Dissipativity for dual linear differential inclusions through conjugate storage functions

Transcription

1 43r IEEE Conference on Decision an Control December 4-7, 24 Atlantis, Paraise Islan, Bahamas WeC24 Dissipativity for ual linear ifferential inclusions through conjugate storage functions Rafal Goebel, Anre R eel, ingshu Hu, Zongli Lin 2 Abstract ools from convex analysis are use to sho ho issipativity properties, expresse in terms of convex storage functions, translate hen passing from a linear ifferential inclusion (LDI) to its ual As special cases, it is shon that a convex, positive efinite function is a Lyapunov function for an LDI if an only if its convex conjugate is a Lyapunov function for the LDI s ual, an that passivity an finite L 2- gain are preserve hen passing from an LDI ith input an output to its ual Also establishe is the uality beteen stabilizability an etectability, incluing stabilizable an etectable issipativity, for ual LDIs Finally, ith examples e sho ho uality effectively oubles the number of tools available for assessing stability an performance of LDIs I INRODUCION Duality is a firmly establishe concept in linear systems theory For example, a matrix is Huritz if an only if its transpose is Huritz he pair (A, B) is stabilizable if an only if (B,A ) is etectable For the transfer function C(sI A) B +D an its ual B (si A ) C +D, the H norms are equivalent Similarly, one is positive real (respectively, strictly positive real) if an only if the other is positive real (respectively, strictly positive real) here are various ays to establish this uality One is to fin a positive efinite matrix P so that 2P establishes the first property an to sho that 2 P establishes the secon property For exponential stability, 2 (A P + PA) < if an only if 2 (AP +P A ) < Also, 2 (A P +PA 2PBB P ) <, hich is equivalent to stabilizability of (A, B), if an only if 2 (AP + P A ) <BB, hich is equivalent to etectability of (B,A )Finally,itcan be shon that 2P satisfies the boune real lemma or the positive real lemma for (A, B, C, D) if an only if 2 P satisfies them for (A,C,B,D ) Not coincientally, the function ξ 2 ξ P ξ is the convex conjugate (in the sense of convex analysis) of the function x 2 x Px hen P = P > In this paper, e ill explore this conjugate relationship an uality for LDIs, at the same time alloing for the possibility of orking ith nonquaratic functions an their conjugates Since the coification of the absolute stability problem in the mile of the tentieth century (see, for example, ]), researchers have looke for Lyapunov functions to ECE Dept, University of California, Santa Barbara, CA {rafal,teel,tingshu}@eceucsbeu Research by AR eel as supporte in part by the ARO uner Grant no DAAD , the NSF uner Grant no ECS , an by the AFOSR uner Grant no F ECE Dept, University of Virginia, P O Box 4743, Charlottesville, VA , USA zl5y@virginiaeu Research as supporte in part by NSF uner grant CMS /4/$2 24 IEEE 27 guarantee exponential stability an/or input-output properties for systems that can be moele as sitching among a (possibly infinite) family of linear systems he classical circle criterion gives necessary an sufficient conition for the existence of a quaratic Lyapunov function that certifies exponential stability in the absolute stability problem Hoever, it is ell knon that a system can be absolutely stable ithout the existence of a quaratic Lyapunov function For example, see 9] In 8] it is note that convex, positively homogeneous of egree to Lyapunov functions alays exist for exponentially stable sitching linear systems In 4] it is shon that, moreover, these functions can alays be taken to be everyhere continuously ifferentiable an smooth except at the origin In orer to give computationally tractable methos to search for Lyapunov functions, researchers have focuse on classes of functions over hich to look In the papers 3], 7], an 3], the authors consier homogeneous polynomial Lyapunov functions an provie linear matrix inequality (LMI) conitions for exponential stability In 2], a matrix conition for a Lyapunov function that is the maximum of positive semiefinite quaratics is outline One of our contributions is shoing that the origin of an LDI is exponentially stable if an only if the origin of the ual LDI is, an that Lyapunov functions for the ual systems are relate through the convex conjugacy hus, the tools for computation of Lyapunov functions mentione above can be applie to the ual system to check stability of the original system In a sense, this observation oubles the number of tools available for assessing stability of LDIs In some cases, there is a vast ifference in the stability range that a certain class of Lyapunov functions can certify for a system as compare to hat it can certify for the system s ual We illustrate this at the en of the paper Further benefits are shon in our companion paper 6], here stability regions of saturate systems are estimate In aition, e present uality results for issipativity hen convex, positively homogeneous storage functions an convex/concave supply rates are use hese results imply, for example, that passivity an finite L 2 gain are preserve hen passing from an LDI ith input an output to its ual Duality of stabilizability an etectability, an stabilizable an etectable issipativity, is establishe as ell Finally, e note that many of the iscrete time counterparts of the results reporte here are also vali In fact, the very efinition of the convex conjugate function is enough to establish the equivalent of heorem 4 in iscrete time hese results ill be reporte elsehere

2 II LDIS AND DISSIPAION INEQUALIIES We are intereste in issipation properties, establishe via convex storage functions, for a linear system ] ] ẋ A B B = 2 x u y C D D 2 an its ual ξ z z 2 = A C B D B2 D2 ξ as ell as a linear ifferential inclusion ] { ] ẋ A B B co 2 y C D D 2 an its ual ξ z co z 2 A C B D B2 D2 i i } m m i= i= ], x u ξ Our most general Lyapunov inequalities ill be of the form V (x) (Ax + B u + B 2 ) γv (x) () k(cx + D u + D 2, u, ), here the function k is convex in its first to arguments, an concave in the thir For such functions, checking () for the vertices of the LDI is sufficient for verify that inequality for all possible matrices in the convex hull For ease of notation, in hat follos e ill usually suppress inices hen consiering LDIs Our assumptions applie to these systems are unerstoo to hol for all matrix vertices Our main results sho that inequalities of the form () can be equivalently restate in terms of inequalities like V (ξ) (A ξ + C ) γv (ξ) (2) +k (, B ξ D, B2 ξ D2 ) hich involve a function V conjugate (in the sense of convex analysis) to V an a k, a convex-concave conjugate of k In many cases of practical interest, verifying these inequality for vertices of the LDI is also sufficient III CONVEX ANALYSIS PRELIMINARIES he stanar reference for the convex analysis material e summarize here is ] Given any function f : R n R, its conjugate function is efine, for ξ R n by f (ξ) = sup {ξ x f(x)} x R n Basic examples are given belo, a more elaborate one conclues this section For a positive efinite matrix P, f(x) = 2 x Px f (ξ) = 2 ξ P ξ ] 27 For any p>, q>, ith p + q =, f(x) = p x p f (ξ) = q ξ q We are mostly intereste in functions that are convex, positive efinite, an positively homogeneous of egree p> From no on assume that f has these properties hen: (i) f (ξ) is finite for every ξ R n (ii) f is a convex, positive efinite, an positively homogeneous of egree q> here /p +/q = (iii) If α p x p f(x) β p x p for some α>, β> (such constants exist for any continuous, positively homogeneous of egree p an positive efinite function), then β q ξ q f (ξ) α q ξ q q q For example, positive homogeneity of f can be verifie irectly from the efinition: f (λξ) = sup x {(λξ) x f(x)} = λ q { sup x ξ (x/λ q ) f(x)/λ q} } = λ q sup x {ξ (x/λ q ) f(x/λ q/p ) = λ q sup x {ξ x f(x)} = λ q f (ξ) since q =q/p A funamental property of convex functions is that the conjugate of f is the function f hat is, (f ) (x) =sup{x ξ f (y)} = f(x) ξ So, for example, the property of f escribe in (ii) above is equivalent to the same property of f Similarly, bouns on f in (iii) are equivalent to those on f A subgraient of a convex function f at x is a vector v R n such that f(x ) f(x)+v (x x) x R n If f is ifferentiable, then f(x) is the unique subgraient at x At points of nonifferentiability, the subifferential f(x), being the set of all subgraients, has more than element A key relationship beteen f an f is: ξ f(x) x f (ξ) his immeiately leas to the folloing observation (the inequality f(x) Ax < shoul be unerstoo as ξ Ax < for all ξ f(x)) When the convex function f an its conjugate f are positive efinite, e have f(x) Ax < x f (ξ) A ξ< ξ Inee, suppose the conition on the left hols Pick any ξ an x f (ξ) hen x, since f (ξ) oul imply ξ minimizes f hus x A ξ = ξ Ax <,

3 as x f (ξ) is equivalent to ξ f(x) A more precise relationship exists for positively homogeneous functions Lemma 3: f(x) =/p an ξ f(x) if an only if f (ξ) =/q an x f (ξ) Given any function g : R n R, its convex hull co g is the greatest convex function boune above by g Uner mil assumptions, for example hen g is finite everyhere (this alays hols if g is positively homogeneous of egree p> an positive efinite) e have { n+ } n+ co g(x) =min λ i g(x i ) λ i x i = x (3) i= i= here the minimum is taken over x i s an λ i s such that n+ i= λ i =, λ i (e enote such set as n ) If co g(x) = n+ i= λ ig(x i ) then co g(x i ) = g(x i ) at each x i ith nonzero λ i Furthermore, if g is ifferentiable at each such x i, then co g(x) = g(x i ) for each such i (in particular, co g is ifferentiable at x) No consier convex functions h j : R n R, j =, 2, l, an efine h(x) = max h j(x) (4) j=,2,l he conjugate function h is the convex hull of the function g(ξ) =min j=,2,l h j (ξ) A similar relationship hols for level sets: those of h are intersections of level sets of h i s hile those of h are convex hulls of level sets of h j s Lemma 32: Consier a positive efinite function h given by (4) hen the conition h(x) =h j (x) h j (x) Ax γh j (x) (5) is necessary an sufficient for h(x) Ax γh(x) to hol for all x R n We conclue ith another example of conjugate functions For positive efinite symmetric Q j, j =, 2, l, let q(x) = max j=,2,l 2 x Q jx (6) It turns out that the conjugate of q, hich is the convex hull of functions ξ 2 ξ Q j ξ, is the same as a function use in 5] for stability analysis (there, q as referre to as the composite quaratic function) Inee, from the efinition of the conjugate function it can be compute that ( l ) q (ξ) = min λ k 2 ξ λ i Q i ξ (7) i= his is exactly the composite quaratic function of 5] Such ual escription of (7) leas to an alternate ay to analyze its properties For example, the function q is strongly convex ith constant ρ, here ρ> is smaller than every eigenvalue of Q i, i =, 2, l (Strong convexity means that q(x) 2 ρ x 2 is convex) his is equivalent to q being ifferentiable an q being Lipschitz continuous ith constant /ρ Numerical examples in Section VII illustrate the use of both q an q in stability an L 2 -gain analysis IV LYAPUNOV INEQUALIIES he subifferential mappings of a pair of conjugate convex functions are inverses of one another Lemma 3 state this more exactly for positively homogeneous functions A key consequence of such symmetry is that a conjugate of a Lyapunov function for a linear system is a Lyapunov function for the ual system heorem 4: Let V : R n R be a convex, positive efinite, positively homogeneous of egree p> function; let A be any matrix hen, the conition is equivalent to V (x) Ax γpv (x) for all x R n (8) V (ξ) A ξ γqv (ξ) for all ξ R n (9) Proof: By positive homogeneity, (8) is equivalent to V ( x) A x γ for all x st V( x) =/p () Inee, given any x (so that V (x) ), consier x = x/s, here s =(pv (x)) /p hen V ( x) =/p, hile V ( x) = V (x) sp hus () becomes s p V (x) Ax γ hich is exactly s (4) Similarly, (9) is equivalent to V ( ξ) A ξ γ for all ξ st V( ξ) =/q () No, () means that ξ A x γ for any element ξ of V ( x) ith V ( x) =/p Such x an ξ can be equivalently characterize by x V ( ξ), V ( ξ) =/q, see Lemma 3 hus () is equivalent to () For V (an automatically V ) positively homogeneous of egree 2, the ecay rates in (8) an (9) are the same Such functions naturally appear in linear systems an LDIs ẋ co {A i } m i= x (2) an their uals ξ co { A } m i ξ (3) i= Belo, e rite L for the set of all convex, positive efinite, an positively homogeneous of egree 2 functions he folloing result can be foun in 8] heorem 42: he origin is asymptotically stable for (2) if an only if there exist γ> an V Lsuch that V (x) Ax γv (x) for all x R n (4) for all A {A i } m i= Corollary 43: he origin for (2) is exponentially stable (ith ecay rate γ) if an only if the origin for (3) is exponentially stable (ith ecay rate γ) A practical consequence of this is that to verify exponential stability of (2) ith a particular computation, one can also carry out that computation ith transpose matrices his can ramatically improve the results; see Example

4 Lemma 32 an its ual interpretation lea to conitions for stability of LDIs, ith Lyapunov functions given by (6) or (7) he first result belo as outline in 2], here it can be euce from Lemma 32, ith h j (x) = 2 x Q jx Corollary 44: Suppose that there exist positive efinite an symmetric matrices Q,Q 2, Q l an numbers λ ijk for i =, 2, m, j, k =, 2, l such that A i Q j + Q j A i l λ ijk (Q k Q j ) γq j (5) k= for all i =, 2, m, j =, 2, l hen V (x) Ax γv (x) x R n,a co{a i } m i=, (6) here V is the maximum of quaratic functions x 2 x Q j x (recall (6)) hrough heorem 4, one obtains a ual result Corollary 45: Suppose that there exist positive efinite an symmetric matrices R,R 2, R l an numbers λ ijk for i =, 2, m, j, k =, 2, l such that R j A i + A i R j l k= λ ijk (R k for all i =, 2, m, j =, 2, l hen R j ) γr j (7) V (x) Ay γv (x) x R n,a co{a i } m i=, (8) here V is the convex hull of quaratic functions x 2 x R jx No consier a control system ẋ co { A B ] ] } m x (9) i i= u Lemma 47: For a function W,letV =cow Suppose W is ifferentiable at every point x ith W (x) =V (x) an at such points, (4) hols (ith W taking the place of subifferential) Suppose that W is finite everyhere hen V is ifferentiable an satisfies (4) V DISSIPAIVIY Here e consier an LDI ith external isturbance ] { ] } m ] ẋ A B x = co, (2) y C D i i= its ual ] { ] } ξ A C = co m ] ξ z B D, (22) i i= an infinitesimal issipation inequalities of the form V (x) (Ax + B) γpv (x) h(cx + D, ) he function V is calle a storage function an the function h is calle a supply rate ] We ill relate issipativity ith storage function V (x) an supply rate h(y, ) for the system (2) to issipativity ith storage function V (ξ) an an appropriate supply rate h (, z) for the system (22) heorem 5: Assume that the supply rate h an the ual supply rate h are positively homogeneous of egree 2 an h(c, ) =sup{c g(, )}, h (, z) =inf{z + g(, )}, for some function g hen, the folloing are equivalent: (a) for all x,, V (x) (Ax + B) γv (x) h(cx + D, ), an its ual system ith output (b) for all ξ,, ] { ] } ξ A m co z B ξ (2) V (ξ) (A ξ + C ) i i= γv (ξ)+h (, B ξ D ) We say that (9) is stabilizable by sitche linear feeback he assumption above is quite mil Consier a positively if there exist m matrices K i such that the system homogeneous of egree 2 function h such that h(c, ) is ẋ co {A i + B i K i } m i= x convex in c for a fixe, concave in for a fixe c, an finite everyhere Let h be given by is exponentially stable We call (2) stabilizable by sitche linear output injection if there exist m matrices L i such that h (, z) =infsup { c + z h(c, )} (23) ξ co { c A i + L i Bi } m ξ i= Such a pair satisfies the assumption, ith the function g given by g(, ) = sup is exponentially stable c { c h(c, )} he same conclusion hols if h is convex/concave as before, an for Corollary 46: he system (9) is stabilizable by some close convex cones K, K 2,ehaveh(c, ) is finite sitche linear feeback if an only if the system (2) is if c K, K 2, h(c, ) =+ if c K, K 2, an stabilizable by sitche linear output injection h(c, ) = if K 2 In both cases, Aitional relationships beteen stabilizability an etectability, hich is strongly relate to stabilization by h(c, ) =supinf {c + z h (, z)} z output injection, ill be given in a later section Finally, e note that restricting our attention to convex Example 52: Suppose that Lyapunov functions oes not limit the breath of LDIs for hich stability can be guarantee h(c, ) = ] ] ] c c Q S M for M = 2 S R 273

5 hen h is convex-concave if an only if Q an R are positive semiefinite If aitionally M is invertible, the ual supply rate h efine by (23) is finite everyhere, an in fact h (, z) = ] ] M 2 z z In hat follos, e ill say that a system is M-issipative if it has a storage function in L supporting some γ an the supply rate h(y, ) ith h efine above Accoring to heorem 5, the system (2) is M-issipative if an only if its ual (22) is M-issipative, ith ] ] I M = M I I I Special cases inclue Passivity By passivity, ] e mean M-issipativity ith I M = See, eg, 2] In this case I M = M hus, passivity for an LDI is equivalent to passivity for its ual Passivity ith extra feeforar By passivity ith extra feeforar, ] e mean M-issipativity ith M = I for some R = R I R In this case, ] R I e have M = an then I M = M hus, passivity ith extra feeforar for an LDI is equivalent to this property for its ual Finite L 2 -gain By L 2 -stable ith gain ] γ>e mean I M-issipative ith M = γ 2 In this case e I get M = γ 2 M Since M-issipativity is not affecte by positive scaling, L 2 -stability ith gain γ for an LDI is equivalent to this property for its ual ] Q Weighte L 2 -gain For M = one obtains ] R R M = Q So, in passing from a system to its ual, the eights on the inputs in the supply rate become inverte eights on the outputs an vice-versa VI SABILIZABLE AND DEECABLE DISSIPAIVIY We no state the promise result, relating the general issipation inequalities allue to in Section II heorem 6: Assume that the supply rate h an the ual supply rate h are positively homogeneous of egree 2 an k(c, u, ) =sup{c g(, u, )}, k (, v, z) =inf sup {v u + z + g(, u, )}, u for some function g hen the conition (a) for all x,, there exists u such that () hols, implies (b) for all ξ,, the inequality (2) hols 274 If furthermore the function g is such that for all x, ξ,, the sup u inf an inf sup u of ξ B u Cx D 2 D u + g(, u, ) are equal, then (a) is equivalent to (b) As before, appropriate convexity assumptions imply that k an k satisfy all the conitions above his is the case in particular hen k(c, u, ) =f(c, u) g() for finite convex functions f an g Hoever, interesting consequences can be obtaine through juicious application of infinite values Example 62: In heorem 6, let B = B, B 2 =, C =, D = D 2 =, k(c, u, ) = if c = hile k(c, u, ) =+ if c Finally, let k (, v, z) equal + if v, if v =an z, an if (v, z) = hen all the assumptions are met, an one obtains the equivalence beteen stabilizability: for all x there exists u such that V (x) (Ax + Bu) γv (x), an etectability: for any ξ, B ξ = V (ξ) A ξ γv (ξ) VII NUMERICAL EXAMPLES Example 7: In 4], an LDI given by co{a,a 2 } ith ] ] a A =, A 2 = /a an a>, as use to sho that existence of a common quaratic Lyapunov function is not necessary for exponential stability he maximal a ensuring existence of such function as foun to be a q =3+ 8=58284, hile the LDI as shon, via analytical methos not leaing to a Lyapunov function, to be stable for all a, ] Here, e sho using Corollaries 44 an 45 that the convex hull function q (7) an the maximum function q (6) can lea to goo estimates of largest a guaranteeing stability With q forme by to quaratics (l =2), the maximal a is 84 With l =3, the maximal a is 895 he three matrices Q i etermine uner a =895 are as follos: ] ] Q =, Q 2 = Q 3 = ] Corresponing ellipsois (points x ith x Q i x =) an their convex hull (points x ith q (x) =) are in the upper plots of Fig Also shon there are irections of ẋ = A x (left) an that of ẋ = A 2 x (right) along the bounary of the convex hull Loer plots of Fig are the ellipsois x Q i x =an their intersection, an the irection of ẏ = A y an ẏ = A 2 y along the bounary of the intersection Example 72: he folloing thir-orer LDI as iscusse in 3] For matrices A = 2 4, M = 2 3, 3 4 2

6 x 2 y x 5 5 A A y x 2 A A 2 Fig y x y Vector fi els an invariant sets let A 2 = A + am ith a>, an consier the LDI ith the state matrix belonging to co{a,a 2 (a)} he maximal a ensuring the existence of a common quaratic function is a q =942 he maximal a that ensures the existence of a common homogeneous forth-orer polynomial Lyapunov function as foun to be a h =757 by 3] By Corollary 43, the stability of the LDI is equivalent to that of the ual LDI escribe by co{a,a 2 (a) }For this ual, e use the metho from 3] to etermine a for hich a common fourth-orer homogeneous Lyapunov function exists We may expect a ifferent estimate, as the conjugate of a fourth-orer homogeneous polynomial is not itself a fourth-orer homogeneous polynomial It turns out that there is no upper boun for a Let A m be the augmente matrix for A an A m2 (a) be the augmente matrix for A 2 (a) Let L(α) be the matrix containing auxiliary parameters (see, page 32 of 3]) hen for each a>, there exist a symmetric positive efinite matrix Q = R 6 6 an parameters α, β R 6 such that QA m + A m Q + L(α) 66Q, QA m2 (a) + A m2 (a)q + L(β) 66Q No numerical problem arises even for a = 2 We also performe stability analysis on these systems using q ith l =2 Stability can be verifie ith such function for a up to 43 Fora = 43, it can be verifie that there exist Q > an Q 2 > satisfying Q A + A Q < 5(Q 2 Q ) Q 2 A + A Q 2 < Q A 2 + A 2 Q < Q 2 A 2 + A 2 Q 2 < 26547(Q Q 2 ) he same algorithm use for the ual LDI shos that there is no upper boun for a Actually, for each a>, there exist Q,Q 2 > an λ ij such that Q j A i + A i Q j <λ ij (Q k Q j ) 53Q j for i, j, k =, 2, j k We teste a up to 2 an no numerical issues occur For a = 8, λ =35473, λ 2 =, λ 2 = 8764, λ 22 = Example 73: We no use q, q to boun the L 2 -gain of an LDI ith input an output Let A, A 2 be as in Example 72, for S = ]let B i = S, C i = S, D i = Corollary 44 suggests that γ is an upper boun for the L 2 -gain, verifie through q (ith l = 2), if there exist Q,Q 2 > ith λ ij >, i, j =, 2, such that A i Q j + Q j A i + Ci C ] i + λ ij (Q j Q k ) Q j B i Bi Q j γ 2 < I for j k heorem 5 an Example 52 sho that γ is a upper boun for the L 2 -gain, verifie through q (ith to quaratics), if the above inequality hols ith B i s sitche to Ci s an vice versa, an A i s to A i s Case : a = he system is quaratically stable With the quaratic Lyapunov function, the gain is estimate as γ 2 =47475 With q, the gain is estimate as γ =8423, the parameters are λ =398, λ 2 = λ 2 =, λ 22 = With q, the gain is estimate as γ =93988 λ = λ 22 =, λ 2 =9784, λ 2 =86882 Case 2: a = he system is not quaratically stable With q, the gain is estimate as γ =348763, the parameters are λ =3365, λ 2 = λ 2 =, λ 22 = With q, the gain is estimate as γ =855 he parameters are λ =5, λ 2 = λ 2 =, λ 22 = Case 3: a = 5 he stability is not confirme ith q With q, the gain is estimate as γ =576956, the parameters are λ =35587, λ 2 = λ 2 =, λ 22 = Case 4: a = 8, γ = REFERENCES ] MA Aizerman an FR Gantmacher Absolute stability of regulator systems Holen-Day Inc, 964 2] S Boy, L El Ghaoui, E Feron, an V Balakrishnan Linear Matrix Inequalities in System an Control heory SIAM, 994 3] G Chesi, A Garulli, A esi, an A Vicino Homogeneous Lyapunov functions for systems ith structure uncertainties Automatica, 39:27 35, 23 4] WP Dayaansa an CF Martin A converse Lyapunov theorem for a class of ynamical systems hich unergo sitching IEEE rans Automat Control, 44(4):75 76, 999 5] Hu an Z Lin Composite quaratic Lyapunov functions for constraine control systems IEEE rans Automat Contr, 48(3):44 45, 23 6] Hu, Z Lin, R Goebel, an A eel Stability regions for saturate linear systems via conjugate Lyapunov functions In Proceeings of the 43r IEEE Conference on Decision an Control, Bahamas, 24 7] Z Jarvis-Wloszek an AK Packar An LMI metho to emonstrate simultaneous stability using non-quaratic polynomial Lyapunov functions In Proc 4st IEEE Conf Decision an Control, Las Vegas, 22, volume, pages , 22 8] AP Molchanov an YeS Pyatnitskiy Criteria of asymptotic stability of ifferential an ifference inclusions enountere in control theory System Control Lett, 3:59 64, 989 9] H Poer an A soi Improving the preictions of the circle criterion by combining quaratic forms IEEE rans Automat Control, 8():65 67, 973 ] R Rockafellar Convex Analysis Princeton University Press, 97 ] JC Willems Dissipative ynamical systems, part I: General theory Arch Rational Mech Anal, 45:32 35, 972 2] JC Willems Dissipative ynamical systems, part II: Linear systems ith quaratic supply rates Arch Rational Mech Anal, 45: , 972 3] AL Zelentsovsky Non-quaratic Lyapunov functions for robust stability analysis of linear uncertain systems IEEE rans Automat Control, 39():35 38, 994