Robust Stability of Polynomials with Annular Uncertainties

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

ROBUST STABILITY OF POLYNOMIALS WITH ANNULAR UNCERTAINTIES F. Perez*, D. Docampo* and C. Abdallah** *RSP Group, ETSI Telecommunicacion, Universidad de ViRO. 36200- ViRO, Spain **BSP Group, EECE Departmem. University of New Mexico. AlImqllerqlle. NM X7 I 3 I, USA

Abstract. In this paper, we formulate and solve the stability problem for complex polynomials whose coefficients lie in annular sets. The results include the previously known case of disks uncertainties. Key Words. Complex polynomials, Annular sets, Winding number, Unit disk .

The following Lemma will be fundamental in deriving our robust stability results .

1. INTRODUCTION The seminal work of Kharitonov (Kharitonov, 1978a) has inspired considerable research in studying the robust stability of polynomials. As is well known, Kharitonov's original theorem restricts the perturbation of each coefficient to a real interval and to be independent from all others. This result was later extended to families of complex interval polynomials (Kharitonov, 1978b). Many proofs for Kharitonov's theorems are now available. In particular, (Minnichelli et al., 1989) used a frequency domain approach due to (Dasgupta et al., 1991) combined with simple geometry to obtain easy proofs of Kharitonov's results. Most of the work on the stability analysis for uncertain polynomials was concentrated on real uncertain polynomials. Notable exceptions are the problems studied in (Chapellat et al ., 1989) and (Y.Li et al., 1992) . While the real-coefficient case is important for many practical systems, it is sometimes desirable to consider complex interval polynomials. This problem appears, for instance, in the analysis of minimum-phase equalizers for communications channels (Perez et al ., 1992b). In many situations, it is reasonable to model uncertain complex-coefficients polynomials as follows: The phase of each coefficient is a random variable uniformly distributed in [0,211') and the modulus is known to lie within some confidence interval. In this context, every complex coefficient is contained in an annulus. It is also plausible to include a fixed polynomial (center polynomial) to account for the known part of the system. In this paper, we find necessary and sufficient conditions for the stability of such a family of annular polynomials. The proposed results can be modified to solve the disk problem, by simply zeroing out the inner radii of the annuli . In Section 2, we describe the uncertain sets and give some useful lemmas. Section 3 gives the solution for the Schur stability case, and Section 4 addresses the Hurwitz stability problem. To illustrate our theorems, we present some examples in Section 5. Our conclusions are given in Section 6.

Lemma 1 Given the annular sets Si C C and a complex constant K, the set n

S= K+ LSi i==O

is an annulus in the complex plane, centered at K with outer radius n

Rout = LRi 1=0

and inner radius n

Rin = max{rl- LRi,O} i=o

.;el

Proof: The proof can be found in (Perez et al., 1992a) .

Definition 1 (Winding number) (Marden, 1989) . Let u : [a, b] -+ C be any closed curve such thal Zo rf- u. We define the Winding Number of u about Zo , N(u, Zo), by

N(u, zo) = [Arg{u(b) - zo} - Arg{u(a) - Zo}]/211' The winding number represents the number of times that a poinl z encircles Zo as it moves from u(a) to u(b) along u. Lemma 2 (Rouche's Theorem) (Marden, 1989). Let 1} : [a, b] -+ C and u : [a, b]-+ C be two closed curves in the complex plane, such that 11}(t) - u(t)1 < 11}(t) - zol, a :S t :S b. Then, • N(1},zo) = N(u,zo) .

2. PRELIMINARIES Throughout this paper, we will deal with annular sets in the complex coefficient space C. We let R + denote the set of all nonnegative real numbers. The annulus Si, centered at 0, with inner radius ri E R+ and outer radius Ri E R+, is described as

Si = {z E C : ri :S Izl :S Ri}.

•

Remark 1 It is important 10 note that given any point bei ., bE [Rin, Rout], t/> E [0,211') it is always possible to find Zi, Zi E Si, for all i = 0, ... , n, for which be i • = K + Ei=o z.. •

3. DISCRETE-TIME CASE

(I) The discrete-time case is easier than its continuous-time counterpart because, the coefficients annuli are mapped onto the complex plane to annuli with radii that are independent of frequency. We will consider annular families of polynomials of the type

We define the sum of two sets in the usual way, i.e.,

(2)

235

We will proceed by contradiction assuming that I in (10) is such that I < n. Since there is one stable polynomial in the family,

n

{c(z)

+ I:aie#'zi

0::; ai ::; ai ::; at ,~i E [O,2".)}

it follows that there exist a: E [ai, an~: E [0,2".),0 ::; i ::; n such that the polynomial Ho(z) = C(z) + L:i=oaiei
(3)

{H(z)}

where i = 0, ··· , n, C(z) is a given complex-coefficients polynomial in z and the family has a constant degree . Next, we specify the definition of stability that will be used throughout the text .

n

n

i=O

.",, 0

I: aiej,pieii8r' = I: aiej~ieii8ri + a;ei4l;ejIBr'

Definition 2 The polynomial H(z), with 0 ::; ai ::; ai::; at'~i E [0, 2,.. ), i = 0, · . . ,n is stable if! all its zeros lie in the region 1), given by

i;"1

Since 1)

=

{z E C : Izl < r < 1}

(4)

We are now ready to state and prove the following lemmas. and noting that the polynomial a;e il 4>;zl encircles the OrIgIn times, it is possible to apply Lemma 2 to determine that the polynomial L::':o a:e#: zi encircles the origin exactly I times as z travels around the circle of radius r. Therefore , the proof is completed by contradiction, since this last polynomial must encircle the origin n times in order to be stable.

Lemma 3 The image of H(z) onto the complex plane, obtained as z traverses along z = re i9 is, for a fixed lI,an annulus centered at C(re i9 ) with outer radius, Rout, given by n

Rout

= I:atri

(5)

1:::;0

Theorem 1 The family 'H. is stable if and only if the following two conditions hold i) The family contains at least one stable polynomial. l > 1 for all 11 E [0,2".) ii) IC~~~'ll < 1 or

and inner radius, Rin, given by n

Rin = max{a,rl- Latri,O}

(6)

,.0

Ict:: I

i;"1

where Rout and Rin were given in (5) and (6), respectively.

Proof: Substituting z = re i9 in the expression for H(z) given in (3), we obtain

Proof: Sufficiency. Suppose that IC(reiB)1 > Rout for every 11 < 2". . From Lemma 3, we have that for every 11, the coefficient set is mapped in the complex plane onto an annulus. We then have

o ::;

n

H(re iB ) = C(re i9 ) + I:airiei
.=0

n

and since the factor eii9 represents only a rotation, we have that the family 'H, for a fixed z = re iB , is the sum of annuli Si( airi, atri) so the problem fits into the conditions of Lemma 1 making C(re i9 )+ l:i=o Si(airi, atri) an annulus with the specified radii.

IC(reiB) 1

.=0 Therefore, a direct application of Lemma 2 shows that the winding number of C( reiB ) is equal to that of C( re iB ) + L::'=o aiei L:~ol atri. Following an argument similar to the proof of the Lemma 5, the polynomial L:i=oaiei
Remark 2 The condition for the annulus to degenerate into a disk for every frequency 0 ::; 11 < 2". is that r' a, < l:~=o ri at. In i;"1

o.

this case, we let Rin =

> Rout> lI:aiei
Lemma 4 If

(7) then, the polynomial C(z) + l::':o aiei
Proof: Since n

Rin::; 1 I: aiei
< 2,..

(8)

':=0

Then, it would be possible to find some unstable polynomial in the family with a root on the circle z = re iB . This contradiction results in the validity of ii). •

it is clear from (7) that n

lC(re iB ) 1 < 1 I: aie#'eiiBr'l.

0::; 11 < 2".

(9)

;=0

Remark 3 Only one of the two conditions present in ii) can hold in any particular case. For instance, suppose that for some (JI the first condition holds, i.e., IC(reiB') 1 < Rin, then, only checking the validity of IC( reiB)1 < Rin, for all (J E [0, 2?r) is necessary. The same can be said for the second condition as described in (Nrez et al., Ig92a). •

Then, from Lemma 2, the winding numbers about 0 for

l:i=oaiei ~i=o

at

Proof: We have already shown that if Rin 1,0 ::; I ::; n, we have

>

For the case considered in (Chapellat et aI., 1989) and (Y.Li et aI., 1992) we have the following Corollary.

<

Corollary 1 Consider the family 'H with Rin = O. Then, 'H is stable ifJ the following two conditions hold i) The family contains at least one stable polynomial.

0 then, for some

ii)

..

rla,

> I:at ri

1~~::} I>

1 for all (J E [0,2,..).

(10) Proof: From the definition of H, it is clear that Rin equals O. The proof is completed by Remark 3. •

_=0 i;"1

236

Remark 4 There is no restriction on the degree of C(z) if the second condition in ii) holds. If the first condition of ii holds, the degree of C(z) can be at most n. If C(z) = 0 condition ii) transforms into R,n # O. •

i) The family contains at least one stable polynomial. ii)

1;(;"') I> 1, ,",'(101)

'Vw E R

where Rout(w) was defined in (12) .

Proof: The proof is similar to the proof of Theorem 1. Note that the only difference is that condition IC(jw)[ < R,n(w) is not present here. Therefore we will concentrate on showing why this condition is not necessary. To see this, suppose that every

4. CONTINUOUS-TIME CASE

member in 1t is stable and there exists some frequency W1 E R such that IC(jW1)[ < R,n(W1) ' Following an argument similar to Remark 3, we conclude that

Here, we will consider annular families of polynomials given by

[C(jw)[

where i = 0, .. ·, n, C(s) is a given complex polynomial and the degree of all members in the family is constant. The sta.bility region will be the open left half complex plane (Hurwitz stability). Lemma 6 The image of H(s) onto the comple", plane, obtained as s travels along the line s = jw, for a fi:r;ed w, is an annulus centered at C(jw) with outer radius given by

ER

(18)

Remark 5 The previous Theorem remains valid if ai = 0, i = 0 .. ·, n in (11) as discussed in (Chapellat et al., 1989) (Y.Li et al., 1992).

(12)

Remark 6 When w goes to infinity, from condition in ii) is necessary that m == deg C( a) 2: n. For the case where m = n, the coefficient of an in C(a), cn , must be such that Icn [ > a;t .

(13)

5. EXAMPLES

n

Rout(w) = l:>tlwl'

< R;n(w), 'Vw

Otherwise, assumption i) would be violated. On the other hand, from Lemma 7, R'n(w·) = 0 which obviously contradicts (18) . The proof is now complete.

• =0

and inner radius given by n

R;n(w)

= ma.x{ajlwl'- }:>tlwl',O}

where

(14) Proof: Letting s we obtain

To show how the theorems given in the text can be transformed into simple graphical tests in the spirit of the work in (Tsypkin and Polyak, 199180) and (Tsypkin and Polyak, 1991b), we present some examples. First, we concentrate on the discrete-time case. Consider the family 1t of uncertain polynomials defined below 2

= jw in the expression for H(s) given in (11),

1t

= {H(z) = C(z) + L:a;ei·'z"
n

where ao E [1,1.5], a1 E [0,0.5] and a2 E [8,10] and where C(z) is the complex polynomial

+ L: a,ei ., (jw)'

H(jw) = C(jw)

,=0

C(z) = 6e i ,,/4 z 3 + 3ei ,,/3 z 2 + 2e i3 ,,/4 z

With a simple identification of terms, we can see that the family 1t is, for a fixed w

1t = C(jw) +

t

5,( ailwl',

,=0

and we are interested in determining if the family 1t is Schur stable (r 1). Applying Lemma 3, we obtain that Rin 6 and Rout = 12. The conditions in Theorem 1 for stability of 1t can be transformed as follows . First, we choose any polynomial in the family and check its stability. If this polynomial is stable we plot C( ei8 ) with (J varying from 0 to 211'. This plot must lie outside the annulus with radii Rin and Rout . In Figure 1, we plot C( ei8 ) using a dashed line. As we can see, the plot enters the annulus, so the family 1t is not stable.

=

at Iwl')

which, according to Lemma 1, is an annulus in the complex plane with the specified radii. • Prior to the statement of the main result of this section, we need the following Lemma. Lemma 7 There exists w· E R, for which R'n(w·) = O. Proof: We will consider here the annuli S,(ailwl', atlwl') that appear in Lemma 6. First note that for w = 0 the inner radius of every 5, is 0, except for So. Then, for this frequency, the set 1t is an annulus centered at C(O) with inner radius aD and outer radius For those frequencies at which

=

15

at .

aD> ailwl', for all i = 1,· ·· ,n

' .'- - - -, \\(;)./ (Q

10

(15)

which follows from condition (14), the inner radius is

,'

;

n

R,n(w)

= aD -

L:atlwl'

(16)

,=0

·5

The expression above becomes 0 for some w· at which n

aD = L:atlw·I'

+5

...

-

,. ...... -.

"

,

·10

(17)

,=0

·15

But this expression will be valid only if (15) holds for w·. Substituting (17) into (15) we see that the condition verifies for every i, i = 1,· . . , n, and then R;n(w·) = O. •

·20 ·20

Theorem 2 The family 1t is stable if and only if the following two conditions hold

·15

·10

·5

10

15

Figure 1: C(re i8 ) and the forbidden annulus

237

Now we consider the continous-time case. Suppose that p is the common degree of the members of the family of polynomials. The test given by Theorem 2 is applied as follows. If the plot goes through 2p quadrants counterclockwise and does of not f;;ie~sect the closed unit disk when w goes from -00 to +00, then the family is stable. Now consider the family

IO,-------r------,------~---___,

;Url)

2

1(.

= {H(~) = C(8) + '£ a..ej~,s', q,. E [0,211'), i =

1 .. ·, 3}

,:::;;0

where a.o E [0.5,1.25], a.) E [0,0 .75] and the real coefficients polynomial

0.2

E [1,2] and C(s) is

-2

C(s) = 9.6s 5 + 41.11s4 + 44 .51s 3 + 34.46s 2 + 12.955 + 3.45

-4

According to the graphical test described above, we plot in Figure 2 the function C(jw)/ R".Aw) for w E (-00,00) and see that it goes counterclockwise through 2p = 10 quadrants. In Figure

-6

-8

3, the same function is plotted in the interval [-2,2] along with the unit disk. Since the graph does not enter the unit disk we conclude that the family 1(. is Hurwitz stable. Note that for the case in which C(~) has real coefficients the test can be simplified since the plot will be symmetric with respect to the real axis.

_IO L-------'-----~---~-------'

10

-5

15

Figure 3: C(jw) / Roul(W) and the forbidden disk

6. CONCLUSIONS We have presented a frequency domain criterion for analyzing the stability of polynomials with annular uncertainties in both the discrete and continuous time domains . The tests so obtained are easily transfromed into a one-plot problem which facilitates the graphical computation of the common margin of perturbations. The main result of this paper shows that for the Hurwitz and Schur cases, the admissible set is transformed into a complex annulus for every frequency. Of course, this idea can be easily extended to deal with other 'V-stability problems. The analysis presented here can be extended to the more general case of linear dependencies on annular sets of parameters . The results in this line of research will be reported shortly. The tests given have found applications in the analysis and design of communications equalizers.

References Chapellat, H., Battacharyya, S., and Dahleh, M. (1989). On the robust stability of a family of disk polynomials. In Proc. IEEE Conf. on Dec. and Control, pages 37-42, Tampa, FL. Dasgupta, S., Parker, P., Anderson, B., Kraus, F., and Mansour, M. (1991). Frequency domain conditions for the robust stability of linear and nonlinear dynamical systems. IEEE Trans . Circ. and Syst., AC-38:389-397. Kharitonov, V. (1978). Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differentsial'nye Uravneniya, 14:2086-2088. Kharitonov, V. (1978). On a generalization of a stability criterion. Izv. Akad. Nauk. Kazach. SSR Ser. Fiz. Mat., 1:53-57. Marden, M. (1989). Geometry of Polynomials. Am. Mat . Soc., New York, 4th edition.

ISO

Minnichelli, R ., Anagost, J ., and Desoer, C. (1989). An elementary proof of Kharitonov's stability theorem with extensions. IEEE Trans . on Auto. Control, AC-34 :995-998.

100 SO

Perez, F., Docampo, D., and Abda.Ilah, C. (1992a). Robust stability of complex polynomials with annular uncertainties. Submitted to Multidimensional Systems and Signal Processing. Perez, F., Docampo, D., and Abdallah, C. (1992b) . Root location and confinement for systems with annular uncertainties. In Proceedings of the Conference on Mathematics in Signal Processing, Warwick, England.

-50 -100

Tsypkin, Y. Z. and Polyak, B. (1991a) . Frequency domain criterion for robust stability of polytope of polynomials. In Battacharyya, S. and Keel, L., editors, Control of Uncertain Dynaimc Systems, pages 491-499. CRC Press, Boca Raton, Fl.

-150 _~L---L---L---L---L---L---L-~

-50

SO

100

150

200

250

Tsypkin, Y. Z. and Polyak, B. (l991b). Frequncy domain criteria for lP-robust stability of continuous linear systems . IEEE Trans . Auto. and Control, AC-36 :1464-1469.

300

Y.Li, Nagpal, K., and Lee, E. (1992) . Stability analysis of polynomials with coefficients in disks . IEEE Trans. on Auto. Control, AC-37:509-513.

Figure 2: C(jw)/Rout(w)

238