A RELAXED SUFFICIENT CONDITION FOR ROBUST STABILITY OF AFFINE SYSTEMS

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1 Доклади на Българската академия на науките Comptes rendus de l Académe bulgare des Scences Tome 60 No SCIENCES ET INGENIERIE Théore des systèmes A RELAXED SUFFICIENT CONDITION FOR ROBUST STABILITY OF AFFINE SYSTEMS Svetoslav Savov Ivan Popchev (Submtted on June ) Abstract A new suffcent condton for robust stablty of lnear tme-nvarant systems s proposed It s based on the proved equvalence between stablty of an augmented system and stablty of the orgnal one Ths helps to solve the problem for the frst system by makng use of certan addtonal matrx varables The suggested approach s less conservatve than the exstng smlar ones whch represent partcular cases of the proposed here suffcent robust stablty condton Key words: uncertan systems robust stablty polynomal Lyapunov functons 1 Introducton Robustness analyss of lnear tme-nvarant systems subjected to parameter uncertanty belongng to some vector set attracts the attenton of the control theory communty for more than three decades In partcular the LMI-based approach has become very promsng snce t admts the usage of parameter-dependent Lyapunov functons As a result less conservatve condtons for robust stablty can be derved (see eg 1 2 ) An LMI approach for robust stablty analyss va Lyapunov functons assocated wth hgher-order state vector dervatves was recently proposed n 3 The objectve of ths paper s to generalze ths approach For the purpose the statespace descrpton of an augmented system s ntroduced It s proved that the stablty of ths system s equvalent to the stablty of the orgnal one The man advantage conssts n the possblty to use some addtonal arbtrary parameter-dependent matrx varables It s shown that the LMI problem can also be convexfed Thus the problem soluton conssts n a check for negatve defnteness of a fnte number of matrces The man result s a suffcent condton for robust stablty whch s proved to be less conservatve than the ones proposed by smlar approaches 2 Prelmnares The followng notatons are used throughout the paper R rq R r H r denote the sets of real r q r r and r r Hurwtz (negatve stable) matrces respectvely and R s the set r 1 real vectors An r q zero block and the r r dentty matrx are denoted 0 rq I r respectvely The set of egenvalues (spectrum) of a square matrx X s σ(x) The set of p n real matrces of rank n p s denoted Z(p n) Some basc for the present research results are gven below Ths work s supported by the Bulgaran Academy of Scences under grant /

2 Theorem 21 The followng statements are equvalent: (a) A H n ; (b) for any gven matrces B Z(p n) and X H p n there exst matrx à H p and nonsngular S R p such that ÃB = BA σ(ã) σ(a) σ(x) B = S B Z(p n) Matrx à s smlar to an upper block trangular matrx wth the off-dagonal block beng arbtrarly chosen; (c) there exsts a postve defnte matrx Π R p such that for any gven B Z(p n) one has L = A T P + P A < 0 P = B T ΠB Proof Let (a) holds and B Z(p n) R R np n and G R p n are arbtrary matrces There exst some p p nonsngular matrces S and T such that S B In In 0 np n = = B T = C C I p n (1) T ÃT 1 A R A RC R = à = 0 p nn (CR + G) CA + GC G Then ÃB = BA B Z(p n) Snce R and G are arbtrary matrces for G = CR X one gets σ(ã) σ(t ÃT 1 ) σ(a) σ(x) e à H p Ths proves (a) (b) Let (b) holds There exsts some postve defnte matrx Π R p such that L = à T Π + Πà < 0 For any B Z(p n) one has 0 > BT LB = A T B T ΠB + B T ΠBA = L whch proves (b) (c) If (c) holds then obvously A H n e (c) (a) Corollary 21 A H n f and only f for any gven matrces B Z(p n) and F Z(m p) there exsts a postve defnte matrx Π R m such that ÃT F T ΠF + F T ΠF à < 0 ÃB = BA Proof It follows from Theorem 21 The followng result s a drect consequence from Corollary 21 Corollary 22 Consder the dynamc systems (2) ẋ = Ax x R n ; (3) µ = õ µ = Bx Rp B Z(p n) ÃB = BA System (2) s asymptotcally stable f and only f for any gven matrces B n (3) and F Z(m p) there exsts a postve defnte matrx Π R m such that v(µ) = µ T F T ΠF µ s a vald Lyapunov functon (VLF) for system (3) e v(µ) > 0 v(µ) < 0 µ 0 If v(µ) s a VLF for system (3) then v(x) = x T B T F T ΠF Bx s VLF for system (2) and vce versa For any nteger f 1 denote m = pf and (4) F = µ = I pã à 2 à f 1 F µ = µ µ (1) µ (2) µ (f 1) = µ 1 µ 2 = ( µ1 µ 1 ) ( ) µ1 E1 0m Π µ (f) E = E Π = 2 Π 0 m 974 S Savov I Popchev

3 where F Z(m p) µ 1 R m µ 2 R 2m µ R m+p E 1 = I m 0 mp E 2 = 0 mp I m E Z(2m m + p) Π R 2m Π R m and µ () denotes the -th total tme dervatve of µ eg µ (1) µ Note that µ 2 = E µ and the total tme dervatve of the canddate for a Lyapunov functon for system (3) s computed as follows: (5) v(µ) = 2µ T F T ΠF µ = 2µ T 1 Π µ 1 = µ T 2 Πµ 2 = µ T E T ΠE µ Defne the followng matrces: bl dag à = Ãd R m U = Ãd I m R m2m and Ā = Āj R mm+p Ā j R p = 1 2 f j = 1 2 f + 1 A = j Ā j = I p j = p otherwse Theorem 22 Consder the followng (m + p) (m + p) symmetrc matrx (6) L = E T ΠE + Ā T K T + KĀ K R m+pm System (2) s asymptotcally stable f and only f for any gven matrces B n (3) and F n (4) there exst a postve defnte matrx Π R m and matrx K such that L n (6) s a negatve defnte matrx Proof Let system (2) be asymptotcally stable For any gven matrces B n (3) and F n (4) there exsts some postve defnte Π R m such that v(µ) = µ T F T ΠF µ s a VLF for system (3) n accordance wth Corollary 2 e à T P + P à = L < 0 P = F T ΠF Consder the matrces K1 K = K R 2mm 2 L = Π+U T KT + KU ÃT = d K1 T + K 1Ãd Π + ÃT d KT 2 K 1 L11 L Π + K 2 à d K1 T (K2 T + K = 12 2) L T 121 L R 2m 22 Let K 1 = Π + ÃT d KT 2 Then L 11 = ÃT d Π + Πà d ÃT d L 22Ãd and L 12 = 0 m For Π = bldag P one has L 11 = bldagl ÃT d L 22Ãd bldagl < 0 There always exsts some approprate matrx K 2 such that L 22 < 0 and L 11 < 0 e L < 0 Snce E Z(2m m + p) and UE = Ā one gets 0 > ET LE = E T ΠE + Ā T K T + KĀ = L K = E T K whch proves the necessty part Let L n (6) be a negatve matrx for some matrces B n (3) F n (4) Π > 0 and K Snce Ā µ = 0 Rm µ 2 = E µ and havng n mnd (5) one has 0 > µ T L µ = µ T (E T ΠE + Ā T K T + KĀ) µ = µt 2 Πµ 2 = v(µ) Ths means that v(µ) = µ T F T ΠF µ s a VLF for system (3) and n accordance wth Corollary 2 v(x) = x T B T F T ΠF Bx s a VLF for system (2) e t s asymptotcally stable Ths proves the suffcency part Comments Theorem 22 proves the equvalence between negatve defnteness of L n (6) and the exstence of matrces B n (3) F n (4) and Π(m) m = pf p n f 1 such that (7) L(m) = ÃT Π(m) + Π(m) à < 0; Π(m) = F T Π(m)F Π(m) > 0 à T B = BA Compt rend Acad bulg Sc 60 No

4 For the partcular case p = n m = nf B = I n (7) becomes (8) L(nf) = A T Π(nf) + Π(nf)A < 0; Π(nf) = F T Π(nf)F Π(nf) > 0 and the condton based on negatve defnteness of the correspondng matrx L n (6) was proposed for f = 2 n 4 and for f 2 n 3 Obvously Corollary 22 and Theorem 22 generalze these results By makng use of Theorem 21 and Corollary 21 one can easly show that nequalty (8) always mples nequalty (7) for arbtrary B Z(p n) p > n and F Z(m p) Therefore the condton proposed by Theorem 22 does not ntroduce any conservatsm for p > n wth respect to the partcular case p = n n the stablty analyss for system (2) 3 Uncertan dynamc systems Consder the set of constant but not exactly known vectors Ω = {α R N : α mn α α max α mn 0 α max = 1 2 N} and the followng uncertan dynamc systems: (9) ẋ = A(α)x x R n A(α) = A 0 + (10) µ = Ã(α)µ µ = B(α)x Rp A(α) RC(α) Ã(α) = C(α)A(α) + G(α)C(α) α A A 0 H n α Ω =1 R G(α) B(α) = I n C(α) where A = 0 1 N are gven constant matrces R R np n C(α) = C 0 + α C R p nn G(α) = G 0 + N α G R p n and R C G = 0 1 N are =1 =1 arbtrary matrces Matrces Ã(α) and B(α) n (10) satsfy the equalty Ã(α)B(α) = B(α)A(α) B(α) Z(p n) Lemma 31 Denote X = C A + G C For any matrx A there exst (a) nteger p > n and real matrces C and G such that X = 0 p nn ; (b) matrces C and G (possbly complex) such that X = 0 p nn for arbtrary p > n Proof (a) Let rank(a ) = r n r = max r = 1 2 N and p n r Consder the sngular value decomposton of matrx A = U Σ V T U R nr V T R r n and Σ R r s a postve dagonal matrx When A s nonsngular then U and V T are n n orthogonal matrces Let Σ V G = C Y Y = U 0 np n r R np n C = T C R C p n r n where C s an arbtrary matrx Then X = C (A Y C ) = 0 p nn Snce A s a real matrx U and V T are also real whch proves asserton (a) for p 2n p > n If p n for any C of rank n and G = C A (C T C ) 1 C T one has also X = 0 p nn (b) Let G = g G where G s some nonsngular matrx Then X = 0 p nn f and 1 only f G C A + g C = 0 p nn whch represents a lnear wth respect to the unknown matrx C equaton and can be put n a compact vector form as K vec(c ) = 0 R n(p n) vec(c ) R n(p n) 976 S Savov I Popchev

5 K = A T 1 G + g I n(p n) K R n(p n) where X Y and vec( ) denote the Kronecker product of matrces X Y and the usual operaton that stacks the columns of the matrx argument ( ) on top of each other respectvely Denote K 1 = A T 1 G It s well known that the spectrum of K 1 s defned as 1 σ(k 1 ) {λ s (A )λ k ( G ) s = 1 2 n k = 1 2 p n} For g = λ σ(k 1 ) matrx K s sngular and vec(c ) 0 s obtaned as an egenvector for ths matrx correspondng to ts zero egenvalue Dependng on the spectrum of A matrces C and G may be real or complex Snce p s arbtrary asserton (b) s proved From Lemma 31 t follows that matrx Ã(α) can always be represented as (111) Ã(α) = Ã0 + (112) α à + =1 j=1 <j α α j à j à = 0 A à = RC 0 np n C 0 A + C A 0 + G 0 C + G C 0 G 0 n 0 np n à j = C A j + C j A + G C j + G jc 0 p n A 0 RC 0 R C 0 A 0 + G 0 C 0 G 0 Remark 1 As far as asymptotcal stablty of system (10) s concerned matrx Ã(α) s Hurwtz only f Ã(0) = Ã0 s Hurwtz snce α = 0 s admssble Havng n mnd matrx à n (1) t follows that σ(ã0) σ(a 0 ) σ( C 0 R G 0 ) By no doubt there always exst matrces C 0 R and G 0 such that Ã0 s Hurwtz for any p > n 4 Robust stablty Consder the matrces n (4) where à = Ã(α) F = F (α) Π = Π(α) Π = Π(α) and matrx (6) wth L = L(α) Ā = Ā(α) Let Γ denotes the set of 2 N vertces of the set Ω The next theorem provdes a suffcent condton for robust stablty of system (9) Theorem 41 Consder (10) Let there exst ntegers p > n f 1 constant matrces R K affne parameter-dependent matrces C(α) G(α) satsfyng C A + G C = 0 = 1 2 N Ã(0) = Ã0 H p and a symmetrc matrx (12) Π(α) = Π 0 + α Π (α) + =1 j=1 <j α α j Π j (α) R m m = pf such that L(α) n (6) s a negatve defnte matrx for all α Γ Then the uncertan system (9) s asymptotcally stable and the polynomal n α VLF of degree d = 2(f +1) gven by (13) v(x α) = x T B T (α)f T (α)π(α)f (α)b(α)x = d d 1 + +d N =0 guarantees ts robust stablty α d 1 1 αd 2 2 αd N N v ( d 1 d 2 d N x) = x T P (α)x Compt rend Acad bulg Sc 60 No

6 Proof Let the above assumpton hold Havng n mnd (6) (11) and (12) one has 0 > f(y α) = y T L(α)y = l00 (y) + α l (y) + =1 y R m+p y 0 α Γ j=1 <j α α j l j (y) Functon f(y α) s multconvex n α and t s negatve on Ω f and only f t takes negatve values for all α Γ Therefore L(α) < 0 α Ω Recall vector µ n (4) Snce 0 > µ T L(α)µ = v(µ) = µ T ÃT (α) P (α) + P (α)ã(α)µ µ 0 α Ω P (α) = F T (α)π(α)f (α) t follows that P (α) s a nonsngular matrx for all α Ω For α = 0 Ω Ã(0) = Ã0 H p by assumpton and obvously P (0) > 0 By contnuty of the egenvalues of P (α) on Ω one has P (α) > 0 α Ω Havng n mnd that µ = B(α)x the above scalar nequalty can be rewrtten as follows: 0 > x T A T (α)p (α) + P (α)a(α)x = v(x α) v(x α) = x T B T (α)f T (α)πf (α)b(α)x > 0 for all nonzero vectors x and uncertan vectors α Ω e system (9) s asymptotcally stable and v(x α) n (13) s a polynomal n α VLF of degree 2(f + 1) guaranteeng ts robust stablty Conclusons A new suffcent robust stablty condton for a class of uncertan systems has been derved The proposed approach generalzes some known results consderng the same problem and s proved to be less conservatve than them REFERENCES 1 Blman P SIAM J Contr And Optmz Ches G AGarull A Tes AVcno Proc Conf on Decson and Contr Hawa USA Ebhara Y DPeaucelle D Arzeler T Hagwara Proc IEEE Conf on Decson and Contr Sevlle Span Olvera M R Skelton Perspectves n Robust Control Lecture Notes n Contr and Informaton Sc 268 Sprnger 2001 Insttute of Informaton Technologes Acad G Bonchev Str Bl Sofa Bulgara 978 S Savov I Popchev