Multiplicative Perturbation Research Articles

Stability Robustness of LQ Optimal Regulators Based on Multirate Sampling of Plant Output

The stability robustness of stable feedback loops, designed on the basis of multirate-output controllers (MROCs), is analyzed in this paper. For MROC-based feedback loops, designed in order to achieve LQ optimal regulation, we characterize additive and multiplicative norm-bounded perturbations of the loop transfer function matrix which do not destabilize the closed-loop system. New sufficient conditions for stability robustness, in terms of elementary MROC matrices, are presented. Moreover, guaranteed stability margins for MROC-based LQ optimal regulators are suggested for the first time. These margins are obtained on the basis of a fundamental spectral factorization equality, called the modified return difference equality, and are expressed directly in terms of elementary cost and system matrices. Sufficient conditions in order to guarantee the suggested stability margins are established. Finally, the connection between the suggested stability margins and the selection of cost weighting matrices is investigated and useful guidelines for choosing proper weighting matrices are presented.

Robust H∞ infinity control in the presence of stochastic uncertainty

This paper considers a robust H infinity state feedback control problem for linear uncertain systems with stochastic uncertainty. The uncertainty class considered in the paper involves uncertain multiplicative white noise perturbations which satisfy a certain variance constraint. A state feedback controller is presented which guarantees a prescribed level of disturbance attenuation for all admissible stochastic uncertainties.

Non-Linear Self-Excited Random Vibrations and Stability of an SDOF System with Parametric Noises

Abstract. The paper presents the solution to the properties of stochastic response of a system with random parametric noises, which is prone to the loss of aerodynamical stability. The system is described by an equation of van der Pol type with the negative linear, and with the positive cubic dampings. The coefficients of the linear damping and of the stiffness include the multiplicative random perturbations, the external excitation being given as a sum of a deterministic function and of an additive perturbation. All three input random processes are supposed to be Gaussian and centered, with the non-zero mutual stochastic parameters, as it corresponds to the properties of real systems. The solution has been based on the method of stochastic linearisation and of the subsequent solution of the Fokker–Planck–Kolmogorov equation in the sense of the first and second stochastic moments for the transient and stationary states. There have been demonstrated several effects, which are typical for systems with parametric noises, differentiating them from the systems with constant coefficients. The principal attention has been devoted to the properties of the spectral density of the response, the character of which changes abruptly with the degree of non-linearity of the damping and of the level of random perturbations. Sommario. La presente memoria studia le proprieta della risposta stocastica di un sistema con eccitazione casuale parametrica, che tende alla perdita della stabilita aerodinamica. Il sistema e descritto mediante un'equazione del tipo di van der Pole con il termine lineare dello smorzamento negativo e il termine cubico positivo. Poicha l'eccitazione esterna e la somma di una funzione deterministica e di una perturbazione additiva, i coefficienti dello smorzamento lineare e della rigidezza comprendono le perturbazioni casuali moltiplicative. I tre processi stocastici di eccitazione sono assunti gaussiani e a media nulla con parametri stocastici incrociati diversi da zero, come si verifica per le proprieta dei sistemi reali. La soluzione e basata sul metodo della linearizzazione stocastica e della successiva soluzione dell'equazione di Fokker-Planck-Kolmogorov studiando i primi e i secondi momenti statistici per gli stati transitori e stazionari. Vengono mostrati diversi effetti, tipici dei sistemi con eccitazione parametrica, differenziandoli dai sistemi a coefficienti costanti. Particolare attenzione e rivolta alle proprieta della densita spettrale della risposta le cui caratteristiche cambiano bruscamente con il grado di non linearita dello smorzamento e del livello di casualita delle perturbazioni.

Multiplicative perturbations of stable and convergent operators

Some familiar classes of stable Hilbert-space operators are studied to determine how they overlap and where the unitary similarity classes of their members lie. Analogous, but less familiar, classes of convergent operators are examined with the same aim. The classes considered are often sets of products M A where M is a given set of diagonal or Hermitian matrices and A is a single matrix. The A's for which M A is a set of stable or convergent operators are sometimes characterized.

Heat kernels of multiplicative perturbations: Holder estimates and Gaussian lower bounds

Heat kernels of multiplicative perturbations: Holder estimates and Gaussian lower bounds

Perturbation theory of abstract cauchy problems and volterra equations

Some multiplicative perturbation theorems and additive perturbation theorems for resolvent families are presented.

Simultaneous stabilization of multiplicatively perturbed plants

Necessary and sufficient conditions are obtained for simultaneous stabilizability of a given linear, time-invariant, multi-input multi-output nominal plant and a multiplicatively perturbed plant, where the multiplicative perturbation is stable. These conditions are derived from the general parity-interlacing-property applicable to any two arbitrary plants and are expressed explicitly in terms of the real-axis poles and zeros of the nominal plant and the perturbation. The class of perturbations for which simultaneous stabilization is achievable are characterized using these conditions. A special class of unknown diagonal perturbations is also considered and simultaneously stabilizing controllers are designed for the nominal plant under any such unknown perturbation.

On robust stabilization regions of unstructured uncertainties

Relations on the robust stabilization regions of four main forms of unstructured uncertainties are investigated. The robust stabilization regions represented by three other perturbations are derived from coprime factor perturbation and additive perturbation, respectively. Furthermore, it is shown that the normalized coprime factor H ∞ robust controller can also be explained in additive and multiplicative perturbations.

Uncertainty modeling and experiments in ℋ/sub ∞/ control of large flexible space structure

Approaches to uncertainty modeling for robust control of large flexible space structures (LFSSs) such as additive or multiplicative perturbations in /spl Hscr//sub /spl infin// do not work very well because of the special properties of LFSS dynamics. We propose the use of a left-coprime factorization (LCF) of LFSS dynamics in modal coordinates in order to get improved stability margins and performance in robust control design. The plant uncertainty is then described as stable perturbations of the coprime factors accounting for modal parameter uncertainty and unmodeled dynamics. Two multivariable /spl Hscr//sub /spl infin// designs based on LCFs of 46th-order collocated and noncollocated models of an LFSS experimental testbed are presented together with simulation and experimental results to illustrate the technique.

Multiplicative perturbations in semigroup theory with the (Z)-condition

In [4] the (Z)-condition was introduced by G. W. Desch and W. Schappacher. This condition on a Banach spaceZ and a generatorA inX ensures thatA(I+B) and(I+B)A generateC0-semigroups, ifB has its range inZ. In this paper we will consider how certain properties of the semigroup generated byA are inherited by the semigroups generated byA(I+B) and(I+B)A. We shall furthermore investigate a related condition, that simplifies certain sufficiency assumptions given previously.

Linear Volterra equations and integrated solution families

We consider the linear Volterra equation $${\text{(VE;}}A{\text{,}}a{\text{)}}u{\text{(}}t{\text{) = }}x {\text{ + }}\int_{\text{0}}^{\text{t}} { a{\text{(}}t{\text{ - }}s{\text{)}}Au{\text{(}}s{\text{)}}ds {\text{for }}t \geqslant {\text{0}}{\text{.}}} $$ HereA is an unbounded closed linear operator in a Banach spaceX anda is a scalar valued function. We study the theory of solution families which are not necessarily exponentially bounded and also, as their generalizations, consider the notion ofn-times integrated solution families for (VE;A, a). These families are characterized in terms of the associated Volterra integral equation $${\text{(VE;}}A{\text{,}}a{\text{)}}_n u{\text{(}}t{\text{) = }}\frac{{t^n }}{{n!}}x {\text{ + }}A{\text{ }}\int_{\text{0}}^{\text{t}} { a{\text{(}}t{\text{ - }}s{\text{)}}u{\text{(}}s{\text{)}}ds {\text{for }}t \geqslant {\text{0}}{\text{.}}} $$ The results are applied to additive and multiplicative perturbation theorems and adjoint problems.

Fringe detection in noisy complex interferograms

A new algorithm to estimate the two-dimensional local frequencies of phase interferometric data is described. With a complex sine-wave model, demonstration is given that a conventional multiple-signal classification (MUSIC) algorithm can be used in spite of multiplicative noise perturbations. A faster algorithm dedicated to the processing of interferograms is developed and a measure of confidence in the estimate is proposed. We studied numerical performances using synthetic fringes. As a result of the frequency estimation, knowledge of the fringe local width and orientation can be applied to restore noisy phase data. Results of a complex phase filter are presented for real interferograms obtained from synthetic aperture radar images.

Connection of Multiplicative or Relative Perturbation in Coprime Factors and Gap Metric Uncertainty

In this paper, it is shown that an uncertain system described by a certain L∞ multiplicative or relative perturbation in its coprinw factors is the same as the one described by a gap or v-gap metric ball. Hence all of the stability robustness results for gap or v-gap metric uncertainty carries over to this type of coprime factor perturbation. Uncertain systems described by H∞ multiplicative or relative perturbations in coprime factors are also studied in this paper. Necessary alld sufficient, conditions for robust stability of it feedback system with coprime factors of both the plant and the controller subjed to simultaneous H∞ multiplicative or relative perturhations are obtained.

Robustness Analysis of Linear Systems with Nonlinear Uncertain Parameters

The present paper studies the stability robustness analysis problem for uncertain state space systems. We propose a new uncertain state space model which allows second-order uncertain parameters. The uncertainties in the model appear in the fonn of combination of “additive perturbation” and “multiplicative perturbation”. It is shown that the model has strongly practical background. Some robustness criteria based on the new model are derived. An example shows the advantages of the proposed approach.

Filtrage avec observation discontinuesur une variété. existence d'une densité régulière

We consider a random signal (a vectorial Markovian process) partially observed on a symmetric Riemannian space. The observation is a cadlag stochastic process given by a multiplicative perturbation...

Perturbation and comparison of cosine operator functions

LetA be a closed linear operator such that the abstract Cauchy problemu″(t)=Au(t), t∈R; u(0)=x, u′(0)=y, is well-posed. We present some multiplicative perturbation theorems which give conditions on an operatorC so that the abstract Cauchy problems for differential equationsu″(t)=ACu(t) andu″(t)=CAu(t) also are well-posed. Some new or known additive perturbation theorems and mixed-type perturbation theorems are deduced as corollaries. Applications to characterization of the infinitesimal comparison of two cosine operator functions are also discussed.

Computation-oriented expression of a non-conservative condition for robust stability of sampled-data systems

A new expression of a non-conservative robust stability condition is derived for sampled-data systems taking into consideration its convenience in numerical computation. Our expression is described using a global maximum of a certain differentiable function, while the conventional expression includes a global maximum of a structured singular value, which is expensive to compute. When the sampling period is long, the advantages of our expression become clear. A possible problem is that the maximized function may be multimodal. However, the conventional expression has the same problem and even a local maximum is shown to give useful information. Our expression is applicable to a fairly large class of systems including the multi-input multi-output additive and multiplicative perturbation models. Using an example, we show how our expression is computed and make a comparison with the conventional expression.

Single-Loop QFT Design for Robust Performance in the Presence of Non-Parametric Uncertainties

This note focuses on the robust performance control design problem in single-loop systems from the viewpoint of the Quantitative Feedback Theory (QFT) method. Unlike the more general setup for QFT which allows arbitrary plant uncertainty description (i.e., value sets or templates), we assume that the plant uncertainty is described by multiplicative perturbation. A set of quadratic inequalities is developed for the robust performance problem in general, and, in particular, for the robust complimentary sensitivity problem. The so-called QFT bounds can be computed in a closed-form from these quadratic inequalities.

Eigenvalue perturbation models for robust control

This paper presents a nonconservative description of the regions containing the eigenvalues of a diagonal matrix, perturbed by the set of all unknown, Euclidean norm bounded matrices. This is extended to real valued block diagonal matrices where each two-by-two block reveals a complex conjugate pair of eigenvalues. A weighting matrix allows one to specify the exact size of the perturbation to each of the eigenvalues. An identical result is obtained for real valued perturbations. This result is motivated by, and applied to, the modeling of frequency and damping perturbations in models of flexible structures. The resulting perturbation description fits within the established H/sub /spl infin// robust control framework and, in certain situations, is less conservative than the more standard additive or multiplicative perturbation models. >

Multiplicative Perturbations of C0-Semigroups and Some Applications to Step Responses and Cumulative Outputs

For a C0-semigroup T(·), we prove a general multiplicative perturbation theorem which subsumes many known multiplicative and additive perturbation theorems, and provides a general framework for systematic study of the perturbation associated with a step response U(·) of a linear dynamical system. If the semivariation SV(U(·), t) of U(·) on [0, t] tends to 0 as t→0+, then the infinitesimal operator As of the pair (T(·), U(·)), as a mixed-type perturbation of the generator A of T(·), generates a C-o-semigroup T-s(.) with parallel to T-s(t)-T(t)parallel to=0(1)(t→0(+)). Furthermore, C0-semigroups S(.) which satisfy ||S(t)-T(t)||=O(t)(t→0(+)) are exactly those mixed-type perturbations caused by Lipschitz continuous step responses. Perturbations related to a cumulative output V(·) are also investigated by using a multiplicative perturbation theorem of Desch and Schappacher. In particular, we show that bounded additive perturbations of A are exactly those mixed-type perturbations caused by Lipschitz continuous cummulative outputs. It is also shown that the generator of T(·) is bounded if and only if SV(T(·), t) is sufficiently small for all small t.