Multiplicative Perturbation Research Articles

Optimal Filtering for Discrete-Time Linear Systems With Multiplicative White Noise Perturbations and Periodic Coefficients

In this technical brief, the problem of the estimation of a remote signal generated by a discrete-time dynamical system with periodic coefficients subject to multiplicative and additive white noise perturbations is investigated. To measure the quality of the estimation achieved by an admissible filter, we introduced a performance criterion described by the Cesaro limit of the mean square of the deviation between the estimated signal <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">zF</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> ) and the remote signal <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> ). The dimension of the state space of the admissible filters is not prefixed. The state-space representation of the optimal filter is constructed based on the unique periodic solution of a discrete-time linear equation together with the stabilizing solution of a suitable discrete-time Riccati equation with periodic coefficients.

On Perturbation of ConvolutedC-Regularized Operator Families

Of concern are two classes of convolutedC-regularized operator families: convolutedC-cosine operator families and convolutedC-semigroups. We obtain new and general multiplicative and additive perturbation theorems for these convolutedC-regularized operator families. Two examples are given to illustrate our abstract results.

Multiplicative Perturbations of ConvolutedC-Cosine Functions and ConvolutedC-Semigroups

We obtain the multiplicative perturbation theorems for convolutedC-cosine functions (resp., convolutedC-semigroups) andn-times integratedC-cosine functions (resp.,n-times integratedC-semigroups) forn∈ℕ. Moreover, we obtain some new results for perturbations onC-cosine functions (resp.,C-semigroups). Some examples are presented.

PERT: A Method for Expression Deconvolution of Human Blood Samples from Varied Microenvironmental and Developmental Conditions

The cellular composition of heterogeneous samples can be predicted using an expression deconvolution algorithm to decompose their gene expression profiles based on pre-defined, reference gene expression profiles of the constituent populations in these samples. However, the expression profiles of the actual constituent populations are often perturbed from those of the reference profiles due to gene expression changes in cells associated with microenvironmental or developmental effects. Existing deconvolution algorithms do not account for these changes and give incorrect results when benchmarked against those measured by well-established flow cytometry, even after batch correction was applied. We introduce PERT, a new probabilistic expression deconvolution method that detects and accounts for a shared, multiplicative perturbation in the reference profiles when performing expression deconvolution. We applied PERT and three other state-of-the-art expression deconvolution methods to predict cell frequencies within heterogeneous human blood samples that were collected under several conditions (uncultured mono-nucleated and lineage-depleted cells, and culture-derived lineage-depleted cells). Only PERT's predicted proportions of the constituent populations matched those assigned by flow cytometry. Genes associated with cell cycle processes were highly enriched among those with the largest predicted expression changes between the cultured and uncultured conditions. We anticipate that PERT will be widely applicable to expression deconvolution strategies that use profiles from reference populations that vary from the corresponding constituent populations in cellular state but not cellular phenotypic identity.

THE CAUCHY PROBLEM FOR CONSERVATION LAWS WITH A MULTIPLICATIVE STOCHASTIC PERTURBATION

We study the Cauchy problem for multi-dimensional nonlinear conservation laws with multiplicative stochastic perturbation. Using the concept of measure-valued solutions and Kruzhkov's entropy formulation, the existence and uniqueness of an entropy solution is established.

Continuous time-delay estimation in Laguerre domain - Revisited

The topic of time-delay estimation from Laguerre spectra of the input and output signal is re-approached. A formal proof of the relationship underlying an earlier suggested subspace continuous time-delay estimation method is given. Further, it is shown that a class of Laguerre polynomials plays a key role in the mathematical modeling of time delays in Laguerre domain. The robustness of the time-delay estimation algorithm is also investigated. It turns out that there is always a certain choice of the Laguerre parameter that renders an exact estimate of the time delay in the face of a finite-dimensional multiplicative dynamical perturbation, provided that the input is a single Laguerre function of any order.

New bounds for perturbation of the orthogonal projection

Based on the singular value decomposition, we obtain both additive and multiplicative perturbation bounds for the orthogonal projection, which improve some existing results. Furthermore, the Q-norm bounds for additive and multiplicative perturbations of the orthogonal projection are also given.

Non-Additive Random Data Perturbation for Real World Data

Privacy preservation is the major concern while real datasets are handled. A specific topic- privacy preserving data mining (PPDM), completely deals with data modification but also limits rule loss. Data perturbation is one of the PPDM techniques, which mostly deals with numerical data and concentrates on the statistical analysis of the data. Perturbation is of two types, additive perturbation and multiplicative perturbation, where generated random data is either added or multiplied with the data, which results in a random modified data. In this paper we have proposed a model in which the perturbation is done by randomization, where the data is generated in intervals based on the level of privacy specified by each customer. Our model is proved by applying classification algorithm on the perturbed data set and the accuracy is still maintained the same.

Transmission Eigenvalues for Elliptic Operators

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-self-adjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.

Multiplicative relative perturbation bounds of eigenvalues for diagonalizable matrices

In this paper, we present the general bounds of multiplicative relative perturbation for diagonalizable matrices, which are the improvement of recent results. The bounds gained here are sharper than those in related literatures. In addition, we obtain the multiplicative relative bounds for normal matrices, which are discussed first.

Finite settling time stabilisation: the robust SISO case

This article deals with the problem of robustness to multiplicative plant perturbations for the case of finite settling time stabilisation (FSTS) of single input single output (SISO), linear, discrete-time systems. FSTS is a generalisation of the deadbeat control and as in the case of deadbeat control the main feature of FSTS is the placement of all closed-loop poles at the origin of the z-plane. This makes FSTS sensitive to plant perturbations hence, the need of robust design. An efficient robustness index is introduced and the problem is reduced to a finite linear programme where all the benefits of the simplex method, such as effectiveness, efficiency and ability to provide complete solution to the optimisation problem, can be exploited.

Non-Fragile Decentralized Control for the Uncertain Singular Large-Scale Systems with Time-Delay

This paper is concerned with the non-fragile decentralized controller design problem for uncertain singular large-scale system with time-delay. Sufficient condition for the controller is expressed in terms of linear matrix inequalities(LMIs). When this condition is feasible, the desired controller can be obtained with additive gain perturbations and multiplicative gain perturbations. Finally, a numerical example is also given to illustrate the effectiveness.

Multiplicative perturbations of the Laplacian and related approximation problems

Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on \({{\mathbb{R}}^{N},\; N\geq1}\) . It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive C0-semigroups of operators that fulfill the Feller property. Accordingly, these semigroups are associated with a suitable probability transition function and hence with a Markov process on \({{\mathbb{R}}^{N}}\) . An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes.

The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices

We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models. The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well-known objects in free probability theory that linearize non-commutative free additive and multiplicative convolution. Furthermore, we uncover a phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. Square root decay of the eigenvalue density at the edge is sufficient to ensure that this threshold is finite. This critical threshold is intimately related to the same aforementioned integral transforms and our proof techniques bring this connection and the origin of the phase transition into focus. Consequently, our results extend the class of ‘spiked’ random matrix models about which such predictions (called the BBP phase transition) can be made well beyond the Wigner, Wishart and Jacobi random ensembles found in the literature. We examine the impact of this eigenvalue phase transition on the associated eigenvectors and observe an analogous phase transition in the eigenvectors. Various extensions of our results to the problem of non-extreme eigenvalues are discussed.

Multiplicative perturbation analysis for QR factorizations

This paper is concerned with how the QR factors change when a real matrix $A$ suffers from a left or right multiplicative perturbation, where $A$ is assumed to have full column rank. It is proved that for a left multiplicative perturbation the relative changes in the QR factors in norm are no bigger than a small constant multiple of the norm of the difference between the perturbation and the identity matrix. One of common cases for a left multiplicative perturbation case naturally arises from the computation of the QR factorization. The newly established bounds can be used to explain the accuracy in the computed QR factors. For a right multiplicative perturbation, the bounds on the relative changes in the QR factors are still dependent upon the condition number of the scaled $R$-factor, however. Some ``optimized'' bounds are also obtained by taking into account certain invariant properties in the factors.

Lyapunov design of adaptive super-twisting controller applied to a pneumatic actuator

AbstractA novel super-twisting adaptive sliding mode control law is derived using Lyapunov function technique. The both drift uncertain term and multiplicative perturbation are assumed to be bounded with unknown boundaries. The proposed approach consists in using dynamically adapted control gains that ensure the establishment, in a finite time, of a real second order sliding mode. The efficacy of the proposed super-twisting control algorithm is evaluated through its application to position control of an electropneumatic actuator.

Wick product in the stochastic Burgers equation: A curse or a cure?

It has been known for a while that a nonlinear equation driven by singu- lar noise must be interpreted in the re-normalized, or Wick, form. For the stochastic Burgers equation, Wick nonlinearity forces the solution to be a generalized process no matter how regular the random perturbation is, whence the curse. On the other hand, certain multiplicative random perturbations of the deterministic Burgers equation can only be interpreted in the Wick form, whence the cure. The analysis is based on the study of the coe-cients of the chaos expansion of the solution at difierent stochastic scales.

Gaussian estimates on networks with applications to optimal control

We study a class of reaction-diffusion type equations on a finite network with continuity assumptions and a kind of non-local , stationary Kirchhoff's conditions at the nodes. A multiplicative random Gaussian perturbation acting along the edges is also included. For such a problem we prove Gaussian estimates for the semigroup generated by the evolution operator, hence generalizing similar results previously obtained in [21]. In particular our main goal is to extend known results on Gaussian upper bounds for heat equations on networks with local boundary conditions to those with non-local ones. We conclude showing how our results can be used to apply techniques developed in [13] to solve a class of Stochastic Optimal Control Problems inspired by neurological dynamics.

Additive and multiplicative perturbation bounds for the Moore-Penrose inverse

In this paper, we obtain the additive and multiplicative perturbation bounds for the Moore-Penrose inverse under the unitarily invariant norm and the Q - norm, which improve the corresponding ones in [P.Å. Wedin, Perturbation theory for pseudo-inverses, BIT 13(1973)217–232].

Disturbance observer for non-minimum phase linear systems

Disturbance observer (DOB) approach has been widely employed in the industry thanks to its powerful ability to attenuate disturbances and compensate plant uncertainties. Motivated by the fact that the application of the DOB approach has been limited to minimum phase systems, we propose a new DOB configuration for non-minimum phase systems. Rigorous analysis for robust stability and performance recovery is presented in terms of a new filter Θ (s). By restricting the plant uncertainty to a multiplicative perturbation, we also present a systematic design methodology for the filter Θ(s) by virtue of the H∞ synthesis technique. An illustrative example of autopilot design is presented to show the effectiveness of the proposed method for which it is not easy to design H∞ controller to achieve some control goals. The simulation results show that the response of the perturbed system in the presence of disturbance can be recovered to the nominal system response in the absence of disturbance.