Passive Network Synthesis Research Articles

Dissipativity, Reciprocity, and Passive Network Synthesis: From the Seminal Dissipative Dynamical Systems Articles of Jan Willems to The Present Day

The dissipativity concept sits at the intersection of physics, systems theory, and control engineering as a natural generalization of passive systems that dissipate energy. It relates the external behavior of systems to their internal state and connects the subjects of optimal control, algebraic Riccati equations, linear matrix inequalities, complex functions, and spectral factorization. Within control, its applications include the analysis and design of interconnected systems (such as cyberphysical systems), robustness, and the absolute stability problem as well as network synthesis (of electrical, mechanical, and multiphysics systems). Dissipativity emerged as a stand-alone concept following the seminal “Dissipative Dynamical Systems” articles of Jan Willems, which drew on the work of Kalman, Youla, Anderson, Yakubovich, Popov, and others. This article details recent developments in the treatment of dissipativity and the related concept of reciprocity for systems that are not necessarily controllable and need not lend themselves naturally to an input–state–output perspective, as is the case for many physical and passive systems. We illustrate these concepts using simple electric circuit and mechanical network examples. We also draw inspiration from the behavioral theory developed by Jan Willems and collaborators, a natural formalism for analyzing physical systems that need not be controllable and characterized in terms of inputs and outputs.

Modal-based synthesis of passive electrical networks for multimodal piezoelectric damping

This work presents a new approach to design an electrical network which, when coupled to a structure through an array of piezoelectric transducers, provides multimodal vibration mitigation. The characteristics of the network are specified in terms of modal properties. On the one hand, the electrical resonance frequencies are chosen to be close to those of the targeted set of structural modes. On the other hand, the electrical mode shapes are selected to maximize the electromechanical coupling between the mechanical and electrical modes while guaranteeing the passivity of the network. The effectiveness of this modal-based synthesis is demonstrated using a free–free beam and a fully clamped plate.

Passivity-Based Control of Robots: Theory and Examples from the Literature

The use of passivity in control theory was motivated by the earlier use of passivity in circuit theory and passive network synthesis, beginning in the 1950s. With the well known analogy between electrical and mechanical systems, it is natural that passivity should play an important role in the control of mechanical systems and, in particular, in the control of robots. Beginning in the 1980s, the passivity property and passivity-based control began to play an increasingly important role in a host of robotics applications, including robust and adaptive control of manipulators, and time-delayed teleoperation. This article provides a historical overview of passivity-based robot control from the authors’ perspective and their contributions in control of Lagrangian and Hamiltonian systems, networked control, teleoperation, and vision-based control.

A compliance modeling method of flexible rotary joint for collaborative robot using passive network synthesis theory

Collaborative robots have become a research focus because of their wide applications. However, the previous compliance design method of the flexible rotary joint for collaborative robot mainly relied on experience of designers, and “trial and error” method is usually adopted, no feasible and systematic theory for the designer to select numerical value and series-parallel connection mode of the springs and dampers for the flexible rotary joint. Thus, developing a feasible compliance modeling theory to guide the design of the flexible rotary joint is a particularly challenging task. The main contribution of this paper is to present a novel and effective compliance modeling theory of the flexible rotary joint for collaborative robot based on electrical and mechanical passive network synthesis, to provide theoretical and systematic guidances for compliance design of the flexible rotary joint. First, inerter element is introduced into the mechanical system, and the compliance of the flexible rotary joint is expressed as an angular velocity admittance function using electrical and mechanical network analogy. Then, by passive network synthesis theory, the three kinds of compliance realization forms of rational function and four-element compliance realization conditions of biquadratic function for the flexible rotary joint are given using inerters, springs, and dampers. Moreover, numerical examples and simulations are conducted to illustrate effectiveness of the proposed compliance realization method. Finally, discussions are given to illustrate advantages of the proposed compliance modeling and design methods compared with the previous method.

The Inerter: A Retrospective

This article provides an introduction and overview of the inerter concept and device. Careful attention is given to the distinction between the inerter as an ideal modeling element and devices that approximate the ideal behavior. The background is given to the formal definition of the inerter as a mechanical one-port with terminal forces proportional to the relative acceleration between them. Four major methods of construction are described and modeled. The discussion focuses particularly on the notion of terminals, the distinction between a device and an effect, sign reversals, back driving in geared systems, the conceptual aspects of the modeling step for inerter embodiments, and the problem of reverse engineering to discover a purpose. The article includes an analysis and discussion of the rotational inerter, a brief review of the ideas of passive network synthesis that led to the inerter concept, and an analysis and discussion of several examples of integrated mechanical devices. It concludes with an imaginary dialogue between the author and an interlocutor on the understanding and purpose of the inerter.

Synthesis of Fractional-Order Biquadratic Immittance Functions

Passive network synthesis, as an important part of circuit and system theory, has been well developed in integer-order circuits. With the development of fractional-order calculus and fractional-order elements, the problem of using fractional-order passive networks to realize fractional-order immittance functions has drawn much attention. In this paper, the realization of a fractional-order biquadratic immittance function is considered. First, the form of a fractional-order biquadratic function and some theorems that could promote later research are introduced. Second, a detailed study for the realization of a fractional-order biquadratic immittance function is shown. Finally, through summarizing the realizability conditions of each network, we have obtained the scope of fractional biquadratic impedance functions that can be realized by this paper.

Passive controller realization of a bicubic admittance containing a pole at [formula omitted] with no more than five elements for inerter-based mechanical control

This paper is concerned with a passive network synthesis problem about the damper-spring-inerter realization of a bicubic admittance containing a pole at s=0 with at most five elements, where the least number of elements for the possible realizations is four. The admittances of many passive mechanisms (controllers) in inerter-based control systems are in this form. A specific four-element realizability that is a parallel connection of a spring and a three-element subnetwork is first solved. By utilizing the realizability constraints based on graph theory, it is proved that only one configuration can cover all the other four-element cases through the discussions of other possible network graphs. By deriving its realizability condition, a necessary and sufficient condition for the four-element realizations can be combined. More generally, the five-element realizability can be investigated. The specific five-element realizability that is the parallel connection of a spring and a four-element subnetwork is solved. By making use of realizability constraints described by network graphs and eliminating impossible configurations, a set of four configurations is found out to cover all the other five-element cases. By investigating their realizability conditions, a necessary and sufficient condition for the five-element realizations can be combined. The results of this paper can reduce the realizability redundancy compared with classical synthesis approaches like Bott–Duffin procedure, which provide a first critical step towards solving the minimal realization problem of such admittances. The results can be applied to the design and optimization of mechanical control systems based on inerters, and in the long term can also contribute to the development of other areas of circuits and systems.

A Passive Network Synthesis from Two-Terminal Measured Impedance Characteristic

A Passive Network Synthesis from Two-Terminal Measured Impedance Characteristic

Realization of Biquadratic Impedances as Five-Element Bridge Networks

This paper studies the passive network synthesis problem of biquadratic impedances as five-element bridge networks. By investigating the realizability conditions of the configurations that can cover all the possible cases, a necessary and sufficient condition is obtained for biquadratic impedances to be realizable as two-reactive five-element bridge networks. Based on the types of the two reactive elements, the discussion is divided into two parts, and a canonical form for biquadratic impedances is utilized to simplify and combine the conditions. Moreover, the realizability result for the biquadratic impedance is extended to the general five-element bridge networks. Some numerical examples are presented for illustration.

Kalman's Last Decade: Passive Network Synthesis [Historical Perspectives

The Art Nouveau splendor of Budapest's Hotel Gellert was the scene for a fascinating day of scientific discourse between an 80-year-old Rudolf Kalman and me on a warm Sunday in July 2010. The focal point of the discussions was the classical theory of passive network synthesis-a beautiful subject that reached its zenith around 1960, only to decline steadily thereafter as an active research interest, and yet a subject that still raises many deep and intriguing questions. Widening conversations and a special session at the Mathematical Theory of Networks and Systems conference in the following days led the late Uwe Helmke to organize and host a workshop on network synthesis in September of that year in Wurzburg (see Figures 1 and 2), which initiated an inspirational series of meetings that has alternated with Cambridge. What was the origin of Kalman's intense interest in this topic?

Kalman's Last Decade: Passive Network Synthesis [Historical Perspectives

Kalman's Last Decade: Passive Network Synthesis [Historical Perspectives

Circuit synthesis of electrochemical supercapacitor models

This paper considers the synthesis of RC electrical circuits from physics-based supercapacitor models that describe conservation and diffusion relationships. The proposed synthesis procedure uses model discretisation, linearisation, balanced model order reduction and passive network synthesis to form the circuits. Circuits with different topologies are synthesized from physical models. Because the synthesized impedance functions are generated by considering the physics, rather than from experimental fitting which may ignore dynamics, this work provides greater understanding of the physical interpretation of electrical circuits and will enable the development of more generalised circuits.

Gradient and passive circuit structure in a class of non-linear dynamics on a graph

We consider a class of non-linear dynamics on a graph that contains and generalizes various models from network systems and control and study convergence to uniform agreement states using gradient methods. In particular, under the assumption of detailed balance, we provide a method to formulate the governing ODE system in gradient descent form of sum-separable energy functions, which thus represent a class of Lyapunov functions; this class coincides with Csiszár’s information divergences. Our approach bases on a transformation of the original problem to a mass-preserving transport problem and it reflects a little-noticed general structure result for passive network synthesis obtained by B.D.O. Anderson and P.J. Moylan in 1975. The proposed gradient formulation extends known gradient results in dynamical systems obtained recently by M. Erbar and J. Maas in the context of porous medium equations. Furthermore, we exhibit a novel relationship between inhomogeneous Markov chains and passive non-linear circuits through gradient systems, and show that passivity of resistor elements is equivalent to strict convexity of sum-separable stored energy. Eventually, we discuss our results at the intersection of Markov chains and network systems under sinusoidal coupling.

Realization of Three-Port Spring Networks With Inerter for Effective Mechanical Control

This note is concerned with the passive network synthesis problem of one-port networks consisting of one inerter, one damper, and at most three springs. To solve the problem, a necessary and sufficient condition is first derived for the realization of a three-port resistive network containing at most three elements, utilizing graph theory and several existing results of $n$ -port resistive networks. By extracting the damper and the inerter, a necessary and sufficient condition is obtained for the realization of one-port networks containing one damper, one inerter, and at most three springs under an assumption that the admittance of three-port networks containing only springs is well-defined. The covering networks are also presented. Based on properties of circuit topology, a realizability condition is derived for the special case when the earlier assumption does not hold. Combining the two conditions when the assumption holds or not, the final realizability condition is obtained.

On the Minimality and Uniqueness of the Bott–Duffin Realization Procedure

This paper establishes some new results on the Bott-Duffin procedure-a fundamental result in both passive network synthesis and linear systems theory. Since its appearance in 1949, this procedure has puzzled circuit and systems researchers as the networks produced contain an apparently extravagant number of elements. In this paper, we prove that the networks obtained by the Bott-Duffin procedure contain the least number of reactive elements (six) and the least number of resistors (two) among all series-parallel networks realizing a certain type of impedance function (a biquadratic minimum function). We further show that the Bott-Duffin procedure, in combination with some simple network transformations, can be used to produce all of the series-parallel networks containing exactly six reactive elements and two resistors which realize any specified biquadratic minimum function. In the course of the argument, we prove several additional interesting results pertaining to series-parallel realizations of minimum functions. The results are motivated by the application of the inerter device to passive mechanical control.

Genetic-based Design of Harmonic-tuned Dual-band GaN HEMT Power Amplifier

Abstract The design of a harmonic-tuned dual-band GaN HEMT power amplifier (1.28 GHz and 2.14 GHz) is presented. To increase the output power and efficiency, the transistor's load impedance at the fundamental frequency, second and third harmonics have been optimally selected. In order to design the output matching network, a new genetic-based software system for automatic synthesis of passive networks has been used. The measured performances of the dual-band amplifier are as follows: Pout = 37.28 dBm, G = 11 dB, PAE = 42.1% at f1 = 1.28 GHz; Pout = 35.7 dBm, G = 9 dB, PAE = 23.7% at f2 = 2.14 GHz.

Synthesis of biquadratic impedances with at most four passive elements

In this paper, we consider the passive network synthesis problem of biquadratic impedances with at most four elements, motivated by the passive mechanical control. In order to solve this problem, a necessary and sufficient realizability condition for no more than three elements is obtained by some topological properties derived previously. Furthermore, the constraints on the possible realizations are used to derive the networks which can cover all the cases, and they are classified as several quartets. Through investigating one of the networks in each quartet, we obtain a necessary and sufficient condition for any biquadratic impedance to be realizable with at most four elements. Finally, the interlocking conditions are illustrated graphically, and numerical examples are given.

Realization of a general all‐pole current transfer function by using CBTA

ABSTRACTIn this article, a general all‐pole current transfer function synthesis procedure using current backward transconductance amplifiers (CBTAs) is proposed. The proposed configuration uses n current backward transconductance amplifiers and n grounded capacitors as the only type of passive elements. The circuit is eligible to realize any all‐pole transfer characteristics with a given strictly Hurwitz (stable) denominator polynomial. Further, it is straightforward to find the values of the passive elements from the coefficients of this polynomial by using the Routh–Hurwitz algorithm as in the realization of a two‐element kind passive network synthesis. In this sense and as far as the author's knowledge, it is the only active structure that can be synthesized like a passive two‐element kind Cauer circuit. The simulations that are performed using PSPICE exhibit satisfactory results coherent with the theory. Copyright © 2011 John Wiley & Sons, Ltd.

New Proposed Algorithms for Nth Order Butterworth Passive Filter Computer-Aided Design

Abstract Modeling techniques are presented which allow C++ program to be used for nth order Butterworth passive LPF design. Synthesis of passive networks (ladder networks) is implemented to design any order Butterworth passive Filter. It is shown how these models can be simply interconnected to produce a complete passive LPF design program. CAD program of nth order Butterworth filter is flexible design approach of filters' frequencies and components. New proposed algorithms are presented to achieve the highest flexibility of design the selected order, components and frequencies. Typical examples, for various orders Butterworth passive LPF are presented. The presented algorithms are developed for fast, simple, accurate and flexible design. Experiments show that this method is fast and capable of generating circuits which are more economical than those generated by traditional design approaches because of the high flexibility of choosing the cheapest components.

Circuit synthesis of passive descriptor systems—a modified nodal approach

AbstractIn this work we consider the problem of multiport passive reciprocal network synthesis by descriptor systems. A numerical method is presented that leads to circuit equations in modified nodal analysis. Copyright © 2008 John Wiley & Sons, Ltd.