Linear Parameter Systems Research Articles

Income and Price Elasticities of Demand in South Africa: An Application of the Linear Expenditure System

AbstractThis paper investigates the expenditure patterns of South African households using detailed cross‐sectional expenditure and price data that varies across region and time. Linear expenditure system parameter estimates are used to calculate income and price elasticities for a number of product categories at different points of the income distribution. We find substantial variation in the price and income elasticities of demand for items across the income distribution, with the bottom quartile being extremely sensitive to increases in the price of food and clothing items, and the top quartile being as sensitive as households in developed countries.

Exact solution to a parametric linear programming problem

We consider the problem of determining the interval hull solution l ∗ to a parametric linear programming (PLP) problem l(x, p) = c T (p)x, where c i (p) can be in general nonlinear functions of p, and x satisfy the constraint A(p)x = b(p), p ∈ p. A new iterative method for determining l ∗ is suggested. The method exploits the concept of the p-solution to a square linear interval parametric (LIP) system, having the parametric form x(p) = L p + a, p ∈ p, where L is a real n × m matrix (n and m are, respectively, the sizes of the square matrix A and vector p), whereas a is an interval vector. A numerical example is provided to illustrate the new approach.

A new method for computing a p-solution to parametric interval linear systems with affine-linear and nonlinear dependencies

We propose a new approach to computing a parametric solution (the so-called p-solution) to parametric interval linear systems. Solving such system is an important part of many scientific and engineering problems involving uncertainties. The parametric solution has many useful properties. It permits to compute an outer solution, an inner estimate of the interval hull solution, and intervals containing the lower and upper bounds of the interval hull solution. It can also be employed for solving various constrained optimisation problems related to the parametric interval linear system. The proposed approach improves both outer and inner bounds for the parametric solution set. In this respect, the new approach is competitive to most of the existing methods for solving parametric interval linear systems. Improved bounds on the parametric solution set guarantees improved bounds for the solutions of related optimisation problems.

Characterization of AE solution sets of parametric linear systems based on the techniques of convex sets

New approach to linear systems with uncertain parameters is proposed. The notion of AE solution sets is extended to the case when the domain of parameters need not be an interval vector. The tools from the theory of convex sets are applied to the characterization of AE solution sets. The description of these sets by systems of inequalities is presented. Some examples are included.

Enclosing the solution set of parametric interval matrix equation A(p)X = B(p)

Consider the parametric matrix equation A(p)X = B(p), where the elements of the matrices A(p) and B(p) depend linearly on a number of uncertain parameters varying within given intervals. We prove that the united parametric solution sets of the matrix equation and that of the corresponding linear system with multiple right-hand sides, although different as sets, have the same interval hull. A generalization of the parametric Krawczyk iteration with low computational complexity for the matrix equation is presented. Some details improving the implementation and the application of this method are discussed. An interval method, designed by A. Neumaier and A. Pownuk for enclosing the united solution set of parametric linear systems with particular dependency structure, is generalized for arbitrary linear dependencies between the parameters and for systems with multiple right-hand sides. A new, more powerful, sufficient condition for regularity of a parametric interval matrix is proven. An important application of the linear systems with multiple right-hand sides is presented as a key methodology for feasibility in computing the interval hull of a class of united parametric solution sets that appear in practical problems.

Undermodelling Detection with Sign-Perturbed Sums

Sign-Perturbed Sums (SPS) is a finite sample system identification method that can build exact confidence regions for the unknown parameters of linear systems under mild statistical assumptions. Theoretical studies of the SPS method have assumed so far that the order of the system model is known to the user. In this paper we discuss the implications of this assumption for the applicability of the SPS method, and we propose an extension that, under mild assumptions, i) still delivers guaranteed confidence regions when the model order is correct, and ii) it is guaranteed to detect, in the long run, if the model order is wrong.

Alignment Distances on Systems of Bags

Recent research in image and video recognition indicates that many visual processes can be thought of as being generated by a time-varying generative model. A nearby descriptive model for visual processes is thus a statistical distribution that varies over time. Specifically, modeling visual processes as streams of histograms generated by a kernelized linear dynamic system turns out to be efficient. We refer to such a model as a System of Bags. In this work, we investigate Systems of Bags with special emphasis on dynamic scenes and dynamic textures. Parameters of linear dynamic systems suffer from ambiguities. In order to cope with these ambiguities in the kernelized setting, we develop a kernelized version of the alignment distance. For its computation, we use a Jacobi-type method and prove its convergence to a set of critical points. We employ it as a dissimilarity measure on Systems of Bags. As such, it outperforms other known dissimilarity measures for kernelized linear dynamic systems, in particular the Martin Distance and the Maximum Singular Value Distance, in every tested classification setting. A considerable margin can be observed in settings, where classification is performed with respect to an abstract mean of video sets. For this scenario, the presented approach can outperform state-of-the-art techniques, such as Dynamic Fractal Spectrum or Orthogonal Tensor Dictionary Learning.

Control With Communications Constraints: Measuring the Instability in Parametric Linear Systems

This paper investigates the instability measure of linear systems defined as the sum of the unstable eigenvalues in the continuous-time case and the product of the unstable eigenvalues in the discrete-time case. The problem consists of determining the largest instability measure in systems depending polynomially on parameters constrained in a semialgebraic set. It is shown that upper bounds of the sought measure can be established via linear matrix inequality feasibility tests. Moreover, a priori and a posteriori conditions for establishing nonconservatism are proposed. Finally, two special cases of the proposed methodology are investigated-the first one concerns systems with a single parameter, and the second one concerns the determination of the largest spectral abscissa and radius. Three applications in control with communications constraints are discussed.

A Sample-Efficient Black-Box Optimizer to Train Policies for Human-in-the-Loop Systems With User Preferences

We present a new algorithm for optimizing control policies for human-in-the-loop systems based on qualitative preference feedback. This method is especially applicable to systems such as lower limb prostheses and exoskeletons for which it is difficult to define an objective function, hard to identify a model, and costly to repeat hardware experiments. To solve these problems, we combine and extend an algorithm for learning from preferences and the Predictive Entropy Search Bayesian optimization method. The resulting algorithm, Predictive Entropy Search with Preferences (PES-P), solicits preferences between pairs of control parameter sets that optimally reduce the uncertainty in the distribution of objective function optima with the least number of experiments. We find that this algorithm outperforms the expected improvement method (EI), and random comparisons via Latin hypercubes (LH) in three simulation tests that range from optimizing randomly generated functions to tuning control parameters of linear systems and of a walking model. Furthermore, we find in a pilot study on the control of a robotic transfemoral prosthesis that PES-P finds good control parameters quickly and more consistently than EI or LH given real user preferences. The results suggest the proposed algorithm can help engineers optimize certain robotic systems more accurately, efficiently, and consistently.

A multi-point parameterized model reduction for large parametric systems by using Krylov-subspace techniques

This paper presents a multi-point parameterized model reduction method based on Krylov-subspace techniques for large parametric systems where the parametric dependence in system matrices can be nonaffine and the parameters are in a wide range. First, a large parametric system is approximated as a weighted combination of different linear parametric systems, which are obtained by Taylor expansion on the different selected parameter expansion points. Then, a numerically stable algorithm based on self-moments and cross-moments is proposed to obtain a reduced parametric system with high fidelity in the full ranges of the parameters. The accuracy of the obtained reduced parametric systems can be improved by applying the multi-point expansions and weighted functions proposed in this paper. Two benchmarks in practical applications are assessed by employing different parameterized model reduction methods. The numerical simulation results verify the effectiveness of the proposed method.

A new algorithm for Chebyshev minimum-error multiplication of reduced affine forms

Reduced affine arithmetic (RAA) eliminates the main deficiency of the standard affine arithmetic (AA), i.e. a gradual increase of the number of noise symbols, which makes AA inefficient in a long computation chain. To further reduce overestimation in RAA computation, a new algorithm for the Chebyshev minimum-error multiplication of reduced affine forms is proposed. The algorithm yields the minimum Chebyshev-type bounds and works in linear time, which is asymptotically optimal. We also propose a simplified \(\mathcal {O}(n\log n)\) version of the algorithm, which performs better for low dimensional problems. Illustrative examples show that the presented approach significantly improves solutions of many numerical problems, such as the problem of solving parametric interval linear systems or parametric linear programming, and also improves the efficiency of interval global optimisation.

Parametric FGLM algorithm

In this paper, we consider the non-trivial problem of converting a zero-dimensional parametric Gröbner basis w.r.t. a given monomial ordering to a Gröbner basis w.r.t. any other monomial ordering. We present a new algorithm, so-called parametric FGLM algorithm, that takes as input a monomial ordering and a finite parametric set which is a Gröbner basis w.r.t. a given set of parametric constraints, and outputs a decomposition of the given space of parameters as a finite set of (parametric) cells and for each cell a finite set of parametric polynomials which is a Gröbner basis w.r.t. the target monomial ordering and the corresponding cell. For this purpose, we develop computationally efficient algorithms to deal with parametric linear systems that are applicable in computing comprehensive Gröbner systems of parametric linear ideals also in the theory of parametric linear algebra to compute Gaussian elimination and minimal polynomial of a parametric matrix. All proposed algorithms have been implemented in Maple and their efficiency is discussed on a diverse set of benchmark polynomials.

Implicit Volterra series interpolation for model reduction of bilinear systems

We propose a new interpolatory framework for model reduction of large-scale bilinear systems. The input–output representation of a bilinear system in frequency domain involves a series of multivariate transfer functions, each representing a subsystem of the bilinear system. If a weighted sum of these multivariate transfer functions associated with a reduced bilinear system interpolates a weighted sum of the original multivariate transfer functions, we say that the reduced system satisfies a Volterra series interpolation (Flagg and Gugercin, 2015). These interpolatory conditions can also ensure the necessary conditions for H2-optimal model reduction (Flagg and Gugercin, 2015; Benner and Breiten, 2012). We observe that, by carefully selecting the weights of the series, the Volterra series interpolatory conditions are transformed to the problem of interpolating a linear system with an affine parameter dependence. Such linear parametric systems can then be reduced by some method for parametric model order reduction.Linear systems where the affine parameter dependence is given as low-rank variation in the state matrix can be mapped into a non-parameterized multi-input multi-output linear system. This allows us to utilize the standard (non-parametric) linear IRKA (Gugercin et al., 2008) for the problem of parameterized/bilinear interpolation. Numerical results show that the approximations are of comparable accuracy to those obtained from the bilinear iterative rational Krylov algorithm (Benner and Breiten, 2012). The proposed approach, however, has the advantage that it reduces the computational costs as it involves computations associated with solving linear systems only.

Analysis of the practical stability and sensitivity of linear dynamical systems with change of phase space measurability

In this work, the models of linear parametric systems of ordinary differential equations with variable measurability of phase space are investigated. The theorems about the practical stability of linear parametrical systems with variable measurability are proved. It is important that the reverse theorem about practical stability of indicated systems is obtained. The algorithms and criteria of analysis of practical stability of linear parametrical systems with variable measurability of phase space at the presence of constantly occurring perturbations are shown. The matrix equations of sensitivity of linear parametrical systems with variable measurability of the phase space are researched. It was investigated that on the basis of methods of practical stability and conditions which satisfied sensitivity matrices it was possible to effectively find the estimations of parameters for an analysis of the system sensitivity with variable measurability of the phase space. Results of given investigations can be successfully applied in the tasks of digital data processing and pattern recognition.

Transformations of interval linear systems of equations and inequalities

Very recently, new results on transformation of interval linear systems and on generalizations of the Farkas lemma to interval systems appeared in the literature. They are by far not obvious since the standard transformations on linear systems are not easily adapted to interval system due to the dependency problem. The aim of this paper was to come up with new possible transformations and to extend the results to more general classes of AE interval systems and to linear parametric systems. We also show that the transformations can help in simplifying the proofs of some characterization theorems.

Method for determining the coupling parameters for complex linear systems with discrete couplings between subsystems

Complex systems whose subsystems interact at finitely many points are considered. The couplings are given by linear homogeneous differential relations.

Bayesian robust linear dynamic system approach for dynamic process monitoring

In this paper, a Bayesian robust linear dynamic system approach is proposed for process modeling. Traditional linear dynamic system (LDS) constructed with Kalman filter is designed by Gaussian assumption which can be easily violated in non-Gaussian modeling situations, especially those with outliers. To deal with this issue, the conventional Gaussian-based Kalman filter is modified with heavy tailed Student's t-distribution so as to deal with the non-Gaussian noise and modeling outliers. Then, a variational Bayesian expectation maximization (VBEM) algorithm is developed for learning parameters of the robust linear dynamic system. For process monitoring, traditional monitoring scheme are discussed and the residual space monitoring mechanism has been improved. To explore the feasibility and effectiveness, the proposed method is applied for fault detection, with detailed comparative studies with several other methods through the Tennessee Eastman benchmark.

Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p)

In this paper, the parametric matrix equation A(p)X = B(p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A(p) and B(p) are known m-by-m and m-by-n matrices respectively, and X is the m-by-n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods.

Paris law parameter identification based on the Extended Kalman Filter

Aircraft structures are commonly subjected to repeated loading cycles leading to fatigue damage. Fatigue data can be extrapolated by fatigue models which are adopted to describe the fatigue damage behaviour. Such models depend on their parameters for accurate prediction of the fatigue life. Therefore, several methods have been developed for estimating the model parameters for both linear and nonlinear systems. It is useful for a broad class of parameter identification problems when the dynamic model is not known. In this paper, the Paris law is used as fatigue-crack-length growth model on a metallic component under loading cycles. The Extended Kalman Filter (EKF) is proposed as estimation method. Simulated crack length data is used to validate the estimation method. Based on experimental data obtained from fatigue experiment, the crack length and model parameters are estimated. Accurate model parameters allow a more realistic prediction of the fatigue life, consequently, the remaining useful life (RUL) of component can be accurately computed. In this sense, maintenance performance could be improved.

A Generalized Class of Pseudomeasurements for Identifying Unknown Parameters of Linear Stochastic Systems

A Generalized Class of Pseudomeasurements for Identifying Unknown Parameters of Linear Stochastic Systems