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Journal of Computer and Systems Sciences International, Vol. 39, No. 2, 2000, pp. 165н172. Translated from Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, No. 2, 2000, pp. 5н12. Original Russian Text Copyright й 2000 by Mailybaev. English Translation Copyright й 2000 by MAIK "Nauka / Interperiodica" (Russia).

STABILITY

On Stability of Polynomials Depending on Parameters
A. A. Mailybaev
Institute of Mechanics, Moscow State University, Michurinskii pr., Moscow, 117192 Russia
Received July 1, 1999

Abstract--Characteristic polynomials with real coefficients that smoothly depend on a vector of real parameters are considered. A constructive approach is proposed that allows one to determine, in a first approximation, the stability domain or a domain with a bounded decrement in the neighborhood of a singular or regular point of its boundary from the information available at this point (the roots and coefficients of the polynomial as well as the first derivatives of the coefficients with respect to parameters). Examples are presented.

INTRODUCTION The study of stability problems for many mechanical and control systems is reduced to the analysis of the roots of the appropriate characteristic equation. Asymptotic stability is achieved when all roots of the characteristic polynomial lie in the left complex halfplane. The polynomials possessing this property are said to be stable. The properties of stable polynomials have been intensively studied in view of their high practical importance. There exist a number of methods for determining the stability, such as the classical Routhн Hurwitz method and a method of D-decompositions [1н3]. These methods allow one to determine either the stability of a specific polynomial or, if the coefficients of the polynomial are known functions of parameters, the stability domain (the set of values of the parameters for which the polynomial is stable). Nevertheless, one meets considerable computational difficulties when constructing the stability domains of complex multiparameter systems because of a high order of polynomials and large computational cost for the determination of the coefficients and roots of the polynomial with many values of the parameters. One meets difficulties of a different type when multiple roots arise, which lead to the nondifferentiability of the roots of the polynomial with respect to the parameters and give rise to singularities (points of nonsmoothness) on the boundary of the stability domain [4]. Therefore, it seems topical to develop further qualitative and quantitative methods for investigating the stability domains of the families of polynomials that depend on many parameters. Qualitative aspects of the structure of a stability domain and its boundary in the space of all polynomials with the leading coefficient equal to unity were studied in [5, 6]. In these papers, the singularities of stabilitydomain boundaries that appear in the case of general position were classified and the tangent cones (linear approximations) to the stability domain at singular

points on its boundary were described up to a nondegenerate change of parameters. In the present paper, we develop a constructive method for determining an approximation of the stability domain in the neighborhood of a point on this boundary in the space n of parameters based on information available at this point (using the values of the roots, the coefficients, and the first-order derivatives of the coefficients of the polynomial with respect to the parameters). We consider the points of the stabilitydomain boundaries of arbitrary type (characterized by roots of arbitrary multiplicity). It is assumed that the leading coefficients of the polynomial may vanish. Then, we extend the results obtained to the case when a domain with a bounded decrement (degree of stability) is considered instead of the domain of stability. The efficiency of the proposed method is demonstrated by two examples from automatic control theory. The results obtained are the development of works [7, 8], where the case of two- and three-parameter families of polynomials of general position were studied. 1. STABILITY DOMAIN OF POLYNOMIAL FAMILIES Consider a polynomial of degree M, P ( , p ) = a M ( p ) + ... + a 1 ( p ) + a 0 ( p ) ,
M

(1.1)

with real coefficients that smoothly depend on the vector p = (p1, ..., pn) of real parameters (a family of polynomials). Suppose that, for p = p0, the polynomial P0() = P(, p0) has the form P0 ( ) = am + ... + a1 + a0 ,
0 m 0 0 0 0 M

m M,
0 m+1

(1.2) =...=

where a j = aj(p0), j = 0, ..., M; a m 0; and a

a = 0. The case m < M is singular, since the leading coefficients of the polynomial vanish. Denote by 1, ..., k

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MAILYBAEV

the distinct roots of polynomial (1.2) and by m1, ..., mk, their multiplicities. When m < M, we assume that P0() has an infinite root 0 = of multiplicity m0 = M н m. The polynomial P(, p) is called stable if all its roots have negative real parts. The set of values of the vector p of parameters for which P(, p) is stable is called the stability domain. Theorem 1. The family of polynomials P(, p) in the neighborhood of the point p = p0 can be represented as P ( , p ) = ( p ) P ( , p ) P ( , p ) ... P ( , p ) , (1.3)
0 1 k

det( A '0 (p0) н B '0 (p0)) = 1. Denoting
0 P ( , p ) = det ( A '0 ( p ) н B '0 ( p ) ) /det A '0 ( p ) , m j P ( , p ) = ( н1 ) j det ( A 'j ( p ) н B 'j ( p ) ) /det B 'j ( p ) ,

j = 1, ..., k ,
m ( p ) = ( н 1 ) 0 det R ( p ) det A '0 ( p ) det B '1 ( p ) ...

det B 'k ( p ) /det Q ( p ) , we obtain

where P ( , p ) = 0 ( 0 ) + ... +
0
0

m

m0 н 1

( 0 ) + 1, (1.4)

P ( , p ) = ( н 1 ) det ( A ( p ) н B ( p ) )
M M н1 = ( н 1 ) det ( R ( p ) ( A ' ( p ) н B ' ( p ) ) Q ( p ) ) M = ( н 1 ) det R ( p ) det ( A '0 ( p ) н1 н B '0 ( p ) ) ... det ( A 'k ( p ) н B 'k ( p ) ) det Q ( p )

P ( , p ) = ( н j ) j +
j

m

mj н 1

( j)

т ( н j)

mj н 1

+ ... + 0 ( j ) ,

j = 1, ..., k .

The coefficients l(j) are smooth (and complex-valued for j ) functions of vector p such that l(j) = 0 for p = p0 and (p) is a smooth nonvanishing function. If i = j , then Pi(, p) = P ( , p ) , where the bar denotes complex conjugation.
j

= ( p ) P ( , p ) ... P ( , p ) .
0 k

Proof. Introduce M т M matrices A(p) and B(p): 0 1 ... 0 A(p) = 0 0 ... 1 нa0 ( p ) нa1 ( p ) ... нa M н 1 ( p 1...0 0 B(p) = 0...1 0 0 ... 0 aM ( p
... ...

The polynomials Pj(, p), j = 0, ..., k, satisfy the hypotheses of the theorem. The equation Pi(, p) = P ( , p ) for i = j follows from the properties of the blocks A 'i (p) = A 'j ( p ) and B 'i (p) = B 'j ( p ) [9]. The theorem is proved.
j

, )

. )

When the coefficients of the polynomial analytically depend on the parameters, Theorem 1 follows from the Weierstrass preparation theorem [10, 11]. Corollary 1. Suppose that m = M and the real parts of all roots of the polynomial P0() are negative (Re j < 0, j = 1, ..., k). Then, the polynomial P(, p) is stable in a neighborhood of the point p = p0 . Corollary 2. Suppose that the real part of a certain root of the polynomial P0() is positive (Re j > 0). Then, the polynomial P(, p) is unstable in a neighborhood of the point p = p0. Corollaries 1 and 2 distinguish the interior points of stability and instability domains of P(, p). Thus, the presence of infinite, zero, and pure imaginary roots of the polynomial P0() is of interest for the local analysis of stability, provided that the remaining roots have negative real parts. In this case, the point p = p0 belongs to the stability-domain boundary.
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It can be readily shown that P(, p) = (н1)M det(A(p) н B(p)). In [9], it was shown that there exists a transformation A(p) н B(p) = R(p)(A'(p) н B'(p))Qн1(p), where R(p) and Q(p) are nonsingular complex M т M matrices smoothly depending on p in the neighborhood of p = p0, such that the matrix A'(p) н B'(p) has a block-diagonal form A'(p) н B'(p) = Diag[ A '0 (p) н B '0 (p), ..., A 'k (p) н B 'k (p)]. The mj т mj blocks A 'j (p) н B 'j (p) correspond to the roots j; i.e.,
m det( A 'j (p0) н B 'j (p0)) = ( н j ) j , j = 1, ..., k, and

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2. TANGENT CONES TO THE STABILITY DOMAIN Denote by bl(j), j = 1, ..., M, the coefficients of the polynomial P( + j , p) that smoothly depend on p; i.e., P ( + j, p ) = b M ( j ) + ... + b 0 ( j )
M

= a M ( + j ) + ... + a1 ( + j ) + a0 .
M

(2.1)

Differentiating (2.1) l times with respect to and setting = 0, we obtain 1 -b l ( j ) = -- P l! -------l
l Mнl

=
=
j

C

l M н taM н t

Mнlнt j

, (2.2)

t=0

k! l C k = -------------------- . l! ( k н l )! For the infinite root 0 = , define bl ( 0 ) = a
Mнl

,

l = 0, ..., M .

(2.3)

est, provided that the real parts of other roots are negative. Let us enumerate the roots j so that 1, ..., 2s are distinct nonzero pure imaginary roots and, additionally, the roots j + s = s , mj + s = mj , j = 1, ..., s, are followed by the zero root 2s + 1 = 0 if it exists. For the remaining roots, we have Re j < 0, j = 2s + 2, ..., k. Let a curve p = p(), 0, emanate from the point p(0) = p0 in the direction of e = dp/d| = 0. The set of direction vectors e of the curves p = p() that lie in the stability domain for > 0 is called a tangent cone to the stability domain at the point p0 [5]. The tangent cone is the first-order approximation to the stability domain in the neighborhood of the point considered. Thus, the local analysis of the stability domain reduces to the determination of the stability conditions for the polynomial P(, p) along the above curve. Theorem 2. In order that the curve p = p(), p(0) = p0, with the direction e = dp/d| = 0 lies in the stability domain for > 0, it is necessary that ( f l ( j ), e ) = ( g l ( j ), e ) = 0, (f (g
mj н 2 mj н 2

When j , we have bl(j) for all l. Since j is a root of the polynomial P0()of multiplicity mj , then b0(j) = ... = b m j н 1 (j) = 0 and b m j (j) 0 for p = p0 . Introduce the differential operator = --------, ..., -------- , p1 p n where the derivatives are taken for p = p0, and denote by (a, b) = a1b1 + ... + anbn the scalar product in n (when a n, we set (a, b) = (Re a, b) + i(Im a, b)). With reg ard to (2.2) and (2.3), the expression for bl(j) is represented as
Mнl

l = 0, ..., m j н 3, (2.6) ( j ), e ) 0

( j ), e ) 0, (f
mj н 1

( j ), e ) = 0,

for infinite, zero, and pure imaginary roots j , j = 0, 1, ..., s, 2s + 1. Theorem 3. If the system of vectors V0 V1 ... Vs V where V j = { f l ( j ), g t ( j ) ; l = 0, ..., m j н 1, t = 0, ..., m j н 2 } , j = 1, ..., s , (2.8)
2s + 1

,

(2.7)

bl ( j ) =

C

l Mнt

Mнlнt j

a

Mнt

,

j 0;

t=0

(2.4)

bl ( 0 ) = a

Mнl

.

V r = { f l ( r ) ; l = 0, ..., m r н 1 } ,

r = 0, 2 s + 1,

Define the n-dimensional real vectors fl(j) and gl(j) corresponding to the roots j as f l ( j ) + i gl ( j ) = bl ( j ) , f
mj н 2

l = 0, ..., m j н 3,
mj н 2

( j) + i g
mj н 1

mj н 2

( j ) = b
0

( j ) / bm j ( j ) ,
0

f

( j ) = Re [ ( b m j ( j ) b
0 mj + 1

mj н 1 0

( j)
2

(2.5)

нb
0

( j )b

mj н 2

( j ) ) / ( bm j ( j ) ) ] ,

is linearly independent, then the tangent cone to the stability domain at the point p = p0 consists of vectors e satisfying conditions (2.6) for infinite, zero, and pure imaginary roots j , j = 0, 1, ..., s, 2s + 1. Theorem 4. Theorems 2 and 3 remain valid under the replacement of the stability condition Re < 0 by the condition Re < н 0 for all roots ; the latter condition implies the boundedness of the decrement (the degree of stability). In this case, the roots j , j = 0, ..., k, of the polynomial P0() must be enumerated so that 0 = ; Re l = н 0, Im l 0, l = 1, ..., 2s (r + s = r , r = 1, ..., s); 2s + 1 = н 0 and Re t < -0, t = 2s + 2, ..., k. When the coefficient in (2.6) falls outside the domain of definition, the corresponding quantity is assumed to be zero (for example, when mj = 1, we set f m j н 2 (j) = fн1(j) = 0). When j = 0, 2s + 1, we have
Vol. 39 No. 2 2000

where b l (j) is the value of bl(j) for p = p0 and i is the imaginary unit. Note that fl(j) = fl(t) and gl(j) = - gl(t) for j = t . As we noted above, the values of the vector of parameters p = p0 for which the polynomial p0() has infinite, zero, and (or) pure imaginary roots are of inter-

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2

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Fig. 1, this half-plane is hatched by horizontal lines. However, the stability domain of P(, p) is given by p2 н p1 > 0, p1p2 > 0 (hatched by vertical lines in Fig. 1). The tangent cone to the stability domain at the point p0 = 0 is determined by the inequalities {e = (e1, e2): e2 н e1 0, e1 0} {e: e2 н e1 0, e2 0}.
0 p

1

Fig. 1. Example of the stability domain in the degenerate case.

p p
3

1

Remark 2. In order to satisfy the condition of linear independence in Theorem 3, it is necessary that the number of parameters n be no less than the number of vectors in (2.7), N = m0 + 2(m1 + ... + ms) + m2s + 1 н s. In the case of general position, when considering arbitrary n-parameter families of polynomials P(, p), the appearance, for a certain value of the vector of parameters p = p0 , of the polynomial P0() = P(, p) of a given type (the type is determined by the multiplicities of the roots 0, 1, ..., s, 2s + 1) is possible only for n N [5, 6]. Hence, when considering polynomial families of the general form, the condition of linear independence of the system of vectors in Theorem 3 is fulfilled in the case of general position. Remark 3. The problem of limit stability, i.e., the stability of the polynomial P(, ) = P0() + P1() for small values of the parameter > 0, is close to the problem of local analysis of the stability domain. The limit stability can be achieved only in the case when the multiplicities of the roots of P0() lying on the imaginary axis are no greater than two [12, 13]. The situation is different in the multiparameter case: a polynomial P0() with imaginary roots of arbitrary multiplicity may correspond to a point on the boundary of the stability domain. In this case, if there exists a zero or infinite root of multiplicity greater than two or an imaginary nonzero root with multiplicity greater than unity, the stability domain approaches the point p = p0 in a narrow tongue; i.e., a tangent cone is degenerate (there is an equality among conditions (2.6)). For example, the stability domain of the polynomial P(, p) = 3 н p32 н p2 н p1 is given by p1 + p2p3 > 0, pj < 0, j = 1, 2, 3 (Fig. 2), while its boundary at the point p0 = 0 corresponding to a three-fold zero root has a singularity of the "break-of-edge" type [4, 8]. Thus, the presence of several parameters results in new effects. This is associated with the fact that the analysis of the cases of general position is the most informative and important [4]. Therefore, when there are multiple roots, one should carry out the stability analysis in the space of parameters of sufficiently large dimension. 3. PROOF OF THEOREMS 2н 4 The idea of the proof consists in the factorization of the polynomial P(, p) into multipliers Pj(, p) using Theorem 1. After that, for each multiplier, we find the conditions of stability along the curve p = p(); then, we rewrite these conditions in terms of the vectors fl(j)
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0

p

2

S

Fig. 2. Stability domain in the neighborhood of the point with a threefold zero root ("break of edge").

bl(j) , and, hence, conditions (2.6) involving the vectors gl(j) are fulfilled automatically. Theorems 2 and 3 allow one to determine, in a first approximation, the stability domain in the neighborhood of the point p = p0 from the information available at this point (the first derivatives of the coefficients of the polynomial P(, p), the parameters calculated for p = p0, as well as the values of the coefficients and the roots of the polynomial P0()). Theorem 4 provides similar information concerning the domain in which the decrement = - Re is greater than a given value 0 . Remark 1. In Theorem 3, the condition of linear independence of vectors is essential. For example, the necessary conditions of Theorem 2 for the family of polynomials P(, p) = 2 + (p2 н p1) + p1p2, p = (p1, p2), in a neighborhood of the point p0 = 0 are rewritten as (f0(1), e) 0, (f1(1), e) 0, where 1 = 0, m1 = 2, and the vectors f0(1) = (0, 0) and f1(1) = (н1, 1) are linearly dependent. These conditions single out the half-plane p2 н p1 0 on the plane of parameters; in

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and gl(j). These steps of the proof are supported by the following three lemmas. Lemma 1. Suppose that the curve p = p(), p(0) = p0, with the direction e lies, for > 0, in the stability domain of the polynomial Pj(, p) (1.4), j {0, 1, ..., s, 2s + 1}. Then, ( 0 ( j ), e ) = ... = ( ( Re
mj н 2 mj н 3

t = 0, 1, ..., s, 2s + 1, polynomials (3.5) and m0 + 2(m1 + ... + ms) cients of polynomials
1

and compare the expressions for (1.4). We obtain a system of m' = + m2s + 1 equations for the coeffiin p, a = (a1, ..., as), and ╡:

B jl ( p, a, ╡ ) = Re ( l ( j ) ( p ) н jl ( a, ╡ ) ) = 0, j = 0, 1, ..., s, 2 s + 1,
2

( j ), e ) = 0, ( j), e ) = 0, (3.1)

l = 0, ..., m j н 1; t = 0, ..., m t н 1.

( j), e ) 0, ( Im ( Re
mj н 1

mj н 2

B rt ( p, a, ╡ ) = Im ( t ( r ) ( p ) н rt ( a, ╡ ) ) = 0, r = 1, ..., s ,

(3.6)

( j ), e ) 0.
m

Proof. By the substitution ╡ = н j (when j = 0, by the substitution ╡ = 1/ and multiplication by ╡ 0 ), we reduce the polynomial Pj(, p) to the form ╡ j+
m mj н 1

In view of (1.4), (3.5), and (3.6), the differentials of 1 2 the functions B jl and B rt at the point p = p0, a = 0, ╡ = 0 are expressed as dB jl = ( Re l ( j ), d p )
1

( j )╡

mj н 1

+ ... + 0 ( j ) .

(3.2) н (
2

When j {0, 1, ..., s, 2s + 1}, the stability of polynomial (3.2) is equivalent to the stability of Pj(, p). Then, conditions (3.1) follow from the expression for the tangent cone to the stability domain of polynomial (3.2) that was obtained in [14]. The lemma is proved. Lemma 2. Let the system of vectors V '0 V '1 ... V 's V '2 where V 'j = { Re l ( j ), Im t ( j ) ; l = 0, ..., m j н 1, t = 0, ..., m j н 2 } , j = 1, ..., s , (3.4)
s+1

j (m j н 1)l b1

+

j (m j н 2)l b2

) d ╡ = 0,
(m j н 1)t

(3.7)

dB rt = ( Im t ( r ), d p ) н

da r = 0 ,

,

(3.3)

V 'r = { Re l ( r ) ; l = 0, ..., m r н 1 } , r = 0, 2 s + 1, be linearly independent. Then, for any vector e that satisfies conditions (3.1) for j = 0, 1, ..., s, 2s + 1, there exists a curve p = p(), p(0) = p0, dp/d = e, such that the polynomials P0(, p()), P1(, p()), ..., Ps(, p()), P2s + 1(, p()) are stable for > 0. Proof. Consider the polynomials P = (1 + ╡ )
0 2 0 m0 н 2

( 1 + ( b1 ╡ + ╡ )
0 2 2 2

where jl is the Kronecker delta. Using the linear independence of vectors (3.3) and a specific form of the terms containing dat in (3.7), we can show that the Jacobian matrix of system (3.6) has the maximum rank m' at the point p = p0, a = 0, ╡ = 0. Hence, equations (3.6) in a neighborhood of the point p = p0, a = 0, ╡ = 0 determine a smooth surface (of codimension r) whose tangent plane is given by equations (3.7). Consider an arbitrary tangent vector of the form dp = ed, da = dd, d╡ = d that satisfies (3.7) and a curve p = p(), a = a(), ╡ = ╡() in the space (p, a, ╡) that is emanated in the direction of the above vector along the surface (3.6). By the construction, the curve p = p() lies in the stability domain of the polynomials Pj(, p) (3.5) for 0 > > ', where (0, ') is a segment of positiveness of the function ╡ = ╡(). Note that, if we make the substitutions dp = ed, da = dd, and d╡ = d j and take into account the arbitrariness of d, b 1 0, and b 2 0, we obtain that equations (3.7) are equivalent to conditions (3.1). Thus, we constructed the curve p = p(), > 0, that lies in the stability domain of the polynomials Pj(, p) and has an arbitrary direction e that satisfies conditions (3.1) for j = 0, 1, ..., s, 2s + 1. The lemma is proved. Lemma 3. Let P(, p) = Q(, p)R(, p), where Q(, p) and R(, p) are polynomials of degree q and r (M = q + r) with complex coefficients that smoothly depend on p:
j

+ ( b2 ╡ + ╡ ) ) ,
j 2 ~ P = ( + ╡ ) mj н 2

2 2~ 2 j j т (~ + ( ia j + b 1 ╡ + ╡ ) + b 2 ╡ + ╡ ) ,

~ = н j, P
2 2s + 1

j = 1, ..., s ,
2m
2s + 1

(3.5)

= ( + ╡ )
2

н2

т ( + (b
t 2

2s + 1 1

╡ + ╡ ) + b

2s + 1 2

╡ + ╡ ).
2 t t 1 t 2

Q ( , p ) = c q ( н j ) + ... + c 0 ,
q

Polynomials (3.5) are stable for arbitrary real aj , b 1 0, b 0, and ╡ > 0. Let us fix the variables b 0, b 0,

R ( , p ) = d r ( н j ) + ... + d 0 ,
r

j 0.

For j = 0, we set Q(, p) = c0q + ... + cq, and R(, p) = d0r + ... + dr. Let d0(p0) 0 and Q(, p0) has a root j
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MAILYBAEV
mj н 1

of multiplicity mj (c0(p0) = ... = c

(p0) = 0, н (c
0 m
j+1

c m j (p0) 0). Then, the system of vectors Vj (2.8) calculated for P(, p) is a nonsingular linear combination of analogous vectors calculated for the polynomial Q(, p) (after substitution of cl for bl(j)) in (2.5)). In this case, conditions (2.6) written for P(, p) and Q(, p) are equivalent. Proof. Multiplying Q(, p) and R(, p) term by term and taking into account (2.1), we obtain the following relations from the equation P(, p) = Q(, p) R(, p): bl ( j ) = bl ( j ) =
0 t

t = [ cm j d 0 d
0

00 mj н 1 н t 0 mj н 2 н t

d 0 + cm j d 1 ) d
0 0 0

] / ( cm j d 0 ) ,
0

02

t=0 0 lнt

l

d

l н t ct

, (3.8)
0 t lнt

t=0

l

(d

ct + c d

),

where d = dt(p0) and c t = ct(p0). By the hypothesis, c0 = ... = c
0 0 mj н 1

0

= 0. Hence, bm j ( j ) = cm j d 0 ,
0 0 0

where f 'l (j) and g 'l (j) denote the vectors (2.5) calculated for the polynomial Q(, p). The assertions of the lemma immediately follow 0 0 from (3.10) in view of the relations c m j 0 and d 0 0. The lemma is proved. Now, we pass to the proof of Theorems 2н4. By Theorem 1, we represent the polynomial P(, p) as a product of the nonzero coefficient (p) and the polynomials Pj(, p) corresponding to the roots j , j = 0, ..., k. The stability of P(, p) is equivalent to the stability of all multipliers Pj(, p). Since the roots of the polynomials are continuous functions of parameters, the multipliers Pj(, p) corresponding to the roots with negative real parts j , j = 2s + 2, ..., k, are stable in a neighborhood of the point p = p0. The stability of the polynomial Pj + s(, p) is equivalent to the stability of Pj(, p), j = 1, ..., s, since Pj + s(, p) = P ( , p ) by Theorem 1. Hence, the stability of P(, p) is determined by the simultaneous stability of the polynomials Pj(, p), j = 0, 1, ..., s, 2s + 1. By Lemma 3, conditions (2.6) written for P(, p) and Q(, p) = Pj(, p), j {0, 1, ..., s, 2s + 1}, are 0 equivalent. Taking into account m j (j) = 1 and
j

b

0 mj + 1

( j) = c

0 0 mj + 1d0

+ cm j d 1 , l = 0, ..., m j н 1.

0

0

(3.9)

bl ( j ) =

t=0

l

d

0 lнt

ct ,

Substituting (3.9) into the expressions for the vectors fl(j) and gl(j) (2.5), we obtain f l ( j ) + i gl ( j ) = bl ( j ) =

t=0

l

d

0 lнt

c

t

= f

t=0

l

d

0 lнt

( f 't ( j ) + i g 't ( j ) ) ,
mj н 2

l = 0, ..., m j н 3,
mj н 2

mj н 2

( j) + i g

( j ) = b (d
0 mj н 2 н t

( j ) / bm j ( j )
0 0 m

m j + 1 (j) = 0, we can rewrite relations (2.6) for Pj(, p) in the form (3.1). By Lemma 1, conditions (3.1) are necessary for the stability of Pj(, p) along the curve p = p(), > 0. Theorem 2 is proved. By Lemma 3, the system of vectors Vj (2.8) calculated for P(, p) is obtained as a result of nonsingular linear combination of analogous vectors calculated for Q(, p) = Pj(, p), which constitute the system V 'j (3.4)
0

mj н 3

= c

mj н 2

+

/ d 0 ) ct / c
0

j

t=0

= f 'm
mj н 3

j

н2

( j ) + i g 'm

j

н2

( j) (3.10)

+

f

t=0

dmj н 2 н t ----------------- ( f 't ( j ) + i g 't ( j ) ) , 00 d 0 cm j ( j ) = ... = f 'm
j

0

mj н 1

н1

( j)

mj н 3

+Re

t ( f 't ( j ) + i g 't ( j ) ) ,

t=0

with regard to m j (j) = 1 and m j + 1 (j) = 0. Hence, the linear independence of vectors (2.7) implies the linear independence of vectors (3.3). Taking into account the equivalence between relations (2.6) and (3.1) and using Lemma 2, we obtain that, under the hypotheses of Theorem 3, for any direction e satisfying the necessary conditions of stability (2.6), there exists a curve p = p(), p(0) = p0, dp/d = e that lies in the stability domain for > 0. Theorem 3 is proved. The condition Re < н 0 for the decrement for P(, p) is equivalent to the stability condition of the polynomial P( н 0, p). In this case, the roots of P( н 0, p) are equal to 'j = j + 0 and Re 'j = Re j + 0 = 0 for j = 1, ..., 2s + 1. Hence, Theorems 2 and 3 can be applied to the analysis of a domain with bounded decrement, provided that P(, p) is replaced by P'(, p) =
0 0

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171
k2 T2 p + 1 x

P( н 0, p) and j, by 'j in this case, the quantities as the coefficients of the P( + 'j н 0, p) = P( +

. It remains to be noted that, bl(j) (2.1)н(2.3) are defined polynomial P'( + 'j , p) = j , p). Theorem 4 is proved.

u

1 p

k1 T0 p2 + T1 p + 1

4. EXAMPLES Let us illustrate the efficiency of the obtained results by several examples from automatic control theory. Consider a closed-loop system consisting of an integrating, oscillatory, and two aperiodic elements connected as shown in Fig. 3. The characteristic equation of such a system is given by [3] ( T 0 + T 1 + 1 ) ( T 2 + 1 ) ( T + 1 ) + kk 1 k 2 = 0 .
2

нk Tp + 1

Fig. 3. Block diagram of the automatic control system.

(a)

k 1.0 0.9

(b)

k 1.0 0.9

Under the assumption that the parameters of the aperiodic elements are prescribed and equal to T = T2 = 1, k = 2, k2 = 1, let us analyze the stability of the system as a function of the three parameters T0, T1, and k1 of the periodic element: P ( , p ) = T 0 + ( 2 T 0 + T 1 )
5 3 2 4

T0 0.4 0.3 0.3 0.2 0.1 T1 0

0.8 0.7

T0 0.4 0.3 0.3 0.2 0.1 0 T1

0.8 0.7

+ ( T 0 + 2T 1 + 1 ) + (T 1 + 2 ) + + 2k1. In the space of parameters p = (T0, T1, k1), consider the point p0 = (0, 0, 1), which corresponds to the system in the absence of the oscillatory element (the corresponding transfer function is reduced to unity). For p = p0, we obtain P 0 ( ) = + 2 + + 2.
3 2

Fig. 4. (a) Approximation of the stability domain and (b) the stability domain determined numerically.

numerically. The numerical results confirm that there is a trihedral angle (4.1) at the point p0 = (0, 0, 1). In many problems of automatic control, it is required to determine the values of the parameters for which the damping in the system exceeds a certain prescribed value, i.e., Re < н 0 for all roots . Denote by S(0) the set of parameters satisfying this condition. Consider a system consisting of a single one-capacitance object connected with a direct digital controller. This system is described by the following characteristic polynomial [2]: P ( , p ) = ( T 1 + 1 ) ( T ' + T k + 1 ) + k 1 k 2 .
2 2

The polynomial P0() has the simple roots 1 = i, 2 = н i, and 3 = н 2, (m1 = m2 = m3 = 1). In addition, there is the infinite root 0 = of multiplicity m0 = 2 (due to a decrease in the degree of the polynomial). The vectors f0(0), f1(0), and f0(1) calculated according to (2.2)н (2.5) are as follows: f 0 ( 0 ) = ( 1, 0, 0 ) , f 1 ( 0 ) = ( 0, 1, 0 ) , f 0 ( 1 ) = ( н 0.2, н 0.4, н 0.2 ) . The system of vectors {f0(0), f1(0), f0(1)} (2.7) is linearly independent. Hence, by Theorem 3, the tangent cone to the stability domain is determined by conditions (2.6) written for the roots 0 and 1: K = { e : ( f 0 ( 0 ), e ) 0, ( f 1 ( 0 ), e ) 0, ( f 0 ( 1 ), e ) 0 } . (4.1)

Let us fix the parameters T1 = 1, T '2 = 0.5, and k1 = 1 and analyze the domain S(1) in the space of parameters p = (Tk, k2). Consider the point p0 = (3/2, 0) at which the polynomial P0() = P(, p0) has the roots 1 = -1 and 2 = н 2 of multiplicities m1 = 2 and m2 = 1, respectively. Let us calculate the vectors f0(1) and f1(1) by formulas (2.2)н(2.5): f 0 ( 1 ) = ( 0, 2 ) , f 1 ( 1 ) = ( н 2, н 2 ) .

Inequalities (4.1) determine, in the space of parameters p = (T0, T1, k1), a trihedral angle (the intersection of three half-spaces) (Fig. 4a), which represents an approximation of the stability domain in a neighborhood of the point p0 = (0, 0, 1). For comparison, Fig. 4b shows the boundary of the stability domain calculated

The vectors f0(1) and f1(1) are linearly independent. Hence, by Theorem 4, the tangent cone to the domain S(1) at the point p0 = (3/2, 0) is given by K = { e : ( f 0 ( 1 ), e ) 0, ( f 1 ( 1 ), e ) 0 } .
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(4.2)

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2

MAILYBAEV

1/8 S(1) 0 1 5/4 f1(1)

f0(1) Tk

ACKNOWLEDGMENTS The work was supportes by the Russian Foundation for Basic Research, project no. 99-01-39 129. REFERENCES
1. Postnikov, M.M., Ustoichivye mnogochleny (Stable Polynomials), Moscow: Nauka, 1981. 2. Solodovnikov, V.V., Tekhnicheskaya kibernetika. Teoriya avtomaticheskogo regulirovaniya. Kn. 1. Matematicheskoe opisanie, analiz ustoichivosti i kachestva sistem avtomaticheskogo regulirovaniya (Technical Cybernetics: Automatic Control Theory, vol. 1, Mathematical Description, Analysis of Stability and Quality of Automatic Control Systems), Moscow: Mashinostroenie, 1967. 3. Fel'dbaum, A.A. and Butkovskii, A.G., Metody teorii avtomaticheskogo upravleniya (Methods of Automatic Control Theory), Moscow: Nauka, 1971. 4. Arnol'd, V.I., Dopolnitel'nye glavy teorii obyknovennykh differentsial'nykh uravnenii (Additional Chapters of Ordinary Differential Equations), Moscow: Nauka, 1978. 5. Levantovskii, L.V., On Singularities of the Boundary of the Stability Domain, Vestn. Mosk. Gos. Univ., Ser. 1: Math., 1980, no. 6. 6. Levantovskii, L.V., Singularities of the Boundary of the Stability Domain, Funkts. Anal. Ego Prilozhen., 1982, vol. 16, no. 1. 7. Seiranyan, A.P. and Mailybaev, A.A., Geometry of Singularities of Boundaries of Stability Domains, Preprint of Inst. Mech. Mosk. Gos. Univ., Moscow, 1997, no. 25. 8. Mailybaev, A.A. and Seyranian, A.P., On Singularities of a Boundary of the Stability Domain. SIAM J. Matrix Anal. Appl., 2000, vol. 21, no. 1. 9. Edelman, A., Elmroth, E., and K┼gstr╞m, B.A., Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils: P. I Versal Deformations, SIAM J. Matrix Anal. Appl., 1997, vol. 18, no. 3. 10. Shabat, B.V., Vvedenie v kompleksnyi analiz. Ch. II (Introduction to Complex Analysis), Moscow: Nauka, 1976. 11. Baumg─rtel, H., Analytic Perturbation Theory for Matrices and Operators, Berlin: Akademie, 1984. 12. Aizerman, M.A. and Gantmakher, F.R., Absolyutnaya ustoichivost' reguliruemykh sistem (Absolute Stability of Control Systems), Moscow: USSR Academy of Sciences, 1963. 13. Barabanov, A.T., Analytical Theory of Limit Stability, Izv. Ross. Akad. Nauk, Teor. Sist. Upr., 1998, no. 3. 14. Levantovskii, L.V., On the Boundary of the Set of Stable Matrices, Usp. Mat. Nauk, 1980, vol. 35, no. 2.

Fig. 5. Domain S(1) with bounded decrement (Re < н1) for the automatic control system.

Taking into account that p = p0 + e + (), we can rewrite expression (4.2) as 2 k 2 + ( p н p0 ) 0 , н 2 ( T k н 3/2 ) н 2 k 2 + ( p н p 0 ) 0 . (4.3)

In the case considered, the domain S(1) can be determined analytically (by applying the RouthнHurwitz condition to the polynomial P( н 1, p)): S ( 1 ) = { p = ( T k, k 2 ) : 0 < k 2 < н 2 T k + 5 T k н 3 } ,
2

which confirms the obtained approximation (4.3) (Fig. 5). Note that, for determining the approximations to the stability domain or to the domain S(0) in the neighborhood of the point p = p0, we needed only the information about the system at this point (the values of the roots, the coefficients of the polynomial, and the firstorder derivatives of this polynomial with respect to parameters, calculated for p = p0). CONCLUSION In this paper, an efficient method is developed for a local quantitative analysis of the stability domain of multiparameter families of polynomials of arbitrary degree. We studied the cases when the leading coefficients of the polynomial vanish and when the polynomial has imaginary roots of arbitrary multiplicity. Similar results are obtained for the domains with a bounded degree of stability. The approximations of the stability domain determined by the proposed method can be used in various problems of stabilization of mechanical and control systems, as well as in optimization problems with stability criterion.

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