Analysis and applications of robust nonsingularity problem usingthe structured singular value

This note uses the structured singular value technique to solve the robust nonsingularity analysis problem for matrices with unstructured and/or quadratically coupled structured uncertainties. With the ready μ-analysis tools, a new approach is proposed for finding a set of uncertainties within which the perturbed matrix keeps nonsingularity. Applications and examples in robust control theory are given to show the usefulness and the practicality of the proposed approach     

具有结构不确定性的线性系统的鲁棒稳定性分析

本文基于一种非线性向量审变换结构不确定线性系统的鲁棒稳定性问题转休成相应参数矩阵的非奇异性问题，得了保证系统鲁棒稳定的充分条件，以及保证系统稳定的参数允许界。本文考虑了摄动矩阵结构上的特点，使得系统的鲁棒稳定区域大大扩大，最后通过实例和已有的结果进行了比较。     

Robust controllability of linear interval systems with multiple control delays

The robust controllability problem for linear interval systems with multiple control delays is studied in this article. The rank preservation problem is converted to the nonsingularity analysis problem of the minors of the matrix in discussion. Based on some essential properties of matrix measures, a new sufficient algebraically elegant criterion for the robust controllability of linear interval systems with multiple control delays is established. A numerical example is given to illustrate the application of the proposed sufficient algebraic criterion.     

Characteristic polynomial assignment for plants with semialgebraic uncertainty: A robust diophantine equation approach

In this paper, we address the problem of robust characteristic polynomial assignment for LTI systems whose parameters are assumed to belong to a semialgebraic uncertainty region. The objective is to design a dynamic fixed‐order controller in order to constrain the coefficients of the closed‐loop characteristic polynomial within prescribed intervals. First, necessary conditions on the plant parameters for the existence of a robust controller are reviewed, and it is shown that such conditions are satisfied if and only if a suitable Sylvester matrix is nonsingular for all possible values of the uncertain plant parameters. The problem of checking such a robust nonsingularity condition is formulated in terms of a nonconvex optimization problem. Then, the set of all feasible robust controllers is sought through the solution to a suitable robust diophantine equation. Convex relaxation techniques based on sum‐of‐square decomposition of positive polynomials are used to efficiently solve the formulated optimization problems by means of semidefinite programming. The presented approach provides a generalization of the results previously proposed in the literature on the problem of assigning the characteristic polynomial in the presence of plant parametric uncertainty. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust stability of state-space models with structureduncertainties

A method for robust eigenvalue location analysis of linear state-space models affected by structured real parametric perturbations is proposed. The approach, based on algebraic matrix properties, deals with state-space models in which system matrix entries are perturbed by polynomial functions of a set of uncertain physical parameters. A method converting the robust stability problem into nonsingularity analysis of a suitable matrix is proposed. The method requires a check of the positivity of a multinomial form over a hyperrectangular domain in parameter space. This problem, which can be reduced to finding the real solutions of a system of polynomial equations, simplifies considerably when cases with one or two uncertain parameters are considered. For these cases, necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems     

Robustness analysis of uncertain linear singular systems withoutput feedback control

The robustness analysis for a linear singular system with uncertain parameters and static output feedback control is considered. The problem is transformed into a robust nonsingularity problem. Based on the linear fractional transformation (LFT) approach, the robustness bounds to preserve regularity, impulse immunity, and stability are found in terms of the structured singular value μ with respect to parametric uncertainties. The LFT approach provides a unified framework for robustness analysis of both uncertain linear continuous/discrete-time singular systems     

Fast robust regression algorithms for problems with Toeplitz structure

The problem of computing an approximate solution of an overdetermined system of linear equations is considered. The usual approach to the problem is least squares, in which the 2-norm of the residual is minimized. This produces the minimum variance unbiased estimator of the solution when the errors in the observations are independent and normally distributed with mean 0 and constant variance. It is well known, however, that the least squares solution is not robust if outliers occur, i.e., if some of the observations are contaminated by large error. In this case, alternate approaches have been proposed which judge the size of the residual in a way that is less sensitive to these components. These include the Huber M-function, the Talwar function, the logistic function, the Fair function, and the ?₁ norm. New algorithms are proposed to compute the solution to these problems efficiently, in particular, when the matrix A has small displacement rank. Matrices with small displacement rank include matrices that are Toeplitz, block-Toeplitz, block-Toeplitz with Toeplitz blocks, Toeplitz plus Hankel, and a variety of other forms. For exposition, only Toeplitz matrices are considered, but the ideas apply to all matrices with small displacement rank. Algorithms are also presented to compute the solution efficiently when a regularization term is included to handle the case when the matrix of the coefficients is ill-conditioned or rank-deficient. The techniques are illustrated on a problem of FIR system identification.     

Robust stability of uncertain singular time-delay systems via LFT approach

The robust stability problem of continuous-time singular systems with multiple state delays and bounded parametric uncertainties is considered. Both commensurate and non-commensurate delays are investigated. On the basis of the linear fractional transformations (LFTs) framework and μ-analysis, a systematic approach is derived to convert the robustness problem to a robust nonsingularity problem. Some explicit conditions are proposed to guarantee the uncertain singular time-delay system being regular, impulse-free and stable independent of delay. Two illustrative examples are given to show the feasibility of the proposed technique.     

Robust output regulation under uncertainties of physical parameters

In this paper the robust output regulation problem is solved for linear time-invariant systems whose matrices are assumed to depend on some parameters, each of which possibly affects all the elements of the matrices describing the system, thus playing the role of a ‘physical’ parameter. The robustness here obtained is the preservation of the output regulation property under perturbations of such parameters. Both the conditions for the existence of a solution and a design procedure of the compensator are given.     

Frequency domain solution to delay-type Nehari problem

This paper generalizes the frequency-domain results on the delay-type Nehari problem in the stable case to the unstable case. The solvability condition of the delay-type Nehari problem is formulated in terms of the nonsingularity of three matrices. The optimal value γ_opt is the maximal γ∈(0,∞) such that one of the three matrices becomes singular. All sub-optimal compensators are parameterized in a transparent structure incorporating a modified Smith predictor.     

Quantitative robustness measure of uncertain time delay systems vianonsingularity analysis

The objective of this paper is to derive an upper bound for the delay time constant such that a linear uncertain time delay system still keeps its asymptotic stability, if it is nominally stable and the delay time constant remains smaller than the upper bound. The problem is solved by using matrix measure and nonsingularity analysis of matrices, and the results are shown to be less conservative than previous results     

Robust ripple-free regulation and tracking for parameter-dependentsampled-data systems

The robust ripple-free asymptotic tracking and disturbance rejection problem is studied for sampled-data systems whose matrices are assumed to depend on some “physical” parameters, each of which possibly affects all the elements of the matrices describing the system, and for the case when only a subset of the scalar outputs must track corresponding exogenous reference signals; ripple-free dead-beat regulation and tracking is also required at the nominal parameters. The necessary and sufficient conditions are given for the existence of a solution which may exist even when no solution exists for wholly independent variations of the entries of the matrices describing the system. A design procedure of both the continuous-time internal model and the discrete-time controller is also outlined     

The dual algebraic Riccati equations and the set of all solutions of the discrete-time Riccati equation

In this paper, two new pairs of dual continuous-time algebraic Riccati equations (CAREs) and dual discrete-time algebraic Riccati equations (DAREs) are proposed. The dual DAREs are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix. Then, in the case of singular matrices, a generalised inverse is introduced to deal with the dual DARE problem. These dual AREs can easily lead us to an iterative procedure for finding the anti-stabilising solutions, especially to DARE, by means of that for the stabilising solutions. Furthermore, we provide the counterpart results on the set of all solutions to DARE inspired by the results for CARE. Two examples are presented to illustrate the theoretical results.     

区间矩阵的鲁棒稳定性判据

基于一种非线性的幂变换,将区间矩阵的鲁棒稳定性问题转化成相应参数矩阵的非奇异性问题,并结合Gershgorin圆盘定理,得到了保证系统鲁棒稳定的充分条件。该判据对矩阵元素没有任何附加要求,简单而且实用。最后通过实例论证了所给判据的有效性。     

Peak-to-peak filtering for periodic piecewise linear polytopic systems

In this paper, the robust peak-to-peak filtering problem for periodic piecewise systems with polytopic uncertainties is investigated. Attention is focused on designing of robust peak-to-peak filters that guarantee the robust asymptotic stability of periodic piecewise filtering error system and satisfy a prescribed peak-to-peak disturbance attenuation level for all admissible uncertainties. A sufficient condition is proposed for robust stability and peak-to-peak performance by employing the constructed time-varying Lyapunov function. Two design approaches based on the Lyapunov function with parameter-dependent matrices and parameter-independent matrices are utilised to solve this problem. An algorithm is proposed to compute the periodic filter parameters governed by non-convex feasibility conditions. Two numerical examples are presented to demonstrate the effectiveness of the proposed techniques.     

Generalizing discriminant analysis using the generalized singular value decomposition

Discriminant analysis has been used for decades to extract features that preserve class separability. It is commonly defined as an optimization problem involving covariance matrices that represent the scatter within and between clusters. The requirement that one of these matrices be nonsingular limits its application to data sets with certain relative dimensions. We examine a number of optimization criteria, and extend their applicability by using the generalized singular value decomposition to circumvent the nonsingularity requirement. The result is a generalization of discriminant analysis that can be applied even when the sample size is smaller than the dimension of the sample data. We use classification results from the reduced representation to compare the effectiveness of this approach with some alternatives, and conclude with a discussion of their relative merits.     

Techniques for exploiting structure in matrix formulae of the sparse resultant

Resultants characterize the existence of roots of systems of polynomial equations, while their matrix formulae reduce the computation of all common zeros to a problem in linear algebra. Sparse elimination theory has introduced the sparse resultant, which takes into account the sparse structure of the polynomials, as expressed by the respective supports or Newton polytopes. Sparse resultant, or Newton, matrices generalize Macaulay’s matrix and define nontrivial multiples of the sparse resultant from which the latter can be recovered. We exploit their structure with the objective to decrease the complexity of constructing such matrices and of computing the exact resultant. Our present study of these structured matrices implies substantial progress in this direction. Subsequent application of fast matrix-by-vector multiplication shall lead to computational complexity bounds that are roughly quadratic in the matrix size, whereas the existing methods have cubic complexity. Our auxiliary techniques for testing exact and numerical nonsingularity of rectangular matrices may be of some independent interest. Finally, we establish complexity bounds, linear in the number of nonzero terms, for polynomial multiplication, under our model of sparseness.     

Control of Markovian jump discrete-time systems with norm boundeduncertainty and unknown delay

This paper studies the problem of control for discrete time delay linear systems with Markovian jump parameters. The system under consideration is subjected to both time-varying norm-bounded parameter uncertainty and unknown time delay in the state, and Markovian jump parameters in all system matrices. We address the problem of robust state feedback control in which both robust stochastic stability and a prescribed H_∞ performance are required to be achieved irrespective of the uncertainty and time delay. It is shown that the above problem can be solved if a set of coupled linear matrix inequalities has a solution     

Multivariable system design by transfer function synthesis

This paper is concerned with the problem of designing a constant output feedback matrix to synthesize the denominator and certain numerator polynomials of the transfer function matrix of a linear multivariable system. The problem is tackled by first using the special case of unity rank feedback and then removing this restriction to cover the case of unconstrained feedback.

Simple expressions relating closed-loop and open-loop transfer functions of multi-variable systems and unity rank feedback are first established. These are then used to develop a recursive synthesis algorithm employing unity rank output feedback. The algorithm uses the pseudo-inverse concept for obtaining least-squares solutions of sots of simultaneous linear equations. The extension from unity rank feedback to unrestricted rank feedback is carried out by considering the feedback matrix as a sum of unity rank matrices. Using this concept and the expressions derived earlier for unity rank feedback, a recursive algorithm is developed for calculating unrestricted rank output feedback matrices. Examples are given to illustrate the synthesis procedure using both unity rank and unrestricted rank feedback matrices.