Analysis Of Uncertain Systems Research Articles

Worst Case Uncertainty Construction via Multifrequency Gain Maximization With Application to Flutter Control

In the analysis of uncertain systems, we often search for a worst case perturbation that drives the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_\infty $ </tex-math></inline-formula> gain of the system to the maximum over the set of allowable uncertainties. Employing the classical technique, an uncertainty sample is obtained, which, indeed, maximizes the gain but only at the single frequency where that maximum occurs. In contrast, this article considers a method to calculate a worst case perturbation that maximizes the gain of a system with mixed uncertainty at multiple frequencies simultaneously. This approach involves a nonlinear optimization that selects the worst case value of the uncertain parameters and the application of the boundary Nevanlinna–Pick interpolation to calculate the dynamic uncertainty sample. Such a perturbation can be used to augment Monte Carlo simulations of uncertain systems, especially if the system has multiple resonance frequencies. The worst case analysis of a flutter control system designed for a small flexible aircraft is provided to demonstrate the applicability of the proposed method.

Research on Surrounding Rock Deformation Prediction During Underground Excavation of Hydropower Station Based on Prototype Monitoring

This paper proposes an improved gray model of equal dimensional innovation based on time-varying parameters and the gray characteristics of small sample and poor information of the safety monitoring data during underground cavern construction, so as to predict and judge the deformation of surrounding rocks in underground caverns and to obtain relatively reliable results in the analysis of uncertain systems. Under certain conditions, the improved gray model of equal dimensional innovation based on time-varying parameters can be used to predict the deformation of surrounding rocks during the excavation of underground caverns, thereby providing mathematical analysis tools for safe construction.

Instability Analysis of Uncertain Systems via Determinants and LMIs

This technical note investigates two instability measures in continuous-time (CT) and discrete-time (DT) uncertain systems, the first given by the spectral abscissa (CT case) or radius (DT case), and the second given by the sum (CT case) or product (DT case) of the unstable eigenvalues. It is supposed that the system depends polynomially on an uncertain vector constrained into a semi-algebraic set. The problem is to determine the largest instability measures over the admissible uncertainties. It is shown that a sufficient condition for establishing an upper bound of the sought measures can be obtained in terms of linear matrix inequality (LMI) feasibility tests by exploiting the determinants of some specific matrices, and that this condition is also necessary under some mild conditions on the semi-algebraic set by using multipliers with degree sufficiently large. Moreover, a condition is provided for establishing the tightness of the found best upper bounds. Lastly, it is shown that in the special case where the semi-algebraic set is an interval, the degree of the multipliers is known a priori. Some numerical examples illustrate the proposed results.

Robust finite-frequency [formula omitted] analysis of uncertain systems with application to flight comfort analysis

In many applications, design or analysis is performed over a finite-frequency range of interest. The importance of the H2 norm highlights the necessity of computing this norm accordingly. This paper provides different methods for computing upper bounds of the robust finite-frequency H2 norm for systems with structured uncertainties. An application of the robust finite-frequency H2 norm for a comfort analysis problem of an aero-elastic model of an aircraft is also presented.

Stability analysis of uncertain systems on time scales

A method for the (in)stability analysis of the zero solution of an uncertain system on a time scale is proposed. The method is based on the matrix-valued Lyapunov function and its Δ-derivative on a time scale

On the Integrity of Uncertain Systems with Structured Uncertainty

The integrity of decentralized controllers with integral action, with respect to loop failures, has been researched relatively well. Important results have been found, with respect to the Relative Gain Array (RGA). However, less has been done on the treatment of the integrity of uncertain systems. This paper builds on the more-recent progress on the analysis of uncertain systems and presents new conditions for the integrity of closed-loop systems. The developed Integral Controllable (IC), Integral Controllable with Integrity (ICI), and Decentralized Integral Controllability (DIC) conditions are derived using a model of uncertainties that allows both unstructured and structured independent uncertainties to be studied. A convex framework for the evaluation of the conditions is also presented. The utility of the conditions are illustrated through application to existing examples from the literature.

A fuzzy-based approach to comprehensive modeling and analysis of systems with epistemic uncertainties

While aleatory uncertainties have successfully been taken into account by the use of probability theory, the modeling of epistemic uncertainties still remains a challenging topic. As a practical approach to solve this limitation, an interdisciplinary methodology to comprehensive modeling and analysis of systems is presented, which allows for the inclusion of uncertainties – in particular of those of epistemic type – from the very beginning of the modeling procedure. This approach is based on fuzzy arithmetic, a special field of fuzzy set theory, which has gained practical relevance after the introduction of the so-called Transformation Method. As practical and illustrative examples of two different areas of the engineering sciences, the fuzzy arithmetical approach to comprehensive modeling and analysis of uncertain systems is applied to the simulation of automotive crash in structural dynamics as well as to the simulation of landslide failure in geotechnical science and engineering. In both applications, epistemic uncertainties are considered which arise from some lack of knowledge, from simplification in modeling as well as from subjectivity in implementation.

Stability analysis of uncertain systems via matrix-valued Liapunov functions

In this paper, a new approach is given that is applied in the investigation of qualitative motion properties of systems whose parameters are prescribed unprecisely (uncertain systems). The general method of investigation is to construct sufficient conditions under which uncertain system motions possess a certain type of stability. In this work the main method of analysis for this class of systems is a generalized direct Lyapunov method based on matrix-valued auxiliary functions.

Time response bounds for linear parametric uncertain systems

Parametric uncertainties in linear time-invariant state-space models introduce perturbations in the eigenvalues and eigenvectors (eigenstructure), which modify the properties of a nominal system. Besides stability by the perturbed eigenvalues, controllability, observability, transient and steady-state values, etc., are some basic properties of a linear system significantly influenced by the perturbed eigenvectors. In this paper, time response analysis of parametric uncertain systems is considered. First the bounds on eigenvectors and parametric uncertainties are established for the perturbed eigenvalues to reside in a set of distinct circular regions. Secondly, analysis of such parametric uncertain systems is discussed by presenting bounds on the perturbed state variables in time coordinate. These bounds are then investigated for norm-bounded and structured uncertainty classifications. Input-output analysis of uncertain systems is illustrated using the initial conditions and input functions.

Robust Stability and Performance Analysis of Uncertain Systems Using Linear Matrix Inequalities

A wide variety of problems in system and control theory can be formulated or reformulated as convex optimization problems involving linear matrix inequalities (LMIs), that is, constraints requiring an affine combination of symmetric matrices to be positive semidefinite. For a few very special cases, there are analytical solutions to these problems, but in general LMI problems can be solved numerically in a very efficient way. Thus, the reduction of a control problem to an optimization problem based on LMIs constitutes, in a sense, a solution to the original problem. The objective of this article is to provide a tutorial on the application of optimization based on LMIs to robust control problems. In the first part of the article, we provide a brief introduction to optimization based on LMIs. In the second part, we describe a specific example, that of the robust stability and performance analysis of uncertain systems, using LMI optimization.

Covariance averaging in the analysis of uncertain systems

The effects of parametric uncertainty in stable state-space systems are analyzed by averaging the state covariance over the statistics of the uncertain parameters. For natural frequency uncertainty, this computation is related to the Fourier transform of the probability density function of the uncertain parameter. Equipartition and incoherence are illustrated for a single-mode oscillator. Averaging over a discrete uncertainty model yields the Bourret design equations while averaging over a Cauchy uncertainty distribution yields the maximum-entropy covariance equation of D.C. Hyland (1982). >