Number Of Unstable Poles Research Articles

Stability of neutral type fractional delay systems and its relation with stability of time‐delay and discrete systems

The bounded-input bounded-output stability of fractional delay systems of neutral type is investigated in this study. The proposed method calculates the number of unstable poles of the system for each delay value. All time-delay intervals in time-delay space where system is stable are precisely determined. The stability of systems with not only indefinitely large values of time-delay, but also with multiple poles on the imaginary axis has been studied. The proposed method is employed for investigating the stability of a fractional delay system when its fractional orders approach to the integer numbers or zero. It is proved that simple case of neutral types and time-delay systems have the same conditions for the independent stability. Two numerical examples are provided to illustrate the proficiencies of the proposed method.

Instability detection of ARMA systems based on AR system identification

The ability of parametric autoregressive (AR) system identification methods to detect the instability of an autoregressive moving average (ARMA) system of an unknown order is investigated. The collection of least squares AR estimators of various orders is shown to have the capacity to detect the instability of the underlying system. Necessary information is not the order of the system but, instead, an upper bound of the number of unstable poles with the maximal magnitude outside the unit circle.

A state space model for cochlear mechanics

The stability of a linear model of the active cochlea is difficult to determine from its calculated frequency response alone. A state space model of the cochlea is presented, which includes a discretized set of general micromechanical elements coupled via the cochlear fluid. The stability of this time domain model can be easily determined in the linear case, and the same framework used to simulate the time domain response of nonlinear models. Examples of stable and unstable behavior are illustrated using the active micromechanical model of Neely and Kim. The stability of this active cochlea is extremely sensitive to abrupt spatial inhomogeneities, while smoother inhomogeneities are less likely to cause instability. The model is a convenient tool for investigating the presence of instabilities due to random spatial inhomogeneities. The number of unstable poles is found to rise sharply with the relative amplitude of the inhomogeneities up to a few percent, but to be significantly reduced if the spatial variation is smoothed. In a saturating nonlinear model, such instabilities generate limit cycles that are thought to produce spontaneous otoacoustic emissions. An illustrative time domain simulation is presented, which shows how an unstable model evolves into a limit cycle, distributed along the cochlea.

µ-Interaction measure for unstable systems

In this paper, we present a method for system stabilisation through independent designs of the decentralised controller. The proposed method extends the practical applicability of the conventional ? interaction measure (?-IM) to unstable systems. The decentralised controller is designed based on a block diagonal approximation that is different from the block diagonal elements, but has the same number of unstable poles as the system. By expressing the ?-IM in terms of the transfer matrix between the disturbances and inputs, we show that the block diagonal approximation can be suboptimally selected by minimising the scaled L∞ distance between the system and the approximation. We present a numerical method for choosing the block diagonal approximation and a simple method for designing the decentralised controller based on the approximation.

On the reproduction of closed-loop experimental data by an LFT perturbed model set

This paper investigates the consistency between frequency domain closed-loop experimental data and a model set with linear fractional unstructured uncertainties. When the number of unstable poles of the transfer function matrices (TFM) in the model set are identical, a necessary and sufficient condition is derived for the reproduction of experimental data. This condition is expressed as the non-positiveness of a quadratic form of the external disturbances and measurement errors, together with a linear constraint. In this derivation, it is not required that all the TFMs in the model set are internally stabilizable by the controller. The condition is also not necessary for the closed-loop system consisting of the nominal model and the controller to be internally stable. It is believed that this condition is theoretically and computationally attractive in model set validation, no matter whether a deterministic or stochastic framework is adopted.

A Design Method of Bilateral Control System with Uncertain Dynamics of Environments.

In the present paper, we examine a design method for master-slave systems of bilateral control systems. In master-slave systems, human operator works to achieve tasks via the master and the slave system. The slave system contacts the environment and works the tasks. According to past studies, when the dynamics of environment is treated as uncertainties, the number of unstable poles of the slave system is required to be equivalent to that of the slave system with the dynamics of the environment. In some cases, the number of unstable poles of the slave system with the dynamics of environment is different from that of the slave system. We propose a simple design method of bilateral control systems such that the number of unstable poles of the slave system with the dynamics of environment is different from that of the slave system without the dynamics of the environment.

Reduced-order controllers for unstable multivariable linear systems that can be balanced

This paper considers the problem of H ∞-stabilization of unstable multivariable linear systems. The major features of the approach are: (1) a reduced-order model is obtained using low-frequency balancing, the approximant will have the same number of unstable poles as the original system, (2) the controller design is accomplished by dynamic output feedback, and (3) sufficient conditions in the form of two algebraic Riccati equations and an upper bound explicitly characterize a H ∞-controller of lower dimensions. At the end, an illustrative example is given to show the simplicity of the procedure.

Robust Stability of Closed-Loop Systems Resulting From Nonsequential MIMO-QFT Design

Considered in this paper is the question of whether a compensator realized by the MIMO-QFT nonsequential method robustly stabilizes the entire plant family. In order to establish our results, first the classic small gain theorem for robust stability is modified to allow uncertain plant families with poles arbitrarily crossing the imaginary axis, or equivalently, plants in which the number of unstable poles does not remain fixed over all uncertainties. The conventional assumption that the number of unstable poles remain fixed over all uncertainties can be quite restrictive, especially, in the case of plants with structured uncertainties. It is shown that to assure robust stability of the closed loop, resulting from the MIMO-QFT nonsequential approach, one more requirement must be added to the procedure. The needed extra condition can be quite naturally incorporated during the execution of the nonsequential technique. As a result, the proposed condition does not significantly alter the basic MIMO-QFT nonsequential procedure.

A priori stability conditions for an arbitrary number of unstable poles

Necessary and sufficient stability conditions that place bounds on the current inputs are derived that guarantee BIBO stability and retain the control authority needed to asymptotically stabilize a system that is subject to physical input constraints. For open-loop stable systems, the stability conditions coincide with the physical constraints, but for unstable systems, tighter limits may be required. Use of these limits in a supervisory role with controllers that have no stability guarantee is discussed and illustrated.

The four-block Adamjan-Arov-Krein problem

This paper is concerned with the mixed sensitivity H ∞ design in the most general, i.e., the four-block, case. This problem involves in a crucial manner the so-called four-block operator Γ, the norm of which is the achievable feedback tolerance. Our objective in this paper is to provide an interpretation for all singular values of Γ. These singular values are given an Adamjan-Arov-Krein interpretation in terms of the L ∞ distance of an L ∞ function to H ∞( l). Intuitively, the singular values of Γ are the various tolerance levels that can be achieved if we allow a various number of unstable poles in the closed loop. We finally provide an upper bound on the number of singular values.

Model reduction for discrete unstable systems based on generalized normal representations

The input and output normal representations of stable systems are generalized for unstable systems. These are then used to reduce the order of unstable discrete-time linear multivariable systems resulting in reduced order models with the same number of unstable poles as the full order model and with an a priori upper bound on the reduction error. The generalized normal representations are then modified to include a frequency weighting resulting in frequency weighted generalized normal representations. These are used to derive frequency weighted reduced order models for unstable discrete systems with the same number of unstable poles as the full order model and with an a priori upper bound on the frequency weighted reduction error.

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.

Robust Feedback Control of SISO-plants with a Known Number of Unstable Poles

Abstract Model uncertainties described in the frequency domain are used in many applications. The present paper deals with the problem of stabilizing SISO-plants when the number of unstable poles is known. It is shown how some flexibility in defining the uncertainty set can be used in order to be able to cope with common model uncertainties such as pure gain mismatch and large phase mismatch at high frequencies. This is an extension to previous work in this field.

A robustness test for distributed feedback systems†

A robustness test is presented for a class of distributed feedback systems. The test does not require the plant and perturbed plant models to have the same number of unstable poles.