We present a necessary and sufficient condition for dissipativeness in the single input / single output case

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(1)ON THE DISSIPATIVITY OF UNCONTROLLABLE SYSTEMS M.K. C ¸ amlıbel Dogus University 81010 Acibadem Istanbul Turkey Tilburg University P.O. Box 90153 5000 LE Tilburg The Netherlands Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands e-mail: K.Camlibel@uvt.nl. Abstract— This paper deals with dissipativity of uncontrollable linear time-invariant systems with quadratic supply rates and storage functions. A definition of dissipativity appropriate for this class of systems is given. We present a necessary and sufficient condition for dissipativeness in the single input / single output case. Keywords: Behaviors, dissipativity, storage functions, controllability, observability.. I. I NTRODUCTION The roots of the theory of dissipative systems can be found in the early papers on electrical circuit theory. In [2], the notion of positive realness was introduced in the context of circuit synthesis. It was shown in this classic paper that a rational function is the driving point impedance of a circuit consisting of a finite number of positive resistors, inductors, capacitors, and transformers if and only if is positive real. Starting in the late fifties and early sixties, positive realness came to play a key role also in systems and control theory, through what we now call positive real (or KYP) lemma. In [5] dissipativity was conceptualized in terms of the storage function and the supply rate. One of the assumptions in the study of dissipativity has almost always been controllability of the system. The very definition of a dissipative system is often given for the controllable case. There is, however, no reason to make the definition of an “energy’-related concept like dissipativity dependent on a concept like controllability not related to energy at all. Even if there is a certain relationship between these two concepts, it should follow from the definitions instead of being imposed. This paper is a step towards a dissipativity theory for uncontrollable systems in a behavioral context, for linear time-invariant systems and quadratic supply rates and storage functions. After giving a definition of dissipativity, we present a necessary and sufficient condition for a (not necessarily controllable) single-input single-output behavior to be dissipative.. . . Jan C. Willems University of Leuven Kasteelpark Arenberg 10 B-3001 Leuven-Heverlee Belgium Jan.Willems@esat.kuleuven.ac.be www.esat.kuleuven.ac.be/ jwillems Madhu N. Belur University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands M.N.Belur@math.rug.nl www.math.rug.nl/ madhu. II. P RELIMINARIES A (linear time-invariant differential) behavior (for a detailed treatment of behavioral theory, we refer to [4]) is the set of solutions, assumed infinitely differentiable for ease of exposition, of a system of linear constant coefficient differential equations. where. .

(2) . is a polynomial matrix. Note that.

(3) "!$# %&' ( *) (1) " The set of all such behaviors will be denoted by + . + is said to be controllable if for any behavior. -, A and. /. there . such exist a 021 and a /.. 3.

(4). ,. 3.

(5) 76 0 for for all 54 and that all 98 0 . It#;:=is<>well-known that a behavior is controllable if %&@? ( is the ?ACB . and only if same for all On the other extreme of controllability are autonomous. ED is autonomous + behaviors. A behavior. /F , GD H

(6) I 3F for all 94 implies that if ED5for

(7) J /F ., A basic result of behavioral theory states that every behavior can be written as a direct sum of a (unique) controllable one and a (non-unique) autonomous one. For obvious reasons, representations of the type (1) are called kernel representations. There are many other ways of representing a behavior. A representation of the type.

(8) K ML (2) K O N with real polynomial matrices is called a latent vari if there exists able representation. In this L case, a latent variable trajectory such that this system of. differential equations is satisfied holds. The variables are called manifest variables. A particular case of this are image representations,.

(9) K ' L N ".

(10)

(11) JPRQ K T VS S U W. whence . It can be shown that a behavior is controllable if and only if it allows an image representation. Let be a behavior with . We call observable from if implies . It turns out that this is the case if and only if there exists a polynomial matrix such that implies Analogously, we call a latent variable representation observable if, whenever satisfy . It can be shown that a controllable (2), then admits an observable image representation. Every behavior also admits an input-output representation. After a reordering, if need be, of the components of the manifest variable , we obtain where denote inputs and outputs (see [4] for precise definitions of these concepts). An input/output partition of corresponds to a kernel of the form. X

(12) D N F D N GFY N$ D N GFY Y ED N /F[ Z N L Y N$ N L Y Y . . D. GFY

(13) J GFY Y. /F2

(14) Z VS S U \ GD]) L$Y^

(15) _L`Y Y. F. . . i. a

(16) cbedf hg'N(i . g. jk' i

(17) l gN

(18) mgNWi " !eo jq

(19) j p , and j , D l is with the properties that is square, n a matrix of (proper) rational functions. We call this an input i output representation of . The number of components is V # " : r < . % . given by. Two-variable polynomial matrices can be used in the theory of dissipative systems in a very effective way. Let denote the set of real two-variable polynomial matrices in the indeterminates and . An element of this set, say , is a finite sum. $steu vtNWwx. v. y. w. | | y %vrNWw H

(20) {z}| y v w ) ~ | bilinear differential form To each such y , we associate the | | HhN E

(21) z | W y E *) ~ h9N(3$s - %9N(3eu Note that is mapping from h 9 ( N to . If y is square then it induces a quadratic. differential form given by. l E

(22) N E *). . l l E 8 . y . The QDF , or simply , is said to be nonnegative along if for all . Bilinear and quadratic differential forms have been studied in detail in [6]. The (related) operators play an important role in this. Define by ,. N7N(N - s u vtN(wxT_ u s vtN(w y/ %vrNW Hw H

(23)

(24) y /m wN;v s u vtN(wxT_ s u vtN(w %vrNWw %vt-w @vtNWw s u vtN(wxT_ s u bybyy y h H

(25) y 6 tNW y , , s u _ s u by Z xh H

(26) Z / 6 . and

(27) y ,

(28) SVS U , and @

(29) q . In Note that y/ PQ @ if and only if y !`# % . fact, y. III. D ISSIPATIVITY. + OF UNCONTROLLABLE SYSTEMS l 7N y ' vrNWwx TSVS U \

(30) K TSVS U ML 'ex vtN(w. The behavior (not necessarily controllable) is said to be dissipative with respect to the storage function if there exists a latent variable representation of and a such that the dissipation inequality. l^ L$ 3Jl - " . L $. N SVS U 2

(31) 2K SVS U L . l^ is holds for all that satisfy called the storage function. When the dissipation inequality is y -lossless. If the holds as an equality, we say that Hx^ vtNWwx and storage function acts on , i.e. if l^¡¢£¤T¥M¦ - (3). for all , then we call the system dissipative with an. observable storage function. Non-negative storage functions are very important in applications, but we will not consider them in the present paper. Our storage functions need not be sign definite. In the sequel, we confine attention to single-input singleoutput behaviors, and (mostly) to observable storage functions. We assume that the behavior (with the manifest variable ) is governed by.

(32) bedf mgNWi § ¨ ' i

(33) § \© ' g (4) © ¨ © ¨ where , , § are scalar polynomials such that and are co-prime and § is monic. For the supply rate, we take ¬ y

(34) «ª ¬® l E

(35) °¯ g±i . Since © and ¨ are coprime, there exist i.e. polynomials ² and ³ such that © ² ¨ ³

(36) ¬") (5) A decomposition of the behavior , defined by (4), into controllable and autonomous parts can be obtained as follows. Let. -´

(37) ¶µ ©. 6 ¨·. ^¹¸

(38) ª § © ©º 6 § ¨'¨ º ® ³ ² 6 º for some polynomial . Define ´

(39) I"!$# ´$' " and ¸¹

(40) "!$# ^¹¸

(41) ´» ¸¹ yields a desired decomposition. In fact, Now, º parametrizes all possible autonomous parts. An alternative and also let. representation for the autonomous part is given by. ¸¹

(42) ½¼$ ¾

(43) ¿ª ² YY SVSS U ® L ³ SVU. and. § ' L5

(44) tÀ ".

(45) ² Y

(46) ² 6 ¨'º. ³ Y

(47) 6 ³ 6 ©º. . where and . Dissipativity of a controllable behavior is a wellunderstood subject. The present paper is an attempt to investigate dissipativity of uncontrollable systems. We begin by recalling well-known results for the controllable case. Before that, we need some nomenclature. Let be para-Hermitian, i.e. . A polynomial is called a symmetric factor of if . It is easy to see that a symmetric factor exists if and only if is real (hence is even) and non-negative for . The following proposition gives an answer to the question when a controllable system is dissipative. The proof follows from propositions 5.2, 5.6, and theorem 6.4 of [6]. Proposition 1: Let be given by (4) and let . The following statements are equivalent. 1) is -dissipative with an observable storage function, 2) is -dissipative, 3) admits a symmetric factor. The main contribution of the present paper is the following theorem which provides a necessary and sufficient condition for the dissipativity of an uncontrollable behavior. be given by (4). Assume that has no Theorem 2: Let roots on the imaginary axis. Then, the following statements are equivalent. 1) is -dissipative with an observable storage function. 2) admits a symmetric factor that is coprime with . ): (by contradiction) Suppose that 1 holds Proof: ( but 2 does not hold. In other words, suppose that is dissipative and all symmetric factors of have a common root with . As is -dissipative, its controllable part is also dissipative, i.e., there exists a such that. 3 Á Â Á

(48) Á 3 Ã Á Á

(49) Ã Ã mÄ@Å Á ÅÆ Á §

(50) ½¬. . y ©'¨ 3 y © ¨. . §. ©'¨ 9 y © ¨ § ¬ÇÈ¯. §. . y. ©'¨ E © ¨ . . y. y @ vt

(51) N(w 6 @vG Éw %vrNW w \Ê Z %v Z mw É ´ @v @vtN(w Ê D @vtF NWw -´hw (6) È and Ê for matrices Z D F some vtN(w .polynomial S \ ´

(52) ´ such that Z SVU , l a , trajectory -´[SVS U ´

(53) For ´ 5

(54) . Consequently, dissipativity N ´ of implies that the bilinear differential form , . and for all ´ such that must \ vanish for all

(55). S ´ [ ´ S ´ SVU

(56) Ë . Therefore, there exist Ì N(ÍÎ Z F SVFU vtN(w such that y @vtN(w 6 @vGÉw %vrNWw

(57) %v Ì %vtN(w ÏÍÏ%vrNWw ª Z ´ hhw w ® ) µ %v 6 ³ %v · , post-multiplication Pre-multiplication by ² e b " d f ¨. © (. m w N h w v

(58) 6 and w

(59) by , and evaluation at result in (7) ² © 6 ³ ¨Ð

(60) § Ò Ñ ÓtÑ Ô ) Ñ

(61) ` b " d f ¨ ©. Ñ Ó Ô N Ò Z for some polynomials and where . The ? above polynomial equation is satisfied if and only if for. B. , the implication. Ô %? H

(62) § 6 ? H

(63) Ç ² 6 ? W© @? 9

(64) ³ 6 ? ¨ %? Ô is nothing but a holds. On the other hand, the© polynomial ' © ¨ ¨ symmetric . To see this, postmultiply b`d"f factor ¨ hw N ©ofmw ( , premultiply (6) by of bedf ¨ mw N © hw W , and evaluate at v

(65) by6 theandtranspose w

(66) . So Ô %?

(67) Õ implies that © @? \¨ 6 ? © 6 ? ¨ %? Ö

(68) Õ . ? W© @? 6 ³ 6 ? ¨ @?

(69) × , this implies Together with ² 6 that ª ¨ 6 ?? © 6 ? ? ® ª ©¨ @%?? ®

(70) >) 6 ³ 6 ² 6 However, the first factor on the left hand side is© nonsingular ¨ due to (5) and the second factor is nonzero as and are coprime. This means the polynomial equation (7) has Ô andthat§ ifare a Y solution then necessarily coprime. ©¨ EDefine © ¨ Ô

(71) Ô . Note that Ô Y is a symmetric factor of Y. Ô N § is coprime. We reach a contradiction as we found and ©'¨ 3 © ¨ which is coprime with § . a symmetric factor of ¯^ÇØ¬ ): Let Ô be a symmetric factor of ©'¨ © ¨ which ( is coprime with § . Define Ù

(72) ª ©¨ ² ® N 6 · ³ , D

(73) µ Ô Ù N Ê %vtN(w ZÖ

(74) µ ² %v \© mw 6 ³ %v ¨ hw ³ %v ² mw · Ù , D hw e) Ù Note that is unimodular due to (5). Straightforward computation yields. Z Z Ï ´ Ê É Ê ´

(75) y ) Indeed, one can check above equation by pre-multiplying Ù and post-multiplying by Ù . Therefore, there both sides by exists an Ú such that y %vtN(w

(76) 6 %vGÉ w Ú %vrNWw \Ê Z %v Z mw É ´ @v @vtN(w Ê @vtNWw ´ hw N l , ´ 8 (8) Û. ´/ ´ . Let ¹ be the autonomous part correspondfor all ºÜ

(77)

(78) Ú ÂÝJÏÞß satisfies, for ing to . We claim that Ð Þ some , l , E 8 (9). if Ý and ß satisfy for all l à ´ H

(79) I for all ´/ ´ , i). ´ and l , Û ´

(80) implies 5 ´ ii) à N ´

(81) , Û N ´ for all , l ´ H

(82) I for all ´/ ´ , iii) á âN ´

(83) I for all ´/ ´ , and iv) á l ¹ 4 for all

(84) J p ¹ ¹. v) á. and hence.

(85) ã

(86) ½ ¹ ´ where ¹ ¹ and. /´ ´ l , E H

(87) l , ¹ ¯ , ¹ N ´ l , ´ ä (iv) å (i) and (iii)

(88) l , ¹ ¯ , , à ¹ N ´ l , ´ e) Û Û. Í F(F %vrNWw § @ v Ì D\F %vrNWw Ê F

(89) % vEíw ì F @vtNWw ì F %vrNWw W. To prove this, take any .Then, we get. Note that (13) and (16) are solvable as soon as (14) and (15) are. Also note that (14) and (15) are solvable if and only if. Í F;D %vrNWw Z D mw § @ v Ì D(D %vrNWw

(90) Í Í F*D Ô § Ì DVD É. Þ. Note that the first summand of the last line can be made arbitrarily large by choosing sufficiently large due to (v). The last summand is already nonnegative due to (8). Together with (ii), these imply that (9) holds for all if we choose sufficiently small. To finish the proof, we will show the existence of and such that (i)-(v) are satisfied. To do so, take. . Þ. Ý. ß. ßH@vtNWw H

(91) ¿ª © ¨ % v % v ®9æ @v æ mw µ © mw 6 ¨ mw · (10) 6 where æ induces a state map for ¹ , the corresponding state model is given by. Ô. Ý. Ý. Í ì Ì Ý&%vrNWw H

(92) ´ @v ì %vrNWw ì %vtN(w ´mw N @vGíw Ý

(93) ´ %v WÊ É ¹ @v Ì ÏÍ ª -Z ´$mmw w \® ). Consider the following partitions. ì %vrNWw Ùkmw H

(94) Ëµ ì D %vtN(w DVD %vtN(w Ì %vrNWw Ùhw

(95) «ª ÌÌ F*D %vtN(w Ù %v ÍÏ@vtN(w 9

(96) «ª ÍÍ F*DVDD @@vtvtNWNWww . Ùmw . Ý&%vtN(w . Ù/@v . (11) (12). ©¨ E © ¨ §

(97) § D§ F. §. . §D § Fë

(98) © ¨ © ¨ ©¨ © ¨ §F. §. y. ¨§. ©¨ ^ © ¨. §. . ´. IV. U NOBSERVABLE. Pre-multiplying (11)-(12) by , post-multiplying by , and eliminating , yields the system of polynomial equations:. Ì F*D @vtN(w ÉÍ DVD @vtN(w Ô 6 w

(99) N Í WD F @vtNWw Ì VF F @vtN(w

(100) @vGíw ì D %vtN(w N Í F;D %vtN(w Ô 6 w § %v Ì DVD @vtN(w ÊD %vrNWw

(101) %vGÉw ì D @vtN(w N. §. Remarks: 1. Note that the roots of are symmetric with respect to imaginary axis. Let where has no symmetric roots with respect to imaginary axis and . Then, there exists a symmetric factor of which is coprime with only if and are coprime. In particular, when has no symmetric roots with respect to imaginary axis (i.e., it has no even factor), the behavior is -dissipative if and only if its controllable part is. 2. In [3] a sufficient condition for the passivity of uncontrollable multiple input / multiple output state space systems is given. For the single input / single output, the condition given in [3] comes down to the requirement that should have no symmetric roots with respect to the imaginary axis. The first remark shows that this is a special case of theorem 2. 3. When the controllable part is lossless, it can be shown is identically zero. Thus, the coprimeness that condition of theorem 2 holds only if is a constant and hence the behavior is controllable. On the other extreme, when is a constant, the controllable part is strictly dissipative, i.e. the dissipation inequality (3) holds with the strict inequality for all nonzero trajectories of . Then, coprimeness condition of theorem 2 readily holds independently on the autonomous part.. ©'¨ C © ¨. ì F @ vtNWw \· N Ì FVDWFF %%vrvrNWNWww \® N Ì Í D\F %vtN(w ) Í F(F %vtN(w ®. ÊD @vtNWw D\ F %vrNWw Ì F( F %vrNWw ÊD/

(102) . are solvable. Clearly, the former equation is solvable as soon as the latter is. As and are coprime, the latter always admits a solution. This end the proof of theorem 2.. î § F ç

(103) æ ' \ ¹ Nè ç

(104) é ç N ¹

(105) ê ç N " é " ê for some matrices and with appropriate sizes, and é é 7 ë Æ 4 . Such an exists since § has is such that. no roots on the imaginary axis. It can be easily checked that (iii)-(v) are satisfied by the choice of (10). Therefore, it remains to show the existence of a satisfying (i)-(ii). We know from [6, proposition 3.2 and proposition 3.5] that there exists satisfying (i)-(ii) if and only if there exist two-variable polynomial matrices , , and such that. (16). (13) (14). (15). STORAGE FUNCTIONS. Theorem 2 deals with observable storage functions, i.e. storage functions that are only functions of the manifest variables. The use of uncontrollable systems and/or unobservable storage functions is of considerable importance. We discuss this in the present section. Example: Consider the system. i

(106) g) . g±i. It follows from theorem 2 that with respect to the supply rate , this system does not have an observable storage function..

(107) Consider however the latent variable representation given by. çD

(108) çF ç F-

(109) I i

(110) ç F íg ). l ç I

(111) D ç F Ó ç D SVS U ç F

(112) Ó ç FF

(113) g F Ïg ç F Ó ç FF Ó ç 6 6. FF Ó ç D ç F. g. çF. An important area of application of the ideas of this article is the area of electrical circuits. I. external port. +. R. C. C. L. R. L. %ñHN(ò . FD ê ñ ó F FD Hò ô F ê ó

(114) p õ ô ö ê ó

(115) õôö. %ñHN(ò ó ê ó ñ

(116) ¬ ê ó ó ò ) ô " . % ñ V N ò ô are then unobservable from %ñHN(ò , The variables ó and hence, the stored energy an unobservable ê ó becomes

(117) õô ö and ó

(118) p ô , storage function. When. then the manifest behavior is controllable. So, there exists an observable storage function. In fact, classical results from electrical circuit synthesis allow to conclude that the port behavior can also be realized using passive elements (resistors and one capacitor, in fact), and with observable branch currents and voltages. However, when and , the port behavior becomes uncontrollable and theorem 2 shows that there does not exist an observable storage function. In fact, in this case, it can be shown that there does not exist a passive synthesis with only one reactive element. So, in this sense, the realization which we. ó

(119) ô. ê ó

(120) õôö. ò± Ë

(121). as its driving point impedance, but not its behavior (which admits, for example, the short circuit response not realized by the resistor). An example of a circuit that does realize this behavior exactly is the above circuit, with , so this uncontrollable behavior is realizable. This leads to two nice open problems: Problem 1: What behaviors are realizable as the port behavior of a circuit containing a finite number of passive resistors, capacitors, inductors, and transformers? It is easy to see that must be single input / single output, and that the transfer function must be rational and positive real. In addition must be passive, but in general with a non-observable storage function, and therefore it is not clear what this says in terms of . Problem 2: Is it possible to realize a controllable single input / single output system with a rational positive real transfer function as the behavior of a circuit containing a finite number of passive resistors, capacitors, and inductors, but no transformers? Note that in a sense this is the BottDuffin problem, the issue being that the Bott-Duffin synthesis procedure usually realizes a non-controllable system that has the correct transfer function (i.e., the correct controllable part), but not the correct behavior. There are standard synthesis procedures known that do realize the correct behavior, but they need transformers.. ÷ , U NVñø ù

(122) ú N. Consider for example the circuit shown, and regard as manifest and the internal branch currents and voltages as latent variables (see [4], pages 10-13, 160-161, and 175176 for a derivation of the equations and the analysis of the controllability and observability properties of this circuit). Of course, this circuit is dissipative with respect to the supply with the internal energy, (in the rate obvious notation) as the storage function. If , this circuit is controllable and observable (in the sense that the branch currents and voltages are observable from the , the port voltage and current). However, when differential equation that governs is. ñ^ò. . ñ Âñ

(123) òEÏò>N . V −. The most classical result of circuit theory is undoubtedly the fact that is the driving point impedance of a circuit containing a finite number of passive resistors, capacitors, inductors, and transformers if and only if is rational and positive real. This result was obtained by Brune [2] in his MIT Ph.D. dissertation. In 1949, Bott and Duffin [1] proved that transformers are not needed. It seems to us that a more ‘complete’ version of this classical problem is to ask for the realization of a differential behavior. This problem is somewhat more general than the driving point impedance problem, because of the existence of uncontrollable systems. For example, a unit resistor realizes the transfer function of the system. . If we allow the storage function to be a function of the latent variables, then this system becomes dissipative. To see this, consider the storage function . So, . Then, . It is easy to verify that this expression is nonnegative for all and if .. l ç

(124) Ó VS S U ç l E 6 l ç

(125) ±g i Ó 6 ¬ï]ð. started from is a minimal one. Of course, all this shows the limited relevance of the classical notion of minimal (controllable and observable) state space representations in the context of physical systems..

(126) Ø¬ N êû

(127) X¬ (N ô

(128) Ø¬ (N ó

(129) Ø¬ + F. . . . V. ACKNOWLEDGMENTS This research is supported by the Belgian Federal Government under the DWTC program Interuniversity Attraction Poles, Phase V, 2002 - 2006, Dynamical Systems and Control: Computation, Identification and Modelling, by the KUL Concerted Research Action (GOA) MEFISTO-666, and by several grants en projects from IWT-Flanders and the Flemish Fund for Scientific Research..

(130) VI. REFERENCES [1] R. Bott and R.J. Duffin, Impedance synthesis without transformers, Journal of Applied Physics, volume 20, page 816, 1949. [2] O. Brune, Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency, Journal of Mathematics and Physics, volume 10, pages 191-236, 1931. [3] A. Ferrante and L. Pandolfi, On the solvability of the positive real lemma equations, Systems & Control Letters, volume 47, pages 211-219, 2002.. [4] J.W. Polderman and J.C. Willems, Introduction to Mathematical Systems Theory, A Behavioral Approach, Springer Verlag, 1997. [5] J.C. Willems, Dissipative dynamical systems Part I: General theory; Part II: Linear systems with quadratic supply rates, Archive for Rational Mechanics and Analysis, volume 45, pages 321-393, 1972. [6] J.C.Willems and H.L. Trentelman, On quadratic differential forms, SIAM Journal of Control and Optimization, volume 36, pages 1703-1749, 1998..

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