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On Reciprocal Systems and Controllability

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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 reci- procity 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 C¸amlibel et al., 2003; Willems, 2004; reciprocity); elasticity (the Maxwell-Betti reciprocal Hughes, 2017d). Indeed, as noted by C¸amlibel 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 reci- procity (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) := fλ 2 C j det(λI−M) = 0g; 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 2 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 2 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 2 Rn×n[ξ]). If ferential equations describing the system. We also prove H 2 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 2 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 C¸amlibel ?; i.e., if w1; w2 2 D+ R; R , then (w1 ? w2)(t) = R 1 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 = fw 2 L1 (R; R ) j R( dt )w=0g;R 2 R [s]: (2.1) 2 Notation and Preliminaries The behavior B is called controllable if, for any two tra- jectories w ; w 2 B and t 2 , there exists w 2 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 2 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 2 . 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 = f(u; y; x)2L1 (R; R ) ×L1 (R; R ) ×L1 R; R j block column and block diagonal matrices with entries dx = Ax + Bu and y = Cx + Dug; 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 2 R ;B 2 R ;C 2 R and D 2 R : (2.2) the partition in block matrix equations (e.g., see (B.5)).
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