DISSIPATIVE SYSTEMS. Lectures by. Jan C. Willems. K.U. Leuven

Transcription

1 DISSIPATIVE SYSTEMS Lectures by Jan C. Willems K.U. Leuven p.1/26

2 LYAPUNOV FUNCTIONS Consider the classical dynamical system, the flow with, the state, and Denote the set of solutions, the vector-field. by, the behavior. p.2/26

3 LYAPUNOV FUNCTIONS Consider the classical dynamical system, the flow with, the state, and Denote the set of solutions, the vector-field. by, the behavior. is said to be a Lyapunov function for if along Equivalently, if p.2/26

4 " $ LYAPUNOV FUNCTIONS Lyapunov function V system trajectory X Let arg min Typical Lyapunov theorem global stability : and for there holds " for p.2/26

5 & & ) ( % & LYAPUNOV FUNCTIONS Refinements: LaSalle s invariance principle. Converse: Kurzweil s thm. LQ theory (matrix) Lyapunov equation. * A linear system is stable if and only if it has a quadratic positive definite Lyapunov function. Basis for most stability results in diff. eq ns, physics, (adaptive) control, system identification, even numerical analysis. p.2/26

6 LYAPUNOV FUNCTIONS Lyapunov functions play a remarkably central role in the field. Aleksandr Mikhailovich Lyapunov ( ) Introduced Lyapunov s second method in his Ph.D. thesis (1899). p.2/26

7 OPEN SYSTEMS Open systems are a much more appropriate starting point for the study of dynamics. For example, input SYSTEM output p.3/26

8 % OPEN SYSTEMS Open systems are a much more appropriate starting point for the study of dynamics. For example, input SYSTEM output What is the analogue of a Lyapunov function for open systems? Dissipative systems. p.3/26

9 THEME A dissipative system absorbs supply. SYSTEM supply p.4/26

10 THEME A dissipative system absorbs supply. How do we formalize this? Physical examples. Conditions for dissipativeness in terms of (state space, transfer function) system representations. Linear-quadratic theory. Leads to important classical notion of positive realness. How did this arise? Direct applications of positive realness: electrical circuit synthesis, covariance generation. Applications to stability, stabilization, and robustness. p.4/26

11 Part I: General Theory p.5/26

12 +,? 4 p.6/26 DISSIPATIVE SYSTEMS: Def n Dynamics: *, : the input, output, and state. 1 0 /. satisfying all Behavior " > = :8;< 89 external behavior.

13 -, DISSIPATIVE SYSTEMS: Def n Dynamics: + + * Let. 5 be a function, called the supply rate. p.6/26

14 p.6/26 DISSIPATIVE SYSTEMS: Def n if is said to be dissipative w.r.t. called the storage function, such that 7" " 4 " ED EF ( "BC "BA along trajectories: " C "GA

15 4 DISSIPATIVE SYSTEMS: Def n or, incrementally, p.6/26

16 4 + p.6/26 DISSIPATIVE SYSTEMS: Def n This is called the dissipation inequality I H defined by.65 7 The function I H + KJ is called the dissipation rate If equality holds: conservative system.

17 DISSIPATIVE SYSTEMS: Def n For power and energy, + power delivered. internal stored energy. Dissipativity rate of increase of stored energy power delivered. p.6/26

18 - 7 7 DISSIPATIVE SYSTEMS: Def n input supply SUPPLY SYSTEM output STORAGE SYSTEM DISSIPATION p.6/26

19 PHYSICAL EXAMPLES Electrical circuit: (potential, current) Dissipative w.r.t. L MON C P M M (electrical power). p.7/26

20 Q S U PHYSICAL EXAMPLES System Supply Storage Electrical circuit RTS QVU voltage current energy in capacitors and inductors p.7/26

21 PHYSICAL EXAMPLES Mechanical device: (position, force, angle, torque) Dissipative w.r.t. M \[ Z M ( M YX WM L MON C (mechanical power) p.7/26

22 Q S U a ` ] U ^ d f U PHYSICAL EXAMPLES System Supply Storage Electrical circuit Mechanical system RTS QVU voltage current R_^ ]gu force, : angle, RTf ed b bc velocity torque energy in capacitors and inductors potential + kinetic energy p.7/26

23 ( M M J i [ PHYSICAL EXAMPLES Thermodynamic system: Heating terminal (Q h,t h ) Working terminal W Thermodynamic Engine (Q c,t c ) Cooling terminal Conservative w.r.t. L MON C hl MON C M Dissipative w.r.t. L MON C p.7/26

24 Q S U a ` ] U ^ d f U ` j U lf j k f U PHYSICAL EXAMPLES System Supply Storage Electrical circuit Mechanical system RTS QVU voltage current R_^ ]gu force, : angle, RTf ed b bc velocity torque energy in capacitors and inductors potential + kinetic energy Thermodynamic internal system j U heat, work energy Thermodynamic system j U heat, temp. entropy etc. etc. etc. p.7/26

25 CONSTRUCTION of STORAGE F NS Given (a representation of ), the dynamics, and given, the supply rate, is the system dissipative w.r.t., i.e., does there exist a storage function such that the dissipation inequality holds? SYSTEM supply Monitor dynamics, power flow. How much energy is stored? p.8/26

26 2? CONSTRUCTION of STORAGE F NS Assume: 1. State space of connected: every state reachable from every other state; 2. Observability: given at most one 4 such that 32 4 Let be an element of, a normalization point for the storage functions, since these are only defined by an additive constant. p.8/26

27 2 = q> = x 4 = q> = z ˆ z ~ m { { x 8 z 4 x z ˆ x ~ m { { x CONSTRUCTION of STORAGE F NS "BA A mon "BC C " A. at time to state " C at time takes the state Notation: := A C Consider the following two state f ns, universal storage f ns:, defined by r>8 =p The available storage: 7" J w+ vw ut+ s r>8 =p x 9 { 9 { ƒ yz_{, defined by qœ; 8Š ; The required supply: 7" w+ w Ž 8 qœ; 8Š ; 9 { 9 { ƒ yz { p.8/26

28 +, [ $ 8 = q> = CONSTRUCTION of STORAGE F NS Note: 0. is an equilibrium, associated with if +, + and +., we can take lim qœ; 8Š ; =p r>8 and then in def. of p.8/26

29 % % CONSTRUCTION of STORAGE F NS SYSTEM input output supply Maximize the supply extracted, starting in fixed initial state available storage. Minimize the supply needed to set up a fixed initial state required supply. p.8/26

30 7" 4 4 $ = q> $ CONSTRUCTION of STORAGE F NS Basic theorem: Let and be given. The following are equivalent: 1. is dissipative w.r.t. (i.e. a storage f n ) 2. for all periodic observability, for all periodic 32 > = :8;< 89 32, equivalently, by r>8 = =p 4. J 8 qœ; 8Š ; p.8/26

31 @ = q> = Q Q k = q> = CONSTRUCTION of STORAGE F NS be given. Basic theorem: Let and Moreover, assuming that any of these conditions are satisfied, then 8 qœ; 8Š ; =p r>8 and are both storage functions, the set of storage f ns is convex, and e aœ Q Ÿ k Q Ÿ e aœ Ÿ Ÿ e aœ š Q k š Q. 8 qœ; 8Š ; r>8 =p In fact, p.8/26

32 @ 7" 4 4 PROOF of the BASIC TH M : is dissipative w.r.t. for all periodic 32 : > = :8;< 89 Use the dissipation inequality (and observability). p.9/26

33 = q> = $ = q> = $ = q> = 4 m J x 4 J z [ = q> = p.9/26 PROOF of the BASIC TH M r>8 =p : sup over non-empty set by reachability. J r>8 =p : r>8 =p (i) (ii) [ A m n [ C and 4 32 Note that by 2.,. 7" x D x F implies [. mn with h n [ Concatenate Then 7" 4 xh z 7" Take the supremum over the left hand side.. : the sup then occurs for r>8 =p Note

34 = q> = = q> = 4 ˆ ~ m { { x ~ 4 ˆ ~ m { { x ~ 4 J ( 4 ˆ ~ m { { x ~ 4 = q> = PROOF of the BASIC TH M =p r>8 satisfies the dissipation inequality: "BC r>8 =p x ª F 7" J w+ vw ut+ s EF x ª F 9 { F { F ƒ yz_{ x ª «D 7" J w+ vw ut+ s EF x ª D 9 { F { F ƒ yz { 7" ED F x ª «D 7" J w+ vw ut+ s D x ª D 9 { D D { ƒ yz { "BA r>8 =p ( 7" D F J p.9/26

35 PROOF of the BASIC TH M The proof with 8 qœ; 8Š ; as a storage function is analogous. p.9/26

36 = q> x 7" J z PROOF of the BASIC TH M Convexity of the set of storage functions: obvious. Bound r>8 = =p J Consider a trajectory implies [ m n. The dissipation inequality " 4 " J Take the supremum of the right hand side. p.9/26

37 = q> = x J z p.9/26 PROOF of the BASIC TH M J r>8 =p Bound. The dissipation inequality [ m n Consider a trajectory implies 7" " 4 " J Take the supremum of the right hand side. 8 qœ; 8Š ; J Bound is proven analogously.

38 -, " 4 A 4 C 4 A 4 C ". SMOOTHNESS To make all arguments rigorous requires certain assumptions. The behavior of + + must have the state property, i.e. 32C 32A C A and "BA " C A 32A C 32C ( denotes concatenation at ). This can be achieved by assuming that the set of admissible input functions is closed under concatenation, and the sol n set of s consists of abs. cont. f ns. p.10/26

39 4 SMOOTHNESS To make all arguments rigorous requires certain assumptions. The behavior of + - +, is locally integrable, i.e. must have the property that "BA "BC 4 32 $ 7" " 4 " ED EF p.10/26

40 7" 4 - = q> SMOOTHNESS To make all arguments rigorous requires certain assumptions. The equivalence of the global and local versions of the dissipation inequality 1. "BA " ED EF ( " C " also requires certain smoothness on and on. Obviously, must be differentiable. While for a given one may simply wish to assume this, for, if needed, =p = r>8 and 8 qœ; 8Š ; this has to be proven. p.10/26

41 7" 4 - ± 2 SMOOTHNESS To make all arguments rigorous requires certain assumptions. 1. "BA " ED F ( " C 4 " Note that assuming 1. for a small behavior (e.g., having, and/or compact support conditions), deducing from there 3., will yield, by integrating, 1. for a large behavior (e.g. with locally integrable s, absolutely continuous s). p.10/26

42 4?? 7" 4 RECAP A system is dissipative 7" ED F ( "BC " A many physical examples of dissipative open systems. storage function for all periodic trajectories. Universal storage functions: the available storage, the required supply. p.11/26

43 Part II: Linear Quadratic Theory p.12/26

44 ( ³ & + ºµ  ºµ Á ¼µ À ¼µ ¾» ¹ Á µ Ç Í D Ê µ Ç Í D È Ë b Ì Á Ë ¾ Ë Ë Ê Ì Ê Ë p.13/26 LINEAR DIFFERENTIAL SYSTEMS These admit many representations: 1. State space representation: ² (, + à ¼ ¼ µ¼6½ µº µ À ¾ Â. Assume (in Part II & III) controllability and Notation: observability. Behavior abs. cont. ¼µ µóò Ì ËÅ µ ÎÐÏÑ Ì µ ÎÐÏÑ Ì Å Ê ÇÉÈ ÄšÅÆ a.e. Çc Â È ÌÔ Çc Ì Çc Æ Ì Çc À È ÌÔ Çc Ì Çc Æ bc satisfying Ä ). ÇÉÈ is unimportant, we will denote Occasionally (when

45 1 Õ aj aø Ý áà áà aî aì Þ Ý Ý ë î àè LINEAR DIFFERENTIAL SYSTEMS These admit many representations: 2. Transfer function Ö / Usual interpretation via exponential or frequency response, or Laplace transform, or differential equation (kernel representation) eüþ Ù ÙÚ eüû Ù ÙÚ with Øß jß åçæ ãä âá factorization, or (image representation) j ë ØÓê Ý é åçæ ãä âè, a left co-prime eí æ Ù ÙÚ Û eí æ Ù ÙÚ with ìß jß åçæ ãä âè àè Ý é åçæ ãä âá ìê, a right co-prime fact. p.13/26

46 " 2 z 4 J LINEAR DIFFERENTIAL SYSTEMS These admit many representations: 3. Impulse response 7" " 2 " z ( " " possibly completed. p.13/26

47 ² C & ÖP ³ ( J Õ 1 ( ï z n ª " ñ z LINEAR DIFFERENTIAL SYSTEMS These admit many representations: 4. Relations among these representations Ö Õ / 1 Ö the Laplace transform of " / ² " ³ ð p.13/26

48 @ 4 % õ ô ô õ A J + ö QUADRATIC SUPPLY RATES, + a quadratic form in., + a q.f. in ( often not relevant, and by observability the properties - as periodicity, or, of and coincide): + ò A ó + ó + + with as important special cases, + and, +, Relevant in electrical circuits (supply rate: mechanics: (supply rate force,velocity voltage,current ), ), scattering repr., etc. p.14/26

49 À LQ THEOREM Theorem: Let Hand + ú ø û ù» ü ý ú ø û ù» be given. The following are equivalent: 1. is dissipative w.r.t.. p.15/26

50 ú ø ü ú ø û ù û ù H ÿ ÿ æ ÿ ê ÿ æ ÿ æ p.15/26 LQ THEOREM À ¾ be given.»» + Hand Theorem: Let The following are equivalent: 2. Behavioral characterizations: 2.1 ÙÚ ee aú e æ aú a Þ þ e ß a Þ for all periodic 2.2 ÙÚ ee aú e æ aú a Þ þ e à àè à æ a e ß a Þ for all with comp. supp. Ý e à àè à æ a e ß a Þ 2.3

51 À ¾ % $ " & % $ LQ THEOREM Theorem: Let and ü ý ú ø û ù» ú ø û ù» be given. The following are equivalent: 3.1 is dissipative w.r.t. with a quadratic storage function. 3.2 Linear matrix inequality (LMI): there exists such that p.15/26

52 4 3 2 $ 4 & $ " : 9 LQ THEOREM. *, / *, / + - ) ( be given. and Theorem: Let The following are equivalent: 4. Frequency-domain characterization % 7 65 " $ % " $ for all denotes the spectrum, the set of eigenvalues of ;: 5. Characterization in terms of impulse response:?? p.15/26

53 % $ " & % $ % &?? > % > > (LMI) The matrix eq n: has become a (the?) key equation in systems and control theory. Note that this (LMI) states exactly that $@? " > > < <= > < <= i.e. that is a (quadratic) storage f n. p.16/26

54 % $ " & % $ A A 5 A 5 (LMI) The matrix eq n: has become a (the?) key equation in systems and control theory. Solution set is convex, compact, and attains its infimum its supremum : 5and = available storage, = required supply. p.16/26

55 % $ " & % $ B & F C UV ON MC L CH F G N KH C I (LMI) The matrix eq n: has become a (the?) key equation in systems and control theory. If, then equivalent to Algebraic Riccati inequality (ARIneq) CED C I F T LM QSR FPO GJI p.16/26

56 B & F C UV ON MC L KH CH F G N C I F C V ON D MC L KH CH F G N C I A A 5 (LMI) If, then equivalent to Algebraic Riccati inequality (ARIneq) CED C I F T LM QXR FWO GJI In fact, there exist sol ns to (ARIneq) there exist sol ns to the Algebraic Riccati equation (ARE) CED C I F T LM QSR FPO GJI In particular, the extreme sol n 5and of (LMI) satisfy (ARE). There exist various other characterizations of. p.16/26

57 Y Y [ [ [ [ [ [ a ` [ [ [ B & B & A 5 PROOF of LQ TH M and (LMI) We will prove the equivalence of the following 10 statements: I. II. III. Z\[ [ [ quadratic (1, page 15) Z [ [ [ for all periodic (3.1, page 15) (2.1, page 15) IV. V. for all for all ]_^ [ [ [ ]_^ (2.2, page 15) of compact support VI. for all of compact support VII. Frequency domain condition (4, page 15) VIII. (LMI) (3.2, page 15) IX. For X. For, solvability of the (ARIneq) (page 16), solvability of the (ARE) (page 16); (2.3, page 15) sol ns. p.17/26

58 Y Y g t > i o g y y y PROOF of LQ TH M and (LMI) I VIII I: Z\[ [ [ VIII: sol n to the (LMI) The difficult part is the following proposition, which we take for granted Proposition: Assume that xzy w v u <= ef f dc b? og prq likj s i lnm j hikj Then this supremum is a quadratic form in y { and y { ~ It follows from the basic th m that inequality, equivalently, the (LMI). satisfies the dissipation p.17/26

59 B & Y Y B ^^ ^^ ^ ^7 7 7 ^^ ^ ^ PROOF of LQ TH M and (LMI) I VIII VIII IX Include VIII IX only in the case. VIII: sol n to the (LMI) IX: sol n to the (ARIneq) Schur complement: Let {. Then ~ 7^ 7^ p.17/26

60 B & Y Y A { 5 ƒ Š ˆ ˆ ( ˆ ˆ ˆ ˆ ( ˆ ˆ ( 0 ˆ ƒ Š ƒ ˆ PROOF of LQ TH M and (LMI) I VIII VIII IX IX X Include IX X only in the case. IX: sol n to (ARIneq) X: sol n to (ARE), sol ns is trivial. To show idea due to C. Scherer., use the following propostion, a clever Proposition: Assume ƒ 0 0 and ( controllable. Then if the ARineq ˆ 0 ( 0 ˆ has a sol n, so does the (ARE) ƒ ( 0 ( 0 Proof: Define ˆ 0 ( R ŒŽ and consider the (ARE) ƒ Œ R ( 0 This is a standard (in the sense that contr.) (ARE) of the theory of LQ optimal control. We assume that it is known that a sol n exists. Now prove by a straightforward calculation that solves the (ARE). Now, there even exist a sol ns and. Hence the infimal and supremal sol s of (LMI) and (ARIeq) solve (ARE). ( ƒ Œ ƒ p.17/26

61 Y Y Y PROOF of LQ TH M and (LMI) I VIII VIII IX IX X VIII II I VIII: sol n to (ARE) II: quadratic Z\[ [ [ Z [ [ [ Trivial. p.17/26

62 Y [ [ [ PROOF of LQ TH M and (LMI) I VIII VIII IX IX X VIII II I I III I: Z [ [ [ III: for all periodic Basic theorem of dissipative systems. p.17/26

63 [ [ [? 3 { $ œ > PROOF of LQ TH M and (LMI) I VIII VIII IX IX X VIII II I I III III VII III: for all periodic VII: Frequency condition Use your frequency domain intelligence. Consider the (complex) periodic inputs. For all there is an associated periodic š = = Calculate with œ 65 and obtain the frequency condition.. p.17/26

64 > a > { a 5 PROOF of LQ TH M and (LMI) I VIII VIII IX IX X VIII II I I III III VII VII IV VII: Frequency condition IV: for all ]_^ [ [ [ Assume that compute? { =?. Use Parseval s theorem to ]_^ <= ~ = p.17/26

65 a ` x x x PROOF of LQ TH M and (LMI) I VIII VIII IX IX X VIII II I I III III VII VII IV IV V VI IV: ]_^ V: c. supp. VI: c. supp. Trivial. p.17/26

66 a ` x [ [ [ a ` a ` PROOF of LQ TH M and (LMI) I VIII VIII IX IX X VIII II I I III III VII VII IV IV V VI VI III VI: c. supp. III: for all periodic Assume the contrary, truncate this periodic sol n after a large number of periods, make the truncation into a compact support sol n, and smooth (e.g. by convoluting with a compact support kernel) in order to obtain a compact support solution that violates VI. p.17/26

67 B & A 5 A PROOF of LQ TH M and (LMI) I VIII VIII IX IX X VIII II I I III III VII VII IV IV V VI VI III That the set of sol s of the (LMI) (and hence of the (ARIneq) for ) is convex and compact is trivial. The inequality follows immediately from the interpretation of5and terms of the available storage and the required supply. in p.17/26

68 A B & A { 5 RECAP A linear differential system with a quadratic supply rate is dissipative there exists a quadratic storage function. Leads linea recta to the (LMI). The set of sol ns of this (LMI) is convex, compact, and attains its infimum 5and its supremum. These correspond to the available storage and required supply. The (LMI) is very closely related to algebraic Riccati inequality and the algebraic Riccati equation. The extreme sol ns of the (LMI) are sol ns of the (ARE) (when ). There is also an explicit condition for dissipativity in terms of the frequency response. p.18/26

69 Part III: Contractivity, Passivity p.19/26

70 ^ Ÿ NON-NEGATIVE STORAGE F NS Do storage functions need be? Since one can always add a constant, one should really ask: Are storage functions bounded from below? We did NOT demand this. The reason is physics: in mechanics (e.g. a mass in an inverse square gravitational field), the energy need not be bounded from below, in thermodynamics, the entropy (often the log of the temp.) need not be bounded from above or below. Nevertheless, in applications (stability, circuit synthesis) storage f n is essential. We will cover the LQ cases of the Ÿ { ^ Ÿ (contractivity) Ÿ { (positive realness) p.20/26

71 e e t A e e p.21/26 CONTRACTIVITY We start with a bit of notation and recalling a couple of definitions: u v t š 5 u B š u t š i 5 u t š A i = complex conjugate.

72 9 š y = š 9 y ª w { CONTRACTIVITY We start with a bit of notation and recalling a couple of definitions: is Hurwitz Ž. 5 Equivalently, of course, all trajectories of y go to zero as w ~ 1. is almost Hurwitz, i 5 2. the eigenvalues on the imaginary axis are semi-simple. Equivalently, of course, all trajectories of on. y are bounded p.21/26

73 a A ± «w v ± 3 3 ± ³?² { ³ i a 5 «š ± p.21/26 CONTRACTIVITY We start with a bit of notation and recalling a couple of definitions: -norm equals. Its «Let u ~ «t ef Ÿ f dc bÿ š J «Hurwitz). A i ( proper, no poles in «u ~ 2 «t ef Ÿ f dc bÿ «Then induced norm of the operator ]\^ ±equals the «~ <= =? = = " = =? i or µ ~ «contractive Call

74 ) ( CONTRACTIVITY Theorem: Consider, controllable & observable, and Ÿ { ^ Ÿ ^ ~ The following are equivalent: 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. p.21/26

75 ) ( B " $ " " " 4 $ " B A CONTRACTIVITY Theorem: Consider, controllable & observable, and Ÿ { ^ Ÿ ^ ~ The following are equivalent: 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 2. The (LMI) has a solution. Equivalently, the supremal sol n. p.21/26

76 ) ( ³ a ½ ³ º ¹m { 5 CONTRACTIVITY Theorem: Consider, controllable & observable, and Ÿ { ^ Ÿ ^ ~ The following are equivalent: 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 3. Behavioral characterization: <= ^ = ^ =? i 5 for all? { ] ^ ¼». This is called half-line dissipativity. Note: upper bound on immaterial, may as well take a. p.21/26

77 ) ( ««µ ± CONTRACTIVITY Theorem: Consider, controllable & observable, and Ÿ { ^ Ÿ ^ ~ The following are equivalent: 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 4. Frequency-domain characterization: is contractive, i.e. ¾. 5. is dissipative w.r.t., and is Hurwitz. p.21/26

78 ) ( CONTRACTIVITY Theorem: Consider, controllable & observable, and Ÿ { ^ Ÿ ^ ~ The following are equivalent: 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 1. diss. w.r.t., with all storage f n bounded from below. p.21/26

79 ) ( " $ " " " 4 $ " B B 5 CONTRACTIVITY Theorem: Consider, controllable & observable, and Ÿ { ^ Ÿ ^ ~ The following are equivalent: 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 2. All solutions of the (LMI) are. Equivalently, the infimal sol n. p.21/26

80 ) ( a ` B CONTRACTIVITY Theorem: Consider, controllable & observable, and Ÿ { ^ Ÿ ^ ~ The following are equivalent: 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 6.,7.,... Various variations, with support, ARineq, (ARE), etc. instead of,, compact p.21/26

81 Ã Ä { { Ã Â Ä { { v { B { Ä " à " Å Â Â Ä " Æ" { Æ Æ { { à PROOF of CONTRACTIVITY TH M Preliminary: The inertia of is defined as the triple  š ÁÀ with of eigenvalues of Of course Btw, the number (counting multiplicity) with respectively real part.. is called the signature of. Recall the following result involving the inertia and the Lyapunov equation ÇÆ ~ Theorem: Assume that Then, and Æ {, and À Æ À satisfy the Lyapunov eq n, with is observable.. p.22/26

82 ) ( A { 5 " " " { { PROOF of CONTRACTIVITY TH M Note that each of the conditions of the th m implies that is dissipative w.r.t. Therefore, we can freely use the LQ th m, and assume the existence of, etc. Ÿ { ^ Ÿ ^ ~ Each sol n of the (LMI) satisfies ~ Since is observable, so is. The inertia theorem therefore implies that all these s are non-singular and have the same number of positive and negative eigenvalues. p.22/26

83 Z ~ ya y Z Z y y 5 Z ya y A B A A PROOF of CONTRACTIVITY TH M By the basic th m on dissipativity, there holds for any storage function, y Hence bounded from below implies bounded from below. Since is non-singular, it is bounded from below if and only if, but since it also non-singular, ~ p.22/26

84 A ½ ³ º ¹m { { È =?? = ³? = ³ ½ ³ º ¹m { { ± Ò p.22/26 PROOF of CONTRACTIVITY TH M is By 2.,. By the inertia th m, therefore, Hurwitz. Assume that 3. does not hold. Then there is such that B ¼» ] ^? ƒ ÍÑ Î Ð ÊÊ Ì Í ÊÊÇÏ Î R ÊÊ Ì Í Ê ÊÇË ± É for, and is Hurwitz, the resulting = =? with, with. Since yields B Consider the input for for = = = ³, and ¼» ]_^? ƒ ÍÑ Î Ð ÊÊ ÌÌ Í ÊÊÏ Î R ÊÊ ÌÌ Í Ê Ê Ë ± É Contradicts dissipativeness.

85 u «= ³ > a Ó w ³ i g ³ a 5 PROOF of CONTRACTIVITY TH M implies dissipativeness. Therefore µ ~ «t ef Ÿ f dc bÿ We need to prove that 3. implies that has no poles, i.e., that is Hurwitz. If this were not the case, choose a (compact support) input that is zero for, and such that yields a with ^ =. Hence for sufficiently large i A ~ v <= ^ = ^ =? Contradicts 3. p.22/26

86 PROOF of CONTRACTIVITY TH M obvious p.22/26

87 B PROOF of CONTRACTIVITY TH M By dissipativeness there exist sol ns inertia theorem they are all. 1. follows. to the (LMI). By the p.22/26

88 Ô PROOF of CONTRACTIVITY TH M use the inertia theorem. p.22/26

89 Ô ya y Z Z y y 5 B PROOF of CONTRACTIVITY TH M follows from y and. p.22/26

90 Ô PROOF of CONTRACTIVITY TH M µ ~ trivial. p.22/26

91 Ô PROOF of CONTRACTIVITY TH M µ ~ p.22/26

92 Õ š t u Õ t A A Y t u Õ t Õ Õ e Õ A B Õ Õ w POSITIVE REALNESS A VERY important notion in system theory: is positive real (p.r.) u ~ numerous equivalent conditions for positive realness. E.g.: i A { and, more intricate, has no poles in 3. the im. axis poles of not a pole of for all are simple, with residue real and u A i 4. is proper, and its limit for is p.23/26

93 š u ~ «" «t u t A ««p.23/26 POSITIVE REALNESS Matrix case: is positive real (p.r.) «. is the Hermitian conjugate of

94 ) ( Ÿ POSITIVE REALNESS Theorem: Consider, controllable & observable, and ~ The following are equivalent: Ÿ { 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. p.23/26

95 ) ( Ÿ B $ " $ B A POSITIVE REALNESS Theorem: Consider, controllable & observable, and ~ The following are equivalent: Ÿ { 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 2. The (LMI) (note: the corresponding storage f n is ) y 7 y ^ has a solution. Equivalently, the supremal sol n. p.23/26

96 ) ( Ÿ ³? a ½ ³ º ¹m { POSITIVE REALNESS Theorem: Consider, controllable & observable, and ~ The following are equivalent: Ÿ { 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 3. Behavioral characterization: <= = = i 5 for all? { ] ^ ¼». This is called half-line dissipativity. p.23/26

97 ) ( Ÿ «POSITIVE REALNESS Theorem: Consider, controllable & observable, and ~ The following are equivalent: Ÿ { 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 4. Frequency-domain characterization: is positive real. 5. is dissipative w.r.t., and is almost Hurwitz. p.23/26

98 ) ( Ÿ POSITIVE REALNESS Theorem: Consider, controllable & observable, and ~ The following are equivalent: Ÿ { 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 1. diss. w.r.t., with all storage f n bounded from below. p.23/26

99 ) ( Ÿ $ " $ B B 5 POSITIVE REALNESS Theorem: Consider, controllable & observable, and ~ The following are equivalent: Ÿ { 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 2. All solutions of the (LMI) are. Equivalently, the infimal sol n. p.23/26

100 ) ( Ÿ a ` B POSITIVE REALNESS Theorem: Consider, controllable & observable, and ~ The following are equivalent: Ÿ { 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. 6.,7.,... Various variations, with support, ARineq, (ARE), etc. instead of,, compact p.23/26

101 ) ( Ÿ POSITIVE REALNESS Theorem: Consider, controllable & observable, and ~ The following are equivalent: Ÿ { 1. diss. w.r.t., with a storage f n that is bounded from below, or, equivalently, non-negative. It is customary to refer to this case as PASSIVITY. p.23/26

102 «B { " { $ 4 POSITIVE REALNESS In the special case ( strictly proper), the (LMI) becomes ~ The fact that solvability of this (LMI) is equivalent to positive realness of «$ 7 5 KYP-lemma (after Kalman, Yakubovich, Popov). is usually called the p.24/26

103 ÞÞÞ Ú Î Í R Î Í R ÞÞÞ Ú Î Í R ÞÞÞ Ú 0 Î Í R ÞÞÞ Ú 7 POSITIVE REALNESS In this case, it is possible to express passivity as a sort of condition on the impulse response: ƒ. â â¾â âøâøâøâ â¾â âøâøâøâù/ ƒ 0 ƒ Í Û R Íàß Í Û R ÍÜÝ ÍÜÛ ÎR Í 0 ƒ ƒ Íàß Í Ý Í Û ÎR 0 ƒ ƒ Í Ý R Íáß 0 Í Ý Í Û R 0 ÍÜÝ Í ƒ * Ö Ö¾Ö ÖØÖØÖØÖ Ö¾Ö ÖØÖØÖØÖÙ+ ƒ Í Ý R 0 Íàß 0 Í ß Í Û R 0 Íàß å ~ ä and all =rã = ^ = [ [ [ for all The proof will not be given. p.24/26

104 " y æ ~ " Ÿ { { Ô " $ y æ { ³ " > ³ { > ³ { { PROOF of the P.R. TH M We give the proof only in the case. It makes some points of independent interest, namely that the choice of inputs and outputs in a system is not something that is fixed. The system eq ns are š y $ÁŸ { y behavior With the new input and output µ Ÿ Ô µ Ÿ Ô the system equations become š y $ÁŸ y " Ÿ beh. Obviously? { 7? ^ 7? ^. p.25/26

105 { Ÿ { Z Z PROOF of the P.R. TH M Define Ÿ Ÿ ^ ^ ~ Note that ~ Ÿ { Ÿ Hence is dissipative w.r.t. with storage function is dissipative w.r.t. with storage function. Conclude that 1. of the contractivity th m 1. of the p.r. th m. 1. of the contractivity th m 1. of the p.r. th m. 2. of the contractivity th m 2. of the p.r. th m. 2. of the contractivity th m 2. of the p.r. th m. and from the relation between and 3. of the contractivity th m 3. of the p.r. th m. p.25/26

106 «««" 4 «4 «" 4 4 è " 4 " «µ ~ PROOF of the P.R. TH M Note that the transfer functions the fractional transformation of and of are related by 7 ~ ç5 Now, for, there holds invertible and 5 7 µ ~ Conclude that ± «positive real Hence, 4. of the contractivity th m 4. of the p.r. th m. p.25/26

107 $ " { B B PROOF of the P.R. TH M We still need to prove that This uses the following Lemma: Assume ~ { and Then is almost Hurwitz ~ Assume that 5. holds. Then, by dissipativeness, the (LMI) has a sol n. By the lemma, Conversely, if 2. holds, then, by the lemma,, whence 2. holds. is almost Hurwitz. p.25/26

108 $ " { B B PROOF of the P.R. TH M We still need to prove that This uses the following Lemma: Assume ~ { and Then is almost Hurwitz ~ Assume that 5. holds. Then, by dissipativeness, the (LMI) has a sol n. By the lemma, Conversely, if 2. holds, then, by the lemma,, whence 2. holds. is almost Hurwitz. I wasn t able to construct a proof of of the lemma in time. In cauda venenum p.25/26

109 ^ «± Ÿ «RECAP Non-negativity of the storage function is important in the analysis of physical systems, and in stability applications. For, positivity of the (all) storage function comes down to the condition Ÿ { ^ Ÿ µ ~ For positivity of the (all) storage function comes down to positive realness of. Ÿ { { Recently, a n.a.s.c. for the existence of a postive storage function in the general LQ case has been published by Trentelman and Rapisarda (SIAM J. Control & Opt., 2003?). p.26/26