Multidimensional Convex Directions for Hurwitz Stability Case

Copyright © IFAC Robust Control Design, Budapest, Hungary, 1997

MULTIDIMENSIONAL CONVEX DIRECTIONS FOR HURWITZ STABILITY CASE A. G. Zheludkov 1

Dept. of Applied Mathematics and Control Theory St.-Petersburg State University St.-Petersburg, 198904, RUSSIA

Abstract: Some properties of the set of Hurwitz polynomials related to the convexity are treated. The concepts of local and global multidimensional convex directions are introduced and investigated . The necessary and sufficient conditions for polynomial families to be a multidimensional convex directions are obtained. Some easely testable special classes of multidimensional convex directions are derived. The possibility of application of the concepts introduced to some robust analysis problems is shown. Keywords: Hurwitz polynomial, robust stability.

8 arg( q( iw» < Isin(2 arg q(iw» I 8w 2w

1. INTRODUCTION The concept of convex direction introduced by A. Rantzer (Rantzer, 1992) received a good deal of attention ((Hinrichsen and Kharitonov, 1994a), (Fu , 1993) etc.) recent years. It appeared to be useful to construct some effective methods of the robust stability analysis.

(1)

where i is the imaginary unit .

o This is the so called Rantzer 's inequality. The next concept to be treated is the concept of local convex direction for single polynomial that was introduced in (Hinrichsen and Kharitonov , 1994a) as follows.

Definition 1. Polynomial q( s) is said to be a convex direction if the stability of polynomials p(s) , p(s) + q(s) implies the stability of their convex hull p(s) + J-Lq(s) J.L E [0; 1] for any p(s) such that degp(s) > deg q(s).

Definition 2 . Polynomial q(s) is said to be a (local) convex direction for some stable polynomial p( s), deg p( s) > deg q( s) if all the roots Sj(J.L) , j = 1, ... ,1 of p(s) + J-Lq(s) , J.L ~ 0 on the punctured imaginary axis iRjO are simple and satisfy Re{sj(J.L)} > O.

Hereafter under the stability of a polynomial its Hurwitz stability is assumed and only real polynomial families are treated . The set of convex direction polynomials admits according to (Rantzer, 1992) the following characterization.

So the nonzero purely imaginary roots of p( s) + J-Lq( s) should be simple and cross the imaginary axis only from the left to the right for w increasing from zero to infinity.

Theorem 1. Polynomial q( s) is a convex direction if and only if it satisfies for any real w ~ 0 the following argument condition

Analitically (Hinrichsen and Kharitonov , 1994 a)

p( s) , q( s) should satisfy P'(iw) q'(iw)) R e ( - - - - - >0 p(iw) q(w)

The author would like to thank the INTAS program (project 93-1034) for the financial support of this work.

1

85

for any w > 0 such that Im

It allows to rewrite Rantzer 's condition as

q(iw) 0 R e--< p(iw)

q(iw) - 0 - , p(iw)

g' h - h' 9 5:. Igh I w

(3)

The concept of (local) convex direction for single polynomial allowed (Hinrichsen and Kharitonov , I994a) to suggest another characterization of convex direction in global case.

(Such a form of Rantzer 's condition was used e.g. in (Fu , 1993)). Hereafter the arguments of some functions are also dropped if it not leads to the ambibuity.

Theorem 2. Polynomial q(s) is (global) convex direction if and only if it is (local) convex direction for any stable p(s) such that deg p( s) > deg q( s).

Let 's first consider two special classes of the families (2) . Class PI -

the polynomial families of the form the linear span of some odd polynomials;

o

Q(a, s) = sk H(a , s)q(s) , where h(a , s) -

Following e.g. (Hinrichsen and Kharitonov , I994b) introduce also n+ I-dimensional vector space of all polynomials ofthe form aaSn+alsn - I+ ... +an and identify each polynomial of the degree n with it's coefficient vector aa , ... , an E Rn+1 providing Pn with the topology induced from Rn+l . Let 's also denote H n - the set of all stable polynomials of the degree n.

In this case the phase derivative of q(iw) equal for any real w to the phase derivative Q( a , iw) and their arguments satisfy arg q( iw) argQ(a , iw) + k2"'" (k - some integer) for all such that arg Q( a, iw) is defined.

Let 's consider the following problem. Given a polynomial family

Evidently this class may be characterized as the set of all families (2) such that for any j , k = 1, m

m

=L

akqk(s) ,

9j (w) _ hj (w) -

(2)

;=k

In the following particular case the conditions to find are especially simple.

Let 's also define d - the degree of Q(a , s) as a maximal degree of qk(S) , k = 1, ... , m.

Lemma 1. If ql (s) and q2( s) are the even and odd polynomials respectively then Q(a , s) is twodimensional convex direction if and only if two polynomials ql(S) + q2(S) , ql(S) - q2(S) are convex directions .

Definition 3. A polynomial family (2) is called a multidimensional convex direction or a convex direction subspace if every polinomial from this family is a convex direction.

Proof. By direct applying of Rantzer condition.

It means geometrically that the intersection of the linear manifold p( s) + Q( a , s) in Pn and the set of Hurwitz polynomials Hn is a convex set for any p( s) which degree is more than m .

o The example of the family satisfying the conditions of the lemma 1 is al(as 2 + b) + a2CS , where ab 5:. 0 - real constants .

Let 's introduce the notations

q(a , iw) = h(a ,w) + ig(a ,w) ,

Theorem 3. Every multidimensional convex direction belongs to one of the classes mentioned above.

m

h(a,w) =

L

akhk(w),

Proof. Let's suppose that the convex direction subspace of the form (2) does not belong to the class PI and prove that it belongs to the class P2 . Let 's fix some wand consider g , h , g' and h' as linear forms of the parameters a. Since the family does not belong to the class PI , the forms g, hare linearly independent almost for all w.

k=1 m

g(a ,w) =

9k (W) hk (W)

Class P2 - two-dimensional families Q(a , s) = aIql(s) + a2q2(s) .

where a = (al . . . a m ) E R m - vector of the arbitrary real parameters. The polynomials qk(S) are linearly independent .

where

= a

It means that Rantzer's conditions for q(s) and Q( a , s) are equivalent for all a E Rn such that argQ(a , iw), argq(iw) are defined. So Q(a , s) is convex direction subspace if and only if q(s) is convex direction.

2. MULTIDIMENSIONAL CONVEX DIRECTIONS . THE GLOBAL CASE.

Q(a , s)

is of

L akgk(w) k=1

are real polynomials. Hereafter they are denoted for brevity m

9 = Lak9k.

It follows from Rantzer 's inequality that

k=1

86

g'(o: ,w)h(o: ,w)::; 0 when g(o: ,w) = 0;

3. MULTIDIMENSIONAL CONVEX DIRECTIONS . LOCAL CASE .

-h'(o: ,w)g(o:,w) ::; 0 when h(o: , w) = 0

Consider again the family (2) and some given polynomial p(s) . Analogously to the global case the concept of multidimensional convex direction for single polynomial may be also introduced.

So the triples of the forms g , g' , hand g, hi , h are linearly dependent and since Q( 0: , s) does not belong to the class PI g~(w)

= 1Igk(W) + 12hk(w)

= 13gk(w) + 14hk(w) where k = 1, m , Ij , j = l,4 - real constants . h~(w)

Definition 4. A polynomial family (2) is called a local multidimensional convex direction for given polynomial p( s) if every polynomial from this family is a local convex direction for p( s).

(4)

Constants li in (4) correspond to almost all w so they may be considered as (rational) functions of w.

Let 's denote p(iw) = hp(w) + igp(w) and q(iw) = hk(W) + i9k(W) , k = 1, m where hp , hk, gp , gk real polynomials.

Let 's choose three polynomials qk(S) , k = l,3 such that their linear span does not belong to PI and denote (gl(W) g2(W) g3(W))* = rl(w) , (hl(W) h2(W) h3(W))* = r2(w) .

Let's also denote A(W) = g~hp - gph~ and for k = I,m

O'k(W)

According to (4) , the derivatives of the vector functions rI, r2 belong to the same plane as rI , r2 for any fixed w . It's easy to see that if two vector functions together with their derivatives belong to the same plane for any fixed w then this plane (its normal vector) does not depend on w. Denoting this normal vector as (J..Ll can obtain at last that

J..L2

J..L3) , one

Theorem 5. If polynomial family Q( 0: , s) belongs to the class PI , Q( 0: , s) = sk H( 0: , S)q( s) then it forms a convex direction subspace for p( s) for even and odd k if and only if the polynomials ±q(s) and ±sq(s) respectively are convex directions for polynomial p( s) .

o

o

It's not hard to derive from (1) the necessary and sufficient conditions for Q( 0: , s) to be twodimensional convex direction in general case considering it as a quadratic inequality with respect to g , h , g' , h' with constraints (4) .

For the class P2 one can obtain the statement that is partially analogous to the theorem 3 of the previous section.

0 for

gh 2: 0

Theorem 6. The linear family Q( 0: , s) that does not belong to the class PI is local multidimensional convex direction for the polynomial p( s) if and only if it belongs to the class P2 and the following conditions are satisfied

0 for

gh::; 0

1. The polynomial glh 2

Indeed , from (3) , (4)

+ (/1 -/4 - I)gh -/3g 2 2: 12h2 + (/1 - 14 + I)gh - 13g 2 2:

Analysing these inequalities one can obtain the following

2. [O'l(W) - A(W)]

+ 1 2: 0

(/1 -/4 - 1)2

when

14

(/1

when

14 - /1 - 1 2: 0

-/1

-

< 0 for

g2hl has no real roots . any real w.

Proof. Involving the notations of the previous section the conditions of the Theorem 2 may be rewritten as

Theorem 4. The family from P2 is multidimensional convex direction if and only if for any w 2: 0 functions 12 , -/3 are nonnegative and

+ 4/2 /3 ::; 0 -/4 + 1)2 + 412/3 ::; 0

gphk)(gmhp - gph m ) _ gkhm - gmhk (g:r,hp - gph:r,)(gkhp - gphk) gkhm - gmhk

Every qk(S) , k = 1, ... , m is assumed to be (local) convex direction for the polynomial p( s) k = l...m is convex direction polynomial for p(s) and the family (2) is considered for only nonzero O:k , k = 1, .. . , m. The direct applying of the analytic conditions from the first section gives the following

Hence any three polynomials from the linear span (2) are linearly dependent and the dimension of (2) does not exceed two. The proof was carried out for almost all w . Using the continuouty of oHn one can easily extend it to the finite number ofthe rest values of w .

12h2

= (g~hp -

g'h - h'g g~hp - gph~ g2 + h 2 < g; + h~

o

for such w that 87

(5)

Let's also consider the case when polynomials h1' h2 ' g1 , g2 satisfy

(6) and

g1

(7)

+ f(w)g2 = gp

where f is some (rational) function. Then the condition to find has the form

Let's fix w . Since the family (2) is symmetric with respect to the zero polynomial then condition (7) is redundant here.

(11)

One can write (5) as

(g'h - h'g)(g~ + h~) -(g~hp - gph~)(g2 + h 2) < 0

3.2 Convex directions for single polynomial in a week sense.

(8)

and (for such w that gmhp - gph m :j:. 0) (6) as

Definition 5. Polynomial q( s) is said to be a (local) convex direction for some stable polynomial p(s), degp(s) > degq(s) in a weak sense if the condition g'h - h'g < g~hp - gph~ g2 + h 2 g; + h;

m-I

am(gmhp - gph m ) = -

L ak(gihp - gph i )

(9)

k=1

Let 's define the following linear forms of a as m-I

ZI

=

L

is satisfied for such w that ghp - hgp

ak(gkhm - gmhk)

Introducing analogously the concept of multidimensional convex direction in a weak sense for single polynomial and repeating the above reasoning as for its "strong" version with the same notations one can obtain the following

k=1

m-I

Z2 =

L

ak[(gA,hp - gphA,)(gmhp - gph m ) -

k=1

Theorem 7. The family Q(a, s) that does not belong to the class P1 is multidimensional convex direction for the polynomial p( s) in weak sense if and only if [O"(w) - -X(w)] :s; 0 for any real w.

(g'mhp - gph'm)(gkhp - 9phk)] Substituting am(gmhp - gph m ) from (9) to (8) we obtain that (8), (9) are equivalent to

ZI(Z2 - -X(w)zd (gmhp - gph m )2

0

<

= O.

o

(10)

These conditions are considerably more simple then the conditions of the theorem 6.

Certainly the forms ZI , Z2 should be dependent for any fixed w in such a way that (10) was satisfied. So if o"j(w) = O"k(W) := O"(w) then z2(a,w) = O"(W)Zl(W) and (10) turns into zf[O"(w) - -X(w)]j (gmhp - gph m )2 < O. It leads to the conclusion that [O"(w) - -X(w)] should be negative for all w. To provide the strict inequality for nonzero a the linear form ZI ( a) must be of only one variable a , that is m-I = 1 and the polynomial gIh 2 - g2hI must not have real roots .

3.3 Properties of local multidimensional convex directions

The following two properties of local multidimensional convex direction Q( a , s) for the polynomial p(s) hold true both for weak and strong version . Property 1. For any q(s) E Q(a, s) the stability of polynomials p( s), p( s )+q( s) implies the stability

Renumerating the aj one can carry out analogously the proof of the sufficiency for the finite number of such w that g2hp - gph2 = O.

of their convex hull.

Let's introduce a function 6(a) as 6(ao) = sup{6 I p(s) + 6Q(ao , s) E Hn} - the structured distance between the polynomial p( s) and the set of unstable polynomials. The property 1 is equivalent to the following

o

3.1 Two particular cases.

Property 2. The function 6( a) is continuous in

Let the family (2) has the form a1H( _s2) + a2sG( _S2) , or, equivalently, q(a, iw) = aIh(w) + ia2g( w). If q1 (s) , q2 (s) are even and odd polynomials respectively then substituting a2 from (9) to (8) , we obtain the conditions to find as

any point where it is finite.

According to the theorem 7 there can exist the linear families Q( a , s) which have the properties 1,2 and do not belong to the classes P1 , P2 . It 's also not hard to construct the examples of the local convex direction subspaces for some given polynomial p( s).

!!. (9ph)' > 0 h hpg

88

unstable ones (otherwise c:«1 - t)a + tao) can not be decreasing anywhere on t E [0 ; 1) by the definition of the function c:( a)) .

4. EXAMPLE OF APPLICATION TO THE PROBLEMS OF ROBUST STABILIZATION. Let 's consider the following robustness maximizing problem . (The formulations of this kind of problems may be found e.g. in (Zhabko and Kharitonov , 1993)) .

Since l1(s) - /2(s) E Q(a , s) then we obtain that Q( a, s) is not a convex direction subspace i.e . contradiction with the conditions of the theorem.

o

Given a polynomial family

Corollary 1. In the conditions of the lemma the function c:( a) does not have local extremum points.

m

K(s , a , c: ) = Po(s)

+ c:P(s) + L akqk(s) ,

(12)

i=k

=

Using lemma 2 and corollary lone can construct the effective numerical methods to solve the mentioned problem.

where PO(S) ,qk(S) k 1, ... , m are given polynomials , degpo(s) = n , n > maxdegqk(s), polynomials qk(S) are linearly independent and pes) is some given polynomial family which maximal degree is less than n . a = (al .. . am) - some constant vector, c: is nonnegative constant. Let 's also suppose that P( s) forms a connected bounded set in Pn and contains zero polynomial as an interior point .

It 's not hard to construct the similar examples of application to the robust analysis problems for the convex direction for single polynomial concept . Some trivial applications of multidimensional convex directions to linear polynomial families stability analysis motivated by Edge Theorem (Bartlett et al. , 1988) are also clear .

The function c:(a) = sup{c:IK(s , a ,c:) E Hn U oHn} defined on the set A = {aIK(s , a, 0) E HnU oHn} is the so called robustness function.

5. REFERENCES

We suppose that the set A is bounded. To guarantee that according to (Hinrichsen and Kharitonov , 1994b) it's enough to assume that (n - maxdegqk(s)) > 2 or that Q(a , s) = L:7:k akqk(s) , a E R m does not contain polynomials with all their roots in the closed left-half plane. Then it's easy to see that the function c:(a) is bounded too. The problem is to find the vector the function c:( a) .

0'0

Bartlett , A.C. , C.V. Hollot and Huang Lin (1988). Root location of an entire polytope of polynomials: it suffices to check the edges. M athem. Control, Signals and Systems pp . 61-71 . Fu, Minyue (1993). Test of convex directions for robust stability. In: Proc. 32-nd CDC. San Antonio, Texas. pp . 502-507 . Hinrichsen, D. and V.L . Kharitonov (1994a) . On convex directions for stable polynomials. Technical Report 309. Bremen University, Institute for Dynamic Systems. Bremen , Germany. Hinrichsen , D. and V.L. Kharitonov (1994b). Stability of polynomials with conic uncertainty. Technical Report 303. Bremen University, Institute for Dynamic Systems. Bremen , Germany. Rantzer , A. (1992) . Stability conditions for polytopes of polynomials. IEEE Trans. AC 37, 79-89 . Zhabko, A.P. and V.L. Kharitonov (1993) . The

maximizing

It 's impossible to imagine some general analitic method to solve this problem but applying some numerical methods one faces a lot of new problems. One of them is whether the maximum that will be found is local or global. If qk(S), k = 1, ... , m form the basis of multidimensional convex direction then the problem to consider possesses some features which make it more attractive for numerical solution .

Lemma 2. Let Q(a , s) = L:7:k akqk(s ) be multidimensional convex direction, 0'0 - the global maximum point of c:(a) . Then for any point a E A the function c:«I-t)a +tao)) is nondecreasing on t E [0 ; 1] .

methods of linear algebra for control problems. St .-Petersburg University Press . St.-

Petersburg.

Proof. Let's suppose the opposite and consider the polynomial families K (s , a, c:( a) - 5)) and also K(s , 0'0, c:(a) - 5)) which are stable for any o ~ 8 ~ €(a) .

Then for sufficiently small 5 there exist such p( s) E P( s) that the convex hull of stable polynomials l1(s) = po(s ) + (c:(a) - 5)p(s) + Q(ii , s), /2(s ) = po(s)+(c(ii)-8)p(s)+Q(ao , s) containes 89