Positive Definite Uncertain Homogeneous Matrix Polynomials: Analysis and Application

BULGARIA ACADEMY OF SCIECES CYBEREICS AD IFORMAIO ECHOLOGIES Volume 9 o 3 Sofia 009 Positive Definite Uncertain Homogeneous Matrix Polynomials: Analysis and Alication Svetoslav Savov Institute of Information echnologies 1113 Sofia E-mail: savovsg@yahoocom Abstract: he roblem of deriving necessary and sufficient conditions for ositive definiteness of a Homogeneous Matrix Polynomial (HMP) defined on the unit simlex is considered he obtained results clearly indicate the sueriority of the roosed here solution aroach over a classical theorem which can be viewed as a articular case of them he main results are alicable for the solution of the stability analysis roblem for a class of uncertain linear systems It is shown that the roved exact conditions generalize and imrove (in sense of conservatism reduction) two recent results aimed at solving the same roblem he resented aroach is illustrated by a comarative numerical examle Keywords: Uncertain HMP M-matrices Polya s theorem dynamic olytoic systems 1 Introduction Stability analysis of linear systems subected to structured real arametric uncertainty belonging to a comact vector set has been recognised as a key issue in the analysis of control systems Usually a quadratic in the state candidate for a LF is ostulated which is either fixed (quadratic stability) or arameter deendent (robust stability) Quadratic stability aroaches lead to conservative results esecially if the uncertainty is known to be constant On the other hand robust stability cannot be directly assessed using convex otimization In order to reduce the ga between quadratic and robust stability attemts for reducing the conservatism of LMI methods have been made for more than a decade Aimed at going beyond arameter-indeendent LFs LMI techniques were roosed to derive 14

quadratic in the state candidates for Lyaunov functions which are affine [6 7 14] quadratic [1] and recently olynomial [ 4 9 13 17] in the uncertain arameter Robust stability is verified through convex otimization roblems formulated in terms of arameterized LMIs which can be efficiently solved by olynomial-time algorithms An imortant result derived in [3] aved the way to necessary robust stability conditions via homogeneous matrix olynomials (HMPs) More accurate results have been obtained at exense of increased comutational effort he stability analysis of uncertain linear systems is based mainly on the owerful Lyaunov s second method It has been roved in [4] that this roblem reduces to the determination of non-conservative conditions for ositive definiteness of a uncertain HMP in this case herefore this becomes a roblem of outstanding imortance he obective of this research is to find comutable less conservative relaxed necessary and sufficient conditions for ositive definiteness of a given HMP in a case when the uncertain vector α belongs to the unit simlex It is actually motivated by several recently obtained results [6 7 13 14] aimed at solving the stability analysis roblem which exhibit some common shortcomings (sources of conservatism) he main contributions are: (i) a based on the theory of M-matrices new necessary and sufficient condition for ositive definiteness of a HMP of degree two is obtained (heorem 3) (ii) aimed at taking some additional advantage from the fact that α is a nonnegative vector some or all airwise inequalities between its entries are also considered which results in new alternative necessary and sufficient conditions (heorems 4 5 and 7) for ositive definiteness of a HMP and stability of a olytoic system and (iii) three generalizing conditions (Lemmas 1 and ) roved to be less conservative in comarison with the available ones are obtained Contrary to the usual ractice the roosed aroach takes into account the contribution of each term of the HMP to the overall ositive definiteness reflecting adequately the various relations between its coefficients Preliminaries revious results oen roblems he following notations will be used in the sequel is the set of ositive integers and x denotes a set of x ositive integers he i-th eigenvalue of a matrix X is λ i(x ) he notations A > ( ) 0 and a > 0 indicate that A is a ositive (semi-) definite matrix and a is a ositive vector A = [ a i ] R n and a = ( a i ) R denote real n n matrix and 1 vector with entries a i and a i resectively he sum of nonnegative scalars α is α Define also the vector sets i n xn { x R : x x= 1} and ω { α = ( αi) R : α = 1} Consider a HMP in α ω of an arbitrary integer degree k > 1 ( k 1)! with χ( k) = k!( 1)! 0! = 1 symmetric matrix coefficients given by 15

k k k 1 (1) Π ( α k) = α α α P R d k = k 1 k k k n 1 Since α = 1 d = 0 1 then the HMP (1) can be equivalently reresented as a HMP of degree d k with χ ( k d) symmetric matrix coefficients d Π ( α k) =Π ( α k d) = α Π ( α k) = () χ ( k d) k k k 1 α α α P 1 k k k α R l l n k = k d 1 l= 1 = = Π where ~ α l and Π l l = 1 χ( k d) denote the lexically ordered monomial k k 1 k α α α and matrix coefficient P k = k d resectively Let k d = 1 kk k 1 τ which makes ossible to write (3) Π ( α k) =Π ( α τ) = αα Π τ τ 1 τ i i χ( τ ) i i 1 where α = α α α τ = τ i = 1 χ( τ) denotes the i-th monomial of i degree τ and obviously ~ α l = α iα and Π l = Π i for some subscrits l i and Define the real uncertain scalar (4) f ( ατ x) = x Π ( α k) x= x Π ( α τ) x= ααc ( x) x x n where c ( x ) = x Π x and the uncertain vector i i τ τ i i χ( τ ) i i χ τ α = ( α ) ( ) 1 R α = α α α τ = τ i = 1 χ( τ) v i i 1 containing all monomials of degree τ hen (4) can be rewritten in a quadratic (with resect to α v ) comact matrix form as (5) f ( α τ x) = α Cx ( ) α C( x) = [ ci ( x)] R χ ( τ ) v v 1 i = ci ( x) = π c~ i i ( x) π i = 05 i where the symmetric matrix C(x) is said to be a Coefficient Matrix (CM) for the uncertain Homogeneous Scalar Polynomial (HSP) f ( α τ x) It is desired to derive conditions under which the HMP in (3) is ositive definite on the comact vector set ω ie Π ( α k) contains only ositive definite matrices or equivalently the strict scalar inequality (6) f ( α τ x) > 0 α ω x xn holds ext consider the following imortant result concerning the analysis of symmetric HMPs defined on ω obtained in [16] τ 16

heorem 1 Let a given HMP in (1) be ositive definite here exists some sufficiently large integer d * * such that for all d d all χ ( k d) matrix coefficients of the HMP in () are ositive definite his result generalizes the famous Polya s theorem [8] for the case of HMPs It is based on the derived in [1] lower bound for d which is roven to be tighter than all reviously obtained ones and reresenting an asymtotically exact condition it rovides a systematic way to decide whether a given HMP is ositive definite Unfortunately this result is a very conservative one with resect to sufficiency due to the following reasons It follows that the above stated roblem has a solution if and only if for some d all coefficients c ( x ) = x P x of the HSP f ( α τ x) are ositive for all x xn It is clear that the scalar inequality in (6) may hold even if some coefficients are not ositive definite his aer is devoted to the roblem of deriving relaxed necessary and sufficient condition for the validity of (6) and is intended to imrove and generalize heorem 1 3 Relaxed analysis for HMPS 31 Case k = In this secial case χ() = 05 ( 1) χ(1) = ; the HMP in (1) and the HSP in (5) can be reresented resectively for d = 0 as follows: (7) Π ( α) =Π ( α τ) = αα Π i = 1 i i i i (8) f ( α 1 x) = α Cx ( ) α C( x) = [ c i ( x)] R Let L denotes the set of real matrices with nonositive off-diagonal entries he set of M-matrices consists of all matrices M L which are ositive stable and is denoted M he next theorem resents some of the numerous roerties of the matrix set M heorem [10 11] A matrix M M if and only if the following equivalent statements hold: (s1) M has an eigenvectorα ω the corresonding to it 1 eigenvalue λ is real and such that 0 < λ Reλ i ( M ) i = 1 ; (s) M exists and its entries are nonnegative (nonnegative matrix); (s3) there exists a vector β > 0 such that M β > 0; (s4) there exists a ositive diagonal matrix D such that M D DM> 0 Positive definiteness of the CM C(x) in (5) is only a sufficient condition for robust stability he next result states that under some assumtion it becomes a necessary one as well heorem 3 Suose that MC(x) L x M M he following statements are equivalent: (i) the HMP in (7) is ositive definite; (ii) for any x there exists i 17

a vector β (x) > 0 such that C(x) β (x) > 0; (iii) C(x) is a ositive definite CM for all x P r o o f Let (i) holds ie Π ( α ) is ositive definite on ω In accordance with the assumtion that MC(x) L x and heorem (s1) there exists some vector α(x) ω such that MC(x) α (x) = λ( x) α( x) x It follows that C(x) α (x) = λ(x) where 1 1 α (x) M (x) MC(x) M x 1 M (x) α and α ( x) C(x) α (x) = λ(x) 1 α ( x) M α (x) > 0 M is a non-negative matrix (heorem (s)) he scalar α is ositive then λ(x) is also ositive by necessity and or equivalently MC( x) β ( x) > 0 x for some β (x) > 0 1 (heorem (s3)) and (ii) follows since M [ MC( x) β ( x)] = C( x) β ( x) > 0 If (ii) holds ie CxM ( ) M β( x) = CxM ( ) β( x) > 0 β ( x ) > 0 x then CxM ( ) M x since CxM ( ) L x For any diagonal matrix D > 0 1 matrix K ( x) = D C( x) M D M In accordance with heorem (s4) there exist some diagonal matrices D ( x) > 0 and D > 0 such that 1 1 D ( xk ) ( xdx ) ( ) DxKxD ( ) ( ) ( x) > 0 x ie 1 1 1 z{[ D ( xd ) CxD ( ) ( x)][ DxM ( ) DDx ( )] 1 1 1 [ DxDMDx ( ) ( )][ D ( xcxd ) ( ) D ( x)]} z> 0 z x 1 1 1 and M D DM > 0 All eigenvalues of D ( x) C( x) D D ( x) are real and let z = z(x) be the eigenvector corresonding to the minimal one λ (x) he above scalar inequality becomes: λ( xz ) ( x)[ Dx ( )( M D DMDx ) ( )] zx ( ) > 0 x λ( x) = λ [ C( x) D ( x)] > 0 x C( x) > 0 x min 1 where D ( x) = [ DD ( x)] is a ositive diagonal matrix Finally (iii) always imlies (i) his result shows also that under some assumtions statements (ii) and (iii) are equivalent necessary and sufficient conditions for ositive definiteness of a HMP of degree two From the theory of ositive matrices it is known that (i) and (ii) are valid only if C(x) L x Due to heorem 3 this assumtion is not necessarily required any more in order to have statements (ii) and (iii) alicable for the analysis of ositive definiteness Although a M-matrix is used for their roof it is not actually resent in them Let α (s) denotes a vector with s arbitrarily selected entries fromα If v (s) is the set of s 1 vectors with entries reresenting an arbitrary nondescending sequence then all ossible systems of s!05s(s 1) airwise inequalities α α α α α( s) i are described by the set of ordered vectors i i 18

α (s) v (s) = 1 s! For fixed the number of all ossible distinct and comatible monomial inequalities α α α α i u v i uv is given by µ ( s) = 05[ s( s 1) s i= 0 i u v ( s i)( s i 1) s 1 s 1 k = i= k ( s i 1)( s i)] For any α (s) the µ (s) scalar inequalities imly µ (s) matrix inequalities ( α iα α uα v ) X iuv 0 where X iuv = X uvi 0 are arbitrary matrices Consider the associated with α (s) v (s) HMP ~ ~ (9) Π ( α ) = α iα ( µ iuv X iuv ) = α iα Π i i = 1 i u v= 1 u v uv i i = 1 i = 1 s! where µ iuv = 1 if α iα α uα v 0 µ iuv = 1 otherwise and µ iuv = 0 if the sign of the monomial difference is indefinite due to s < Consider the HMP (10) Π ( α ) =Π ( α ) Π ~ ( α ) = α α Π Π = Π Π i i i i i i = 1 i = 1 s! he next theorem rovides an alternative necessary and sufficient robust stability condition heorem 4 he HMP in (7) is ositive definite on ω if and only if for any s there exist s! µ (s) arameter matrices in (9) such that all s! HMPs in (10) are ositive definite on ω P r o o f Let Π ( α ) > 0 α ω In accordance with heorem 1 there exists some ositive integer d such that all coefficients of the HMP d α Π( α) = Π( α d ) of degree d are ositive definite hen the following imlication holds: d Π ( α ) > 0 α α Π ( α ) =Π ( α d ) Π ( α d ) > 0; Π ( α d ) > 0 α For any s and there always exist some aroriate µ (s) ositive semidefinite matrices in (9) such that all coefficients of the HMP α Π ( α) d are ositive definite matrices which guarantees that Π ( α ) > 0 α Let the converse be true ie Π ( α ) > 0 α and consider an arbitrary vector α For any s there exists some vector α (s) such that the s common entries of α and α (s) reresent one and the same nondescending sequence Having in mind (3) and (4) one has 19

Π ( α ) 0 0 <Π ( α ) Π( α ) for this articular α but since it has been arbitrarily chosen it follows that Π ( α ) >0 α and inequality () holds In other words the function f ( α 1 x) is a ositive if and only if there exist s! µ (s) matrices in (8) such that f( α1 x) x [ Π ( α)] x= x Π ( α) x= α [ C( x) C ( x)] α = α C ( x) α > 0 (11) α x ~ where C ( x ) and C (x) = 1 s! are the CMs for the HMPs in (9) and (10) resectively If s = 1 then v(1) Ø µ ( 1) = 0 and (11) reduces to the trivial condition f ( α 1 x) > 0 Consider an arbitrary matrix coefficient defined in (10) and denote Π =Π Π Π ; Π = X 0 i i i i i iuv µ = 1 iuv Π = X 0 i = 1 s! i iuv µ = 1 iuv Remark 1 For any α (s) one has µ iuv = µ uvi his means that all sums Π i i are comosed of distinct matrices (the same refers to all Π i ) since if µ iuv = 1 then X iuv articiates only in the sum Π i and it aears only once more time but now as X uvi = X iuv in the sum Π uv Resective conclusions are made when µ iuv = 1 heorems 4 and 5 give rise to the following new robust stability conditions Lemma 1 For an integer s and s! µ (s) arameter matrices in (8) the following statements are distinct sufficient conditions for validity of the inequality in (11): (i) there exist s! M-matrices M and ositive vectors β = ( β ) R such that MC( x) L x; C( x) β > 0 x = 1 s! (1) β ( π Π ) > 0 i= 1 = 1 s!; = 1 i i i c (ii) there exist s!05(1) scalars i such that (13) c ( x) c x i ; C = [ c ] > 0 = 1 s! i i i P r o o f he roof of the first statement follows easily from heorem 3 (ii) and heorem 4 If (13) holds then inequality (5) is valid since 0 < α C α α C( x) α α x 0

As the dimension of the selected vector α (s) increases one obtains more and more relaxed conditions and the maximal effect is achieved when the whole vector α is selected ie for s = he next lemma states a condition which eliminates the awkward deendence of all CMs in (5) on vector x Lemma Let α m denotes the minimal entry of the -th ordered vector α () = 1! Consider the! µ ( ) arameter matrices in (9) chosen for any as follows: (14) Π = Π Π if Π Π 0 c ( x) = c = 0 i< (15) i i i i i i i ( Π =Π Π λ Π Π ) I 0 i i i min i i if λ ( Π Π ) = c < 0 c ( x) = c < 0 i< min i i i i i (16) Π ii = Π ii Π ii λ min ( Π ii Π ii ) I 0 c ( x) = λ ( Π Π ) = c i m ii min ii ii ii For this choice (11) holds if and only if C = [ c i ] > 0 where 05(1) 1 of the entries of C are defined in (14) (15) and (16) and ( cmm = λ min Π mm Π mm ) P r o o f Having in mind Remark 1 all equalities in (14) (15) and (16) are ossible since s = and µ iuv = µ 0 For any the entries of the CM uvi C (x) are equal to some scalars c i excet for the entry c ( x) = x ( Π Π ) x since µ = 1 uv ie Π = 0 aking mm mm mm mm mmuv c as above is obligatory since c ( ) = c mm x mm C 0 mm is an admissible case hen inequality (11) holds if and only if α α > α which is a necessary and sufficient condition for ositive definiteness of the considered HMP due to heorem 4 Finally from heorem 3 (iii) it follows that this condition is equivalent to C > 0 since C L 3 Positive definite HMP of arbitrary degree k A monomial ~ α l = α iα of an arbitrary even degree is said to be even if i = otherwise it called an odd one Consider the µ even monomials of variable degree r given by r r r r r 1 i f 1 i τ 1 α = α α α α α ; r = 01 τ 1 µ = χ( r ) r = 0 1

Any scalar inequality αi α i αi α α( s) imlies µ monomial inequalities of the form γ 1 γ (17) αα = α ααα = αα; f g h g h f i f f i h g where γ = τ r 1 = 1 3 5 τ 1and α α α are some monomials of f h g degree τ Any vector α (s) ν (s) defines µ s = 05s(s 1) airwise inequalities (17) Finally all s! vectors α (s) determine s! µ s systems of such monomial inequalities which corresond to all ossible sets of airwise inequalities involving the entries of vector α (s) For a given α (s) ν (s) any odd monomial in (17) serves as an uer bound for at least one even monomial ie (18) α α α ; f η 1 g h f h g η χ( τ) gh gh and any even monomial is a lower bound for at least one odd monomial For a given s consider the associated with some vector α ( s) ν( s) HMP of degree τ (19) χτ ( ) Π ( α τ) = ( α α α ) X R ; α α α = 1 s! f h g fgh n f g h f g h g h χ( τ ) with µ s arbitrary ositive semidefinite matrix coefficients X fgh Assume that of the even monomials in () are lexically ordered as follows: τ α = α i= 1 For τ > 1 any of the rest χ( τ ) even i i monomials can be reresented as a roduct of two distinct monomials of degree τ ie α = α α uv which makes ossible the definition of the HMP (0) t u v χ ( τ ) χτ ( ) Π ( α τ) = ( α α α ) X = 0 R α = 1 s! 0 t u v tuv0 n t= 1 u v u v χ( τ ) with χ( τ ) arbitrary symmetric matrix coefficients Consider the HMP (1) Π ( α τ ) = Π ( α τ ) Π ( α τ ) Π ( α τ ) = 0 = αα Π = 1 s! i i χ( τ ) i i where Π ( α τ ) is defined in (3) the associated with it Homogeneous Scalar Polynomial (HSP) in α and x x n h ( α τ x) = x Π ( α τ) x= αα c ( x) i i i i χ( τ ) () c ( x) = x Π x = 1 s! and the HSP in α i i

(3) h ( α τ) = αα c ; c I Π = 1 s! i i i i i i χ( τ ) where Πi denotes the resective symmetric matrix coefficient of the HMP in (1) corresonding to the monomial α iα and vector α ( s) ν( s) Consider the quadratic inα v matrix reresentation of the -th HSP in (10) (4) h ( α τ) = α C α C = C R = 1 s! v v χτ ( ) he entries of the CM C are the χ (τ ) scalar valued coefficients of the HSP in (3) heorem 5 Consider a given HMP (1) of arbitrary degree he following statements are equivalent: (a) there exist scalars d and s such that d k = τ 1 s s![ µ s χ( τ ) ] arameter matrices in (19) and (0) and s! χ(τ ) scalars c i defined in (3) such that all s! CMs in (4) are ositive definite; (b) Π ( α k) is a ositive definite HMP P r o o f Let assertion (a) holds Consider an arbitrary given vector α ω For any s 1 there exists some subscrit such that the s common entries of the vectors α and α ( s) ν( s) reresent one and the same sequence of nondescending scalars Having in mind (1) () (3) and the HMPs in (19) (0) and (1) the following matrix and scalar inequalities are valid: Π ( α k) =Π( α τ) Π ( α τ) h ( α τ x) h ( α τ) = α C α x x v v n Since all CMs are ositive definite by assumtion and vector α has been arbitrarily chosen it follows that (1) is a ositive definite HMP In the secial case when s = 1 then ν(1) is an emty set µ s = 0 and Π ( α k) =Π ( α τ) = =Π ( α τ) Π ( α τ) Positive definiteness of the single CM C 01 1 is a sufficient condition for ositive definiteness of the HMP his roves (a) (b) Let (b) be valid According to heorem 1 there exists some d such that all χ ( τ ) matrix coefficients Πi in (3) are ositive definite Let X = 0 = 1 s! in this case It will be shown that (a) holds for arbitrary tuv0 integer s Let s = 1 hen there always exist some aroriate scalars ci 1 in (3) such that C 1 is a ositive definite CM If s > 1 consider an arbitrary vector α ( s) ν( s) and the associated with it monomial inequality in (18) which imlies 1 3

f η gh α X α α Π ; Π = X f fgh g h gh gh fgh f η gh X 0 g h = 1 s! fgh ~ Let the coefficient matrices in (19) are chosen such that Π gh = Π gh for all subscrit airs (g h) in (18) his choice guarantees that θ () s (number of distinct o odd monomials in (17)) matrix coefficients in (1) become ~ Π gh = Π gh Π gh = 0 and θ () s (number of distinct even monomials in (17)) e matrix coefficients are Π ff > Π ff he rest of the χ (τ ) [ θ () s θ () s ] matrix o e coefficients are not affected and hence they remain ositive definite For a given s the integers θ () s and θ () s are fixed for all = 1 s! here always exist o e c some aroriate scalars i such that the resective -th CM is a ositive definite one he vector α (s) has been arbitrarily chosen which roves that (b) (a) for any s heorem 4 can be viewed as a generalization of heorem 1 which overcomes its most imortant shortcoming the unnecessarily hard requirement for ositive definiteness of all matrix coefficients in () At the same time the equivalence of the necessary and sufficient conditions (6) and (11) for ositive definiteness of a given HMP of arbitrary degree revealed by heorem 4 makes ossible to achieve flexibility by means of an aroriate choice for the arameter matrices in (19) and (0) ext it will be shown how these facts become very imortant for getting relaxed exact solution conditions when the discussed below alication roblem is faced 4 Alication: Exact stability conditions for uncertain systems Consider the uncertain linear system x& = A( α ) x A( α) = αi Ai R n α ω i= 1 where all matrices A are fixed and Hurwitz (negative stable) he stability analysis i roblem for this class of uncertain systems is: determine necessary and sufficient conditions under which the olytoe A { A( α) : α ω } contains only Hurwitz = matrices heorem 6 [15] he following statements are equivalent: (a) A is a Hurwitz olytoe; (b) there exists a HMP Π( α s) R in (1) of degree s s = b 1 b = 05n(n 1) such that n 4

(5) Π ( α k) = { A ( α) Π ( α s) Π ( α s) A( α)} > 0 α ω he significance of this result roved by means of the fundamental result obtained in [5] consists in the determination of a tighter than some reviously derived uer bound for the degree s of the HMP via which the robust stability of a given olytoic system can be analyzed Eg this bound was determined as s = b n in [4] for HMPs and s = n in [9] or s = b n 1 in [17] for general matrix olynomials heorem 6 rovides a necessary and sufficient condition for stability of a class of uncertain systems exressed in terms of the ositive definiteness of a HMP heorem 5 rooses a new relaxed asymtotically exact condition via which it can be tested he next theorem summarizes them heorem 7 he following statements are equivalent: (i) assertion (a) of heorem 5 holds for the HMP Π ( α k) in (5); (ii) A is a Hurwitz olytoe P r o o f It is entirely based on heorems 5 and 6 and due to self-evidence is omitted heorem 7 reresents a generalization of two recent results resented below heorem 8 [6 7] Let d = 0 and s = 1 A HMP Π( α1) of degree one assures robust stability of A ie validity of the matrix inequality in (5) if the single CMC 1 in ( 4) is ositive definite heorem 9 [13] he olytoe A is Hurwitz stable if and only if there exist a HMP Π ( α s) and some sufficiently large integer d such that all matrix d coefficients of the HMP α Π( α k) are ositive definite where Π( α k) denotes the HMP in (5) Although heorem 7 and heorem 9 rovide distinct but asymtotically equivalent conditions their sufficiency arts differ substantially For one and the same HMP Π ( α s) via which the robust stability of A is analyzed the stated conditions are based on heorems 5 and 1 resectively heorem 9 treats the roblem as the solution of a rather conservative set of more all less isolated LMIs where only the sign of the coefficient matrices is significant On the contrary heorem 7 takes into account their lower bounds and the various relations between them ut in a Comact Matrix form (CM) It is clear that if robust stability of A is concluded via heorem 9 the same refers to heorem 7 but not vice versa since all s! CMs may be ositive definite when some or even all matrix coefficients Π i< are not ositive definite heorem 1 is based on the assumtion that the i uncertainty vector is non negative while heorem 5 is aimed at taking some additional advantage from this fact aking into account the resence of monomial inequalities makes ossible to get some relaxations Eg if Π i< is a ositive semi-definite matrix then this fact can be reflected by the choice the resective arameter matrix in (7) which leads to i X i = Π i for Π = 0 and i 5

Π =Π Π Π ie this aroach reflects the contribution of each ositive ii ii i ii semi-definite term α α Π i = Πi min( Pi ) I i to the overall HMPs ositive definiteness For X i λ 0 an advantage is obtained even if Π i is a signindefinite matrix 5 umerical examle Examle Consider a olytoe A and a HMP Π ( α 1) = α Π described by the vertices: 06895 4137 478 A 1 = 98500 315 0197 131990 13987 15169 07056 856 35 A = 48496 1344 04 ; 3 56000 83 7168 07880 01970 0955 Π 1 = 01970 05910 03940 0955 03944 07880 03864 00560 0110 Π = 3 00560 0390 040 0110 040 03360 6 For d = 0 the HMP in (5) is Π ( α ) = 3 i = 1 i 3 i= 1 i i 0376 0343 04836 A = 080 05616 0184 19968 14976 1480 01404 0031 00156 Π = 0031 0064 0031 00156 0031 00780 α iα Πi and let λ i denotes the minimal eigenvalue of Πi It is desired to find whether the HMP Π ( α ) is ositive definite on the unit simlex he resective eigenvalues of the matrix coefficients are comuted as: λ11 = 013966 λ = 00017 λ 33 = 005448 λ1 = 0084 λ13 = 000143 λ3 = 0001998 ie for d = 0 the olytoe is not stable by heorem 9 Consider the case when s = 1 Since there does not exist a scalar c 11 λ1 such that the single CM C1 is ositive definite the considered HMP does not assure robust stability in accordance with heorem 8 Let s = and α() = ( α α ) For the single 3 arameter matrix in (19) chosen as = X = X = Π I one gets λ 31 3 3 X 31 3 3 3 λ3 331 = P 33 X = P X Π =Π = I Π 3 Π 3

Consider the two ossible cases: = 1 α α3 α3 αα 3 λmin( Π331) = 0 0864 = λ331 > λ33 = α 3 α α αα 3 λmin( Π ) = 0 08137 = λ > λ Let c 11 = c1 = 0 < λ1 he resective CMs C and C are: 1 λ11 0 λ13 λ11 0 λ13 C 1 = 0 λ λ3 > 0 ; C = 0 λ λ3 > 0 λ 13 λ3 λ331 λ 13 λ3 λ33 According to heorem 7 A is robustly stable via the considered HMP Π( α 1) with d = 0 Robust stability by heorem 9 has been concluded for d = his simle examle illustrates the advantages of heorem 7 over heorems 8 and 9 Acknowledgements his research work is suorted by the Bulgarian Academy of Sciences under Grant o 010077/007 References 1 B e r n s t e i n D W H a d d a d Robust Stability and Performance Analysis for State Sace System via Quadratic Lyaunov Functions SIAM Journal of Matrix Analysis and Al 405 005 o 09-8 B l i m a n P -A A Convex Aroach to Robust Stability for Linear Systems with Uncertain Scalar Parameters SIAM Journal of Control and Otimization 4 004 o 6 016-04 3 B l i m a n P -A An Existence Result for Polynomial Solutions of Parameter-Deendent LMIs Systems and Control Letters 51 004 o 3-4 165-169 4 C h e s i G A G a r u l l i A e s i A V i c i n o Polynomially Parameter-Deendent Lyaunov Functions for Robust Stability of Polytoic Systems: An LMI Aroach IEEE rans Automatic Control 50 005 o 3 365-379 5 F u l l e r A Conditions for a Matrix to Have Only Characteristic Roots with egative Real Parts Journal of Math Analysis and All 3 1968 71-98 6 G e r o m e l J R K o r o g u i Analysis and Synthesis of Robust Control Systems Using Linear Parameter Deendent Lyaunov Functions IEEE rans Automatic Control 51 006 o 1 1984-1988 7 G r m a n L D R o s i n o v a V V e s e l y A K o z a k o v a Robust Stability Conditions for Polytoic Systems Journal of Syst Science 36 005 o 15 961-973 8 Hardy G J Littlewood G Polya Inequalities Cambridge UK Cambridge University Press 195 9 Henrion D D Arzelier D Peaucelle J Lasserre On Parameter-Deendent Lyaunov Functions for Robust Stability of Linear Systems In: Proc 43rd IEEE Conf on Decision and Control Paradise Island Bahamas Vol 1 004 887-89 10 Horn R C Johnson oics in Matrix Analysis Cambridge University Press 1991 11 M i n c H onnegative Matrices John Wiley and Sons 1988 1 P o w e r s V B R e z n i c k A ew Bound for Polya s heorem with Alications to Polynomials Positive on Polyhedra Journal of Pure Al Algebra 164 001 o 1 1-9 7

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