Stable Poles Research Articles

Orthogonal Rational Functions: A Transformation Analysis

Finite impulse response (FIR) models are among the most basic tools in control theory and signal processing and are routinely used in almost all fields of application. The connections to orthogonal polynomials are well known. However, infinite impulse response (IIR) models often provide much more compact descriptions and in many cases give improved performance. The objective of this paper is to present a simple framework for the derivation and analysis of orthogonal IIR transfer functions, which are directly related to orthogonal rational functions. Orthogonality simplifies approximation analysis and leads to improved numerical properties. The basic idea is to use a fractional transformation to map the problem to a new domain, where an FIR description is most appropriate. This FIR representation is then mapped back to the original domain to give an orthogonal IIR representation. It is then straightforward to extend many results for FIR models to IIR model structures with arbitrary stable poles; i.e., properties of orthogonal polynomials are easily generalized to orthogonal rational functions. Much of the theory to be presented is classical, e.g., Laguerre and Kautz functions, and we will make use of well-known results in orthogonal filter theory. However, our main contribution is to present a uniform and transparent theory which also covers more novel results that have mainly been presented in the signals, systems, and control literature in the last decade.

Computer code for rational approximation of frequency dependent admittance matrices

This paper deals with the problem of approximating with rational functions a matrix whose frequency-dependent elements have been obtained from calculations or from measurements. Based on a previously developed technique (vector fitting), a set of callable routines have been written in the MATLAB language. These routines allow for rational approximation with a common set of stable poles, automatic selection of initial poles, passivity enforcement, and creation of an equivalent electrical network which can be imported into ATP-EMTP. Usage of sparse arithmetic permits the computer code to handle large systems. The methodology is demonstrated by application to a frequency-dependent network equivalent of a radial distribution network (phase domain), for which the accuracy is validated in both the frequency domain and the time domain. The computer code is in the public domain and is available from the author.

Computer Code for Rational Approximation of Frequency Dependent Admittance Matrices

This paper deals with the problem of approximating with rational functions a matrix whose frequency-dependent elements have been obtained from calculations or from measurements. Based on a previously developed technique (vector fitting), a set of callable routines have been written in the MATLAB language. These routines allow for rational approximation with a common set of stable poles, automatic selection of initial poles, passivity enforcement, and creation of an equivalent electrical network which can be imported into ATP-EMTP. Usage of sparse arithmetic permits the computer code to handle large systems. The methodology is demonstrated by application to a frequency-dependent network equivalent of a radial distribution network (phase domain), for which the accuracy is validated in both the frequency domain and the time domain. The computer code is in the public domain and is available from the author.

Preserving positive realness through discretization

It is shown that a discrete positive real transfer function is obtained from a positive real continuous one of relative order zero being strictly stable poles via discretization by a sampler and zero-order hold device provided that the direct input-output transmission gain is sufficiently large. Also, a discrete positive real transfer function may be obtained from a stable continuous one of relative order zero and high direct input-output gain which possess simple complex conjugate critically stable poles even in the case that this one is not positive real. For that purpose, the use of an appropriate phase lag or phase lead compensating network for the continuous transfer function may be required to ensure positive realness of the discrete transfer function.

Investigating the Stability Characteristics of Natural-Circulation Boiling Water Reactors Using Root Loci of a Reduced-Order Model

Linear stability analysis of a natural-circulation boiling water reactor (BWR) and the underlying thermal-hydraulic subsystem is performed using a reduced-order BWR dynamic model. The root-locus method is used to examine the stability of the system. The relation between the poles of the system and the physical processes causing the instabilities is investigated. For a natural-circulation thermal-hydraulic system, the two types of instabilities (type-I and type-II oscillations) can clearly be attributed to the dynamics of different types of pressure drops. However, it is not possible to associate these instability types with certain poles of the system.The root loci of a reactor with weak void reactivity feedback and those of the thermal-hydraulic system behave similarly: The same pole pair remains the least stable one as the operating conditions move from the type-I instability region to the type-II region. In the case of a reactor with strong void reactivity feedback, an exchange in the stability of two pole pairs is found: The least stable pole pair in the type-II region is not the same as in the type-I region.

Enforcing Passivity for Admittance Matrices Approximated by Rational Functions

A linear power system component can be included in a transient simulation as a terminal equivalent by approximating its admittance matrix Y by rational functions in the frequency domain. Physical behavior of the resulting model entails that it should absorb active power for any set of applied voltages, at any frequency. This requires the real part of Y to be positive definite (PD). We calculate a correction to the rational approximation of Y that enforces the PD criterion to be satisfied. The correction is minimal with respect to the fitting error. The method is based on linearization and constrained minimization by quadratic programming. Examples show that models not satisfying the PD criterion can lead to an unstable simulation, even though the rational approximation has stable poles only. Enforcement of the PD criterion is demonstrated to give a stable result.

Enforcing passivity for admittance matrices approximated by rational functions

A linear power system component can be included in a transient simulation as a terminal equivalent by approximating its admittance matrix Y by rational functions in the frequency domain. Physical behavior of the resulting model entails that it should absorb active power for any set of applied voltages, at any frequency. This requires the real part of Y to be positive definite (PD). We calculate a correction to the rational approximation of Y which enforces the PD-criterion to be satisfied. The correction is minimal with respect to the fitting error. The method is based on linearization and constrained minimization by quadratic programming. Examples show that models not satisfying the PD-criterion can lead to an unstable simulation, even though the rational approximation has stable poles only. Enforcement of the PD-criterion is demonstrated to give a stable result.

PID×(<i>n</i>-2)/2 Stage PDA Cascade Robust Controllers

This paper presents a design technique for the two-degree of freedom control system using PID×(n-2)/2 stage PDA as a cascade robust controller for a class of nth order plants. It is intended to satisfy both transient and steady state response specifications based on root locus approach. This controller can be used instead of a conventional PID controller for the higher order plant to obtain better performances. By this design technique, the controlled system is approximated as a second order system, and the desired performances are satisfied. For the plant with stable poles including the case with small dead time, the closed-loop poles are located on the left half of the s-plane and the controlled system become robustly stable. The cascade robust controller gain is adjusted to meet faster responses with a small or no overshoot. Robustness properties given by the controllers proposed in this paper have also been demonstrated by illustrative examples.

Slow stable open-loop poles: to cancel or not to cancel

Stable plant poles can either be shifted by feedback, or they can be approximately cancelled by a controller. While it is known that the cancelling design impacts on the transients due to input disturbances, here we show that the alternative, i.e., the shifting design, impacts on closed-loop robustness. In this paper we consider control loops with integral action and show that shifting slow stable real poles, rather than cancelling them, increases the l 1, H ∞, and the point-wise in frequency norms of the complementary sensitivity. Thus, the choice to either cancel or shift any one stable pole emerges as a design issue that requires a deliberate trade-off between input disturbance response versus output disturbance response and robustness.

Haunting Transcendence: The Strategy of Ghosts in Bataille and Breton

An examination of the trope of the ghost in Andre Breton's Nadja (1928) and Georges Bataille's Histoire de l'ceil (1928) [1] enables one characterize both the political urgency of the authors' aesthetic projects in the late 1920s and 1930s and the public acrimony that erupted between them during that time. [2] The trope of the ghost was resonant for them in part because of its powerful and influential utilization by Marx and Engels, who begin their Manifesto with the declaration A specter is haunting Europe-- the specter of Communism. They continue: It is high time that Communists should openly, in the face of the whole world, publish their views, their aims, their tendencies, and meet this nursery tale of the specter of Communism with a manifesto of the party itself (8). The publication of the communists' beliefs, which haunted the capitalistic structure of Europe, made the individual communist plain and visible. Many ghosts circulate in the novels of Breton and Bataille, but neither writer trusts these specters anchor a Marxist enterprise. Instead, they use the ghost as a way of considering the boundaries of conventional identity. Both find that the economic crisis of capitalism is bound up in sexual and gender factors that are difficult channel into a single political movement. By examining the definitions, operations, and formulations of ghosts in the fiction of Bataille and Breton, one sees how differently they configure the political value of art. Whereas Breton uses the metaphor of the ghost dramatize the subject's dissolution as he tries escape ideology, Bataille uses the ghost examine the ideological terms through which both subjectivity and the threat of subjective dissolution are realized. The ghosts in Breton reflect a hopeful potential for art transcend psychiatric formulation and economic ideology. Bataille's ghosts deny transcendence and reflect a trace of the old sense of the word gho st: to wound, tear, pull pieces (OED). Breton formulates the identity of Nadja thus: Who am I? If this once were rely on a proverb, then perhaps everything would amount knowing whom 'haunt' (Nadja 11). Nadja, whom the narrator initially haunts and who eventually haunts him, demonstrates the central points of Breton's manifestoes. In his preface for a reprint of the Manifesto of Surrealism, he writes: simply believe that between my thought, such as it appears in what material people have been able read that has my signature affixed it, and which the true nature of my thought involves in something but precisely what do not yet know, there is a world, an imperceptible world of phantasms. (ix) The believing subject is stranded between two ideal points of a cause--effect sequence: the author and the text, or the author's intention and the text in which the intention is communicated. For Breton, the phantoms circulating in the area between the ostensibly stable poles of author and novel afford an ironic self-recognition. Breton's subject becomes a shuttlecock, batted between the cause and effect of the author and the text (or the text and the author). Although the notion of an author and a text may blur in the subject's confusion, this does not reflect merely a reversal of agency, but instead a complication of the preconditions necessary in speaking of Breton's subject. Subjectivity, or the self-recognizing me, relies on this tension between the I (or the public, authoring self) and the text. For Breton, phantoms fluctuating between these formerly stable poles of subjective intent and textual product represent symptoms be monitored in probing the subject. This erosion of subjective certainty ironically corresponds with a determination maintain the subject's autonomy in the face of stifling convention. Breton is not abandoning agency, but instead attempting reject the drastically determining nature of political, social, and economic ideologies. …

Fracture of a Superplate: The Proximal Cause of the Hawaiian-Emperor Seamount Bend?

—A change in plate motions of the Pacific basin - as inferred from the abrupt change in the strike at 43 Ma of the Hawaiian-Emperor Seamount chain - is modeled as the direct consequence of a plate fracture associated with the separation of Australia and Antarctica during the late Eocene. The postulated fracture was that of a large, and for the most part oceanic plate composed of the present Australian and Pacific plates. Volcanism in the Tasman Sea region with emphasis on the Balleny Islands hotspot is the proposed cause of the fracture, while plate tectonics in general is modeled in association with a mesospheric Rayleigh-Benard convection mode whose specific kinematics are determined from the global distribution of hotspots. Mesospheric convection is assumed to descend along a network of polygon boundaries shared by neighboring convection cells, and to approach those boundaries as radially advecting flow arising from hotspot plumes. Tractions due to the resulting shear strain across the asthenosphere both stabilize and drive the plate motions. The model yields unique and stable Euler poles for the motions of arbitrary plates, and specifically for the postulated pre-43 Ma ‘superplate’ and for the Pacific and Australian plates of present day to within about 10° to 20° (central angle) of observation. Plate tectonic torque magnitudes due to the underlying Benard convection are computed and found to be comparable to, or potentially larger than torques due to lithospheric density gradients, subsidence and subduction which may act in concert with, but are not necessarily required by the model. The analysis evaluates a unified lithosphere-bulk mantle approach to plate tectonics and addresses an ambiguity in the potential causes of plate motions.

SYNTHESIS OF TWO-INPUT PSS BASED ON THE H∞ CONTROL THEORY

In the standard mixed sensitivity weighting strategy for H∞ controllers, the designer can formulate desired requirements (desired loop shape) using frequency domain weighting functions. The ∞all pass∞ property of the optimal H∞ control laws ensures that the singular value Bode plots will precisely conform to those specified by the weighting functions. However, there is a serious drawback to this property; namely, the resulting standard H∞-PSS always cancels the stable poles of the plant (pole-zero cancellation). It is known that the cancellation of lightly damped poles can lead to poor robust stability and performance. In the case of multi-input H∞ controller, it becomes much more difficult to achieve good robust stability and performance. In this paper, a design of H∞-PSS is proposed to prevent the pole zero cancellation phenomenon and increase the damping of weakly damped modes. The design method consists of applying the bilinear transform to a stable poorly damped nominal plant in order to transform it into a fictitious unstable plant suitable for the standard mixed sensitivity design approach. A combination of additive and multiplicative uncertainty representation was used to achieve the robust stability for a wide range of operating conditions. The PSS designed based on the proposed approach was compared with those based on the standard H∞ approach and the conventional method. Simulation results show that the proposed PSS gives better performance and is more robust than the standard H∞-PSS and the optimally tuned conventional PSS.

VSS theory-based disturbance estimation scheme for MIMO systems and its applications

Based on variable structure systems (VSS) theory, this paper presents a new method of estimating the disturbances (or system nonlinearities and any model uncertainties) for continuous-time multi-input multi-output (MIMO) minimum phase (with respect to the relation between the disturbance and the output) dynamical systems. The designed robust method requires only the input and output measurements of the system. Even for MIMO systems under the assumption that the partial states directly affected by the disturbances do not directly appear in the outputs (when it is considered in the state space), the disturbances can still be estimated by applying the proposed formulation. The estimated disturbances are then employed to construct a robust state observer. Further, the estimated disturbances and the state observer are applied to synthesize a controller to place the desired stable poles and to cancel the disturbances. A design example and simulation results are presented to show the practicality and effectiveness of the proposed algorithm.

Analysis of Hysteretic Damping Using Analytic Signals

The purpose of this technical note is to present a time-domain formulation for linear hysteretic damping. The integro-differential equations that govern the dynamics of structures with linear hysteretic damping are transformed into ordinary differential equations in analytic signals—that is, complex-valued signals in which the real and imaginary parts are a Hilbert transform pair. The poles of this type of system show radial symmetry in the complex plane, determining that for each stable pole in the left-hand half of the complex plane, there is an “unstable” pole in the right-hand half. The impulse response functions of these unstable poles are bounded but noncausal. To illustrate the formulation, the analytic impulse response of a single-degree-of-freedom oscillator with linear hysteretic damping is obtained. The response of the structure to any loading signal can be obtained using time convolution of this impulse response function and the corresponding analytic excitation signal.

Variable structure system theory based disturbance identification and its applications

Based on the VSS theory, this paper presents a new method of estimating the disturbance for minimum-phase (with respect to the relation between the disturbance and output ) continuous-time dynamical systems with arbitrary relative degrees. By using the transfer function method, first, the higher order filter of the disturbance is estimated, then, the lower order filter of the disturbance is estimated inductively. Eventually, the disturbance is estimated. The estimated disturbance is employed to construct a VSS-type state observer. Further, the estimated disturbance and the state observer are applied to a controller to place the desired stable poles and to cancel the disturbance. A design example and simulation results are presented to show the practicality and effectiveness of the proposed algorithm.

TIME-DOMAIN ANALYSIS OF LINEAR HYSTERETIC DAMPING

Two linear-hysteretic-damping models that provide energy dissipation independent of the deformation frequency, are studied in this paper: a hysteretic Kelvin element and a hysteretic Maxwell element. Both models use the Hilbert transform and yield integro-differential equations for the equations of motion of structures when real-valued signals are utilized in the formulation. It is shown that the use of analytic (complex-valued) signals allows the transformation of these integro-differential equations into differential equations with analytic input signals and complex-valued coefficients. These differential equations show both stable and unstable poles. A technique for the solution of these differential equations is presented; it consists of a conventional modal decomposition of the state-space equations and the integration of the differential equations forward in time for the modal co-ordinates associated with stable poles, and backwards in time for the modal co-ordinates associated with unstable poles. Some numerical examples are presented to illustrate the characteristics of the models and the proposed analysis technique.

Robust pole-placement technique for plants with ellipsoidally uncertain parameters

A technique is proposed for synthesizing pole-placement controllers capable of assigning the closed-loop poles of a family of plants inside a specified design region. The design is applicable to plants characterized by uncertain parameters that are known to lie inside an ellipsoid. Ellipsoidal parametric uncertainty descriptions are particularly useful because they arise naturally in the context of parameter estimation, and hence have the advantage of being quantifiable using experimental input-output data. The control-design problem is posed as a minimax program that is shown to be convex, and that can be solved using numerical techniques with global convergence guarantees. Two reliable techniques for relating regions of stable poles to regions of allowed variability for the coefficients of the characteristic polynomial are discussed. The robust pole-placement design technique is illustrated with an example.

Quantum electrodynamics at large distances. I. Extracting the correspondence-principle part.

In quantum electrodynamics a classical or correspondence-principle part of the S matrix is normally factored out in order to obtain a remainder that can be treated perturbatively without the occurrence of infrared divergences. However, this separation, as usually performed, introduces spurious large-distance effects that produce an apparent breakdown of the important correspondence between stable particles and poles of the S matrix, and, in fact, lead to violations of the correspondence principle and to incorrent results for computations in the mesoscopic domain lying between the atomic and classical regimes. A computational technique is developed here that allows valid results to be calculated in this domain. It is shown in this article, and in paper II, in the context of a special example, how to extract a distinguished part of the S matrix that meets the requirements of the correspondence principle, of the pole-particle correspondence, and of infrared finiteness. In paper III the terms of the perturbatively treated remainder are shown to vanish, relative to this distinguished part, in the appropriate macroscopic limits. Thus this work provides both a needed computational technique and a confirmation in quantum electrodynamics of the validity of the correspondence principle and of the pole-particle correspondence, at least in the special case treated here in detail.

Compensation of uncertain continuous systems by using the internal model control principle

In this paper the design of compensators for uncertain continuous plants is investigated. The standard derived compensators are based on the application of the internal model control (IMC) method. The required a priori knowledge on the plant is rather weak, namely, an upper bound of the plant relative order, the numbers of the strictly unstable and critically unstable plant poles being integrators and upper and lower bounds of the amplitude-versus-frequency plot over the low frequency band in the case of minimum-phase open-loop systems. If the open-loop system has unstable zeros and/or poles then the above bounds are required to be known for a modified magnitude plot which substitutes the unstable zeros (poles) by stable poles (zeros) which are their complex-conjugate reflections on the left-hand plane. An absolute upper bound of the open-loop phase plot is obtained on a finite frequency interval which allows the closed-loop system to guarantee a prescribed relative stability in many practical situations. The method is dependent on the alternative design of phase lead/lag classical compensators and to indirect adaptive control situations where the adaptive identifier is used for the parametrization of the adaptive controller.

Stable interaction control and coulomb friction compensation using natural admittance control

AbstractA new control system design formulation is presented for achieving high‐performance, guaranteed‐stable impedance control. While the bandwidth of the resulting controller is no higher than alternative techniques, the new formulation significantly improves performance when Coulomb friction is present in the system. The technique requires a careful choice of the target impedance. The resulting feedback compensators are causal and have stable poles, although they are often non‐minimum phase. General rules for controller synthesis are derived. Experimental performance results are presented for a two‐mass system with internal compliance and Coulomb friction. Results demonstrate that the technique is successful in rejecting internal friction force disturbances while maintaining a passive driving‐point impedance. © 1994 John Wiley & Sons, Inc.