On Reciprocal Systems and Controllability

On reciprocal systems and controllability

Timothy H. Hughes a

aDepartment of Mathematics, University of Exeter, Penryn Campus, Penryn, Cornwall, TR10 9EZ, UK

Abstract

In this paper, we extend classical results on (i) signature symmetric realizations, and (ii) signature symmetric and passive realizations, to systems which need not be controllable. These results are motivated in part by the existence of important electrical networks, such as the famous Bott-Duﬃn networks, which possess signature symmetric and passive realizations that are uncontrollable. In this regard, we provide necessary and suﬃcient algebraic conditions for a behavior to be realized as the driving-point behavior of an electrical network comprising resistors, inductors, capacitors and transformers.

Key words: Reciprocity; Passive system; Linear system; Controllability; Observability; Behaviors; Electrical networks.

1 Introduction Practical motivation for developing a theory of reciprocity that does not assume controllability arises from This paper is concerned with reciprocal systems (see, electrical networks. Notably, the driving-point behavior e.g., Casimir, 1963; Willems, 1972; Anderson and Vong- of an electrical network comprising resistors, inductors, panitlerd, 2006; Newcomb, 1966; van der Schaft, 2011). capacitors and transformers (an RLCT network) is nec- Reciprocity is an important form of symmetry in physi- essarily reciprocal, and also passive, 2 but it need not be cal systems which arises in acoustics (Rayleigh-Carson controllable (see Çamlibel et al., 2003; Willems, 2004; reciprocity); elasticity (the Maxwell-Betti reciprocal Hughes, 2017d). Indeed, as noted by Çamlibel et al. work theorem); electrostatics (Green’s reciprocity); and (2003), it is not known what (uncontrollable) behav- electromagnetics (Lorentz reciprocity), where it follows iors can be realized as the driving-point behavior of an as a result of Maxwell’s laws (Newcomb, 1966, p. 43). RLCT network. In addition, an RLCT network need not Special cases of reciprocal systems include reversible possess an impedance function, so the conventional def- systems, as arise in thermodynamics; and relaxation inition of a reciprocal system as one with a symmetric systems, such as viscoelastic materials (Willems, 1972). transfer function is inappropriate for such networks. In addition, reciprocity is a property of important classes of electrical, mechanical and structural systems, The purpose of this paper is to address the aforemen- such as lightly damped flexible structures (Ferrante and tioned limitations with the theory of reciprocity. The Ntogramatzidis, 2013). Our focus in this paper is on paper is structured as follows. In Section 3, we review linear reciprocal systems. In contemporary systems and the classical theory of reciprocal systems in more detail, arXiv:1712.09436v2 [cs.SY] 4 Apr 2018 control theory, a linear reciprocal system is typically with a particular focus on passive and reciprocal sys- defined as a system with a symmetric transfer function. tems, to highlight the limitations of the existing theory A fundamental result in systems and control theory and the contributions of this paper. In Sections 4 and states that if the transfer function is also proper, then 5, we proceed to develop a theory of reciprocal systems the system possesses a so-called signature symmetric which addresses these limitations. Sections 6 and 7 are realization (see Willems, 1972; Anderson and Vongpan- then concerned with systems that are both reciprocal itlerd, 2006; Fuhrmann, 1983; Youla and Tissi, 1966). and passive, such as RLCT networks. The main results However, this result is subject to one notable caveat: are summarised in the following two paragraphs. the system is assumed to be controllable. In Section 4, we provide a formal definition of reciprocity (Definition 5), which was first proposed by Newcomb (1966). The main advantage of this definition Email address: [email protected] (Timothy H. Hughes). 1 c 2018. This manuscript is made available under the 2 A system is passive if the net energy that can be extracted CC-BY-NC-ND 4.0 license http://creativecommons.org/ from the system into the future is bounded above (this bound licenses/by-nc-nd/4.0/ depending only on the past trajectory of the system).

Preprint submitted to Automatica 17 July 2018 is that it does not assume the existence of a symmetric then spec(M) := {λ ∈ C | det(λI−M) = 0}; and if, in transfer function. This is particularly fitting in the con- addition, M is symmetric, then M > 0 (M ≥ 0) indi- text of electrical networks as these need not possess an cates that M is positive (non-negative) definite. A ma- impedance function. We then provide a 2-part theorem trix Σ ∈ Rn×n is called a signature matrix if it is diagonal which we call the reciprocal behavior theorem. In part 1 and all of its entries are either 1 or −1. A V ∈ Rn×n[ξ] (Theorem 7), we provide necessary and sufficient con- is called unimodular if det (V ) is a non-zero constant ditions for a system to be reciprocal in terms of the dif- (equivalently, V is nonsingular with V −1 ∈ Rn×n[ξ]). If ferential equations describing the system. We also prove H ∈ Rn×n(ξ), then H is called positive-real if H is ana- ¯ T that, for any given reciprocal system, it is possible to lytic in C+ and H(λ) + H(λ) ≥ 0 for all λ ∈ C+. permute the system’s variables to obtain a system with The (k-vector-valued) locally integrable functions a proper symmetric transfer function. Part 2 (Theorem loc k 9) then proves the existence of a signature symmetric re- are denoted L1 R, R (Polderman and Willems, alization for any given system with a proper symmetric 1998, Defns. 2.3.3, 2.3.4), and we equate any two lo- transfer function (irrespective of controllability). cally integrable functions that differ only on a set of measure zero. The (k-vector-valued) infinitely dif- Section 6 contains another 2-part theorem: the passive ferentiable functions with bounded support on the and reciprocal behavior theorem. Part 1 (Theorem 17) left (resp., bounded support on the right, bounded provides necessary and sufficient algebraic conditions for k k support) are denoted D+ R, R (resp., D− R, R , a system to be passive and reciprocal in terms of the dif- k D R, R ). The convolution operator is denoted by ferential equations describing the system. This theorem k also answers the first open problem posed in Çamlibel ?; i.e., if w1, w2 ∈ D+ R, R , then (w1 ? w2)(t) = R ∞ T et al. (2003) in the more general setting of multi-port −∞ w1(τ) w2(t − τ)dτ. networks: it is shown that a behavior B is realizable as A main contribution of this paper is to develop a theory the driving-point behavior of an RLCT network if and of reciprocal systems which doesn’t assume controllabil- only if B is passive and reciprocal. Part 2 (Theorem 18) ity, observability, or the existence of a transfer function. then proves the existence of a passive and signature sym- This is relevant to electric networks which can possess metric realization for any given passive system with a uncontrollable and unobservable internal modes, and proper symmetric transfer function. The results in this whose driving-point currents and voltages need not ad- section build on earlier results in (Hughes, 2017c,a) on here to the conventional system theoretic input-output systems which are passive but not necessarily reciprocal. view. The natural framework to formalise these issues The extension to consider passive and reciprocal systems is the behavioral approach (Polderman and Willems, is by no means trivial, and depends on a number of sup- 1998). Accordingly, the remainder of this section con- plementary lemmas that are provided in Section 7 and tains relevant definitions and results on behaviors. Appendix B. Finally, the proofs in the paper, together with existing results in the literature, provide an algo- We consider behaviors (systems) defined as the set of rithm for constructing an RLCT network realization of weak solutions (see Polderman and Willems, 1998, Sec- an arbitrary given reciprocal and passive behavior. This tion 2.3.2) to a differential equation: is illustrated by two examples in Section 8. loc q d p×q B = {w ∈ L1 (R, R ) | R( dt )w=0},R ∈ R [s]. (2.1) 2 Notation and Preliminaries The behavior B is called controllable if, for any two tra- jectories w , w ∈ B and t ∈ , there exists w ∈ B We denote the real and complex numbers by R and C, 1 2 0 R and t ≥ t such that w(t) = w (t) for all t ≤ t and and the open and closed right-half plane by C+ and C+. 1 0 1 0 w(t) = w (t) for all t ≥ t (Polderman and Willems, If λ ∈ C, then λ¯ denotes its complex conjugate. The poly- 2 1 nomials, rational functions, and proper (i.e., bounded at 1998, Definition 5.2.2). From (Polderman and Willems, infinity) rational functions in the indeterminate ξ with 1998, Theorem 5.2.10), B in (2.1) is controllable if and only if rank(R(λ)) is the same for all λ ∈ . real coefficients are denoted R[ξ], R(ξ), and Rp(ξ). The C m×n matrices with entries from R (resp., R[ξ], R(ξ), We pay particular attention to state-space systems: m×n m×n m×n Rp(ξ)) are denoted R (resp., R [ξ], R (ξ), m×n(ξ)), and n is omitted if n = 1. We denote the loc n loc n loc d Rp Bs = {(u, y, x)∈L1 (R, R ) ×L1 (R, R ) ×L1 R, R | block column and block diagonal matrices with entries dx = Ax + Bu and y = Cx + Du}, H1,...,Hn by col(H1 ··· Hn) and diag(H1 ··· Hn); dt d×d d×n n×d n×n and we will use horizontal and vertical lines to indicate A ∈ R ,B ∈ R ,C ∈ R and D ∈ R . (2.2) the partition in block matrix equations (e.g., see (B.5)). m×n m×n m×n T If H ∈ R , R [ξ], or R (ξ), then H denotes Here, we call the pair (A, B) controllable if Bs is con- its transpose, and if H is nonsingular (i.e., det (H) 6= trollable; and we call the pair (C,A) observable if −1 m×n 0) then H denotes its inverse. If H ∈ R , then (u, y, x) ∈ Bs and (u, y, xˆ) ∈ Bs imply x = xˆ (Pol- rank(H) denotes its rank; and if G ∈ Rm×n(ξ), then derman and Willems, 1998, Definition 5.3.2). These m×m normalrank(G) := maxλ∈C(rank(G(λ))). If M ∈ R , concepts are equivalent to the well known algebraic

2 conditions for controllability/observability of a pair of for k = 0, 1, 2,.... 3 Note that the equivalence of matrices (see Polderman and Willems, 1998, Chapter 5). transfer functions (condition (i)) is necessary but not sufficient. E.g., let B = 0, C = 1 and D = 1, so We also consider behaviors obtained by transforming D + C(ξI − A)−1B = 1 for all A ∈ . If A = −1, then and/or eliminating variables in a behavior B as in (2.1). R (u, y) ∈ B(u,y) if and only if there exists k ∈ such that For example, associated with the state-space system Bs s 1 R (u,y) −t (u,y) in (2.2) is the corresponding external behavior Bs = y(t) = u(t) + k1e . But if A = 0, then (u, y) ∈ Bs if {(u, y) | ∃x with (u, y, x) ∈ Bs}. More generally, if T1 ∈ and only if there exists k2 ∈ R such that y(t) = u(t)+k2. p1×q pn×q R ,...,Tn ∈ R are such that col(T1 ··· Tn) ∈ Rq×q is a nonsingular real matrix, and m is an integer 3 Signature symmetric realizations of symmet- satisfying 1 ≤ m ≤ n, then we denote the projection of ric transfer functions B onto T w,...,T w by 1 m The following fundamental result in systems and control theory states that any given controllable system with a B(T1w,...,Tmw) = {(T w,...,T w) | ∃(T w,...,T w) proper symmetric transfer function has a so-called sig- 1 m m+1 n nature symmetric realization. such that w ∈ B}. Lemma 3 Let Bˆ in (2.3) be controllable. Then the following are equivalent. A representation for the behavior B(T1w,...,Tmw) can be obtained by the so-called elimination theorem (see Ap- 1. Qˆ−1Pˆ is symmetric. pendix A). In particular, by eliminating the state vari- 2. There exists Bs as in (2.2) and a signature matrix d×d ˆ (u,y) ables x from Bs, we obtain a behavior of the form Σi ∈ R such that (i) B = Bs ; (ii) (A, B) is controllable; (iii) (C,A) is observable; and (iv) AΣi = T T T ˆ loc n loc n ˆ d ˆ d ΣiA , ΣiC = B, and D = D . B={(u, y)∈L1 (R, R ) ×L1 (R, R ) | P ( dt )u=Q( dt )y}, P,ˆ Qˆ ∈ n×n[ξ], Qˆ nonsingular and Qˆ−1Pˆ proper. (2.3) R PROOF. If Bˆ in (2.3) is controllable, then there exists Bs as in (2.2) which satisfies (i)–(iii) in condition 2 (see More specifically, from (Hughes, 2016, Sections 2 and 4) Hughes, 2017c, Appendix D). Furthermore, D + C(ξI − we have the following lemma on behavioral realizations. A)−1B = (Qˆ−1Pˆ)(ξ), and it is then easily verified that if A, B, C and D are as in condition 2 then Qˆ−1Pˆ is sym- Lemma 1 Let Bs be as in (2.2) and A(ξ):=ξI−A. There exist polynomial matrices P,ˆ Q,ˆ Y, Z, U, V, E, F, G where metric. This proves that 2 ⇒ 1. The proof of 1 ⇒ 2 then follows from (Willems, 1972, Theorem 6) (alternatively, " #" # " # YZ −DI −C −Pˆ Qˆ 0 see Anderson and Vongpanitlerd, 2006; Fuhrmann, 1983; 1. = ; Youla and Tissi, 1966; Reis and Willems, 2011). This UV −B 0 A −E −FG proof proceeds by first showing that, if Aˆ ∈ Rd×d, Bˆ ∈ " # YZ Rd×n, Cˆ ∈ Rn×d and Dˆ ∈ Rn×n are such that Dˆ + 2. is unimodular; and ˆ ˆ −1 ˆ ˆ ˆ UV C(ξI − A) B is symmetric, (A, B) is controllable, and ˆ ˆ 3. G is nonsingular. (C, A) is observable, then there exists a nonsingular symmetric P ∈ Rd×d such that P Aˆ = AˆT P , CˆT = ˆ T Furthermore, if conditions 1–3 hold and B is as in (2.3), P Bˆ and Dˆ = Dˆ . Note that, with the notation Vô = (u,y) ˆ d−1 d−1 then Bs = B, and we say that (A, B, C, D) is a realiza- col(Cˆ CÂ...ˆ CÂˆ ) and Vˆc = [Bˆ AˆB...ˆ Aˆ Bˆ], ˆ ˆ ˆ ˆ ˆ T tion of (P, Q). Also, if B is as in (2.3), then there exists then P Vc = Vo , whereupon P can be computed from the B as in (2.2) and polynomial matrices Y,Z,U,V,E,F ˆ T ˆ T ˆ ˆ T −1 s explicit formula P = Vo Vc (VcVc ) (Anderson and and G satisfying conditions 1–3. Vongpanitlerd, 2006, Section 7.4). Since P is symmetric, d×d ˆ then there exists a signature matrix Σi ∈ R and a Remark 2 For a given behavior B as in (2.3), al- d×d T gorithms for computing a realization (A, B, C, D) nonsingular T ∈ R such that P = T ΣiT . We then let A := T ATˆ −1,B := T B,Cˆ := CTˆ −1 and D := Dˆ. for (P,ˆ Qˆ) (i.e., a state-space system Bs such that (u,y) Bs = Bˆ) are described in (Fuhrmann et al., 2007, Section 4.7) and (Hughes, 2016, Section 4). Such be- Of particular interest are controllable systems with havioral realizations are not unique. Indeed, it is easily proper symmetric transfer functions that are positive- shown from (Hughes, 2017a, Note A.3) that (A,ˆ B,ˆ C,ˆ Dˆ) real. These arise as the impedances of electrical networks containing resistors, inductors, capacitors and is another realization for (P,ˆ Qˆ) if and only if (i) ˆ ˆ ˆ −1 ˆ −1 D + C(ξI − A) B = D + C(ξI − A) B; and (ii) there 3 In fact, by the Cayley Hamilton theorem, it can be shown dˆ×d d×dˆ exist matrices T1 ∈ R and T2 ∈ R such that that these two conditions hold if and only if they hold for i i k k CA T1 = CÂˆ for i = 0, 1, 2,..., and CÂˆ T2 = CA i = 0, 1, . . . , d and k = 0, 1,..., dˆ.

3 transformers (RLCT networks). In fact, such systems behavior is described by the state-space realization have a particular physical relevance, since every known √ physical system with a non-symmetric positive-real √ − 2 −2 0 0 −√3 0 0  0 impedance actually contains active components (see 0 0 0 − 2 0 0 0 dx √ √ q  Ferrante et al., 2016). A second fundamental result √0√ 0 0 0 2 − 3  3  =   x +  2  i in systems and control theory is that any controllable dt  3 2√ 0 0 0 0    0 0 − 2 0 0 0  0  system with a proper symmetric positive-real transfer √ 1 0 0 3 0 0 −2 √ function has a so-called passive and signature symmetric 2 h √ q 3 1 i realization, in accordance with the following lemma. v = − 2 0 0 0 √ x + i, where 2 2 Lemma 4 Let Bˆ in (2.3) be controllable. Then the fol- √ q √ √ x = col( √v1 3 v √v3 2 i 3i 2i ) lowing are equivalent: 2 2 2 6 3 4 5 6 ˆ−1 ˆ 1. Q P is positive-real and symmetric. This realization satisfies conditions (iv) and (v) of 2. There exists Bs as in (2.2) and a signature matrix Lemma 4, but is neither controllable nor observable. d×d ˆ (u,y) Σi ∈ R such that (i) B = Bs ; (ii) (A, B) is controllable; (iii) (C,A) is observable; The aforementioned RLCT networks indicate the impor- tance of removing the assumptions of controllability, ob- " # " #T −A −B −A −B servability, and existence of a proper symmetric transfer (iv) + ≥ 0; and function from Lemmas 3 and 4. This is the objective of CD CD this paper. Theorem 9 (resp., 18) generalizes Lemma 3 T T T (v) AΣi = ΣiA , ΣiC = B, and D = D . (resp., 4) to systems that need not be controllable. Also, Theorems 7 and 17 extend the results to systems that do not necessarily possess a proper symmetric transfer function. In particular, Theorem 17 provides necessary and sufficient conditions for a behavior to be realizable PROOF. See (Willems, 1972, Theorem 7). by an RLCT network, thereby answering the first open problem in Çamlibel et al. (2003). To conclude this section, we discuss some recent develop- ments in the literature on uncontrollable systems, and we contrast these with the results in the present paper. Mo- Using the reactance extraction approach, any realiza- tivation for developing a theory of reciprocity that does tion of the form of Lemma 4 gives rise to an RLCT net- not assume controllability was provided in the behav- work whose impedance is equal to Qˆ−1Pˆ (see Anderson ioral literature in Çamlibel et al. (2003); Willems (2004). and Vongpanitlerd, 2006). However, Lemma 4 contains Indeed, as previously noted, Çamlibel et al. (2003) stated several notable assumptions that are not satisfied by an open problem that we solve in this paper: what be- many RLCT networks. First, the theorem assumes the haviors are realizable as the port (driving-point) behavior existence of a proper symmetric transfer function, yet of a circuit containing a finite number of passive resis- not all RLCT networks possess a proper impedance (see tors, capacitors, inductors and transformers? This ques- Hughes, 2017c, Section 3). Second, the theorem assumes tion concerns (not necessarily controllable) systems that the system is controllable, but not all RLCT networks are both passive and reciprocal. There have since been have controllable driving-point behaviors. Examples in- papers that have considered the question of uncontrol- clude the famous Bott-Duffin networks and their sim- lable passive systems (e.g., Hughes, 2017c), and uncon- plifications (see Hughes and Smith, 2014, 2017; Hughes, trollable (cyclo)-dissipative systems (e.g., Pal and Belur, 2017d). One such network is provided in Fig. 1, whose 2008). 4 But no papers have considered uncontrollable reciprocal systems. For example, consider the behavior ˜ loc loc d B := {(i, v) ∈ L1 (R, R) ×L1 (R, R) | ( dt + 1)i = d dv 1 di3 i = 4v ( +1)( +v)}. It has been shown in (Hughes, 2017c,b) v3 = 2 dt 8 8 dt dt i1 i8 that B˜ can be realized as the driving-point behavior of an electrical network containing resistors, inductors, trans- v = 3 di2 v = 1 di3 i = 3 dv5 2 4 dt 3 6 dt 5 dt formers and gyrators (an RLCTG network). The present i2 v7 = i7 paper provides the first proof that (i) this behavior has i3 = i5 i 2 dv4 i dv6 4 = 3 dt 6 = 2 dt 4 i6 A system is cyclo-dissipative if it has a (not necessarily i7 i non-negative) storage function with respect to some supply 4 rate. It is shown in (Hughes, 2017a) that a system is passive (in the sense of Definition 13 of the present paper) if it is Fig. 1. Bott-Duffin realization of the driving-point behavior d d2 d d d2 d cyclo-dissipative with respect to the energy supplied to the ( dt + 1)( dt2 + dt + 1)i = ( dt + 1)( dt2 + dt + 4)v system, and the associated storage function is non-negative.

4 a signature symmetric realization; and (ii) it can be re- Remark 6 Our objective in this paper is to develop alized without gyrators (i.e., by an RLCT network). a concept of reciprocity that is consistent with the existence of signature symmetric realizations, and the In fact, as discussed by Willems (2004), the subject of driving-point behaviors of RLCT networks. We note uncontrollable reciprocal systems is related to a subtle that a behavior is reciprocal if and only if its control- yet significant question in the development of the theory lable part is reciprocal (this follows from Definition 5 of uncontrollable (cyclo)-dissipative systems: whether to and Lemma 11). In particular, it will follow from The- allow unobservable storage functions. In particular, both orems 9 and 17 that whether a system has a signature Çamlibel et al. (2003) and Pal and Belur (2008) define symmetric realization depends only on its controllable (cyclo)-dissipativity in terms of the existence of an ob- part, and whether the driving-point behavior of an servable storage function. Yet, in Willems (2004, Sec- electric network can be realized without gyrators also tion VI), it is demonstrated that systems that are not depends only on its controllable part. (cyclo)-dissipative in accordance with this definition can nevertheless possess an unobservable storage function. The next theorem shows that any given reciprocal sys- Moreover, unobservable storage functions arise in elec- tem B can be transformed into a system of the form of trical networks. In fact, if we consider an uncontrollable (2.3) that is also reciprocal (condition 3 in Theorem 7). behavior with a state-space realization that satisfies the In addition, a necessary and sufficient condition for reci- signature symmetry of condition (v) of Lemma 4, then it procity is provided in terms of the polynomial matrices can be shown that this realization is both uncontrollable P and Q (condition 2 in Theorem 7). and unobservable. It can also be shown that any RLCT realization of an uncontrollable behavior necessarily has Theorem 7 (Reciprocal behavior theorem, part 1) an unobservable storage function, corresponding to the Let B be as in (4.1). The following are equivalent: energy stored in the network’s inductors and capacitors. 1. B is reciprocal. 2. PQT = QP T . Given the aforementioned issues with the question of un- 3. There exist real matrices T ∈ r×n and T ∈ observable storage functions, the approach in this paper 1 R 2 (n−r)×n such that (i) col(T T ) is a permutation is aligned with Hughes (2017c). That paper provided a R 1 2 ˆ (col(T1i −T2v),col(T1v T2i)) theory of passivity that does not assume controllability matrix; and (ii) B := B or observability (and also removes other alternative as- takes the form of (2.3) and Qˆ−1Pˆ is symmetric. sumptions prevalent in the literature). We refer to that Remark 8 A well known result in behavioral theory is paper for results pertaining to passivity. In this paper, that any behavior B of the form of (4.1) necessarily has our focus is on developing the theory of (not necessarily an input-output partition. However, condition 3 of The- controllable) reciprocal systems. orem 7 is not a trivial application of this result. Specifi- cally, in the definition of a reciprocal system (Definition 4 Reciprocal behaviors 5), the system’s variables are partitioned into two sets, with an equal number of variables in each set (in the con- Following the motivation outlined in the previous sec- text of electrical networks, these two sets correspond to tions, our focus in this paper is on systems of the form: the driving-point currents and voltages). Condition 3 of Theorem 7 implies that if the system is reciprocal then loc n loc n d d it is possible to choose as input a subset of the variables B={(i, v)∈L1 (R, R ) ×L1 (R, R ) | P ( dt )i=Q( dt )v}, from one of the sets together with the complementary with P,Q ∈ n×n[ξ], normalrank([P −Q]) = n. (4.1) R variables from the other set. Note from the example in (Hughes, 2017c, Remark 11) that this need not be true The driving-point behavior of any passive electrical cir- if the system is not reciprocal. cuit necessarily has the above form, where i denotes the driving-point currents and v the corresponding driving- We will also show that the system Bˆ in condition 3 of (u,y) point voltages (see Hughes, 2017b). We note that the Theorem 7 has a state-space realization Bˆ = Bs with partitioning (i, v) need not be an input-output partition the properties described in the next theorem. in the sense of (Polderman and Willems, 1998, Defini- tion 3.3.1). Specifically, Q need not be nonsingular, and Theorem 9 (Reciprocal behavior theorem, part 2) if Q is nonsingular then Q−1P need not be proper. In Let Bˆ be as in (2.3). Then the following are equivalent. this more general setting, it is not possible to define a ˆ reciprocal system as a system whose transfer function 1. B is reciprocal. is symmetric. Instead, we adopt the following definition 2. There exists Bs as in (2.2) and a signature matrix d×d ˆ (u,y) from (Newcomb, 1966, Definition 2.7). Σi ∈ R such that (i) B = Bs ; and (ii) AΣi = T T T ΣiA , ΣiC = B, and D = D . Definition 5 (Reciprocal system) Let B be as in (4.1). B is called reciprocal if, for any given The two-part reciprocal behavior theorem (Theorems 7 n n (ia, va), (ib, vb) ∈ B ∩ (D+ (R, R ) × D+ (R, R )), then and 9) is proved in Section 5. Then, in Sections 6–7, we vb ? ia = ib ? va. consider behaviors that are both reciprocal and passive.

5 n d Remark 10 We emphasise that Lemma 3 is concerned Then zˆ2 ∈ D− (R, R ), i2(t0 − τ) = (M(− dt )zˆ2)(τ) and only with controllable systems, whereas Theorem 9 is d v2(t0 − τ) = (N(− )zˆ2)(τ) for all τ ∈ R. Thus, applicable to any system of the form of (2.3), irrespective dt ˆ of controllability. Note that, if B in (2.3) is not control- R ∞ d T d (v2 ? i1)(t0) = (N(− )zˆ2)(τ) (M( )z1)(τ)dτ, and lable, and B in (2.2) satisﬁes Bˆ = B(u,y), then (A, B) −∞ dt dt s s R ∞ d T d cannot be controllable, so Lemma 3 does not apply. (i2 ? v1)(t0) = −∞(M(− dt )zˆ2)(τ) (N( dt )z1)(τ)dτ.

5 Reciprocity and signature symmetric realiza- It follows from (van der Schaft and Rapisarda, 2011, tions Section 2.2) that

The purpose of this section is to prove the reciprocal (v ?i −i ?v )(t ) = R ∞ zˆ (τ)T ((N T M−M T N)( d )z )(τ)dτ. behavior theorem, parts 1 and 2 (Theorems 7 and 9). 2 1 2 1 0 −∞ 2 dt 1 We first present the following lemma on the so-called controllable and autonomous parts of a behavior. Since t0 is arbitrary, then we conclude that B is reciprocal if and only if the above integral is zero for all Lemma 11 Let B be as in (4.1). The following hold: n n z1 ∈ D+ (R, R ) and zˆ2 ∈ D− (R, R ). In particular, if T T ˜ ˜ n×n N M = M N, then B is reciprocal. Conversely, note 1. There exist F, P, Q, U, V ∈ R [ξ] such that n ˜ ˜ that if the above integral is zero for all z1 ∈ D+ (R, R ) (i) P = F P and Q = F Q; and n T T d and zˆ2 ∈ D− (R, R ), then (N M − M N)( dt )z1 ≡ 0 " ˜ ˜# n P −Q for all z1 ∈ D (R, R ) (since otherwise the integral is (ii) is unimodular. T T d UV strictly positive with zˆ2 = (N M − M N)( dt )z1). It then follows from (Polderman and Willems, 1998, Sec- ˜ ˜ n×n Also, if F, P, Q, U, V ∈ R [ξ] satisfy (i)–(ii); tions 2.5.6 and 3.3) that N T M = M T N. loc n loc n ˜ d Bc := {(i, v) ∈ L1 (R, R ) × L1 (R, R ) | P ( dt )i = ˜ d 4 ⇒ 3. First, bring col(MN) into column proper Q( dt )v}; and loc n loc n d form. In other words, let U be a unimodular matrix with Ba := {(i, v) ∈ L1 (R, R )×L1 (R, R ) | P ( dt )i = d d d Q( )v and U( )i = −V ( )v}, " # dt dt dt M then (i, v) ∈ B ⇐⇒ there exist (i1, v1) ∈ Bc and U =: W, (i2, v2) ∈ Ba with i = i1+i2 and v = v1+v2. N 2. There exist M,N ∈ Rn×n[ξ] such that (i) PM = QN; and L (ii) rank(col(MN)(λ)) = n for all λ ∈ C. in which the leading coefficient matrix W of W has Also, if M,N ∈ n×n[ξ] satisfy (i)–(ii), then (i, v) ∈ full column rank (see Wolovich, 1974, Section 2.5). R L B ∩ (D ( , n) × D ( , n)) if and only if there Next, partition W compatibly with col(MN) as + R R + R R L L L L exists z ∈ D ( , n) such that i = M( d )z and v = W = col(W1 W2 ), let r denote the rank of W1 , + R R dt permute the columns of W L so the first r columns are N( d )z. In particular, (i, v) ∈ B . 1 dt c linearly independent, and then permute the rows so the first r rows are linearly independent. This gives n×n permutation matrices T = col(T1 T2) ∈ R and PROOF. This requires only minor modifications to the n×n r×(n−r) proof of Lemma 17 in (Hughes, 2017c). S = [S1 S2] ∈ R and an X ∈ R such that     Mˆ 11 Mˆ 12 T1 0 PROOF OF THEOREM 7 (see p. 5). We let M and " #     " # Mˆ Mˆ 21 Mˆ 22  0 −T2 M h i N be as in Lemma 11, and we will show the equivalence =   :=   U S S ˆ  ˆ ˆ    1 2 of conditions 1–3 to the additional condition: N N11 N12   0 T1  N T T     4. M N = N M. Nˆ21 Nˆ22 T2 0 Specifically, we will prove 1 ⇐⇒ 4 ⇐⇒ 3 ⇐⇒ 2. is in column proper form, and its leading coefficient ma- 1 ⇐⇒ 4. From Lemma 11, there exist z1, z2 ∈ L L n trix col(Mˆ Nˆ ) takes the form D+ (R, R ) such that     d d d d Mˆ L Mˆ L X T 0 i1=M( dt )z1, i2=M( dt )z2, v1=N( dt )z1 and v2=N( dt )z2. 11 11 1 " ˆ L#  ˆ L ˆ L    " L# M M21 M22   0 −T2 W1 h i Now, consider a fixed but arbitrary t ∈ , and let =   =   S1 S2 0 R ˆ L  ˆ L ˆ L    L N N N   0 T1  W  11 12    2 ˆ L ˆ L zˆ2(t) = z2(t0 − t) for all t ∈ R. N21 N21X T2 0

6 ˆ L where M11 is nonsingular. It is then easily verified that PROOF OF THEOREM 9 (see p. 5). That 2 ⇒ 1 fol- Mˆ T Nˆ − Nˆ T Mˆ = (US)T (M T N − N T M)(US) = 0. lows from Theorem 7, noting from the proof of Lemma ˆ−1 ˆ −1 We will show that Mˆ L is nonsingular, and it follows 3 that (Q P )(ξ) = D + C(ξI − A) B, which is sym- that NˆMˆ −1 is proper (see Rapisarda and Willems, metric. To see that 1 ⇒ 2, note initially that if Bˆ is con- 1997, Section 2). We then let Pˆ := [PT T QT T ] and trollable then the result follows from Lemma 3 and The- 1 2 orem 7. Otherwise, following (Hughes, 2017a, Notes A.1 Qˆ := [QT T −PT T ], we recall that PM = QN, and 1 2 and A.3) and (Polderman and Willems, 1998, Corollary ˆ ˆ ˆ ˆ it is then easily verified that P M = QN. This implies 5.2.25), we can construct a realization (A,˜ B,˜ C,D˜ ) of that Qˆ is nonsingular with Qˆ−1Pˆ = NˆMˆ −1, which (P,ˆ Qˆ) such that (C,˜ A˜) is observable; and is symmetric since NˆMˆ −1 = (Mˆ −1)T Mˆ T NˆMˆ −1 = (Mˆ −1)T Nˆ T Mˆ Mˆ −1 = (Mˆ −1)T Nˆ T . Finally, with " # " # i1 := T1i, v1 := T1v, i2 := T2i and v2 := T2v, then it A˜ A˜ B˜ h i ˜ 11 12 ˜ 1 ˜ ˜ ˜ is easily shown that Bˆ takes the form indicated in the A = , B = , C = C1 C2 , 0 A˜ 0 present theorem statement. 22 To complete the proof of the present implication, it re- n L ˜ ˜ ˜ ˜ mains to show that if z ∈ R and Mˆ z = 0 then z = 0. where (A11, B1) is controllable, (C1, A11) is observable, −1 −1 To see this, we denote the column degree of the jth col- and Qˆ Pˆ(ξ) = C˜1(ξI − A˜11) B˜1 + D, which is sym- ˆ umn of col(Mˆ Nˆ) by dj, and we note that the entry in metric. It then follows from the proof of Lemma 3 that L T L L T L ˜ ˜T the ith row and jth column of (Mˆ ) Nˆ − (Nˆ ) Mˆ there exists a symmetric P such that P A11=A11P , ˆ ˆ ˜T ˜ T is the coefficient of ξdi+dj in the entry in the ith row and C1 =P B1, and D=D . Now, let jth column of Mˆ T Nˆ − Nˆ T Mˆ , which is necessarily zero. n ˆ L ˆ L   Now, let z ∈ R satisfy M z = 0. Then M11[IX]z = 0. A˜ A˜ 0 ˆ L 11 12 Since M11 is nonsingular, it follows that there exists   h i n−r Aˆ :=  0 A˜ 0  , Cˆ := C˜ C˜ 0 , w ∈ R such that z = col(−XI)w. But  22  1 2 ˜T ˜T A12P 0 A22 " # h i −X  ˜    0 = I 0 ((Mˆ L)T Nˆ L − (Nˆ L)T Mˆ L) w B1 P 0 0 I     Bˆ :=  0  , and S :=  0 0 I . " #     h i −X C˜T 0 I 0 ˆ L T ˆ L ˆ L 2 = (M11) N N w. 11 12 I Since (A,˜ B,˜ C,D˜ ) is a realization for (P,ˆ Qˆ), then so too ˆ L ˆ L ˆ L k Since M11 is nonsingular, then [N11 N12]z = 0. It fol- is (A,ˆ B,ˆ C,Dˆ ) (this follows from Remark 2, as CÂˆ = lows that col(Mˆ L Nˆ L)z = 0. But col(Mˆ L Nˆ L) has full [CÃ˜k 0] for k = 0, 1, 2,...). Also, SAˆ = AˆT S and SBˆ = column rank as col(Mˆ Nˆ) is in column proper form, and CˆT . Finally, as P is symmetric, there exists a real matrix we conclude that z = 0. T R and a signature matrix Σ˜ i such that P = R Σ˜ iR, and n×n we define T and Σ (partitioned compatibly) as 3 ⇒ 4. Let M,ˆ Nˆ ∈ R [ξ] be such that the columns i of col(Mˆ Nˆ) are a basis for the right syzygy of [Pˆ −Qˆ] (see Willems, 2007, p. 85). Similar to before, we find     R 0 0 Σ˜ 0 0 that Mˆ is nonsingular and NˆMˆ −1 = Qˆ−1Pˆ, which is i  1 1    symmetric. Also, there exists a unimodular U such that T :=  0 √ I − √ I , and Σi :=  0 −I 0 .  2 2    0 √1 I √1 I 0 0 I " # " #" # 2 2 M T T 0 0 T T Mˆ U = 1 2 . T T N 0 −T T 0 Nˆ T −1 2 1 Then S = T ΣiT , and A := T ATˆ , B := T Bˆ, C := CTˆ −1 satisfy the conditions of the present theorem. 2 This follows from (Willems, 2007, pp. 84–85), noting ˆ Remark 12 Note that the proofs of Lemma 3 and The- from the definition of B and B that the columns of the orem 9 provide an algorithm for the construction of the matrix on the right hand side of the above equation span realization (A, B, C, D) and the signature matrix Σi in the right syzygy of [P −Q]. It can then be verified that Theorem 9. Specifically, P in the proof of that theorem T T T ˆ T ˆ ˆ T ˆ U (M N − N M)U = M N − N M = 0. Since U is can be obtained from the explicit formula in the proof nonsingular, this implies that M T N − N T M = 0. of Lemma 3, whereupon R and Σ˜ i can be obtained from 3 ⇐⇒ 2. The proof is analogous to 4 ⇐⇒ 3. 2 an eigenvalue decomposition for P .

7 6 Passive and reciprocal behaviors 1. B is passive and reciprocal. 2. (P,Q) is a positive-real pair and PQT = QP T . In this section, we present our main results concerning 3. B is the driving-point behavior of an RLCT network. passive and reciprocal systems. We define passivity in accordance with (Hughes, 2017c, Definition 5) as follows. In our final theorem, we generalize Lemma 4 to systems that need not be controllable. Definition 13 (Passive system) B in (4.1) is called Theorem 18 (Passive and reciprocal behavior passive if, given any (i, v) ∈ B and any t0 ∈ R, there ˆ exists a K ∈ R (dependent on (i, v) and t0) such that, if theorem, part 2) Let B be as in (2.3). Then the fol- ˜ ˜ t1 ≥ t0 and (i, v˜) ∈ B satisfies (i(t), v˜(t)) = (i(t), v(t)) lowing are equivalent. R t1 ˜T for t < t0, then − i (t)v˜(t)dt < K. ˆ t0 1. B is passive and reciprocal. 2. There exists B as in (2.2) and a signature matrix R t1 T s Remark 14 Here, − ˜i (t)v˜(t)dt is the net energy d×d t0 Σi ∈ R such that supplied to the system between t0 and t1, and the bound (u,y) (i) Bˆ = Bs ; K is necessarily non-negative (as the integral is zero " # " #T when t = t ). Thus, Definition 13 formalises the con- −A −B −A −B 1 0 (ii) + ≥ 0; and cept that a system is passive if the net energy that can CD CD be extracted from the system into the future is bounded T T T above (this bound depending only on the past trajec- (iii) ΣiA = A Σi, ΣiB = C , and D = D . tory of the system). It is shown in Hughes (2017a) that The two-part passive and reciprocal behavior theorem this definition is consistent with the existence of a non- (Theorems 17 and 18) is proved in Section 7. The proofs negative quadratic state storage function with respect can be combined with existing results in the literature to to the energy supplied to the system. obtain a passive and reciprocal realization for any given The following concept of a positive-real pair was intro- passive and reciprocal system of the form of (2.3), and duced by Hughes (2017c), where it was shown that B in to obtain an RLCT realization for an arbitrary given (4.1) is passive if and only if (P,Q) is a positive-real pair. passive and reciprocal system of the form of (4.1). This will be illustrated by two examples in Section 8. Definition 15 (Positive-real pair) Let P,Q ∈ Rn×n[ξ]. We call (P,Q) a positive-real pair if: 7 Proof of the passive and reciprocal behavior ¯ T ¯ T (a) P (λ)Q(λ) + Q(λ)P (λ) ≥ 0 for all λ ∈ C+; theorem (b) rank([P −Q](λ)) = n for all λ ∈ C+; and n T T The purpose of this section is to prove Theorems 17 (c) if p ∈ R [ξ] and λ ∈ C satisfy p(ξ) (P (ξ)Q(−ξ) + Q(ξ)P (−ξ)T ) = 0 and p(λ)T [P −Q](λ) = 0, then and 18. These two theorems will be proved in reverse p(λ) = 0. order. First, we prove the following result, which uses the supplementary lemmas in Appendix B. Remark 16 Note that, in contrast with reciprocity, it is possible for the controllable part of a system to be Lemma 19 Let Bˆ in (2.3) be passive and reciprocal. (u,y) passive yet for the system itself to not be passive. E.g., Then there exists Bs as in (2.2) such that Bˆ = Bs and let Bs be as in (2.2) with B = 0, C = 1 and D = 1, so the following properties both hold: D+C(ξI−A)−1B = 1 for all A ∈ R (i.e., the controllable 1. there exists X ∈ Rd×d such that X > 0 and part of the system is independent of A). From Remark " # (u,y) −XA − AT XCT − XB 2, if A = −1, then for any given (u, y) ∈ Bs , there ≥ 0; and R t1 R t1 T T exists k1 ∈ such that − (uy)(t)dt = −(u(t) + C − B XD + D R t0 t0 k2 k2 d×d k1 −t 2 1 −2t 1 −2t0 2. there exists a symmetric nonsingular S ∈ R such 2 e ) + 4 e dt ≤ 8 e , so this system is passive. T T T (u,y) that SA = A S, SB = C , and D = D . But if A = 0, then there exists (u, y) ∈ Bs with u = t − k2 = −y for all t ∈ , in which case − R 1 (uy)(t)dt = 2 R t0 k2 2 (t − t ), so this system is not passive. PROOF. We will prove this first for the case in which 4 1 0 D + DT > 0, and then for the general case. In the following theorem, we state necessary and suffi- T ˆ ˆ ˆ cient algebraic conditions for B in (4.1) to be passive and Case (i): D + D > 0. We let A, B, C,D and S be reciprocal. We also show that these conditions are equiv- as in the proof of Theorem 9, and we let A = A,ˆ B = Bˆ alent to B being realizable by an RLCT network, thus and C = Cˆ. From that proof, condition 2 of the present (u,y) solving the first open problem in Çamlibel et al. (2003). theorem statement holds. Also, Bˆ = Bs is passive and Theorem 17 (Passive and reciprocal behavior (C,A) is detectable. Thus, from Lemma B.2, there exists d×d theorem, part 1) Let B be as in (4.1). The following K ∈ R such that K > 0 and Υ(K) ≥ 0, where Υ(K) −1 are equivalent: is as in (B.1). It can then be verified that X := K satisfies condition 1 of the present theorem statement.

8 ˆ " T T # Case (ii): general case. Let P1 := P and Q1 := −X˜A˜ − A˜ X˜ C˜ − X˜B˜ Qˆ, and consider the following three statements (c.f., ≥ 0; and C˜ − B˜T X˜ D˜ + D˜ T Hughes, 2017a, proof of Theorem 13): 2. there exists a symmetric nonsingular S˜ ∈ d×d such ni×ni R (R1) Pi,Qi ∈ R [ξ] where (Pi,Qi) is a positive- ˜ ˜ ˜T ˜ ˜ ˜ ˜T ˜ ˜ T −1 that SA = A S, SB = C , and D = D . real pair and Qi Pi is proper and symmetric. ˜ ˜ d×d (R2) Pi,Qi are as in (R1), Pi is nonsingular, and Since X > 0, then there exists a nonsingular R ∈ R −1 such that X˜ = R˜T R˜. As S˜ is symmetric and nonsingu- limξ→∞((Qi Pi)(ξ)) = diag(Iri 0). −1 T −1 (R3) Pi,Qi are as in (R1), and either ni = 0 or lar, then so too is (R˜ ) S˜R˜ . By considering an eigen- −1 limξ→∞((Qi Pi)(ξ)) = I. value decomposition, we conclude that there exists a sig- d×d nature matrix Σi = diag(I −I) ∈ R , a diagonal ma- By (Hughes, 2017c, Theorem 7) and Theorem 7 of trix 0 < W ∈ d×d, and an orthogonal matrix V ∈ d×d this paper, P ,Q satisfy condition (R1). Then, using R R 1 1 (i.e., V T = V −1), such that (R˜−1)T S˜R˜−1 = V Σ WV T . Lemmas B.4 and B.5 (see Appendix B), we construct i Here, ΣiW = W Σi is a diagonal matrix containing the P2,...,Pm, Q2,...,Qm such that condition (R1) is sat- ˜−1 T ˜ ˜−1 isﬁed, n ≤ n , and deg (det (Q )) ≤ deg (det (Q )), eigenvalues of (R ) SR , which are necessarily real i i−1 i i−1 ˜−1 T ˜ ˜−1 for i = 2, . . . , m; and since (R ) SR is symmetric. Now, let

(1) if, for i = k − 1, (R2) is not satisfied, then (R2) h i h i −1 Yˆ = −Aˆ −Bˆ := V T R˜ 0 −A˜ −B˜ V T R˜ 0 . is satisfied for i = k, and if Pk−1 is singular then Cˆ Dˆ 0 I C˜ D˜ 0 I nk < nk−1 (Lemma B.4); and (2) if, for i = k−1, (R2) is satisfied but (R3) is not, then Then (A,ˆ B,ˆ C,ˆ Dˆ) is a realization for (P,ˆ Qˆ), and deg (det (Qk)) < deg (det (Qk−1)) (Lemma B.5).

This inductive procedure terminates in a ﬁnite number of T h i Yˆ = R˜−1V 0 −X˜ A˜ −X˜ B˜ R˜−1V 0 , steps with polynomial matrices Pm and Qm that satisfy 0 I C˜ D˜ 0 I conditions (R1)–(R3). An example is given in Section 8. Next, we consider the following three statements: which implies that Yˆ + Yˆ T ≥ 0. Next, let

(S1) There exist polynomial matrices Ai(ξ):=ξI − Ai, 1/2 h ˆ î 1/2 −1 Yi,Zi,Ui,Vi,Ei,Fi, and Gi, with Gi nonsingular, and Y = −A −B := W 0 −A −B W 0 . " #" # " # CD 0 I Cˆ Dˆ 0 I Yi Zi −Di I −Ci −Pi Qi 0 = , Ui Vi −Bi 0 Ai −Ei −Fi Gi Then (A, B, C, D) is also a realization for (P,ˆ Qˆ). Also, where the leftmost matrix is unimodular. with the notation G := R˜−1VW −1/2, then GT SG˜ = di×di (S2) The matrix Xi ∈ satisfies Xi > 0 and −1/2 T −1 T −1 −1/2 −1/2 −1/2 R W V (R˜ ) S˜R˜ VW = W ΣiWW = " T T # −1 −1 −Ai Xi−XiAi Ci −XiBi Σi A = G AG,˜ B = G B˜, and C = CG˜ . Thus, Ωi(Xi) := ≥ 0. T T Σ A = GT SÃG˜ = GT A˜T SG˜ = AT Σ ,Σ B = GT S˜B˜ = Ci−B Xi Di+D i i i i i T ˜T T T di×di G C = C , and D = D , which implies that (S3) There exists a symmetric Si ∈ R such that 1/2 T T T diag(−Σi I)Y is symmetric. Note that W is diagonal SiAi = Ai Si, SiBi = Ci and Di = Di . 1/2 since W is, and partition W compatibly with Σi as From case (i) and Lemma 1, there exist real matrices 1/2 W = diag(F1 F2). Also, partition Y and Yˆ compati- A ,B ,C ,D ,X and S such that (S1)–(S3) hold m m m m m m bly with diag(−Σi I) = diag(−III) as follows: for i = m. Then, using Lemmas B.4 and B.5, we find that there exist real matrices A ,B ,C ,D ,X and S such i i i i i i ˆ ˆ ˆ −1 hF1 0 0i −A11 −A12 −B1 F1 0 0 that (S1)–(S3) hold for i = m − 1,..., 1. Since P = P1 ˆ ˆ ˆ −1 Y = 0 F2 0 −A21 −A22 −B2 0 F 0 . 0 0 I ˆ ˆ ˆ 2 and Q = Q1, then letting A = A1,B = B1,C = C1,D = C1 C2 D 0 0 I D1,S = S1 and X = X1, we obtain a state-space real- (u,y) ization Bˆ = Bs with the required properties. 2 Then, let

−1 h−Aˆ −Bˆ i h −1 i ˆ F2 0 22 2 F2 0 Z11:= − F1A11F1 ,Z22:= 0 I ˆ ˆ PROOF OF THEOREM 18 (see p. 8). That 2 ⇒ 1 C2 D 0 I h −1 i h−Aˆ i −1 follows from Theorem 9 and (Hughes, 2017c, Theorem ˆ ˆ F2 0 F2 0 21 Z12:= − F1 [A12 B1] , and Z21:= 0 I ˆ F1 . 13), noting that condition 3 of that theorem holds with 0 I C1 X = I. To see that 1 ⇒ 2, consider the realization in Lemma 19. From that theorem, (P,ˆ Qˆ) has a realization Since diag(−Σi I)Y = diag(−III)Y is symmetric, ˜ ˜ ˜ ˜ then we conclude that Z11 and Z22 are both symmetric, (A, B, C, D) with the following properties: T T and Z12 = −Z21. Thus, Y +Y = diag(2Z11 2Z22), and ˜ d×d ˜ 1. there exists X ∈ R such that X > 0 and to complete the proof it remains to show that Z11 ≥ 0

9 T and Z22 ≥ 0. To prove this, we recall that Yˆ +Yˆ ≥ 0, so whence vb,1 ? ia,1 − ib,1 ? va,1 = 0. 2 ⇒ 3. We will show the following: " # " #T −Aˆ22 −Bˆ2 −Aˆ22 −Bˆ2 n ˆ ˆT (a) If λ0 ∈ + and z ∈ satisfy Q(λ0)z = 0, then −A11−A11 ≥ 0, and + ≥ 0. C C Cˆ2 Dˆ Cˆ2 Dˆ Qz = 0. (b) There exists a nonsingular matrix T ∈ Rn×n and a unimodular matrix Yˆ such that Since Z11 and Z22 are both symmetric, then their eigenvalues are all real. Now, let λ < 0, and let z be a " # real vector with Z z = λz. Then zˆ := F −1z satisfies h i h i T 0 11 1 P −Q = Yˆ Pˆ −Qˆ , ˆ T ˆ ˆT T T −1 −A11zˆ = λzˆ. Thus, zˆ (−A11 − A11)zˆ = 2λzˆ zˆ ≤ 0. 0 (T ) ˆ ˆT Since (−A11 − A11) ≥ 0, then we conclude that zˆ = 0. It follows that the eigenvalues of Z11 are all real and where Pˆ and Qˆ have the compatible partitions non-negative, whence Z11 ≥ 0. A similar argument then shows that Z22 ≥ 0, and completes the proof. 2 " # " # Pˆ11 0 Qˆ11 0 Pˆ = and Qˆ = , PROOF OF THEOREM 17 (see p. 8). That 1 ⇐⇒ 0 I 0 0 2 follows from (Hughes, 2017c, Theorem 9) and Theorem 7 of the present paper. ˆ ˆ−1 ˆ and where Q11 is nonsingular, Q11 P11 is symmetric, If B takes the form of Bˆ in (2.3), then from Theorem 18 and (Pˆ11, Qˆ11) is a positive-real pair. it follows that B is passive and reciprocal if and only if B (c) With Pˆ and Qˆ as in (b), then the limit has a state-space realization with the properties outlined 11 11 ˆ−1 ˆ in condition 2 of that theorem. From (Anderson and limξ→∞((1/ξ)(Q11 P11)(ξ)) exists and is non- Vongpanitlerd, 2006, Sections 4.4 and 9.4), this holds if negative definite. Also, with the notation K := ˆ−1 ˆ ˜ ˆ and only if B is the driving-point behavior of an RLCT limξ→∞((1/ξ)(Q11 P11)(ξ)), P (ξ) := P11(ξ) − −1 network. It remains to consider the case in which B does Qˆ11(ξ)Kξ, and Q˜(ξ) := Qˆ11(ξ), then Q˜ P˜ is proper not take the form of Bˆ in (2.3), i.e., P,Q in (4.1) are such and symmetric and (P,˜ Q˜) is a positive-real pair. that Q is singular or Q−1P is not proper. Now, let P,˜ Q˜ and K be as defined in (c). It follows 3 ⇒ 1. That B is passive follows from (Hughes, from Theorem 18 and (Anderson and Vongpanitlerd, 2017b, Theorem 6). It remains to show that B is recip- 2006, Sections 4.4 and 9.4) that there exist RLCT net- rocal. As explained in (Hughes, 2017b, Section 2), any works N1 and N2 whose driving-point behaviors take given RLCT network corresponds to a cascade loading the form {(i, v) ∈ Lloc , n˜ ×Lloc , n˜ | P˜( d )i = of two networks: (i) N , in which all of the elements (re- 1 R R 1 R R dt 1 ˜ d loc n˜ loc n˜ sistors, inductors, capacitors and transformers) are re- Q( dt )v} and {(i, v) ∈ L1 R, R ×L1 R, R | di moved and every single element port is replaced with K dt = v}, respectively. Next, let T be as in (b), parti- −1 an external port; and (ii) N2, which contains each of tion T and T compatibly with Pˆ as T = col(T1 T2) the elements in the original circuit (disconnected from and T −1 = [Tˆ Tˆ ], and consider the behavior corre- each other). Furthermore, the driving-point behaviors 1 2 5 sponding to the set of locally integrable solutions to of N1 and N2 are both reciprocal. Now, let B and ˜ d d B be fixed but arbitrary reciprocal behaviors, and let  0 0 0 0 P˜( ) 0 0 −Q˜( )  i  ˜ dt dt v (i) (col(ia,1 ia,2), col(va,1 va,2)) ∈ B,(ia,3, va,3) ∈ B, 0 0 0 0 0 I 0 0 vã  0 I −T T −T T 0 0 0 0  ˜  1 2  v˜b  (col(ib,1 ib,2), col(vb,1 vb,2)) ∈ B, and (ib,3, vb,3) ∈ B; T 0 0 0 −I 0 0 0  ˜  =0, (7.1)  1  ia  (ii) ia,3 = −ia,2, va,3 = va,2, ib,3 = −ib,2, and vb,3 = T2 0 0 0 0 −I 0 0  ˜   ib  d v˜ vb,2; and (iii) ia,1, ia,2, va,1, va,2, ib,1, ib,2, vb,1, and vb,2 0 0 0 0 K dt 0 −I 0 a1 have compact support on the left. Then it suffices to 0 0 I 0 0 0 −I −I vã2 show that v ? i = i ? v . To prove this, note b,1 a,1 b,1 a,1 which is the driving-point behavior of the RLCT network that, since B and B˜ are reciprocal, then in Fig. 2. We then let v ? i + v ? i − i ? v − i ? v = 0, " #" # b,1 a,1 b,2 a,2 b,1 a,1 b,2 a,2 Yˆ 0 IZ and vb,3 ? ia,3 − ib,3 ? va,3 = 0, U := , with 0 I 0 I " # 5 To see this, note initially that the behavior of the network −Q˜(ξ)TˆT P˜(ξ)+Q˜(ξ)Kξ 0 Q˜(ξ) −Q˜(ξ) Z(ξ) := 1 , N1 has the same form as the driving-point behavior of a transformer (Anderson and Newcomb, 1966). It is then eas- 0 0 I 0 0 ily verified that the driving-point behaviors of resistors, inductors, capacitors and transformers are all reciprocal, and and it is clear that U is unimodular. Then, following Ap- d so too are the driving-point behaviors of N1 and N2. pendix A, we pre-multiply both sides in (7.1) by U( dt ),

10 ˆT T ˆT T ˜ia we note that T1 T1 = I and T1 T2 = 0, and we find that the driving-point behavior of N is the set of locally i Na1 vã1 ˜ integrable solutions to the differential equation dia1 vã1 = K dt vã v T T 0 T1 T2 i      Na2 vã2 " d d d d #" # v −˜ia = −T1 0 0 vã ˜ ˜ ˜ ˆT −˜i − v d d (P ( ) + Q( )K )T1 −Q( )T i  b   T2 0 0 ˜b  P˜( )˜ia2 = Q˜( )vã2 Yˆ dt dt dt dt 1 = 0. dt dt ˜i T2 0 v b = 0

v˜b But from (b) and (c), the leftmost matrix in this equation d is equal to [P −Q]( dt ). In other words, the driving- Fig. 2. RLCT network realization of the behavior in (7.1). point behavior of N is B. It remains to show conditions (a)–(c). To show condition and it follows that (a), we let Pˆ := P − Q and Qˆ := P + Q. Since (P,Q) "ˆ ˆ ¯ T ˆ ˆ ¯ T ˆ ˆ ¯ T # ˆ ˆ ¯ T ˆ ˆ ¯ T P11(λ)(Q11(λ)) +Q11(λ)(P11(λ)) Q11(λ)(P21(λ)) is a positive-real pair, then Q(λ)Q(λ) − P (λ)P (λ) = ≥0. ¯ T ¯ T ˆ ˆ ¯ T 2(P (λ)Q(λ) + Q(λ)P (λ) ) ≥ 0 for all λ ∈ C+. Now, P21(λ)(Q11(λ)) 0 n T ˆ suppose λ0 ∈ C+ and w ∈ C satisfy w Q(λ0) = 0. Then −wT Pˆ(λ )Pˆ(λ¯ )T w¯ ≥ 0, which implies that ¯ T 0 0 This implies that Pˆ21(λ)(Qˆ11(λ)) = 0. Since this rela- wT Pˆ(λ ) = 0. But this implies that wT [P −Q](λ ) = ˆ 0 0 tionship holds for all λ ∈ C+, and Q11 is nonsingular, 0, whence w = 0 since (P,Q) is a positive-real pair. We then we conclude that Pˆ = 0. ˆ 21 conclude that Q(λ) is nonsingular for all λ ∈ C+, and so I − (Qˆ−1Pˆ)(λ)((Qˆ−1Pˆ)(λ¯))T ≥ 0 for all λ ∈ . Next, it follows from (Hughes, 2017a, proof of Lemma C+ ˆ ˆ ˆ This implies that (Qˆ−1Pˆ)T is bounded-real in accor- D.3 condition 1) that P22 is unimodular since (P, Q) is dance with (Youla et al., 1959, Definition 16), and so a positive-real pair. Accordingly, we let Qˆ−1Pˆ is bounded-real by (Youla et al., 1959, Corollary " # 7(c)). It then follows from (Youla et al., 1959, proof I Pˆ ˆ −1 12 of Theorem 7) that, if λ ∈ and w ∈ n sat- Y = Y , 0 C+ C 0 Pˆ −1 −1 22 isfy (I − (Qˆ Pˆ)(λ0))w = 0, then (I − Qˆ Pˆ)w = 0. n Now, let λ0 ∈ C+ and z ∈ C satisfy Q(λ0)z = 0. −1 1 −1 and we find that Yˆ is unimodular and [P −Q] has the Then (P +Q) (λ0)Q(λ0)z = 2 (I −(P +Q) (λ0)(P − −1 form indicated in condition (b). Finally, it is easily shown Q)(λ0))z = 0, whence (I − (P + Q) (P − Q))z = 0, 1 −1 that (Pˆ , Qˆ ) is a positive-real pair, and PˆQˆT −QˆPˆT = and so Qz = 2 (P + Q)(I − (P + Q) (P − Q))z = 0. 11 11 Yˆ −1(PQT − QP T )(Yˆ −1)T = 0 so Qˆ−1Pˆ is symmetric. To show condition (b), we let r := normalrank(Q), and 11 11 we first show that there exists a nonsingular matrix T = The proof of condition (c) follows from (Hughes, n×n r×n T col(T1 T2) ∈ R with T1 ∈ R such that QT2 = 0. 2017a, Proof of Lemma D.4), noting in addition that Accordingly, we let the columns of W ∈ n×(n−r)[ξ] be a ˜−1 ˜ ˆ−1 ˆ R (Q P )(ξ) = (Q11 P11)(ξ) − Kξ, which is symmetric basis for the right syzygy of Q (see Willems, 2007, p. 85), ˆ−1 ˆ since Q11 P11 and K are symmetric. 2 and we pick a fixed but arbitrary λ0 > 0. Then W (λ0) ∈ n×(n−r) R has full column rank and Q(λ0)W (λ0) = 0, 8 Examples whence QW (λ0) = 0 by condition (a). We then let T be T a nonsingular matrix whose final n−r rows are W (λ0) . First, consider the problem of realising the driving-point T n×r T behavior of the Bott Duffin networks in Fig. 1. From Next, note that QT1 ∈ R and normalrank(QT1 ) = normalrank(QT T ) = r. Then, by considering the upper Lemma 1, we find that this corresponds to the set of T solutions to the differential equation: echelon form for QT1 (see Hughes, 2017c, Note A4), we n×n T ˆ obtain a unimodular Y ∈ R such that Y QT =: Q 2 2 ( d + 1)( d + d + 1)i = ( d + 1)( d + d + 4)v. (8.1) takes the form indicated in condition (b), where Qˆ11 ∈ dt dt2 dt dt dt2 dt Rr×r[ξ] is nonsingular. Now, let Pˆ := YPT −1. It is We begin by finding a B as in (2.2) and matrices X,S ∈ then easily shown that (P,ˆ Qˆ) is a positive-real pair since s d×d as in Lemma 19, and we then find a passive and re- (P,Q) is. Accordingly, we consider a fixed but arbitrary R ˆ ˆ ciprocal realization for this behavior as in Theorem 18. λ ∈ C+, we partition P compatibly with Q as Finally, we obtain an RLCT network realization from this passive and reciprocal realization using results in " # Anderson and Vongpanitlerd (2006). This realization Pˆ11 Pˆ12 Pˆ = , procedure works in the general case, and relies on algo- Pˆ21 Pˆ22 rithms for: 1. computing a state-space realization for a

11 v1c = i1c v1d = i1d i1 i3

i1c i1d v1 v3 i v1 0 0 010 −1 i1  v2   0 0 001 −1i2  00 0 1 0 v3 0 0 0001 i3 vˆ 0 0 0 1 0 i i  =   i  −  2 −i  −10 0000 v  a = 0 vˆ  4 √√617 20 i   a   a 1a 0 0 0 4− 4 − 1a −i   0 −1 0 00 0 v  vˆ vˆ = √617 √617  i  b    b   1b   4  1b  −i   1 1 −100 0 v   1 1 4 0 0  v2 c c v  ic  − √617 v1c  a −      id √√617 20 v1d  0 0 − − 0 0  −  √4 617  v v i b i c b c i i − dvc d = c dt vˆ1a vˆ1b v v − did d = c dt di i di1a i v 1b 1a v1a = − dt 1b 1b = − dt v

vd id v˜1a v˜1b

ib 0 0 −1 −1 vb 4 id  00 0 1   vd  0 0 0 1 4 = v˜ − √617 i v 10 0 0 −i  4   e     e  v˜ 0 0 02  1a 4− i1a  vf  1 −1 0 0  −if  v˜  √617  v˜ 1b = 4  i1b   0 0 00 4   ˆi  √617  v  2a    2a ve vf = 0 ˆ   1 2 0 0 0 v   i2b   4 −4 4  2b  − − 0 0  √4 617 √4 617 √4 617  d i ( dt + 1) e = v v ( d + 1)v 2a ˆ 2b dt e i2b ie if ˆ i2a Fig. 4. RLCT network realization example 2. ˜ dv ˜ dv i2a i 2a i2b 2b 2a = − dt i2b = − dt Then, following Lemma B.1, we let Υ(K) and AΥ(K) 0 0 0 ˜ ˜ ˜ ˜i a v a be as in (B.1) with A, B and C substituted for A, B and 2 0 0 √√617+4  2 ˜  = − 4  v  i2b √617 2b C, and we obtain symmetric matrices v   i  2c 0 √√617+4 0   2c 4 −  √617  1 h 1 −1 −1i 1 h 2 −1 0i K1 = −1 5 5 , and K− = −1 5 0 , i2c 4 −1 5 9 9 0 0 0

v2c = i2c where K1 > 0, K− ≥ 0, Υ(K1) ≥ 0, Υ(K−) ≥ 0, and ¯ spec(AΥ(K−)) ∈ C−. Here, K1 is obtained by comput- Fig. 3. RLCT network realization example 1. dx ing the available energy for the system dt = Ax+Bi, v = Cx + Di to obtain the matrix X in Lemma B.1. Also, behavior (see Remark 2); 2. computing the available en- K− can be obtained by computing the available energy ergy of a passive system (see (Hughes, 2017a, Remark dˆx T T T T for the system dt = A ˆx − C u, y = −B ˆx + D u (see 15)); 3. computing the null space and column space of Lemma B.1). Next, note from Lemma B.1 that there ex- a real matrix; 4. computing a Cholesky decomposition ists α > 0 such that, for any given 0 < ≤ α, then of a positive-definite matrix; 5. computing an eigenvalue K := K1 + (1 − )K− satisfies K > 0, Υ(K) ≥ 0, and ¯ decomposition of a symmetric matrix; and 6. solving spec(AΥ(K)) ∈ C− . In this case, Lyapunov and Sylvester equations (as in Anderson and Vongpanitlerd, 2006, Theorems 3.7.3 and 3.7.4). 3K− + K1 1 h 11 −7 −3i K = = −7 35 15 , We first obtain a state-space realization for the behav- 4 48 −3 15 27 ior in (8.1), and we transform this into controller staircase form (see Polderman and Willems, 1998, Corollary and it can be verified that K > 0, Υ(K) ≥ 0, and ¯ 5.2.25). In this case, we find that B has a state-space spec(AΥ(K)) ∈ C−. Now, following the proof of Theo- (i,v) realization B = Bs , where Bs is the set of solutions to rem 9, we augment the matrices A,˜ B˜ and C˜ to obtain an (i,v) unobservable state-space realization B = Bˆs , where dx ˜ ˜ ˜ Bˆs corresponds to the set of solutions to the equation dt = Ax + Bi, v = Cx + Di, where h 0 1 0 i h0i A˜ = −4 −1 0 , B˜ = 1 , C˜ = [−3 0 1] ,D = 1. 0 0 −1 0 dxˆ ˆ ˆ ˆ dt = Axˆ + Bi, v = Cxˆ + Di, where

12 0 1 0 0 0 −1 ˆ −1 ˆ ˆ ˆ −4 −1 0 0 ˆ 1 ˆ A = G AG, B = G B, and C = CG, which gives A = 0 0 −1 0 , B = 0 , C = [−3 0 1 0] , 0 0 0 −1 1  −4 4  −1 √4 −2 √4  1  617 617 √4 and, as before, D = 1. Then, in Lemma B.2, we let √−4 √4 √−4 4  4 −1 0   617  ˜ ˜ ˜ ˜ ˜ ˜ A =  617 617 617  ,B =  −1  , and A11, A21, A22, B1, B2 and C1 be obtained by partitioning  2 0 0 0  4 −4 4 4 √ ˆ ˆ ˆ ˜ √ √ 0 −1− √ 4 617 A, B and C (here, A11 is formed from the ﬁrst three 4 617 617 617 rows and columns of Aˆ, and so forth), and we ﬁnd that KT = 1 [7 − 11 − 9] solves the Sylvester equation in 12 18 C = BT Σ . These satisfy the conditions of Theorem 18. 1 i that lemma, and ∇ = 2 solves the Lyapunov equation Ψ(∇) = 0 in that lemma. Accordingly, we obtain Next, we use the results in (Anderson and Vongpan- itlerd, 2006, Sections 9.2 and 9.4) to obtain an RLCT network which realizes the behavior in (8.1). We let  11 7 1 7  48 − 48 − 16 18 7 35 5 11 ˆ − 48 48 16 − 18  " # " # K =  1 5 9 1  , DC 1 0 − 16 16 16 − 2  M := , and Σ = , 7 11 1 16 −B −A 0 −Σi 18 − 18 − 2 9

T which satisfies Kˆ > 0 and −Kˆ AˆT − AˆKˆ − (Kˆ CˆT − and we conclude that M + M ≥ 0 and ΣM is symmetric. It follows that M takes the form Bˆ)(D +DT )−1(CˆKˆ −BˆT ). We then let Xˆ = Kˆ −1. Also, following Remark 12, we obtain the matrix " T # M11 −M M = 21 , −3 −3 0 0 M M ˆ −3 0 0 0 21 22 S = 0 0 0 1 , 0 0 1 0 3×3 2×2 where M11 ∈ R and M22 ∈ R are symmetric, and which satisfies SÂˆ = AˆT Sˆ and SˆBˆ = CˆT . We have M11,M22 ≥ 0. In this case, we have thus obtained a state-space realization Bs as in (2.2) and d×d  −4  matrices X,S ∈ as in Lemma 19. " 1 0 # 1 −1 √ R −1√ 0 4 617 M = √ √√ , Next, using a Cholesky decomposition we obtain 11 −4 617−20  617−20  √4 √4 0 0 √ 617 617 4 617 √ √ √ 1 −2 0  357 3 20 4 3  −4 4 −4 √ − √ − √ M21 = √ √ √ , and 7 119 3 357 119 4 617 4 617 617 6 2 3  0 √ − √ √  √ 0 √√ ˆ T  17 17 2 17  √ h 617+4 i X = R R, with R =  √  . M = 617+4 √ , 8 3 22 √ 0 4  0 0 √  4 617  3 3 4  617 3 0 0 0 4 where the factorization for M11 is obtained by (i) finding a matrix T ∈ 3×2 such that the columns of M T Then, following the proof of Theorem 18, we compute R 11 span the column space of M11; then (ii) computing a an eigenvalue decomposition of (R−1)T SRˆ −1 to obtain T Cholesky decomposition for T M11T . The factorization of M22 can be found similarly. Finally, using (Anderson 1√ 0 0 0  and Vongpanitlerd, 2006, Sections 9.2 and 9.4), we find 617+13 −1 0 0 0 0 0 0 that B is realized by the RLCT network in Fig. 3. Σ = 0 −1 0 0 ,W =  32  , and i 0 0 1 0 0 0 1√ 0  0 0 0 1 617−13 Our final example considers the behavior B in (4.1), with 0 0 0 √ √32  √ √ √ √ √ √  2 7 −2 2 617−13 7 −2 2 617+13 1 1 −1 0 0 0 √ √ √4 √ √ √4 51 51 617 51 51 617 0 ξ 0 −1 ξ2+1 ξ2  √ √√ √ √√  P (ξ) = ,Q(ξ) = ,  5 7 617−13 3 7 617+13  ξ+1 1 0 ξ+2 ξ−1 2ξ+1 √ √ √ − √ √ √  2 17 2 34 4 617 17 2 34 4 617  V =  √ √ √ √  .  1 43 617−295 1 43 617+295   √ − √ √ √ √ √  for which Q is singular. We use this example to illustrate  4 3 4 6 4 617 2 3 4 6 4 617   √ √ √ √  both the proof of Theorem 17 and the inductive proce- 11 617+185 11 617−185 1 √ √ 1 √ √ dure described in Lemma 19. Again, the realization pro- 4 4 2 4 4 2 617 4 2 617 cedure works in the general case. In addition to the previously listed algorithms, it relies on the computation of Following the proof of Theorem 18, let G := R−1VW −1/2, an upper echelon form for a polynomial matrix.

13 d d First, following the proof of Theorem 17, we obtain ma- to the diﬀerential equation ( dt + 1)ie = ( dt + 1)ve can T ˆ 3×3 trices T = [T1 T2] and Y ∈ R [ξ], where be obtained by the method outlined in the ﬁrst example.

" 0 0 −1# 9 Conclusions −1 1 1 h1 0i h i −ξ2−1 − ξ− 0 T1 = 0 1 ,T2 = −1 , Yˆ (ξ) = 2 2 , 0 0 1 1 This paper developed a theory of reciprocal systems 1−ξ − 0 2 which does not assume controllability. Necessary and sufficient algebraic conditions were established for a sys- and we find that tem to be reciprocal, both in terms of the high order differential equations describing the system, and in terms PT −1 = Yˆ P1 0 , and QT T = Yˆ Q1 0 , where 0 1 0 0 of a state-space realization. Analogous results were ob- 1 1 1 1 3 3 ξ2+ξ+ ξ2+ ξ+ −1 tained for systems that are both passive and recipro- P1(ξ)= 2 2 2 ,Q1(ξ)= 2 2 2 . −ξ3−ξ2−ξ−1 −ξ−1 −ξ3−2ξ2−2ξ−1 0 cal. Notably, we answered the first open problem in (Çamlibel et al., 2003) by proving that a behavior is re- Here, for any given λ > 0, then T ∈ R3×3 is a nonsingular alizable as the driving-point behavior of an RLCT network if and only if it is passive and reciprocal. matrix such that T2 is a basis for the right nullspace of Q(λ). Also, Yˆ and Q1 are obtained by computing an upper echelon form for QT T . It then follows from the A The elimination theorem proof of Theorem 17 that B is realized by a network of ˆ loc n1 loc n2 Let B = {(w1, w2) ∈ L1 (R, R ) × L1 (R, R ) | the form shown in Fig. 2, where Na,1 is a short circuit, d Rˆ( )col(w1 w2)}. From (Polderman and Willems, and Na,2 is a network whose driving-point behavior is dt d d 1998, Theorem 6.2.6), there exists a unimodular U with the set of solutions to P1( dt )i1 = Q1( dt )v1. −1 " # Next, note that limξ→∞((Q P1)(ξ)) = diag(1 0), R1 0 1 URˆ = , (A.1) which is singular. Thus, (P1,Q1) satisfies conditions (R1)–(R2) on p. 9, but not condition (R3). Then, follow- R2 M2 ing Lemma B.5, we find that lim ( 1 (P −1Q )(ξ)) = ξ→∞ ξ 1 1 where the rightmost matrix is partitioned compat- diag(0 1), and accordingly we let K = 1, Q2 = P1, and ibly with col(w1 w2), and M2 has full row rank. Then, from (Polderman and Willems, 1998, Theorem 1 3 3 1 0 0 ξ2+ ξ+ −1− ξ ˆ P2(ξ) = Q1(ξ) − P1(ξ) 0 ξ = 2 2 2 2 . 2.5.4), B is the set of locally integrable solutions to −ξ3−2ξ2−2ξ−1 ξ2+ξ d d d R1( dt )w1 = 0 and M2( dt )w2 = −R2( dt )w1. Now, loc n1 d −1 let B := {w1 ∈ L1 (R, R ) | R1( dt )w1 = 0}. In this case, limξ→∞((Q2 P2)(ξ)) = diag(1 0), so again Since M has full row rank, then it is easily shown 1 −1 2 we apply Lemma B.5. Here, lim ( (P Q )(ξ)) = n1 ξ→∞ ξ 2 2 that for any w1 ∈ D+ (R, R ) there exists w2 ∈ d d diag(0 1), and we let K = 1, Q3 = P2, and n2 D+ (R, R ) such that M2( dt )w2 = −R2( dt )w1, whence n1 n2 (w1) n1 (B∩ˆ (D+ ( , )×D+ ( , ))) = B∩D+ ( , ). 1 1 1 1 R R R R R R 2 2 (w ) P (ξ) = Q (ξ)−P (ξ) 0 0 = 2 ξ +ξ+ 2 2 ξ +ξ+ 2 . But it may not be the case that Bˆ 1 = B (see, e.g., 3 2 2 0 ξ 3 2 3 2 −ξ −ξ −ξ−1 −ξ −ξ −ξ−1 (w1) Polderman, 1997, Example 2.1). If Bˆ = B, then w2 is called properly eliminable (Polderman, 1997). From Next, we find that (P3,Q3) satisfies condition (R1) on p. (Polderman, 1997, Example 3.1), if Bs is as in (2.2), 9, but not condition (R2). Thus, following Lemma B.4, then x is properly eliminable. Also, the internal currents ˆ ˆ ˆ 2×2 we let T = [T1 T2] and W ∈ R [ξ], where and voltages in any given RLCT network are properly eliminable (Hughes, 2017b, Section 6). 1 ˆ 1 ˆ 1 1−ξ − 2 q×q T1 = [0] , T2 = −1 and W (ξ) = , Finally, if B is as in (2.1) and T ∈ R is a nonsingular −2ξ2−2 −1−ξ real matrix, then it is easily shown that B(T w) = {z ∈ Lloc ( , q) | (RT −1)( d )z=0}. and we find that 1 R R dt B The passive and reciprocal behavior theorem, P Tˆ = W −1 P4 0 ,Q (Tˆ−1)T = W −1 Q4 1 , 3 0 0 3 0 −2 supplementary lemmas

ˆ 2×2 where P4(ξ) = Q4(ξ) = ξ + 1. In this case, T ∈ R is In this appendix, we present four supplementary lemmas used to prove the results in Sections 6 and 7. In the ﬁrst a nonsingular matrix such that Tˆ2 is a basis for the right two lemmas, for any given symmetric K ∈ d×d, we let nullspace of P3, and the matrices W and Q4 are obtained R ˆ−1 T from the upper echelon form for Q3(T ) . It can then T T T −1 T be veriﬁed that B is realized by a network of the form of Υ(K) := −KA −AK−(KC −B)(D+D ) (CK−B ), T T T −1 T Fig. 4. Here, a network realization for the set of solutions and AΥ(K) := A −C (D+D ) (B −CK). (B.1)

14 (u,y) T T T T −1 Lemma B.1 Let Bs be as in (2.2), and let Bˆ := Bs that AΥ(K) = AΥ(K−) + ZC (D+D ) C, and T T T −1 be passive, (C,A) be observable, D + D > 0, and letting Aˆ12 := T1ZC (D+D ) CTˆ2, we find that d×d Υ(K),AΥ(K) be as in (B.1). Then there exists K ∈ R such that K > 0, Υ(K) ≥ 0, and spec(AΥ(K)) ∈ C−. " T T −1 ˆ ˆ # T −1 A1+T1ZC (D+D ) CT1 A12 TAΥ(K) T = . 0 A PROOF. Since (C,A) is observable then there exists 2 X ∈ Rd×d such that X > 0 and −AT X − XA − (CT − T −1 T T T −1 XB)(D + D ) (C − B X) = 0 (see Hughes, 2017c, Thus, spec(AΥ(K)) = spec(A1+T1ZC (D+D ) CTˆ1) −1 d×d Theorem 13). Now, let K1 := X ∈ R , so K1 > 0 ∪spec(A2). Since spec(A1) ∈ C−, then there exists a 0 < T T −1 and Υ(K1) = 0. Thus, from (Hughes, 2017a, Theorems α < 1 such that spec(A1+T1ZC (D+D ) CTˆ1) ∈ 10 and 11), there exists K− ≥ 0 such that Υ(K−) = 0, C− for all 0 < ≤ α. For any such , then K := K satis- spec(AΥ(K−)) ∈ C−, and K− ≤ K1 (here, the available fies the conditions of the present theorem statement. 2 dx T T T energy Sa for the system dt = A x−C u, y = −B x+ T T d D u satisfies Sa(x0) = x0 K−x0 for all x0 ∈ R ). Now, (u,y) Lemma B.2 Let Bs be as in (2.2), and let Bˆ := Bs let be a fixed but arbitrary real number in the interval be passive, (C,A) be detectable (i.e., col(C λI − A) has 0 < < 1, and let K := (1−)K− +K1. Since K1 > 0 T full column rank for all λ ∈ +), D+D > 0, and Υ(K) and (1 − )K− ≥ 0, then K > 0. Also, C be as in (B.1). Then there exists K ∈ Rd×d such that K > 0 and Υ(K) ≥ 0. Υ(K) = (1−)Υ(K−) + Υ(K1) T T −1 + (1−)(K−−K1)C (D+D ) C(K−−K1) ≥ 0, PROOF. By the observer staircase form (see Hughes, d×d and so Υ(K) ≥ 0. To complete the proof of the present 2017c, note D2), there exists a T ∈ R such that theorem, we will show that there exists 0 < α < 1 such that spec(AΥ(K)) ∈ C− for all 0 < ≤ α. To see this, " ˜ # " ˜ # −1 A11 0 B1 −1 h i note that Z := K1 − K− satisfies Z ≥ 0 and T AT = ,TB = ,CT = C˜1 0 , A˜21 A˜22 B˜2 T T T −1 −ZAΥ(K−) − AΥ(K−) Z = ZC (D + D ) CZ. with (C˜1, A˜11) observable. As (C,A) is detectable, then Next, let T ∈ d×d be nonsingular with TA (K )T T −1 = ˜ R Υ − it is easily shown that spec(A22)∈C−. Now, let diag(A1 A2) where spec(A1) ∈ C− and spec(A2) ∈ jR (here, the rows of T1 span the stable left eigenspace ˜ T ˜ loc n loc n loc d of AΥ(K−) ), and partition T compatibly as T = Bs = {(u, y, x˜)∈L1 (R, R ) ×L1 (R, R ) ×L1 R, R | col(T T ). Then the row space of T is spanned by the 1 2 2 dx˜ = A˜ x˜ + B˜ u and y = C˜ x˜ + Du}, left Jordan chains corresponding to the imaginary axis dt 11 1 1 ˜ ˜ ˜ ˜T ˜ ˜ eigenvalues of AΥ(K−). Consider one such Jordan chain: Υ(K) := −KA11 − A11K ˜ ˜T ˜ T −1 ˜ ˜ ˜T − (KC1 − B1)(D + D ) (C1K − B1 ), T T T z1 AΥ(K−) = jωz1 , and T T T −1 T and A˜ ˜ (K˜ ) = A˜ − C˜ (D + D ) (B˜ − C˜1K˜ ). T T T T Υ 11 1 1 zk AΥ(K−) = jωzk + zk−1 (k = 2, 3,...,N). (u,y) It follows from (Hughes, 2017c, Note D3) that B˜s = Then AΥ(K−)z¯1 = −jωz¯1 and AΥ(K−)z¯k = −jωz¯k + (u,y) z¯k−1 (k = 2, 3,...,N). Thus, for k = 1, Bs , which is passive, whence from Lemma B.1 there d˜×d˜ ˜ exists K11 ∈ R such that K11 > 0, Υ(K11) ≥ 0, and T T T −1 spec(A˜ ˜ (K11)) ∈ −. Since, in addition, spec(A˜22) ∈ zk ZC (D+D ) CZz¯k Υ C T T C−, then by (Anderson and Vongpanitlerd, 2006, Theo- = z (−ZAΥ(K−)−AΥ(K−) Z)z¯k = 0, (B.2) k rem 3.7.4) there exists a unique real K12 which satisfies the Sylvester equation whence CZz¯1 = 0. This implies that (−ZAΥ(K−) − T T AΥ(K−) Z)z¯1 = 0, so AΥ(K−) Zz¯1 = −ZAΥ(K−)z¯1 = ˜ T T ˜ A22K12 + K12AΥ˜ (K11) jωZz¯1. It follows that C(Zz¯1) = 0 and A(Zz¯1) = T ˜ ˜ T −1 ˜T ˜ AΥ(K−) (Zz¯1) = jωZz¯1, and so Zz¯1 = 0 since (C,A) = −A21K11 − B2(D + D ) (B1 − C1K11); is observable. Next, note that (B.2) holds for k = 2, and similar to before we find that Zz¯2 = 0. Proceeding and there exists a non-unique ∇ > 0 which satisfies T by induction, we obtain Zz¯k = 0, whence zk Z = 0 (k = 1, 2,...,N). Since the vectors z1 ... zN span the Ψ(∇) := −∇A˜T − A˜ ∇ − KT K−1Υ(˜ K )K−1K row space of T , then T Z = 0. Thus, by partitioning 22 22 12 11 11 11 12 2 2 T −1 T −1 T −1 T −1 −(B˜ −K K B˜ )(D+D ) (B˜ −K K B˜ ) ≥ 0. Tˆ := T compatibly with T as Tˆ = [Tˆ1 Tˆ2], noting 2 12 11 1 2 12 11 1

15 It can then be verified that Remark 20 With Pk,Qk,Pk−1 and Qk−1 as in the above lemma, then the driving-point behavior " #" #" −1 # d d I 0 K11 0 IK K12 Pk−1( )i = Qk−1( )v can be realized by a transformer K:=T −1 11 (T −1)T >0, dt dt T −1 terminated on a network with driving-point behavior K12K11 I 0 ∇ 0 I d d Pk( )î = Qk( )ˆv (see the final example in Section 8). " # dt dt Υ(˜ K11) 0 and Υ(K) = T −1 (T −1)T ≥ 0. 2 Lemma B.5 Let Pk−1,Qk−1 satisfy (R1)–(R2) for 0 Ψ(∇) i=k−1, with mk := nk−1 −rk−1 > 0. The following hold. 1. There exists 0 < K ∈ Rmk×mk such that diag(0 K) = Remark B.3 It is easily shown that K in Lemma B.2 1 −1 limξ→∞( ξ Pk−1Qk−1(ξ)). satisfies spec(AΥ(K))=spec(AΥ˜ (K11))∪spec(A22)∈C−. 2. Let Pk(ξ) := Qk−1(ξ) − Pk−1(ξ)diag(0 Kξ), The final two lemmas concern the decomposition in the and Qk := Pk−1. Then (R1) holds for i = k; proof of Lemma 19. We refer to that proof for the defi- deg (det (Qk)) < deg (det (Qk−1)); and there exist nition of statements (R1)–(R3) and (S1)–(S3). ˆ rk−1×mk ˆ mk×rk−1 ˆ mk×mk D12 ∈ R , D21 ∈ R , D22 ∈ R such that Lemma B.4 Let Pk−1,Qk−1 satisfy (R1) for i = k−1, and let nk := normalrank(Pk−1), mk := nk−1 − nk, and −1 " # rk := rank(limξ→∞(Qk−1Pk−1(ξ))). The following hold. ˆ −1 Irk−1 D12 lim (Qk Pk(ξ)) =: Dk = . (B.4) nk−1×nk−1 ξ→∞ ˆ ˆ 1. There exists a nonsingular T ∈ R ; uni- D21 D22 nk−1×nk−1 ˜ mk×mk modular W ∈ R [ξ] and Q22 ∈ R [ξ]; ˜ nk×mk Q12 ∈ R [ξ]; and Pk,Qk satisfying (R1) and (R2) for i=k, with 3. Let Ak,Bk,Ck,Dk satisfy (S1) for i = k; partition Bk,Ck compatibly with Dk as Bk = [Bˆ1 Bˆ2], Ck = ˆ ˆ " # " ˜ # col(C1 C2); and let Pk 0 −1 T Qk Q12 WPk−1T = ,WQk−1(T ) = . (B.3) 0 0 0 Q˜22 " −1 −1 # Ak − Bˆ1Cˆ1 Bˆ2K − Bˆ1Dˆ 12K Ak−1 := , 2. Let Ak,Bk,Ck,Dk satisfy (S1) for i = k; and let ˆ ˆ ˆ ˆ ˆ −1 ˆ −1 −1 −1 T D21C1 − C2 D21D12K − D22K Ak−1:=Ak, Bk−1:=[Bk 0]T , Ck−1:=(T ) col(Ck 0), and D :=(T −1)T diag(D 0)T −1. Then: " # " −1# k−1 k Bˆ1 0 −Cˆ1 −Dˆ 12K (a) (S1) holds for i = k−1. Bk−1 := , and Ck−1 := . −Dˆ I 0 K−1 (b) Let Xk satisfy (S2) for i = k; and let Xk−1 := 21 Xk. Then (S2) holds for i = k−1. (c) Let S satisfy (S3) for i = k; and let S := S . k k−1 k Then: Then (S3) holds for i = k−1. (a) (S1) holds for i = k−1. (b) Let Xk satisfy (S2) for i = k; and let Xk−1 := −1 PROOF. Condition 1 follows from (Hughes, 2017a, diag(Xk K ). Then (S2) holds for i = k−1. T −1 Lemma D.3, condition 1), noting that T Qk−1Pk−1T = (c) Let Sk satisfy (S3) for i = k; and let Sk−1 := −1 −1 −1 −1 diag(−Sk K ). Then (S3) holds for i = k−1. diag(Qk Pk 0), so Qk Pk is symmetric since Qk−1Pk−1 is. To see condition 2a, we let Ak,Yk,Zk,Uk,Vk,Ek,Fk and Gk be as in (S1) for i = k. Following (Hughes, 2017a, Lemma D.3, proof of condition 2), we let −1 T −1 T PROOF. First, note that Qk−1Pk−1 = Pk−1(Qk−1) T T  ˜  implies that Pk−1Q = Qk−1P , and hence " # " # Yk Q12 Zk " # k−1 k−1 Y Z W −1 0 T T 0 P −1 Q = QT (P T )−1. Conditions 1 and 2 then k−1 k−1  ˜  k−1 k−1 k−1 k−1 :=  0 Q22 0  . U V 0 I   0 I follow from (Hughes, 2017a, Lemma D.4, conditions 1 k−1 k−1 −1 −1 Uk 0 Vk and 2), as Qk Pk(ξ) = Pk−1Qk−1(ξ) − diag(0 Kξ), so −1 −1 Qk Pk is symmetric since Pk−1Qk−1 and diag(0 Kξ) It can be verified that each of the above matrices is uni- are. For condition 3a, we let Ak,Yk,Zk,Uk,Vk, Ek,Fk modular. Also, with Ak−1(ξ) := ξI − Ak−1, Ek−1 := and Gk be as in (S1) for i = k. Following (Hughes, 2017a, −1 T [Ek 0]T , Fk−1 := [Fk 0]T , and Gk−1 := Gk, it can Lemma D.4, proof of condition 3), we partition the two be verified that (S1) holds for i = k − 1. matrices on the left-hand side of (S1) compatibly as The proof of condition 2b follows from (Hughes, 2017a,

Lemma D.3, proof of condition 2(c)): with R := "Yˆ11 Yˆ12 Zˆ1# " −I −Dˆ12 −Cˆ1# −1 T ˆ ˆ ˆ ˆ ˆ I ˆ diag(IT ), then Ωk−1(Xk−1) = R diag(Ωk(Xk) 0)R. Y21 Y22 Z2 and −D21 −D22 −C2 , (B.5) ˆ ˆ ˆ ˆ ˆ Finally, condition 2c is straightforward to check. 2 U1 U2 V − B1 −B2 0 Ak

16 and we let pear in IEEE Trans. on Automatic Control, DOI: 10.1109/TAC.2018.2819656. ˆ ˆ ˆ  ˆ  Hughes, T. H., 2017b. Passivity and electric circuits: " #  Y11 Y12 Z1 0  I D12 0 0 Yk−1 Zk−1 Yˆ Yˆ Zˆ 0 Dˆ21 Dˆ22+Kξ 0 I a behavioral approach. IFAC JournalsOnline, Pro- = 21 22 2  . ˆ ˆ ˆ  Bˆ Bˆ −I 0 ceedings of the 20th IFAC World Congress, Toulouse Uk−1 Vk−1 −U1 −U2 −V 0 1 2 0 I 0 −I Dˆ21 Dˆ22+K(1+ξ) 0 I 50 (1), 15500–15505. Hughes, T. H., 2017c. A theory of passive linear systems It can be verified that each of the above matri- with no assumptions. Automatica 86, 87–97. ces is unimodular. Then, with Ek−1 := col(Fk 0), Hughes, T. H., Sept. 2017d. Why RLC realizations of certain impedances need many more energy storage Fk−1(ξ) := col(Ek(ξ) 0) + col(ξUˆ2(ξ) I)[0 K], and G :=diag(G I), we find that (S1) holds for i=k−1. elements than expected. IEEE Trans. on Automatic k−1 k Control 62 (9), 4333–4346. The proof of condition 3b is identical to (Hughes, Hughes, T. H., Smith, M. 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