Showing papers in "Journal of The Franklin Institute-engineering and Applied Mathematics in 1994"

Continuous action set learning automata for stochastic optimization

Abstract: The problem of optimization with noisy measurements is common in many areas of engineering. The only available information is the noise-corrupted value of the objective function at any chosen point in the parameter space. One well-known method for solving this problem is the stochastic approximation procedure. In this paper we consider an adaptive random search procedure, based on the reinforcement-learning paradigm. The learning model presented here generalizes the traditional model of a learning automaton [Narendra and Thathachar, Learning Automata: An Introduction, Prentice Hall, Englewood Cliffs, 1989]. This procedure requires a lesser number of function evaluations at each step compared to the stochastic approximation. The convergence properties of the algorithm are theoretically investigated. Simulation results are presented to show the efficacy of the learning method.

An overview of bifurcation, chaos and nonlinear dynamics in control systems

Abstract: This article offers a brief review of the fundamental concepts of bifurcation and chaos in nonlinear dynamical and control systems. Both the time-domain and frequency-domain versions of the classical Hopf bifurcation theory are studied in detail. Generalized (or degenerate) Hopf bifurcation is also discussed. Theoretical analysis and potential applications of the bifurcation theory in power systems are introduced. Meanwhile, chaos and the route to chaos from period-doubling bifurcations are described. In particular, chaos and bifurcations in feedback control systems and adaptive control systems are addressed. Because a nonlinear control system is by nature a very complex nonautonomous dynamical system due essentially to the involving of the control input, understanding and utilizing the rich dynamics of nonlinear control systems have an important impact in the modern technology. It calls for new effort and endeavor devoted to this scientific and engineering challenge.

Chua's oscillator: A compendium of chaotic phenomena

Abstract: Chua's oscillator is the only real physical object known to date in which chaotic behavior has been observed experimentally and numerically, and proved rigorously. In summarizing the chaotic phenomena observed so far from the oscillator we emphasize its universality by showing how the dynamical phenomena from other 3D oscillators can be reproduced by using Chua's oscillator. The possibility of its use as an elementary cell in cellular neural networks is briefly discussed.

Circuits for voltage tuning the parameters of Chua's circuit: Experimental application for musical signal generation

Abstract: It is well known that Chua's circuit can exhibit numerous bifurcation phenomena and attractors by tuning one or more circuit parameters. These properties can be used to synthesize sounds with complex frequency spectra for musical purposes. Investigations of Chua's circuit for sound synthesis and music composition have produced specifications for tuning multiple circuit parameters to obtain specific classes of sounds. This paper presents a multiple control of Chua's circuit in which each parameter of the circuit can be varied by an external control voltage. The design of the subcircuits and the experimental results are shown. Programmable computer software was designed to provide fast multiple-parameter control of the analog Chua's circuit for use in music composition and live performance. The basic circuit discussed in this paper appeared in a series of live music performances at Expo '93, Seoul, Korea.

A review of chaos and nonlinear dynamics in phase-locked loops

Abstract: This paper reviews recent results on nonlinear dynamics and chaos in phase-locked loops (PLLs). A phase-locked loop is a versatile integrated circuit device mainly used in communication systems. When it is locked, the dynamics obey mostly linear theory, and it is analyzed traditionally by using the transfer function. However, when it is out-of-lock or when it is in the process of locking, the dynamics become nonlinear and the analysis becomes fairly difficult. In fact, various phenomena inherent to nonlinear systems such as complex bifurcations and chaos can occur in phase-locked loops. In this paper, we will present first, what is a phase-locked loop, and then derive the nonlinear differential equation describing the PLL. We will proceed to nonlinear analysis of PLLs including traditional lock range, pull-in range and chaos, etc.

Parameter plane control design for a two-tank chemical reactor system

Abstract: In this paper, we consider control of a stirred-tank chemical reactor system with a transportation lag using a parameter plane method. Due to wide acceptance of proportional- plus-integral-plus-derivative (PID) control in the chemical process industries, a PID controller is used for this chemical reactor system. Based on two stability equations and using the PID gains as the adjustable parameters, the stability boundary and boundaries with a specified gain margin (G.M.), phase margin (P.M.), damping ratio and damping factor are shown in the first quadrant of parameter planes. The set of all possible gains to maintain the chemical reactor system's stability, and at the same time, to make the system having a prespecified amount of G.M., P.M., damping ratio and damping factor is found. The performance for some representative PID gains chosen in the parameter planes are checked by their Nyquist plots and Bode diagrams. The advantage of using this approach is that the effects of parameters on the stability and performance of the control system can be easily seen in the parameter space, and this can help control engineers to ease the design work by properly adjusting the parameters to obtain the desired system performance.

Bifurcation of power electronic circuits

Abstract: This paper studies various bifurcations of periodic orbits in power electronic circuits: cyclic fold bifurcations, period-doubling bifurcations, and bifurcations due to Poincare map discontinuities. We focus on circuits operating under closed-loop control and/or containing nonlinear reactive components. Section III contains an exploration of cyclic fold bifurcations and the associated resonant jump phenomenon in circuits containing saturable reactors. Section IV gives a comprehensive overview of period-doubling phenomena in closed-loop DC-DC conversion circuits. We study circuits with homeomorphic and unimodal Poincare maps, those that period-double a single time and those that period-double repeatedly in a cascade to chaos. This section ends with a result relating non-genericity of a period-doubling bifurcation to halfwave orbital symmetry. An interesting feature of power electronic circuits is that they may have Poincare maps that are continuous but not everywhere differentiable, or discontinuous. In Section V we study, in detail, bifurcation behavior in a thyristor controlled VAR compensator, understood in terms of Poincare map discontinuities. We show that Poincare map discontinuities are due to jumps in circuit switching times. We show how map discontinuities lead to steady state jump phenomena, and distinguish between transient behavior related to switch time jumps and steady state bifurcations. The paper ends with an Appendix, in which concepts underlying cyclic fold bifurcations for the case of a continuous but not everywhere differentiable map are developed.

Pole assignment for systems modelled by bond graph

Abstract: In this paper, the pole assignment problem is considered for linear systems modelled by bond graphs. A procedure for the formal determination of the controllability matrix is proposed. This matrix is used to transform the state and control matrices into a controllability form. It allows us to formally assign the poles of the system.

Chaos control: How to avoid chaos or take advantage of it

Abstract: The purpose of this paper is to review the most interesting control concepts developed for chaotic circuits and systems. In most cases the goal of the control procedure is in converting chaotic behavior into a periodic or constant motion. There also exist specific tasks where chaotic behavior is advantageous—due to extreme sensitivity to changes in initial state it is possible to direct trajectories to targets much faster than without control. Tracking of specific chaotic trajectories might be useful in communication schemes—where chaos is used to hide useful information. The control concepts described include: (i) system parameter variation, (ii) chaotic oscillation absorber, (iii) entrainment, (iv) error feedback methods, (v) time-delayed feedback, and (vi) methods for stabilizing unstable periodic orbits: occasional proportional feedback and sampled input waveform methods. Potential strengths and weaknesses of these methods are discussed.

Interactive exploration of a chaotic oscillator for generating musical signals in real-time concert performance

Abstract: A chaotic oscillator has a broad range of state space from which musicians can induce a variety of sounds. An application of such an oscillator for music performance and composition opens a door to a series of new projects in the music community. Chua's oscillator has been explored for musical signal generation in the context of music performance and composition. Further, an exploration of the parameter space of the oscillator has been assisted with evaluations of associated sound outputs guided by auditory perception. The application of Chua's oscillator in a music performance system requires several stages of research: (1) understanding basic principles of the chaotic system in terms of mathematical descriptions of the system and hardware configurations, (2) understanding the output signals in terms of vector space descriptions as well as understanding the auditory signal outputs in relation to the vector space descriptions, and (3) incorporating the system in a musical performance context. While the abundance of papers available support the first stage of research, the second stage of research requires alternative ways of listening to the auditory signals with respect to the chaotic system behaviors. For the third stage methods and tools have been invented for exploration of the chaotic system, a simulation of the system in computer has been implemented, and a peripheral performance system has been configured. In order to nourish auditory perception an efficient and intuitive way of interacting with the oscillator is essential. One of the most interesting preparations for this research was the design of an interactive graphical software interface, the manifold interface . This interface was extensively used for exploring parameter regions for precompositional activity and for sending control signals to both analog and simulated versions of the oscillator in real-time performance.

An introduction to fractals and their applications in electrical engineering

Abstract: We present a very brief history of fractals, describe their generation, their characteristics, and their relation with chaos. We point to where and why fractals and chaotic systems are commonly found in nature, which implies that they should be good candidates for modeling different types of real world signals. We then list a number of diverse research areas in electrical engineering where fractals and fractal-based techniques have found applications. Finally, we present in some detail applications of fractals in signal processing, more specifically in the areas of digital image modeling, synthesis, and compression.

Huffman-type codes for infinite source distributions

Abstract: A new sufficient condition is given for an infinite source distribution to share a minimum average codeword length code with the geometric distribution. Thus, some new examples of parametric families of infinite source distributions can be optimally encoded by Huffman-type codes.

Bifurcations and chaos in electric power systems: Numerical studies

Abstract: In the first part of this paper, we show via computer simulation the existence of chaos in two electric networks over a range of loading conditions. These two power systems, which were derived and simplified from physical power networks, are four-dimensional and 11- dimensional respectively, and both exhibit a period-doubling transition to chaos. The existence of chaos is confirmed though Lyapunov exponents, power spectra, and inspection of Poincare maps. The effect of system damping factors on bifurcations and complicated behaviors is also examined. Our studies indicate that some phenomena not appearing in small test systems can appear in larger systems. These simulations favor the claim that bifurcation and chaos can occur in a real power system, motivating a future investigation of the nature, extent, and significance of these phenomena in realistic power system models. We present a survey of existing work on the application of continuation methods to realistic power system models— which may contain thousands of nonlinear equations, some with hard limits on certain state variables. An emphasis is placed on the package CPFLOW, to illustrate, on a 3493-bus power system, the computational requirements and difficulties to be overcome in attempts to develop tools for realistic power system bifurcation and chaos studies. In addition to the review, this paper presents some new work in the interpretation of dynamical phenomena observed in power systems using qualitative results about bifurcation and chaos theory.

A control study of a kneeless biped locomotion system

Abstract: In this paper, a control study on a three-degree-of-freedom kneeless biped locomotion system is performed. Based on the biped dynamics and joint trajectories of the walking gait, a robust adaptive control scheme consisting of a control law and an adaptation law was used. The control law has the structure of the inverse dynamics servo but uses estimates of the dynamics parameters in the computation of torques which propel the biped. The adaptation law uses the tracking error to compute the parameter estimates for the control law. To improve the convergence of the estimated parameters, we modify the timing of applying the adaptation by incorporating a dead-zone operation. Our simulation results show that the adaptive control technique can be effectively used for the biped locomotion control. The joint tracking errors can be made acceptably small and the performance is also robust despite relatively large deviations in the initial estimates of the system parameters.

Genetic algorithms for solving mixed-discrete optimization problems

Abstract: This paper presents the applications of genetic algorithms to nonlinear constrained mixed-discrete optimization problems that occur in engineering design. Genetic algorithms are heuristic combinatorial optimization strategies. Several strategies are adopted to enhance the search efficiency and reduce the computational cost. The effectiveness, robustness, and fast convergence of modified genetic algorithms are demonstrated through the results of several examples. Moreover, genetic algorithms are more capable of locating the global optimum.

New algorithms for 3D surface description from binocular stereo using integration

Abstract: In this paper, we formulate and develop an approach which integrates different modules (feature extractor, matching and interpolation) involved in stereo. We study the integration process at the finest resolution when (i) the precomputed edge map is the only line field driving the model, (ii) the line fields are computed interactively by the feature extracting module of the model, and (iii) when both the interactive line field computation module and the precomputed line field modules are present. This integration process being computationally intensive, we develop a multiresolution stereo integration approach. The energy function for each module at different resolutions is constructed and minimized in an integrated manner yielding a dense disparity map. A new energy function for the matching module is proposed. Experimental results are presented to illustrate our approach.

Fusion methods for multiple sensor systems with unknown error densities

Abstract: Consider that the sensor Si, i = 1,2,…,N, outputs y (i) ϵ R ; d , according to an unknown probability density pi(y(i)|x), corresponding to an object with parameter xϵ R d . For the system of N sensors, S1, S2,…,SN, a training l-sample (x1,y1), (x2,y2),…, (x1,y1) is given where yi = (y(1)i,y(2)i,…,y(N)i) and y(j)i is the output of Sj in response to input xi. The problem is to estimate a fusion rule ƒ: R Nd ↦ R d , based on the sample, such that the expected square error I(ƒ) = ∫ [x − ƒ(y (1) ,y (2) ,…,y (N) )] 2 p(y (1) ,y (2) ,…,y (N) |x)p(x)dy (1) dy (2) …dy (N) dx is to be minimized over a family of fusion rules F based on the given l-sample. Let ƒ ∗ ϵ F minimize I(ƒ). In general, ƒ∗ cannot be computed since the underlying probability densities are unknown. Using Vapnik's empirical risk minimization method, we show that if F has finite capacity, then under bounded error condition, for sufficiently large sample, f can be obtained such that P[I( f )−I(f ∗ )>epsiv;] for arbitrarily specified e > 0 andδ, 0

Calibration of camera parameters using vanishing points

Abstract: A new technique for calibrating the intrinsic and extrinsic parameters of a video imaging system, based on vanishing points is presented. The intrinsic parameters of the camera refers to the focal length of the lens, the physical dimensions of each pixel and the exact position of the optical center on the image grid, as well as the radial distortion of the lens. The extrinsic parameters refer to the position and orientation of the camera described in a predefined frame of reference called the world coordinate system. The proposed method requires only one view of a specially designed test object, or two distinct views of a solid cube. Other objects rich in parallel lines can be used as well. The two views must be generated by subjecting the object to an arbitrary composite three-dimensional (3D) displacement within the field of view of the camera. It is not required to know the exact 3D motion parameters; however, it is necessary to know the initial position and orientation of the cube with respect to the world coordinate system. An intrinsic parameter that is often not addressed is the radial distortion factor. It remains fixed for each lens and thus has been considered a part of the lens specification. The camera model used in this paper incorporates the radial distortion factor; however, the basic equations become highly nonlinear (eighth order). Two cameras connected to the same digitizer hardware have been calibrated to experimentally verify and illustrate the feasibility of the proposed new technique.

Model reduction of SISO systems by Routh expansion and balancing method

Abstract: Among various model reduction methods for single-input single-output (SISO) systems, the balancing method has recently received much attention due to its many attractive properties. However, it is often criticized that the method fails to give a reduced-order model without having steady-state response error for a step input. To remedy this drawback, we propose in this paper a new reduction procedure. It utilizes the Routh canonical expansion to decompose the full SISO system, and then applies the balancing method to reduce the order of a decomposed subsystem. A reduced-order model is then synthesized from the first Routh expansion coefficients and the reduced-order subsystem. The reduced-order model obtained shares good properties that are inherent to the Routh approximation method and the balancing truncation method.

Nonlinear dynamics and chaos in Sigma-Delta modulation

Abstract: One-bit oversampled or Sigma-Delta analog-to-digital conversion has been the subject of much research attention in recent years, finding use in a wide variety of applications. Theoretical analysis of this conversion method has been complicated by the nonlinear nature of the process, precluding the application of standard linear circuit analysis methods. In recent years a number of researchers have applied theory and techniques of nonlinear dynamics to the study of Sigma-Delta modulation. This paper provides a summary of the results thus obtained.

Abc (adventures in bifurcations and chaos): a program for studying chaos

Abstract: “Adventures in Bifurcations and Chaos” (ABC) is a user-friendly program for IBM-compatible PCs which has been designed for exploring bifurcations and chaos in the Chua oscillator paradigm. ABC calculates and draws equilibrium points, eigenvalues, eigen-spaces and trajectories. A two-dimensional projection of the three-dimensional dynamics is shown in a state space plot while the corresponding time waveforms are simultaneously displayed in a time-domain window. The software includes an extensive database of sets of initial conditions and parameters which produce many dynamical behaviors which have been reported for Chua's oscillator: equilibrium points, bifurcation sequences, periodic orbits, homoclinic and heteroclinic orbits and a plethora of chaotic attractors.

A priori error estimates for Corrington's Walsh function method

Abstract: Corrington introduced a Walsh function method for solving differential equations. In this paper, sufficient conditions for the convergence of Corrington's method are derived. Also, a priori error estimates are given in L2 norm for the approximation of the highest order derivative appearing in the differential equation. The solution to the differential equation may be recovered by integrating this high order derivative.

On the Schur-Cohn minors and inner determinants and a new stability table

Abstract: Recently, a new stability table for discrete time systems has been proposed in Hu, Systems Control Lett., Vol. 21, 1993 which is based on a sequence of symmetric polynomials associated with the system under test. This table is equivalent to the Marden-Jury Table but with only approximately half the size. In this paper, a modification of the above table is derived such that the modified table is equivalent to the Modified Jury Table. This allows us to obtain the Schur-Cohn minors and inner determinants of the system under test from the modified table directly. Besides the significant simplification in the stability test of 1 -D systems, this development is expected to gain more benefit in application to multidimensional system stability analysis.

A test for independence based on the Information Energy

Abstract: A statistic is proposed to test the hypothesis of independence in multidimensional contingency tables using the concept of Information Energy introduced by Onicescu. Its asymptotic power is calculated as well as the exact power function for the spike alternative.

Multiobjective flow control in delay constrained telecommunication networks

Abstract: The multiobjective flow control problem is addressed through a Pareto optimality approach. The performance objective of each individual class is the maximization of throughput under a delay constraint and users share the network on a processor sharing basis. The formulation of the problem leads to a multiobjective problem which is solved via a transformation that leads to a multicriteria linear program. The solution method is outlined by a thorough study of a simple example. Pareto optimal solutions are compared on the basis of fairness given different configurations and traffic characteristics.

Transfer function matrix identification from input—output frequency response data

Abstract: A new formulation of transfer function matrix identification in frequency domain is introduced. It reduces the problem to a simple linear least square problem. It is shown that such a system identification problem is a special case of a matrix interpolation problem and much insight can be obtained by examining its algebraic characteristics. A new approach is proposed to determine the transfer function matrix of a multi-input and multi-output system from the input—output data. It eliminates the common assumption in the literature that the frequency response of the system is given. Its efficiency and practicality is superior to the existing methods, where the solution is obtained by solving a nonlinear least square problem using mathematical programming techniques. The simplicity of the new procedure makes it a viable candidate for real time implementation where systems can be identified on-line. Unmodeled dynamics can also be better characterized.

Cumulant-based parametric multichannel FIR system identification methods

Abstract: In this paper, “least squares” and recursive methods for simultaneous identification of four nonminimum phase linear, time-invariant FIR systems are presented. The methods utilize the second- and fourth-order cumulants of outputs of the four FIR systems of which the common input is an independent, identically distributed (i.i.d.) non-Gaussian process. The new methods can be extended to the general problem of simultaneous identification of three or more FIR systems by choosing the order of the utilized cumulants to be equal to the number of systems. To illustrate the effectiveness of our methods, two simulation examples are included.

Chaotic behavior in digital filters

Abstract: In this paper we give an appraisal of the current status of the research on chaotic behavior in digital filters with two's complement (modulo 2) overflow nonlinearity. The digital filters have an abundance of chaotic behaviors. We speculate that the very special and peculiar features of chaos will be used in the near future for solving challenging engineering problems such as secure communications, associative memory, and control problems.

A new robust stability test for a class of linear systems with uncertainties and multiple time delays

Abstract: This paper addresses the robust stability problem for the linear multiple time- delay system subjected to nonlinear time-varying perturbations. By means of complex Lyapunov equations, norm and matrix measure inequalities, a new approach is proposed to solve such problems. Several new criteria, delay-dependent or delay-independent, are derived to guarantee the asymptotic stability of the uncertain multiple time-delay systems. The 7stability decaying rate of such systems is also investigated. The main feature of the developed results is that it is not necessary to solve any troublesome Lyapunov equation even though the Lyapunov stability test is used. Furthermore, the generality of these results is assured by comparing with those presented in the literature.

Model reduction of nonsquare linear MIMO systems using multipoint matrix continued-fraction expansions

Abstract: This paper deals with the multipoint Cauer matrix continued-fraction expansion (MCFE) for model reduction of linear multi-input multi-output (MIMO) systems with various numbers of inputs and outputs. A salient feature of the proposed MCFE approach to model reduction of MIMO systems with square transfer matrices is its equivalence to the matrix Pade approximation approach. The Cauer second form of the ordinary MCFE for a square transfer function matrix is generalized in this paper to a multipoint and nonsquare-matrix version. An interesting connection of the multipoint Cauer MCFE method to the multipoint matrix Pade approximation method is established. Also, algorithms for obtaining the reduced-degree matrix-fraction descriptions and reduced-dimensional state-space models from a transfer function matrix via the multipoint Cauer MCFE algorithm are presented. Practical advantages of using the multipoint Cauer MCFE are discussed and a numerical example is provided to illustrate the algorithms.