Stable Poles Research Articles

Conditions to Preserve Stability of Zeros on Sampling

When a continuous-time system is sampled by use of a zero order hold, all stable poles are transformed into the unit circle. However, there is no simple relation between the zeros of a continuous-time system and its sampled version. This paper has derived sufficient conditions which guarantee that a continous-time system with zeros in the left half plane is transformed to a sampled system with zeros inside the unit circle for all sampling periods. The criteria shown in this paper are represented by coefficients of the partial fraction expansion of a continuous-time transfer function.

The Linear Constrained Regulation Problem for Some Linear Continuous-Time Singular Systems

Abstract A major issue in modem control system design is direct integration of technological and security constraints in the control law. In general, the respect of constraints can be obtained by constructing a stabilizing controller which makes positively invariant a domain included in the set defined by the constraints. In most of practical control problems, state and/or input constraints generate convex polyhedra in the state space. This paper establishes some necessary and sufficient algebraic conditions for positive invariance of a convex polyhedra w.r.t linear continuous-time singular systems. These results can be considered as an extension of the positive invariance relations for linear normal systems. These results are then applied to solve certain constrained control problem, when the inputs are constrained to belong to a compact polyhedron. It is shown that for a controllable singular system with a sufficient number of stable finite poles, it is always possible to construct a stabilizing state feedback control, u(t) = Fx(t) , guaranteeing the closed-loop positive invariance of the' polyhedron determined by the feedback matrix F. A numerical example is given.

Discrete-time pole placement with stable controller

The classical polynomial approach to pole placement involves the computation of a particular solution of a given Diophantine equation. The solution corresponds to a proper controller whose order is fixed by the number of nonminimum phase zeros of the plant transfer function, by its order, and by the desired closed-loop characteristic polynomial. This controller might be unstable for some desired closed-loop characteristic polynomials, even though the plant can be stabilized by a stable controller. Now, unstable controllers are known to deteriorate closed-loop system performances, compared with stable ones. Hence, we propose a procedure to build, for a specific class of plants, a stable pole placement controller when the classical pole placement controller is unstable. The properties of both stable and unstable controllers are compared via simulations. It appears that, for the considered plants, the stable controller improves at least one of the following properties, compared with its unstable counterpart: step response transient, gain margin, measurement noise rejection. The results obtained with our procedure are also compared with the results obtained via Middleton's approach, which consists in modifying the desired closed-loop poles so as to obtain a stable and minimum phase controller with the classical pole placement method. Finally, our pole placement design with stable controller is applied to solve a particular simultaneous partial pole placement problem.

Stable Zeros of Sampled Systems

When a continuous-time system is sampled by use of a zero order holder, all stable poles are transformed into the unit circle. However, there is no simple relation between the zeros of a continuous-time system and its sampled version.In this paper, a necessary and sufficient condition for stable zeros of a sampled system is presented when a continuous-time system has a strictly proper and rational transfer function. The criterion derived in this paper is represented in terms of coefficients of a continuous-time transfer function and of a sampling period. Further, this paper gives a necessary and sufficient condition which ensures that all zeros of a sampled system are inside the unit circle for all sampling periods.

A counterexample to a simultaneous stabilization condition for systems with identical unstable poles and zeros

A paper published in this journal in 1984 (“Some results on simultaneous stabilization”, Vol. 5, pp. 205–208) presents a necessary and sufficient condition under which plants with identical unstable poles and zeros, but differing by their stable poles and zeros, can be simultaneously stabilized (Theorem 3, p.207). We show by a counterexample that the condition presented in that paper is sufficient but not necessary.

Backward linear predictor and overfit techniques for estimating rational impulse responses

The authors present a fast and robust algorithm, based on Prony's method, which improves the identification of time invariant linear systems with pure real poles in the transfer functions. Previous investigations have more or less avoided this aspect of broad-band signals and the phenomenon of pole splitting. The authors supplement overfit techniques and backward predictor formulation with a new method (in this context), based on an inspection of the coefficients related to the stable poles in the sum of exponentials. The capabilities of the proposed algorithm are investigated by means of Monte Carlo simulation and compared with the widely used KT algorithm of Kumaresan and Tufts. For the purpose of on-line identification they assume the system order to be known, although they do comment on some order detection strategies. The modified identification tool yields improved robustness (for instance concerning the problem of pole splitting) and a smaller bias, as illustrated by a typical third order system.

The Effect of Adding Zeroes to Feedforward Controllers

The effect of adding zeroes to two types of feedforward controllers, stable pole zero cancelling (SPZC) controllers and zero phase error tracking (ZPET) contollers, is discussed. When there are uncancellable zeroes in the feedback system, additional zeroes in the feedforward controller can reduce the tracking and/or the contour error. For multi-axis contouring with mismatched axis dynamics, a ZPET controller cancels the phase error and provides good contouring accuracy. With a SPZC controller which does not cancel the phase lag, the axes can be “matched” for equal phase lag by adding zeroes to the feedforward controller.

Sequential design of linear quadratic state regulators with prescribed eigenvalues and specified relative stability

This paper considers the problem of optimal regulator design of linear multivariable systems with prescribed pole locations and/or poles corresponding to specified relative stability. A sequential method based on the frequency-domain optimality condition is proposed for achieving the desired pole assignment and determination of the corresponding quadratic performance index. This design method enables the retention of some stable open-loop poles and the associated eigenvectors in the closed-loop system. An illustrative example is provided to demonstrate the effectiveness of the proposed method.

Wiener filter design using polynomial equations

A simplified way of deriving realizable and explicit Wiener filters is presented. Discrete-time problems are discussed in a polynomial equation framework. Optimal filters, predictors, and smoothers are calculated by means of spectral factorizations and linear polynomial equations. A tool for obtaining these equations, for a given problem structure, is described. It is based on the evaluation of orthogonality in the frequency domain, by means of canceling stable poles with zeros. Comparisons are made to previously known derivation methodologies such as completing the squares for the polynomial systems approach and the classical Wiener solution. The simplicity of the proposed derivation method is particularly evident in multistage filtering problems. To illustrate, two examples are discussed: a filtering and a generalized deconvolution problem. A new solvability condition for linear polynomial equation appearing in scalar problems is also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Domain Structures of Co-Zr-Nb Amorphous Main-Pole Film and Characteristics of Perpendicular Magnetic Recording

Domain structures of Co-Zr-Nb amorphous film for use as the main poles of single-pole perpendicular magnetic recording heads have been observed using the Bitter method. A narrower Co-Zr-Nb main pole with a low H k showed a disordered domain structure near its tip in the remanent state. This in turn caused low and unstable sensitivity of the narrow-track single-pole head. Rotation magnetization of major domains is dominant in the magnetization of stable main poles. A stable and highly sensitive single-sided single-pole head was obtained by using a shorter main pole with an aspect ratio of about 1; this pole had an ordered and stable domain structure even in the remanent state.

Multivariable Nyquist self-tuning: a general approach

An earlier paper showed that the generalised Nyquist approach, although intrinsically a frequency-response approach, can be deployed to produce an effective multivariable self-tuning methodology. The results, however, were confined to the case of plants with open-loop stable poles and characteristic transfer functions whose branch points are all stable. The purpose of the present paper is to develop the framework that applies to the general case. In particular, consideration is given to fractional representations for the branches of the characteristic transfer functions, and algorithms are developed for both the computation of these representations and their implementation within a self-tuning configuration. Consideration is also given to issues of stability. The results of the paper are illustrated by means of numerical examples and the paper is concluded by a design study based on the open-loop unstable chemical reactor.

Stable pole placement direct adaptive control for systems with arbitrary zeros

This paper extends a recently proposed adaptive regulation algorithm to the general case where a non–zero bounded reference signal is allowed. The desired control objectives are asymptotically satisfied in the sense that the estimated plant together with the adaptive controller yield a characteristic polynomial arbitrarily close to the desired one. Furthermore, in the case of a zero reference signal, the regulation objective is asymptotically achieved.

Robust pole placement indirect adaptive control

AbstractThe problem of designing a robustly stable pole placement indirect adaptive controller in the presence of output disturbances and unmodelled dynamics is addressed. The key features of such a design are the following. (1) The unknown parameters are estimated by a normalized least‐squares algorithm with a dead zone to provide the stability robustness with respect to bounded disturbances and ‘small’ unmodelled dynamics. (2) The estimated model controllability is ensured by modifying the control law over a finite time. The modification involved consists of adding an internal impulse excitation and ‘freezing’ the controller parameters.

Unstable rational function approximation

The problem is considered of approximating a transfer function with stable and unstable poles by a lower-order transfer function with the same number of unstable poles. A method is suggested and compared with alternative known methods based on Hankel-norm approximation. Examples suggest that no one method always outperforms other methods in terms of minimizing the approximation error.

Poles-zeros placement and decoupling in dicrete LQG systems

A design algorithm is introduced to synthesise a discrete time LQG optimal controller subject to design constraints requiring (i) complete and arbitrary stable poles placement, (ii) some zeros assignment, and (iii) input-output decoupling. The zeros placement is partially used to deal with the deterministic reference tracking and disturbances rejection problems. In the paper, the Wiener-Hopf technique is employed and two weighting matrices are shaped by the inverse optimal control method, so that the controller is optimal with respect to the chosen weighting matrices and achieves the above three goals simultaneously.

The Carathéodory-Fejér method for recursive digital filter design

A new technique for rational digital filter design is presented which is based on results in complex function theory due to Takagi, Krein, and others. Starting from a truncated or windowed impulse response, the method computes the unique optimum rational Chebyshev approximation with a prescribed number of stable poles. Both phase and magnitude are matched. Deleting the noncausal (unstable) part of the Chebyshev approximation yields a stable approximation of specified order (M,N) which is close to optimal in the Chebyshev sense. No iteration is involved except in the determination of an eigenvalue and eigenvector of the Hankel matrix of impulse response coefficients. In this paper the algorithm is specified and practical examples are discussed.

The stability and instability of partial realizations

A finite or infinite sequence of real numbers is said to be stable if it admits minimal realization by a stable linear system. It is shown that the preservation of stability as such a sequence is truncated or extended is not a generic property even among stable sequences. This is of interest for identification of linear systems from partial data and for partial realization of random systems, in which cases it constitutes a negative result. Certain finite sequences have infinitely many minimal realizations each having different stability properties. In this case, a graphical criterion in the spirit of the Nyquist criterion is derived to exploit this lack of uniqueness in order to determine whether one can achieve a stable pole placement by a judicious choice of partial realization.

Formant estimation with linear prediction

Although the vocal tract transfer function for vowels is approximately all‐pole, the resulting vowel waveform is not. We have investigated the effects of using an all‐pole analysis technique (linear prediction) on real and synthetic vowels. Formant estimation accuracy and pole location stability were studied as a function of the length and placement of the speech segment analyzed, its pitch, and the analysis technique used. The synthetic speech consisted of a glottal waveform model [G. Fant, STL‐QPSR 1 (1979)] with time varying formant damping. Covariance and autocorrelation linear prediction was performed over closed glottis phases, entire pitch periods, and randomly placed 20–25 ms frames. Closed glottis phases for real speech were identified from simultaneous laryngograph measurements and verified by inverse filtering. Covariance linear prediction over the closed glottis phase gave the most accurate formant estimates, stable pole locations, and little or no degradation with increasing pitch. Other anal...

Subdominant critical indices for the ferromagnetic susceptibility of the spin-1/2 Ising model

Starting from the high-temperature series for the susceptibility of the spin-1/2 ferromagnetic Ising models-square, planar triangular, SC, BCC, FCC, diamond, and HSC in four, five and six dimensions-the authors analyse the analytical structure near the critical point of the susceptibility by writing it as a Laplace transform in the variable log (1-w/wc), where w=tanh( beta J). The interest of the method is that the coalescing singularities which sit at w=wc are spread out and can be analysed separately, the first subdominant critical indices, appearing as stable poles of Pade approximants. They first recover the results for the two-dimensional models, with a high accuracy: gamma =1.74995 and gamma s=0.757. The most stable results in three dimensions are obtained for the diamond lattice: gamma =1.2506+or-0.0015 and gamma s=0.42+or-0.11. The other lattices give 0.10<gamma s<0.52. However, in all three-dimensional cases the amplitude of the subdominant singularity is less than a few per cent of the leading one. Therefore the uncertainties are large, but the subdominant singularity stays closer to the value Gamma -1=0.25 than to the field theoretical predicted one gamma s=0.75.

Optimum design of low-pass switched-capacitor ladder filters

A general technique for realizing Bruton's LDI transformation in low-pass switched-capacitor ladder filters is presented, which results is switched-capacitor integrators that are insensitive to parasitic capacitances. It is shown that the correct filter response may be obtained by realizing only the poles of the LDI transformed LC network. Realization of these stable poles is reduced to the problem of solving a system of nonlinear equations. The design of a third-order Chebyshev low-pass filter is presented as an example.