Showing papers in "IEEE Transactions on Circuits and Systems in 1980"

Device modeling via nonlinear circuit elements

Abstract: Two basic approaches to device modeling are presented. The physical approach consists of 4 basic steps: 1) device physics analysis and partitioning, 2) physical equation formulation, 3) equation simplification and solution, and 4) nonlinear network synthesis. The black-box approach consists also of 4 basic steps: 1) experimental observations, 2) mathematical modeling, 3) model validation, and 4) nonlinear network synthesis. Each approach is ilustrated with 2 examples: Gunn Diode and SCR for the physical approach and Hysteretic Inductor and Memristive Device for the black-box approach. While the techniques for carrying out the first 3 steps in each approach presently involve more art than science, a unified theory for carrying out the last step (nonlinear network synthesis) is beginning to emerge. In particular, the universe of all lumped nonlinear circuit elements can now be classified into algebraic and dynamic elements via a completely logical axiomatic approach. Contrary to what is the case in linear circuit theory, it is shown that an infinite variety of basic algebraic and dynamic elements will be needed in the eventual formulation of a unified theory on device modeling. Consequently, these elements are given a complete and in-depth treatment in this paper. This material can also be regarded as a self-contained survey of the state-of-the-art on nonlinear network synthesis.

Stability analysis of 2-D systems

Abstract: A polynomial stability criterion for 2-D systems is taken as a starting point for introducing a frequency dependent Lyapunov equation. The Fourier analysis of its matrix solution leads to an infinite dimensional quadratic form which provides a Lyapunov function for the global state of the system. The Fourier coefficients are explicitly obtained as the sum of series involving the system matrices. The convergence of these series constitutes a necessary and sufficient stability condition, which generalizes the analogous condition for 1-D systems.

Constructive stability and asymptotic stability of dynamical systems

Abstract: In an earlier paper, the authors presented an algorithm for constructing a Liapunov function for a dynamical system. In this paper, we present theorems which allow the algorithm to be used In proving the asymptotic stability of dynamical systems, both difference and differential equations. The notion of an asymptotically stable set of matrices is introduced, and is shown to be a sufficient condition for the algorithm's termination in a finite number of steps. The instability stopping criterion is strengthened and the efficiency of the algorithm is improved in a number of ways. We investigate the tightness of our method by applying it to two-dimensional systems for which necessary and sufficient conditions for stability are known.

Improved circuits for the realization of switched-capacitor filters

Abstract: New topologies for switched-capacitor filters are presented. The circuits are based on a pair of complementary integrators and have transfer functions independent of parasitic capacitances between any node and ground. Design equations are given for low-pass, bandpass, high-pass, and notch biquads, as well as ladder simulation filters. The ladder simulation filters are scaled for optimum dynamic range. Discrete prototypes of both cascade and ladder simulation filters are used to verify the theory.

Computation of the periodic steady-state response of nonlinear networks by extrapolation methods

Abstract: The problem of computing the periodic steady-state response can be formulated as solving a nonlinear equation of the form z = F(z) where F(z) Is the solution vector for the nonlinear network after one period of integration from the initial vector z The convergence of the sequence y_0 , y_1 , \cdots generated by Y_{r+1} = F(y_r) can be accelerated by extrapolation methods This paper presents a unified analysis of three extrapolation methods: the scalar and vector \epsilon -algorithms and the minimum polynomial extrapolation algorithm The main result of the paper is the theorem giving conditions for quadratic convergence of the extrapolation methods To obtain this result the methods are studied for linear problems (where F is a linear function) and the error propagation properties are investigated For autonomous systems a function called G similar to F can be defined In order to obtain quadratic convergence from the extrapolation methods, the derivatives of F and G must be Lipschitz continuous The appendixes give sufficient conditions for the Lipschitz continuity A discussion of practical problems related to the implementation of the extrapolation methods is based on the convergence theorem and the error analysis The performance of the extrapolation methods is demonstrated and compared with other methods for steady-state analysis by four examples, two autonomous and two nonautonomous Extrapolation methods are very easy to implement, and they are efficient for the steady-state analysis of nonlinear circuits with few reactive elements giving rise to slowly decaying transients

Dynamic nonlinear networks: State-of-the-art

Abstract: This paper surveys the state-of-the-art of the qualitative aspects of nonlinear RLC networks. The class of networks being surveyed may contain multi-terminal and multi-port RESISTORS, INDUCTORS, AND CAPACITORS, as well as dc and time-dependent voltage and current sources. The concepts of impasse points and local solvability are introduced and shown to be of fundamental importance in modeling a physical network. Simple criteria are given which guarantee the existence of a global state equation. General theorems are presented for identifying or testing whether a dynamic nonlinear network possesses one or more of the following basic qualitative properties: 1. No finite-forward-escape-time solutions. 2. Local asymptotic stability of equilibrium points and observability of operating points. 3. Eventual uniform-boundedness of solutions. 4. Complete stability and global asymptotic stability. 5. Existence of a dc or periodic steady-state solution. 6. Unique steady-state response and spectrum conservation. The hypotheses of most of these theorems are couched in graph- and circuit-theoretic terms so they can be easily checked, often by inspection. Special efforts are made to state the concepts and results in a form that can be easily understood and used by the nonspecialist. Moreover, each concept and property is profusely illustrated with carefully conceived examples, and intuitive explanations so as to make this paper both motivating and somewhat self-contained. Extensive references are provided to facilitate researchers interested in conducting future research on the many unsolved problems in dynamic nonlinear networks.

Foundations of feedback theory for nonlinear dynamical systems

Abstract: We study the fundamental properties of feedback for nonlinear, time-varying, multi-input muldt-output, distributed systems. The classical Black formula is generalized to the nonlinear case. Achievable advantages and limitations of feedback in nonlinear dynamical systems are classified and studied in five categories: desensitization, disturbance attenuation, linearizing effect, asymptotic tracking and disturbance rejection, stabilization. Conditions under which feedback is beneficial for nonlinear dynamical systems are derived. Our results show that if the appropriate linearized inverse return difference operator is small, then the nonlinear feedback system has advantages over the open-loop system. Several examples are proyided to illustrate the results.

Considerations for high-frequency switched-capacitor ladder filters

Abstract: Monolithic filters using switched-capacitor techniques are beginning to be used in commercial audio frequency communication circuits [1], [2], [10]. The fact that these filters are MOS compatible, high precision and low power makes them very attractive in these filter applications. The usefulness of switched-capacitor filters will be greatly increased if the frequency range of these filters can be extended above the voice band. The purpose of this paper is to present the theoretical considerations as well as practical circuits which are useful to the designer attempting to implement high-speed filters.

Block-shift invariance and block implementation of discrete-time filters

Abstract: This paper describes the general class of linear, time-invariant multivariable systems that can be used in block implementations of time-invariant discrete-time filters. Explicit relations between the properties of the block processor and the properties of the implemented filter are derived, including an explicit expression for the matrix transfer function of the block processor in terms of the single-input, single-output filter transfer function. These properties and relations are independent of the form of realization of the block processor. It is shown that all irreducible state-space realizations of the block processor can be derived by a simple procedure from a simple realization of the required filter transfer function.

Exact design of switched-capacitor bandpass filters using coupled-biquad structures

Abstract: A method is presented for the design of switched-capacitor bandpass filters using the bilinear z -transform. The filters are realized in low-sensitivity coupled-biquad structures and employ biquad circuits which are completely insensitive to stray capacitances between any node and ground. Geometrically symmetric (both all-pole and finite-transmissionzeros) as well as general-parameter filters are treated.

Chaotic motion in nonlinear feedback systems

Abstract: New criteria are found which imply the existence of chaos in R^n . These differ significantly from criteria previously reported in the mathematics literature, and in fact our methods apply to a class of systems which do not satisfy the hypotheses of the usual theorems on chaos in R^n . The results are stated in such a way as to preserve the flavor of many well-known frequency-domain stability techniques. The results provide easily verifiable criteria for the existence of chaos in systems which are of dimension greater than one.

Dynamics of the Van der Pol equation

Abstract: This paper is a review of approaches to understanding "chaotic" dynamics in the forced Van der Pol equation. In addition, it discusses the phenomena of entrainment and phase locking from the point of view of dynamical systems theory. There are two principal regions of the parameter space where chaotic motion has been analyzed. The first occurs in a nearly linear system near resonance. Here one uses the method of averaging to initially reduce the problem to a two-dimensional one. This two-dimensional problem is analyzed by using bifurcation theory and topological methods. The appearance of homoclinic orbits in the averaged equations signals the presence of more complicated dynamics for the original problem. The second region of parameter space one examines is the one in which the Van der Pol equation describes a relaxation oscillation. In this situation one can approximate the dynamics by the iteration of a noninvertible one-dimensional mapping. This process is described together with the use of symbolic dynamics in parametrizing the resulting limit sets.

A best approximation framework and implementation for simulation of large-scale nonlinear systems

Abstract: A conceptual and mathematical framework is presented for optimally approximating a large-scale continuous-time-parameter nonlinear dynamical system S_C by a continuous-time-parameter model \hat{S}_C as well as a discrete-time-parameter model \hat{S}_D , which can be readily simulated respectively on an analog and on a digital computer. A reproducing kernel Hilbert space approach in appropriate weighted Fock spaces is used in the problem formulation and solution. Assuming that the input-output map of the system S_C can be represented by a Volterra functional series V_t , belonging to a Fock space F_{\underline{\rho}}(E) , the input-output maps for the simulators \hat{S}_C and \hat{S}_D are obtained as "best approximations" in F_{\underline{\rho}}(E) for the entire (untruncated) series V_t . Each of these models has the following features: (a) It is adaptive because it is based on a set of test input-output pairs which can be incorporated in the system by on-line multiplexing, (b) it is optimal in the sense of being a projection in a Hilbert space of nonlinear operators, (c) it is easily implementable by means of a set of interconnected linear dynamical systems and zero-memory nonlinear functions of single variables, and (d) unlike polynomic (truncated Volterra series) approximations, it constitutes a global approximation and thus is valid under both small- and large-signal operating conditions.

On linear shift-variant digital filters

Abstract: The paper considers filters described by linear shift-variant difference (LSV) equations. We present the notion of a generalized transfer function and discuss the frequency characteristic of a shift-variant digital filter in terms of the generalized transfer function. A method is presented for determining LSV difference equations from a certain class of impulse responses, and vice versa. In addition, some properties of the impulse response and the generalized transfer function of a shift-variant system are investigated in the present work.

Reduction and expansion of live and safe marked graphs

Abstract: In this paper it is proved that certain parallel, series, and circuit reductions and expansions on a marked graph do not change the number of its equivalence classes of live and safe markings. The reductions are aseful for the analysis of marked graphs, and the expansions can be used for synthesizing live and safe marked graphs.

Simplifications and clarifications on the paper 'An algebra of transfer functions for distributed linear time-invariant systems'

Abstract: In this note we first point out some simplifications in some results of our paper mentioned above [1]. Second, we prove that the algebra of transfer functions \hat{\cal B}(\sigma_0) , introduced in the paper, is in fact the quotient ring of the ring \hat{\cal Q}_{_}(\sigma_0) with respect to the multiplicative system \hat{\cal Q}_{_}^{\infty} (\sigma_0) defined in this note. The analogy between \hat{\cal B}(\sigma_0) , seen as the quotient [\hat{\cal Q}_{_}(\sigma_0)][\hat{\cal Q}_{_}^{\infty} (\sigma_0)]^{-1} , and the algebra of proper rational functions C_p (s) seen as the quotient [\Re (\sigma_0)][\Re^{\infty} (\sigma_0)]^{-1} (where \Re_0 (\sigma_0 ) is the ring of proper rational functions analytic in Re s \geq \sigma_0 , and \Re^{\infty} (\sigma_0) is the multiplicative system of such functions tending to a nonzero constant as |s| \irightarrow \infty ), is fully developed and supports the claim that \hat{\cal B} (\sigma_0 ) is a natural extension of the algebra of proper rational functions to distributed systems. These algebraic developments have been found most useful in applications [11], [12].

Realizations of a state-space model from two-dimensional input-output map

Abstract: This paper discusses the problem to realize an internal description from two-dimensional input-output map. First, for a class of Roesser's model such that the denominator of the transfer function can be expressed by product of two polynomials of a variable, structure decomposition is carried out from the standpoint of separate local controllability and observability, and it is shown that the model is minimal if and only if it is separately locally controllable and observable. Next, the realization problem is formulated as the one of matching an infinite sequence of constant parameters which is equivalent to the input-output map and the model is used for the matching. For both cases where the matching is required over a finite number of terms of the sequence and It is required over the whole sequence, the existence conditions and the uniqueness of realizations are examined and an algorithm for finding a minimal realization within the model is presented.

Passive cascaded lattice digital filters

Abstract: Energy and power analogies are used for the analysis of cascaded (possibly time-varying) lattice digital filters. This approach leads to the requirements for passivity of the sections, and a procedure for the removal of zero-input parasitic oscillations, along with several other results.

State-space realization and inversion of 2-D systems

Abstract: In this paper the state-space realization results of [1] for causal 2-D systems are generalized to a much larger class of 2-D systems. We introduce a generalized notion of a state-space realization for which the state can still be recursively evaluated. The results include a realization method for a class of NSHP filters. In the second part we introduce inverse 2-D systems with inherent delay. Some results concerning existence of an inverse with inherent delay for a 2-D system will be given. It will be shown that, in general, a causal 2-D system cannot have a causal inverse (with inherent delay). Furthermore, it will be shown that a causal 2-D system always has an inverse with inherent delay in the larger class of 2-D systems mentioned above.

Finite word effects in block-state realizations of fixed-point digital filters

Abstract: Finite word effects are examined for a class of block realizations that are derived from a state-space realization of the transfer function. For this class of block realizations it is found that the dynamic range of the state variables is independent of block length, roundoff noise decreases with block length and, if the filter is realized in normal form, all autonomous limit cycles can be eliminated.

Hierarchical stability and alert state steering control of interconnected power systems

Abstract: A state-space model of an interconnected power system having both generator and load nodes is proposed. The resulting system of equations is interpreted as the degenerate limit of a singularly perturbed system. The model is used to derive a condition for the (local) asymptotic stability of an equilibrium. This condition decomposes in an intuitive way for subsystems interconnected via a backbone network. The model is used to formulate the problem of steering the power system from a postdisturbance alert state to a secure state, and a solution to the steering problem is also proposed.

Switched-capacitor filter design using cascaded sections

Abstract: This paper describes a number of simple (first-, second-, or third-order) switched-capacitor circuits. These can be used as simple self-contained filters, or as filter sections in the cascade realization of a higher order transfer function. All these sections are free from parasitic effects, and all can be designed directly in the sampled-signal domain. Low-pass, high-pass, bandpass, and all-pass circuits are discussed, with and without finite nonzero transmission zeros.

An approach to programmable CTD filters using coefficients 0, + 1, and -1

Abstract: A delta modulation-like sampled analog filter structure for realizing low-pass filters is described. The filter uses only the coefficients 0, + 1 , and -1 and can be fabricated as a programmable CCD filter. Interpolation and decimation are employed to increase the accuracy of the delta modulation. It is shown that with this scheme a given response can be realized arbitrarily closely. Examples are included.

Insensitivity of a class of nonlinear compartmental systems to the introduction of arbitrary time delays

Abstract: We examine a class of nonlinear compartmental systems and show that the introduction of arbitrary time delays, including nonconstant ones, in the action of one compartment on another, does not affect system properties such as stability, boundedness, and positivity. The results arise from a combination of techniques introduced separately for the studies of linear time-delay systems and nonlinear systems without delay and are directly testable. Some examples and estimates for the degree of stability are given.

Yield optimization for arbitrary statistical distributions: Part I-Theory

Abstract: This paper generalizes certain analytical formulas for yield and yield sensitivities so that design centering and yield optimization can be effectively carried out employing given statistical parameter distributions. The tolerance region of possible outcomes is discretized into a set of orthotopic cells. A suitable weight is assigned to each cell in conjunction with an assumed uniform distribution on the cell. Explicit formulas for yield and its sensitivities w.r.t. nominal parameter values and component tolerances are presented for linear cuts and sensitivities of these cuts based upon approximations of the boundary of the constraint region. To avoid unnecessary evaluations of circuit responses, e.g., integrations for nonlinear circuits, multidimensional quadratic interpolation is performed. Sparsity is exploited in the determination of these quadratic models leading to reduced computation as well as increased accuracy.

Yield maximization and worst-case design with arbitrary statistical distributions

Abstract: We describe a method by which a variety of statistical design problems can be solved by a linear program. We describe three key aspects of this approach. 1) The correspondence between the level contours of a given probability density function and a particular norm, which we shall call a pdf-norm. 2) The expression of distance in this norm from a given set of hyperplanes in terms of the dual of the pdf-norm. 3) The use of a linear program to inscribe a maximal pdf-norm-body into a simplicial approximation to the feasible region of a given statistical design problem. This work thus extends the applicability of a previously published algorithm, to the case of arbitrary pdf-norms and consequently to a wide variety of statistical design problems including the common mixed worstcase-yield maximization problem.

Spatial-domain design of two-dimensional recursive digital filters

Abstract: A technique using linear algebraic projection is proposed to design two-dimensional (2-D) recursive digital filters that best approximate a desired input/output relationship in terms of total weighted squared error. A 2-D difference equation representation is used. Examples of first-quadrant and asymmetric half-plane filters are presented and compared with other spatial-domain designs. One of the main advantages of the proposed method is that the solution is obtained directly with no need for iterations. >

The colored branch theorem and its applications in circuit theory

Abstract: The colored branch theorem (Minty 1960 [1]) is a result in graph theory, which essentially says that the existence (resp., nonexistence) of a certain loop immediately implies the nonexistence (resp., existence) of a certain cutset. Its relevance and use in circuit theory, however, has only recently been recognized. Since it is expected that many more applications in circuit theory will follow, the theorem is interpreted and proved in a network setting. Many graph-theoretic corollaries are derived, which may facilitate later use. It is illustrated that many results in circuit theory can he simplified or given a simpler proof using this theorem and its corollaries.

Yield optimization for arbitrary statistical distributions: Part II-Implementation

Abstract: A suggested test problem for proposed algorithms in yield optimization is described in detail. The problem is a current switch emitter follower (CSEF) circuit originally described by Ho, which includes a transmission line. The ideas presented in Part I of this paper [1] are applied to this circuit in order to obtain an optimal statistical design. Production yield is maximized taking into consideration statistical distributions of circuit parameters and realistic correlations between transistor model parameters. Nonlinear programming employing the analytical formuias for yield and its sensitivities is used to provide optimal nominal values for the circuit parameters. Different design specifications are assumed and corresponding optimal designs are obtained.

An algorithm for single-row routing with prescribed street congestions

Abstract: The single-row routing approach for layout has attracted a great deal of interest and is in a position to become one of the fundamental routing methods for high density multilayer printed wiring boards (PWB's). A specific development has recently been accomplished on this approach [12], namely: Necessary and sufficient conditions for optimum routing have been obtained. Nonetheless, there still remains a fundamental problem to be overcome, that is, to develop an algorithm to find the optimum solution. The present paper derives an alternate set of necessary and sufficient conditions for the same problem. These are easy to check and are tailored for algorithm development. An efficient algorithm in the special cases of upper and lower street congestions up to two has been proposed. These special cases are particularly of interest in the design of practical PWB's.