2006 IEEE 24th Convention of Electrical and Electronics Engineers in Israel

Generalized Nyquist Criterion and Generalized

Bode Diagram for Analysis and Synthesis of

Uncertain Control Systems

Arie Nakhmani, Michael Lichtsinder, and Ezra Zeheb, Fellow, IEEE

Abstract- Uncertainties in control systems models often haveto be taken into account in their analysis and/or design.Negligence of such uncertainties is often unjustifiable and is doneonly due to lack of methods to treat the uncertainties. Thepresented work is concerned with analysis and design of intervaluncertainty control systems, with regard to clustering of polesinside a simple symmetric bounded contour F. We extend thewell known Nyquist and Mikhailov stability theorems to F -stability tests of uncertain systems, defined by their generalizedBode envelopes. Also, using generalized definitions and theoremswe solve the design problem of a controller which ensuresclustering of closed loop poles of an interval uncertain family oftransfer functions inside such prescribed F -region.

Index Terms-Stability Theorems, Uncertain Systems, BodeEnvelopes, Controller Synthesis.

I. INTRODUCTION

THE wide exploration of frequency methods in controltheory began with the seminal works of H.Nyquist [1] andH.W.Bode [2]. Nyquist developed the well-known stabilitycriterion, called on his name. Later, in 1946, W.Frey gave the"unstable plant" version of Nyquist criterion [3]. In 1938Mikhailov has formulated the similar criterion for polynomialstability in frequency domain. Bode, among other things,showed the usefulness of gain-phase diagrams in analysis andsynthesis of control systems. Their works were followed bymany researchers, and many extensions and generalizationshave been found since then.

At the beginning of 70's of past century, M.Lichtsinderpublished numerous papers (in Russian), for example [4],about generalization of frequency methods and Nyquistcriterion for F-stability analysis. He used an alternativeformulation of Nyquist criterion, that counts the number of

Manuscript received August 30, 2006.A. Nakhmani is with the Department of Electrical Engineering, Technion,

Israel Institute of Technology, Haifa 32000, phone: 972-4-8295734; e-mail:anry otechunix.technion.ac.il.

M. Lichtsinder is with the Department of Aerospace Engineering,Technion, Israel Institute of Technology, Haifa 32000.

E. Zeheb is with the Department of Electrical Engineering, Technion,Haifa 32000, and Jerusalem College of Engineering, Jerusalem (e-mail:zeheb oee.technion.ac. il).

encirclements by Nyquist plot of (-1+Oj) point in complexplane with the help of counting intersections of Nyquist plotwith real axis ray (-,1). Some of these results derivedindependently in [5]. Lichtsinder also reformulated hiscriterion for using Bode diagrams (generalized Bode theorem).

In this work we will take further the research ofM.Lichtsinder and will generalize some well-known stabilitytheorems for F-stability and uncertain systems. The similar to[6] analytical form of Mikhailov criterion can be useful in F-stability analysis of uncertain systems.

In this paper we assume that given uncertain system G(s) isrational, with interval coefficients, and we can build thegeneralized Bode envelope for G(s), as explained in [7], [8],and in section III of this paper. We will use the generalizedtheorems for analysis. Finally, we will present a newtechnique for controller synthesis for uncertain systems. Amore detailed version including the proofs will be publishedelsewhere.

II. BASIC DEFINITIONS

Every control system has its performance specifications. Ingeneral, these specifications are given in time domain, but inthis work we will assume that it is possible to find thetransformation from time specifications to some closed andbounded region F in frequency domain. In other words, if allpoles in closed loop are placed inside F region, then allstability and performance demands are satisfied.

Let's define F as arbitrary, bounded, simple connected, andsymmetrical (with respect to real axis) region in complexplane. The boundary of this region is complex function1(6) =aF (see Fig. 1).The parameterization of F(O): [0, 2ff) -> C is given by

F(O) = R(O)e'6- A (1)where 0 is generalized frequency argument, R(0) E R', and-eE R is a real point inside the F region.

Additional definitions:1. Generalized frequency response is defined by complex

function G(F(0)), where G(s) is a transfer function of

linear time invariant (LTI) system in open loop; F(0) is

1-4244-0230-1/06/$20.00 ©2006 IEEE 250

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generalized frequency.

Fig. 1. Definition of F contour in complex plane.

2. Generalized phase response

o(O) = XG(F(O)) .

is defined by

3. Generalized logarithmic amplitude response is defined by

A(O)[dB] = 20 log ( G(F(O))).

4. The plot of generalized frequency response in complexplane is called Generalized Nyquist diagram, and the plotof generalized phase and logarithmic amplitude is calledGeneralized Bode diagram (GBD).

5. The G(s) is F-stable if all its poles placed inside the givenF region.

6. #p°O - The number of G(s) poles inside (outside) the F

region.

7. # PIC<out) - The number of G(s) (1 + G(s)) poles inside

(outside) the F region.8. # Zl°n(out) - The number of G(s) zeros inside (outside) the F

region.9. The set ( is defined

O-{A0(0) 13kEZ, 0(01)=180 +360°k}, that is if

0(0,)Ee 0, then the function b(O) has an intersection

with one of horizontal lines 180 +360°k ;ke Z at O1point.

10. The positive (negative) intersection of180 +360°k ;ke Z lines with b(O) is such intersection

point O,, the phase b(O) is rising (falling) from below

upwards (from above downwards), as the frequencyargument increases from 0 to J.

11. The crossing index is defined by:0.5 , if (0, = 0 or 0, =;T) and (raising b) and 0(0,) E e-0.5, if (0, = 0 or 0, =;T) and (falling b) and 0(0,) E e

ci(oi) = 1 , if 0 < 0, <T and (raising b)and 0(0S)e (

-I , if 0< 0, < T and (falling b) and 0(0,) E e

O , else

12. The G(s) crossing indexes sum is defined by:

cis{G(s)} = , ci (Oi )i A(Oi)>OdB

for all intersection points O, .

the frequency argument Oe [0,f] and not Oe [0,2ff] in

GBD.* For sake of simplicity, in further discussion we will use thewords "gain" and "phase" instead of "generalizedlogarithmic amplitude response" and "generalized phaseresponse".

* The frequency argument 0 is measured in radian, andphase is measured in degrees.

* G(s) is assumed to be in minimal form, without poles/zeroscancellations.

III. GENERALIZED BODE ENVELOPES

For left hand plane and unit circle the analytical techniquesfor building Bode envelopes described in [7] and [8].Unfortunately, it is not easy to generalize these techniques forarbitrary F region. Nevertheless, it is possible to use similartechnique with grating (by choosing dense enough grid for 0)to get an approximation to generalized Bode envelope (GBE).

Let's define the polynomial interval family

D(s, a) = ans' + an_ s'-l +... + aO , where a is a coefficientsvector, and Vi: a < a, < a-1. Suppose there is no degree

reduction, that is 0 X [a, an] (if degree reduction is present,

then we will test 3 different cases: an =0, an > 0 and an <0.The GBE is the envelope that includes all 3 cases).

For any complex s1, the value set of D(s1, a ) is a convexpolygon. Its vertices are given by affine mapping of D(s1, a)for extreme values of coefficients in vector a [9]. Fromgeometric point of view, the maximal distance from the origin,the minimal and maximal phase is turn out to be in vertices.The minimal distance can be measured to vertices, or toclosest to origin polygon side, or it can be 0 (see Fig. 2).

max ngle

distance min angle

mindistance

Fig. 2. Polygonal value set of polynomial, and extreme distance and anglecomputation from the origin.

We will parameterize F(0), as usual (1). After choosing the

grid for 0, we will find the appropriate value set for everyF(0), and plot the maximal/minimal distance and phase as

function of 0. This plot is called "generalized Bodeenvelope" (GBE) ofpolynomial D(s,a).

For interval transfer functionRemarks* Due to the symmetry of the F region, it is sufficient to use

G(s,aoc8) = N(s,/,) - bmsm + b. 1sm-l +... + boD(s,a) ansn + assnI +... + a

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a =(ao0...a, ,=(bo,....b) and Vi:.jia.<a-1:b i. b1 . b. We assume that a and ,6 vectors are

independent, so we can build the GBE of N(s ,8) andD(s, a) independently. From those two diagrams we canbuild the GBE of G(s, a, 8) [7]:

.max {G} = Q {N}- mrn {D} 2in {G} = O IN}-i max {D} (2

A {G}[dB] = An,, {N}[dB] -Amn {D}[dB]Amn {G}[dB] =Amin N}[dB] - Amax D}[dB]

where 0 is phase, and A is gain.

IV. GENERALIZATION OF MIKHAILOV CRITERION

Mikhailov criterion gives the necessary and sufficientconditions for stability without explicit solution ofcharacteristic equation, by examination of frequency response(of characteristic polynomial) plot in complex plane. Theextended and analytical version of this theorem can be foundin [6]. In this section we will generalize Mikhailov theoremfor F-stability case and for systems with uncertainty.Theorem 1

Let D(s) = ansn + an-is -I + ...+ ao be real polynomial with

an .0, and let o(6) XD(F(6)) be continuous plot of

D(F(O)) phase (6e[O, z]). Denote by O<62<o3<...<6ml <)

the roots of equations system (moD(,F(0.))}= (LRe {D(F(0 ))} < 0

equivalently, the roots of d(XO )E) ) and O1 = 0;,, = z .

mIf D(s) has no roots on the boundary F(0) = aJF, then:

Zin = (00(0f- 0(0)) /180' =2n ci(tYi) (4)i=l

where # Z,,n is number of D(s) zeros inside the F region.In particular case, when # z;n = n , the polynomial is F-stable.We may expand the previous result for uncertain

polynomials described by GBE.Theorem 2Suppose the uncertain family of polynomials of order n isgiven by GBE.1. If the value set of polynomial family excludes the origin

point for all 0e [0, z], then the whole family has an

identical number of roots inside F and also

¢>max (0) = ¢>min (0) and ¢>max (;T) = ¢min (r) . In this case, theidentical number of roots equals

(Omax(T) ->Omax (0))/180 (or (0An()-Omin (0))/180 ).2. The family is F-stable iff the value set of polynomial

family excludes the origin for all 0 e [0, z] and

bmax ( O)bmax (0) 180 n (or bmin (if) i (0) 180 n ).Remark

The condition that the value set of uncertain polynomial

family excludes the origin point for all 0e [0., i] is equivalentto bounded gain envelope, and equivalent to condition of nozeros on the boundary aF for the whole family.

V. GENERALIZATION OF NYQUIST THEOREM

For the arbitrary open loop transfer function G(s) (notnecessarily strictly proper or proper), we define the closedloop with unity feedback, so the closed loop transfer function

is (S) . Our goal is to find the number of closed loopI + G(s)

poles outside the given F region, from the knowledge of # poL(or # zL ) and GBD of G(s) in open loop.Theorem 3 (Generalized Bode Theorem)

If C(s) = -b m +.+b0 transfer function

D(s) sn+a snl +... +ahas no zeros and poles on the boundary arF, then

# p = max(m, n) - 2 c is{G} - # pOL= max(m, n)-2cis{1 G} #z .°ZO =

= max(m, n) - 2 c is{G} - n + # 00°ut =max(m, n) - 2cis{1 /G} - m +#zzo

(5)

and 2cis{G}+#pPL = 2cis{11 GI +#z L (6)where cis{ * } is the crossing indexes sum for transfer function

In particular case, when m<n, the system is F-stable inclosed loop iff 2cis{G} = # pjO° or 2cisl1 G} = n-rm + # zj.Now we will determine the number of closed loop poles of

interval uncertain family, given by GBE.Theorem 4Suppose the gain envelope of given interval family (of ordern) is bounded, that is 3M<o, VOe [0,;r]:

-M < Amin (). Amax (O) <M (the definitions of Amax and

A i are given in section III).The whole family has the same number of closed loop poles(with unity feedback) iff there is no coincidence betweenintervals of 0 that satisfy: Amax 2 OdB and An < OdB, andintervals of 0 where the phase family crosses one of180 (mod 360°) lines. In other words,

{0: Amax(0) > 0[dB] and An (O) < 0[dB]m}nn{o: 180'(mod360)e [0min(0)b0)J } = 0

If the condition from the above satisfied, then the wholefamily has the same cis, and the number of closed loop polesinside F region can be calculated with the help oftheorem 3. Ifthe number of poles equals n, then the whole family is F-stable in closed loop.

VI. CONTROLLER DESIGN

In previous sections we have described the techniques foranalysis of open and closed loop by its GBD or GBE. Now weneed to design suitable serial controller to get all closed loop

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poles inside the given F region. From previous sections weknow, that all we need is to get 2cis equals to number of openloop poles outside F. The plots of generalized phase and gainof controller and plant are added, like in regular Bodediagrams, so we need to choose controller in such way that thesum diagram (plant + controller) will lead to needed cis. Tofind such controller some graphical methods can be explored.For example, we may choose the needed plant+controllerGBD, and then will find the controller diagram by subtraction(of plant diagram). From GBD of controller we mayapproximate its transfer function by numerical methods. Thisexperimental method may require much iteration, beforedesirable satisfactory controller is found, so in this work wewill present more systematic way for controller design.The plant is given by G(s) transfer function, or by GBD:

{gain A(0), phase ¢)(S)}. If the gain of G(s) is bounded, thenit is always possible to get the positive gain (in dB) forplant+controller, by choosing high enough controllers gain.The plant+controller that have positive gain for all 0e [0,;i]is defined as class I system. The controller phase denoted byP (0), and m is a number of open loop poles outside F,divided by 2. By theorem 4, for class I controller design isenough to concentrate on two end-points b(0) and 0((z) .Theorem 5If the controller fulfills the following condition:T(z) = 360°m + 5(0) - o(5z) + T(0) (8)or # Zcontroller - # Pcontroller = 2m + ( 0(0) - 0(/T)) /180 (9)(where # Zcontroller and # Pcontroller is the number of controllerzeros and poles inside F), then the system with controller is F-stable in closed loop (with unity feedback) for controllers gainhigh enough (that gives class I system).

If for some engineering reason the provided solution is notsatisfactory, or not applicable, then we suggest using class IIor class III design, described bellow.

Class II system is described by plant+controller gain thatbegins above the OdB (for 0 = 0), and ends below OdB (for0 = iT ) with single crossover point at 02 .

Class III system is described by plant+controller gain thatbegins below the OdB (for 0 = 0), and ends above OdB (for0 = iT ) with single crossover point at 91 .

Theorem 61. The class II controller has to satisfy the followingconditions:* If b(0) = 0° and m E Z, then T(0) =0°.* If q(0) = 0° and m XZ, then (0) = 180°. (10)

* Ifq(0) = 180° and mEZ, then (0) = 180°.

* Ifq(0) = 180° and m XZ, then T(0) = 0°.360°m-180° +o°)+T(°)- 0(2) < T(02) < (I l)

< 360°m + 180° +(0)+ T(0) -5(02)2. The class III controller has to satisfy the followingconditions:* If 0(;)o0 and me X, then T(if)=360°k, ke EZ.

* If 0(;r)o ) and mx X, then T(;) =180° +360°k, k c Z.

* If 0(if)e ® and meZ, then T() =180° +360°k, kc Z.

* If 0(if)e ® and m& Z, then T() = 36O°k, ke Z. (12)

(13)(.r) + T( ())- 360°m -180°+ -(0)) < T(01) <

<0(;T)+T(T)-360'm+180' -0(01)RemarkIf all conditions of previous theorems are satisfied bycontroller, then the controllers gain has to be chosen to get acorrect class.

Generalizing previous theorems to uncertain systems caseis straightforward.Theorem 7Let define an interval family of order n by GBE with boundedgain (the same phase end-points b(0) and 0(;f) )

{Amax (0) [dB] A4mi (0) [dB],Imax (0) I Ornin (0)}, and let define

The controller by its GBD {Acontroller(0)[dB], T(0)}.1. Class I controller satisfies:T (;f) = 360°m + 0 (0) - 0(ZT) +T (O)where b0(0) -min (0) = Omax (O) and (;fT)Am (z) =max('f)-

Also, it satisfies:{O: A + AOoller > O[dB] and Amin + Aontroller <[dB] }

max cotolen{o: 180 (mod 360 ) E [0.in +Tn, max + T]} = 0 (14)

The (9) equation is also true.2. Class II controller satisfies (14),(10) and

360 m -180 + O (0) +T (0) )i5 1(2 )< T (2 )<

< 360'm + 180 + (o) + T (o) -bm. (02)3. Class III controller satisfies (14),(12) and

360m -180° + Oz(0)+T(0)- im (O1) < T(01) << 360°m+180° +z(0)+TP(0)-0m (O1)

ExampleThe transfer functions interval family is given by

G(s) = 2+[ ,lls+[0 1 and the F region iss3 + [15,1 6]s2 + [63, 65]s + [50,5 1]

given by disk with radius R=20 and with A = 23 displacementto the left from the origin (17() = 20ejo - 23 ).The GBE of G(s) denominator can be seen in Fig. 3.The gain is bounded, so, by theorem 2, we know the numberof G(s) open loop poles inside F is the same for the wholefamily, and this number equals((t5max (;T) Omax (0))/ 180 = (540 -180 )/ 180 = 2 .The number of poles outside F is 1, so we need m=cis=0.5 toget F-stable system in closed loop.The GBE of G(s) is given in Fig. 4. It can be seen that gainenvelope is always negative, so we have cis{G}=O. Let's usethe theorem 5 to design the controller. We choose T(0) = 0°,then T (;r) = 360°m + 0(0) - 05(;) + T (0) = 180° . The suitablecontroller with such phase characteristics is a PD controller.

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cis{HG}=0.5 (see Fig. 5). J5p

Generalized Bode Envelope D -10 A . . .1001 1 100 ------------l-------------r------------l-------------r------------l-----------------------------T---------------------

E

X<42

20 o 0 0 115 2 2 35 3 280 l

oo,5,U40 -- ~F>

_ 500 0=___-----------|

Lo maxm 400 | - - - - ------

--m-

vD - -------------- neralized requency rgum ent

100 0 0 5 1 1.5 2 2 5 3 Fig. 4. Generalized Bode envelope of G(s)Generalized Frequency Argument 0 Generalized Bode Envelope

35! !!Fig. 3. Generalized Bode envelope of s3+[15,16]s2±[63,65]s+[50,51] n 30 V .

25_

max Cont1irolFr6VII. CONCLUSION E 15

This work gives a new important way of looking at robust ,75 1 0

control systems analysis and design. The technique for 5

controller design avoids the need in phase or gain margin A mn+ACntrollerusage and can assure the stability and step response c os 1 15 2 2.5 3

specifications at the same time for the whole family, definedby GBE. The future research can continue in many ways, toinclude MIMO systems, systems with delays, or with othertypes of uncertainty. 350 ---- ------_----7---_

[1]H. Nyquist, ~Regeneration Theory", Bell System Technical Journal, N 250|-----------L------'d'me+-1-------------------------+--------------------------Vol.11, 1932. O /

[2] H.W. Bode, "Network Analysis and Feedback Amplifier Design", Van cDso200 .Nostrand Company, 1945.10

criteria", Brown Boveri Rev., Vol. 33., 1946. Generalized Frequency Argument e[4] M. Lichtsinder, "Generalized Method of Automatic Control Systems Fi.5GerazdBoenvlpofHsG)TyeIeig

Synthesis by Exponential-Frequency Response", (in Russian), g.5GerazdBoenvlpofHs()-TyeIesnKuybishev, 1974.

[5] M. Vidyasagar, R.K. Bertschmann, and C.S. Sallaberger, "SomeSimplification ofthe Graphical Nyquist Criterion", IEEE Trans. onAutomatic control, Vol.33, no.3, 1988.

[6] L.H. Keel, S".P. Bhattacharyya, "A Generalization of Mikhailov'sCriterion w~ith Applications", Proc. of the American Control Conference,Chicago, Illinois, June 2000.

[7] A. Levkovich, E. Zeheb, N. Cohen, "Frequency Response Envelopes ofa Family of Uncertain Continuous-Time Systems", IEEE Trans. onCircuits and systems, 1995.

[8] N. Cohen, A. Levkovich, P. de Oliveira, E. Zeheb, "Frequency ResponseEnvelopes of a Family of Uncertain Discrete-Time Systems", CircuitsSystems Signal Processing,t20u3.

[9] S.P. Bhattacharyya, H. Chapellat, L.H. Keel, "Robust Control: TheParametric Approach", Prentice Hall, 1995.

n

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Generalized Bode EnvelopeFor example H(s)=2(s+20) gives the GBE with needed

m

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