Monetary system dynamics

This is the author accepted manuscript. The final version is available from IFACModern economies are prone to persistent increases in debt and recurrent financial crises. We present a systems and control perspective on two hypotheses which aim to explain these dynamics: 1. the growth imperative - which investigates the conditions under which capitalist economies necessarily exhibit growing debt; and 2. the financial instability hypothesis - which proposes mechanisms underlying the tendency of capitalist economies to experience financial crises.This research was conducted while the author was the Henslow research fellow at Fitzwilliam College, University of Cambridge, U.K., supported by the Cambridge Philosophical Society, http://www.cambridgephilosophicalsociety.org

Passivity and electric circuits: a behavioral approach

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.We present the element extraction approach to electric circuit analysis and synthesis using the behavioral framework. Explicit descriptions are obtained of the behavior and the driving-point behavior of a general circuit comprising resistors, inductors, capacitors, transformers, and gyrators (an RLCTG circuit). It is shown that the internal currents and voltages are always properly eliminable to obtain a driving-point behavior which is the set of locally integrable (weak) solutions to a linear differential equation. We also review a recently introduced trajectory level definition of passivity, and we show that a behavior is passive if and only if it is the driving-point behavior of an RLCTG circuit.This research was conducted while the author was the Henslow research fellow at Fitzwilliam College, University of Cambridge, U.K., supported by the Cambridge Philosophical Society, http://www.cambridgephilosophicalsociety.org

Behavioral realizations using companion matrices and the smith form

This is the author accepted manuscript. The final version is available from Society for Industrial and Applied Mathematics via the DOI in this record.Classical procedures for the realization of transfer functions are unable to represent uncontrollable behaviors. In this paper, we use companion matrices and the Smith form to derive explicit observable realizations for a general (not necessarily controllable) linear time-invariant be- havior. We then exploit the properties of companion matrices to efficiently compute trajectories, and the solutions to Lyapunov equations, for the realizations obtained. The results are motivated by the important role played by uncontrollable behaviors in the context of physical systems such as passive electrical and mechanical networks

On the optimal control of passive or non-expansive systems

This is the author accepted manuscript. The final version is available from IEEE via the DOI in this record.The positive-real and bounded-real lemmas solve two important linear-quadratic optimal control problems for passive and non-expansive systems, respectively. The lemmas assume controllability, yet a passive or non-expansive system can be uncontrollable. In this paper, we solve these optimal control problems without making any assumptions. In particular, we show how to extract the greatest possible amount of energy from a passive but not necessarily controllable system (e.g., a passive electric circuit) using state feedback. A complete characterisation of the set of solutions to the linear matrix inequalities in the positive-real and bounded-real lemmas is also obtained. In addition, we obtain necessary and sufficient conditions for a system to be non-expansive that augment the bounded-real condition with new conditions relevant to uncontrollable systems

A theory of passive linear systems with no assumptions

This is the author's accepted versionFinal version available from Elsevier via the DOI in this recordWe present two linked theorems on passivity: the passive behavior theorem, parts 1 and 2. Part 1 provides necessary and sufficient conditions for a general linear system, described by a set of high order differential equations, to be passive. Part 2 extends the positive-real lemma to include uncontrollable and unobservable state-space systems.This research was conducted in part during a Fellowship supported by the Cambridge Philosophical Society , http://www.cambridgephilosophicalsociety.org

On the internal signature and minimal electric network realizations of reciprocal behaviors

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.In a recent paper, it was shown that (i) any reciprocal system with a proper transfer function possesses a signature-symmetric realization in which each state has either even or odd parity; and (ii) any reciprocal and passive behavior can be realized as the driving-point behavior of an electric network comprising resistors, inductors, capacitors and transformers. These results extended classical results to include uncontrollable systems. In this paper, we establish new lower bounds on the number of states with even parity (capacitors) and odd parity (inductors) for reciprocal systems that need not be controllable

A Survey of Classical and Recent Results in RLC Circuit Synthesis

This is the final version of the article. Available from the publisher via the link in this record.The motivation provided by mechanical network synthesis to make a fresh attack on certain questions in circuit synthesis will be briefly recalled. The classical early work on RLC synthesis, beginning with the works of Foster and Cauer and culminating in the Bott-Duffin construction, will be explained in a tutorial manner. Recent work on RLC synthesis by Jiang and Smith and Hughes and Smith will be introduced. The proof in T.H. Hughes and M.C. Smith, 2014, " On the minimality and uniqueness of the Bott-Duffin procedure " , IEEE Trans. Aut. Contr., (to appear), showing the surprising result that the Bott-Duffin construction for a biquadratic minimum function is the simplest possible among series-parallel circuits, will be explained.This work was supported by the Engineering and Physical Sciences Research Council under Grant EP/G066477/

On Connections between the Cauchy Index, the Sylvester Matrix, Continued Fraction Expansions, and Circuit Synthesis

This is the final version of the article. Available from the publisher via the link in this record.A fundamental result in circuit synthesis states that the McMillan degree of a passive circuit’s impedance is less than or equal to the number of reactive elements in the circuit. More recently, Hughes and Smith connected the individual numbers of inductors and capacitors in a circuit to a generalisation of the Cauchy index for the circuit’s impedance, which was named the extended Cauchy index. There is a close connection between the Cauchy index of a real-rational function and many classical algebraic results relating to pairs of polynomial functions. Using this connection, it is possible to derive algebraic constraints on circuit impedance functions relating to the precise numbers of inductors and capacitors in that circuit. In this paper, we first present these algebraic constraints. We will then show a relationship between the extended Cauchy index and properties of continued fraction expansions of real-rational functions, which we use to provide insight into circuit synthesis procedures.This work was supported by the Engineering and Physical Sciences Research Council under Grant EP/G066477/1

Why RLC Realizations of Certain Impedances Need Many More Energy Storage Elements than Expected

It is a significant and longstanding puzzle that the resistor, inductor, capacitor (RLC) networks obtained by the established RLC realization procedures appear highly nonminimal from the perspective of linear systems theory. Specifically, each of these networks contains significantly more energy storage elements than the McMillan degree of its impedance, and possesses a non-minimal state-space representation whose states correspond to the inductor currents and capacitor voltages. Despite this apparent non-minimality, there have been no improved algorithms since the 1950s, with the concurrent discovery by Reza, Pantell, Fialkow and Gerst of a class of networks (the RPFG networks), which are a slight simplification of the Bott- Duffin networks. Each RPFG network contains more than twice as many energy storage elements as the McMillan degree of its impedance, yet it has never been established if all of these energy storage elements are necessary. In this paper, we present some newly discovered alternatives to the RPFG networks. We then prove that the RPFG networks, and these newly discovered networks, contain the least possible number of energy storage elements for realizing certain positive-real functions. In other words, all RLC networks which realize certain impedances contain more than twice the expected number (McMillan degree) of energy storage elements

Behavioral realizations using companion matrices and the smith form

Classical procedures for the realization of transfer functions are unable to represent uncontrollable behaviors. In this paper, we use companion matrices and the Smith form to derive explicit observable realizations for a general (not necessarily controllable) linear time-invariant behavior. We then exploit the properties of companion matrices to efficiently compute trajectories, and the solutions to Lyapunov equations, for the realizations obtained. The results are motivated by the important role played by uncontrollable behaviors in the context of physical systems such as passive electrical and mechanical networks.This is the author accepted manuscript. The final version is available from SIAM via https://doi.org/ 10.1137/14099191