therefore necessary. 6. ROBUST AND ADAPTIVE CONTROL Robust and adaptive control are two complementary ways to deal with process· uncertainty. Robust control gives a fixed gain regulator which Is designed to be Insensitive to specific parameter variations. Adaptive control deals with uncertainty by reducing It through parameter estimation. The adaptive control laws are normally derived using the certainty equivalence principle. This means that a model Is determined and a regulator Is designed as If the model was exact. The fact that this procedure may lead to difficulties when the estimated model Is Inaccurated Is now well understood. From this point of view It seems attractive to combine robust and adaptive control. This can be done In many different ways. One possibility Is to use a robust control design as the underlying design method In an adaptive system. This would Intuitively be better than to use a certainty equivalence principle. The parameter estimation will also reduce plant uncertainty and thereby make the robust design easier. To use robust control In this way Is difficult because the Horowltz design method In It present form Is difficult to Implement as an analytic procedure which can be used on line. The approach thus requires development of anal ytlc versions. The approach suggested by (Dayle and Stein, 1981) may be an alternative. See also (Gawthrop 1985). There are however other ways to combine robust and adaptive control. Another possibility Is to apply parameter adaptation of a basic robust design. See (Yanlv, 1986b). The Idea Is to modify the parameters but not the structure of the original robust design when tighter uncertainty sets are Identified on-line. A robust gain scheduling procedure can be developed as follows. Divide the total plant parameter uncertainty set luto subseets. Design off-line a robust controller for each subset. Use an on-line estimator to determine In what set the system currently Is, and apply the appropriate controller. This method has certainly been used a long time In flight control. Problems of switching between controllers must be solved. The global stablllty problem must also be considered. Robust feedback can also be combined with adaptive prefllterlng. It Is clear from the examples In Section 5 that It Is easier to design a suitable robust feedback compensator than a prefllter. Adaptive control has also been very successful In tuning feedforward galns where there are no problems with closed loop stability. By having a fixed parameter robust compensator Inside the loop, stability Is easily analyzed. The closed loop system exhibits less variations than the original plant. The adaptive prefllter thus has an easier task to find the parameters for a feedforward compensator wblch gives tbe de.lred response to command signals. An adaptive prefllter also slmpllfle. the feedback design.

7. REFERENCES AstrOm, K.J. (1979): Simple self-tuners I. CODEM: LUTFD2I(TFRT-7184) (TFRT-7184)/1-63). Dept. of Automatic Control, Lund Inst. of Technology, Sweden. AstrOm K.J . (1985): Adaptive Control - a Way to Deal with Uncertainty: In Ackerman, J "Uncertainty and control", Lecture Notes In Control and Information Sciences, 7Q, Berlin: Springer Verlag. Bode, H.W. (1945) : Network Analysis and Feedback Amplifier DeSign, Van Nostrand, New York. Dayle, J.C . and G. Stein (1981): Multlvarlable Feedback Design: Concepts for a Classical/Modern Synthesis. IEEE Trans. Auto. Control. 4-16. Gawthrop P.J. (1985): Comparative Robustness of Non-adaptive and Adaptive Control. lEE Control 85, 1. Gera, A. and I.M. Horowltz (1980): Optimization of tbe Loop Transfer Function. Int. J. Contr., ;n, 389-398. Gutman P.O., and L. Neumann (1985): Horpec - an Interactive Program Package for Robust Control Systems Design. Proc. 2nd IEEE Control System Society Symp. on Computer Aided Control System Design. Santa Barbara, Ca., Marcb 13-15. Horowltz, I.M. (1963): Synthesis of Feedback Systems. Academic Press, New York. Horowltz, I.M. (1973): Optimum Loop Transfer Function In Single-loop Mlnlmum-pbase Feedback Systems. Int. J. Contr., !§, 97-113. Horowltz, I.M. (1975): A Synthesis Theory for Linear Time-Varying Feedback Systems with Plant Uncertainty. IEEE T-AC, ~ 454-464. Horowitz, I.M. (1979): Quantitative Synthesis of Uncertain Multiple Input-output Feedback Systems. Int. J. Control 30, 81. Horowltz, I.M. and M. Sldl (1972): Synthesis of Feedback Systems wltb Large Plant Ignorance for Prescribed Time-domain Tolerances. Int. J. Contr., !§, 287-309. Horowltz I.M. and U. Shaked (1975): Superiority of Transfer Function over State-Variable Metbods In Linear Time-Invariant Feedback System Design. IEEE Trans. Auto. Control, 84-97. Horowltz, I.M. and M. Sldl (1978): Optimum Syntbesls on Non-minimum Phase Feedback Systems with Plant Uncertainty. Int. J. Contr., l!, 361-386. Horowltz, I.M. and Y. K. Llao (1984): A limitation of Non-minimum Phase Feedback Systems. Int. J. Contr., !Q, 1003-1013. Saberl, A. (1985): Simultaneous Stablllzatlon with almost Disturbance Decoupllng Part I: Uniform renk systems. Proc 24th IEEE COC Fort Lauderdale, 401-421. Yanlv, O. and I.M. Horowltz (19868): A Quantitative Design Method For MIMO Linear Feedback Systems Having Uncertain Plants. Int. J. Contr., ~ 401-421. Yanlv, 0 . , P.O. Gutman and L. Neumann <1986b): An algorithm for Adaptation of a Robust Controller to Reduced Plant Uncertainty. 2nd IFAC Workshop on Adaptive Systems In Control and Signal Processing, Lund, Sweden, July 1-3, 1986.