Multiplicative Perturbation Research Articles

A note on the perturbation bounds of eigenspaces for Hermitian matrices

Let H be a Hermitian matrix, and H ˜ be its perturbed matrix. In this paper, both additive and multiplicative perturbations of eigenspace are studied, and some absolute and relative bounds are presented, which are better than the existing bounds in some sense.

New results on robust control design of discrete-time uncertain systems

The paper considers the problem of robust control redesign of a class of linear discrete-time systems with norm-bounded parametric uncertainties and controller-gain perturbations. In both cases of additive and multiplicative perturbations, the control objective is to design a resilient feedback-stabilisation schemes using guaranteed cost control and H∞ control approaches. The problem under consideration is cast as convex optimisation over linear matrix inequalities which led to the derivation of necessary and sufficient conditions for closed-loop robust quadratic stability. Three system examples are provided to illustrate the theoretical developments.

Multiplicative perturbations of constant-domain evolution equations

I prove a variation-of-constants formula and an existence theorem for multiplicative perturbations of nonautonomous linear equations, in the constant-domain, nonparabolic case (CD-systems).

The H.INF. Control for an Inverted-Double Pendulum System Consisting of Elastic Links Connected in Series

A stabilizing control technique for an inverted-double pendulum system consisting of two elastic links connected in series is developed, using a reduced-order model considering only the first vibration mode of the upper elastic beam, and applying an H∞ controller design method for a mixed sensitivity problem that can consider multiplicative perturbations at the input port and is characterized by a settling function. It is found that although the coupling effect of different natural frequencies between the two flexible beams makes the stabilization problem more difficult, it is possible to stabilize the system by a feedback control using only the system’s output signals that include the vibration signal of the upper elastic beam. Furthermore, it is concluded that the more different the lowest natural frequencies of the two elastic beams are, the more easily the system can be stabilized.

Extended Libor market models with stochastic volatility

This paper introduces stochastic volatility to the Libor market model of interest rate dynamics. As in Andersen and Andreasen (2000a) we allow for non-parametric volatility structures with freely specifiable level dependence (such as, but not limited to, the CEV formulation), but now also include a multiplicative perturbation of the forward volatility surface by a general mean-reverting stochastic volatility process. The resulting model dynamics allow for modeling of non-monotonic volatility smiles while explicitly allowing for control of the stationarity properties of the resulting model dynamics. Using asymptotic expansion techniques, we provide closed-form pricing formulas for caps and swaptions that are robust, accurate, and well-suited for both model calibration and general mark-to-market of plain-vanilla instruments. Monte Carlo schemes for the proposed model are proposed and examined.

Analysis of multiframe super-resolution reconstruction for image anti-aliasing and deblurring

Novel theoretical results for the super-resolution reconstruction (SRR) are presented under the case of arbitrary image warping. The SRR model is reasonably separated into two parts of anti-aliasing and deblurring. The anti-aliasing part is proved to be well-posed. The ill-posedness of the entire SRR process is shown to be mainly caused by the deblurring part. The motion estimation error results in a multiplicative perturbation to the warping matrix, and the perturbation bound is derived. The common regularization algorithms used in SRR are analyzed through the discrete Picard condition, which provides a theoretical measure for limits on SRR. Experiments and examples are supplied to validate the presented theories.

Some New Perturbation Bounds for the Generalized Polar Decomposition

The changes in the unitary polar factor under both multiplicative and additive perturbation are studied. A multiplicative perturbation bound and a new additive perturbation bound, in which a different measure of perturbation is introduced, are presented.

Multiplicative perturbations of C-regularized resolvent families.

This paper presents two new theorems for multiplicative perturbations of C-regularized resolvent families, which generalize the previous related ones for the resolvent families.

The H.INF. Control for an Inverted-Double Pendulum System Consisting of Elastic Links Connected in Series

A stabilizing control technique for an inverted-double pendulum system consisting of elastic links connected in series is investigated using a reduced-order model considering only the first strain vibration mode of the upper elastic beam, and applying the H-infinity controller design method for the mixed sensitivity problem which has multiplicative perturbation at the output port and is characterized with a settling function. It is found that although the coupling effect of the different natural frequencies between two flexible beams make the stabilization problem more difficult, even a feedback control using only the output signal based on the strain vibration on the upper elastic beam is possible in stabilizing the system. Furthermore, it is concluded that the more different the lowest natural frequency of each elastic beam is, the more easily the controllability is realized.

A Note on Multiplicative Backward Errors of Accurate SVD Algorithms

Multiplicative backward stability results are presented for two algorithms which compute the singular value decomposition of dense matrices. These algorithms are the classical one-sided Jacobi algorithm, with a stringent stopping criterion, and an algorithm which uses one-sided Jacobi to compute high accurate singular value decompositions of matrices given as rank-revealing factorizations. When multiplicative backward errors are small, the multiplicative perturbation theory for the singular value decomposition developed in the last decade can be applied to get high accuracy bounds on the errors of the computed singular values and vectors.

Periodic Forcing and Goal Oriented Control of Chaos

We study control of chaos by periodic modulation without feedback in discrete maps. We show that period-p forcing shifts and splits the naturally occurring period-p windows (and their multiples) and leads to the stabilization of chaos, and that for multiplicative sinusoidal perturbations of the logistic map stabilization can be done at a desired value of the control parameter and at a desired periodicity, simply by varying the amplitude of forcing, provided the modulated system remains bounded.

Absolute stabilization related to circle criterion: An LMI-based approach

A method is proposed for synthesizing output feedback controllers for nonlinear Lur' e systems. The problem of designing an output dynamic controller for uncertain-free systems and systems subject to multiplicative norm-bounded perturbations in the linear part were proposed respectively. The procedure is based on the use of the absolute stability, through the circle criterion, and a linear matrix inequalities (LMI) formulation. The controller existence conditions are given in terms of existence of suitable solutions to a set of parameter-dependent LMIs.

Well-posedness of an infinite system of partial differential equations modelling parasitic infection in an age-structured host

We study a deterministic model for the dynamics of a population infected by macroparasites. The model consists of an infinite system of partial differential equations, with initial and boundary conditions; the system is transformed in an abstract Cauchy problem on a suitable Banach space, and existence and uniqueness of the solution are obtained through multiplicative perturbation of a linear C0-semigroup. Positivity and boundedness are proved using the specific form of the equations.

Perturbation of Eigenpairs of Factored Symmetric Tridiagonal Matrices

Suppose that an indefinite symmetric tridiagonal matrix permits triangular factorization T = LDL t . We provide individual condition numbers for the eigenvalues and eigenvectors of T when the parameters in L and D suffer small relative perturbations. When there is element growth in the factorization, then some pairs may be robust while others are sensitive. A 4 × 4 example shows the limitations of standard multiplicative perturbation theory and the efficacy of our new condition numbers.

Time dependent Desch-Schappacher type perturbations of Volterra integral equations

In this paper, we study time dependent multiplicative perturbations and unbounded additive perturbations of the Volterra integral equations. Some Desch-Schappacher type perturbation theorems, which generalize previous related results, are established by new and concise approaches.

The Integrated Density of States for Some Random Operators with Nonsign Definite Potentials

We study the integrated density of states of random Anderson-type additive and multiplicative perturbations of deterministic background operators for which the single-site potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Hölder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the L p -theory of the spectral shift function for p⩾1 ( Comm. Math. Phys. 218 (2001), 113–130), and the vector field methods of Klopp ( Comm. Math. Phys. 167 (1995), 553–569). We discuss the application of this result to Schrödinger operators with random magnetic fields and to band-edge localization.

Stability and robust stabilization to linear stochastic systems described by differential equations with markovian jumping and multiplicative white noise

In this paper we consider linear controlled stochastic systems subjected both to white noise disturbance and Markovian jumping. Our aim is to provide a mathematical background in order to give unified approach for a large class of problems associated to linear controlled systems subjected both to multiplicative white noise perturbations and Markovian jumping. First we prove an Itô type formula. Our result extends the result of Ref. [24], to the case when the stochastic process x(t) has not all moments bounded. Necessary and sufficient conditions assuring the exponential stability in mean square for the zero solution of a linear stochastic system with multiplicative white noise and Markovian jumping are provided. Some estimates for solutions of affine stochastic systems are derived, and necessary and sufficient conditions assuring the stochastic stabilizability and stochastic detectability are given. A stochastic version of Bounded Real Lemma is proved and several aspects of the problem of robust stabilization by state feedback for a class of linear systems with multiplicative white noise and Markovian jumping are investigated.

Perturbations of existence families for abstract Cauchy problems

In this paper, we establish Desch-Schappacher type multiplicative and additive perturbation theorems for existence families for arbitrary order abstract Cauchy problems in a Banach space: u (n) (t) = Au(t) (t > 0); u (j) (0) = x j (0 ≤ j ≤ n - 1). As a consequence, we obtain such perturbation results for regularized semigroups and regularized cosine operator functions. An example is also given to illustrate possible applications.

A System Theoretic Analysis of Automotive Vehicle Dynamics and Control

SUMMARYThis paper investigates vehicle dynamics and control from a systems theoretic perspective. Based on improved understanding of MIMO vehicle systems, a four-wheel-steering robust model following control strategy including a virtual vehicle model, a feed-forward internal model and a feedback element is proposed for improving the dynamic performance of the vehicle. The controller gain matrices are determined by a joint eigenstructure assignment and mixed-sensitivity H∞ control technique to satisfy both time- and frequency-domain specifications. The purpose is to improve disturbance rejection performance at low frequencies as well as to enhance robust stability against unstructured multiplicative output perturbations at high frequencies. The performance and robustness are assessed by numerical simulation.

Noise-Induced Transitions in a Barotropicβ-Plane Channel

The concepts of multiplicative stochastic perturbations and noise-induced transitions are applied to a quasigeostrophic b-plane model of barotropic flow over topography. The spectral three-component low-order representation of this configuration yields the Charney‐DeVore (CDV) model. The externally prescribed damping of the system is allowed to scatter around a mean value. The stochastic representation of the damping term leads to a multiplicative stochastic forcing. The Fokker‐Planck equation and the stochastic differential equation of the low-order CDV model are solved numerically. It is found that the qualitative behavior of the system is a function of the multiplicative noise level. In particular, the effect of multiplicative noise is not simply a smoothing of the probability density function, as it would be for pure additive noise. Rather, multiplicative noise leads to the high-index state being favored over the low-index state. The concept of noise-induced transitions explains this behavior. The noise-induced transition of the stochastic low-order model is confirmed by numerical integrations of a corresponding gridpoint model with many more degrees of freedom than the spectral model. It is suggested that the statistics of the unresolved physical processes could be an important factor in understanding the behavior of midlatitude large-scale atmospheric dynamics.