Singular perturbations of linear systems with multiparameters and multiple time scales

AbstractIn this paper, an alternate approach to the method of asymptotic expansions for the study of a singularly perturbed, linear system with multiparameters and multiple time scales is developed. The method consists of developing a linear, non-singular transformation that enables one to transform the original system into an upper triangular form. This process of upper triangularization will enable us to investigate (i) stability and (ii) approximation of solutions of the original system in terms of the overall reduced system and the corresponding boundary layer systems

Multiparameter estimation perspective on non-Hermitian singularity-enhanced sensing

Describing the evolution of quantum systems by means of non-Hermitian generators opens a new avenue to explore the dynamical properties naturally emerging in such a picture, e.g. operation at the so-called exceptional points, preservation of parity-time symmetry, or capitalising on the singular behaviour of the dynamics. In this work, we focus on the possibility of achieving unbounded sensitivity when using the system to sense linear perturbations away from a singular point. By combining multiparameter estimation theory of Gaussian quantum systems with the one of singular-matrix perturbations, we introduce the necessary tools to study the ultimate limits on the precision attained by such singularity-tuned sensors. We identify under what conditions and at what rate can the resulting sensitivity indeed diverge, in order to show that nuisance parameters should be generally included in the analysis, as their presence may alter the scaling of the error with the estimated parameter.Comment: 16 pages with appendices. Comments are more than welcom

Robust stability of discrete time systems under parametric perturbations

Cataloged from PDF version of article.Stability robustness analysis of a system under parametric perturbations is concerned with characterizing a region in the parameter space in which the system remains stable. In this paper, two methods are presented to estimate the stability robustness region of a linear, time-invariant, discrete-time system under multiparameter additive perturbations. An inherent difficulty, which originates from the nonlinear appearance of the perturbation parameters in the inequalities defining the robustness region, is resolved by transforming the problem to stability of a higher order continuous-time system. This allows for application of the available results on stability robustness of continuous-time systems to discrete-time systems. The results are also applied to stability analysis of discrete-time interconnected systems, where the interconnections are treated as perturbations on decoupled stable subsystems

Nonlinear zero-sum differential game analysis by singular perturbation methods

A class of nonlinear, zero-sum differential games, exhibiting time-scale separation properties, can be analyzed by singular-perturbation techniques. The merits of such an analysis, leading to an approximate game solution, as well as the 'well-posedness' of the formulation, are discussed. This approach is shown to be attractive for investigating pursuit-evasion problems; the original multidimensional differential game is decomposed to a 'simple pursuit' (free-stream) game and two independent (boundary-layer) optimal-control problems. Using multiple time-scale boundary-layer models results in a pair of uniformly valid zero-order composite feedback strategies. The dependence of suboptimal strategies on relative geometry and own-state measurements is demonstrated by a three dimensional, constant-speed example. For game analysis with realistic vehicle dynamics, the technique of forced singular perturbations and a variable modeling approach is proposed. Accuracy of the analysis is evaluated by comparison with the numerical solution of a time-optimal, variable-speed 'game of two cars' in the horizontal plane

Multiparameter actuation of a neutrally-stable shell: a flexible gear-less motor

We have designed and tested experimentally a morphing structure consisting of a neutrally stable thin cylindrical shell driven by a multiparameter piezoelectric actuation. The shell is obtained by plastically deforming an initially flat copper disk, so as to induce large isotropic and almost uniform inelastic curvatures. Following the plastic deformation, in a perfectly isotropic system, the shell is theoretically neutrally stable, owning a continuous manifold of stable cylindrical shapes corresponding to the rotation of the axis of maximal curvature. Small imperfections render the actual structure bistable, giving preferred orientations. A three-parameter piezoelectric actuation, exerted through micro-fiber-composite actuators, allows us to add a small perturbation to the plastic inelastic curvature and to control the direction of maximal curvature. This actuation law is designed through a geometrical analogy based on a fully non-linear inextensible uniform-curvature shell model. We report on the fabrication, identification, and experimental testing of a prototype and demonstrate the effectiveness of the piezoelectric actuators in controlling its shape. The resulting motion is an apparent rotation of the shell, controlled by the voltages as in a "gear-less motor", which is, in reality, a precession of the axis of principal curvature.Comment: 20 pages, 9 figure

Stability and stabilization of one class of three time-scale systems with delays

summary:A singularly perturbed linear time-invariant time delay controlled system is considered. The singular perturbations are subject to the presence of two small positive multipliers for some of the derivatives in the system. These multipliers (the parameters of singular perturbations) are of different orders of the smallness. The delay in the slow state variable is non-small (of order of $1$). The delays in the fast state variables are proportional to the corresponding parameters of singular perturbations. Three much simpler parameters-free subsystems are associated with the original system. It is established that the exponential stability of the unforced versions of these subsystems yields the exponential stability of the unforced version of the original system uniformly in the parameters of singular perturbations. It also is shown that the stabilization of the parameters-free subsystems by memory-free state-feedback controls yields the stabilization of the original system by a memory-free state-feedback control uniformly in the parameters of singular perturbations. Illustrative examples are presented

Computing all Pairs (λ,μ) Such That λ is a Double Eigenvalue of Α + μΒ

Double eigenvalues are not generic for matrices without any particular structure. A matrix depending linearly on a scalar parameter, Α + μΒ, will, however, generically have double eigenvalues for some values of the parameter μ. In this paper, we consider the problem of finding those values. More precisely, we construct a method to accurately find all scalar pairs (λ,μ) such that Α + μΒ has a double eigenvalue λ, where Α and Β are given arbitrary complex matrices. The general idea of the globally convergent method is that if μ is close to a solution, then Α + μΒ has two eigenvalues which are close to each other. We fix the relative distance between these two eigenvalues and construct a method to solve and study it by observing that the resulting problem can be stated as a two-parameter eigenvalue problem, which is already studied in the literature. The method, which we call the method of fixed relative distance (MFRD), involves solving a two-parameter eigenvalue problem which returns approximations of all solutions. It is unfortunately not possible to get full accuracy with MFRD. In order to compute solutions with full accuracy, we present an iterative method which returns a very accurate solution, for a sufficiently good starting value. The approach is illustrated with one academic example and one application to a simple problem in computational quantum mechanics. Copyright 2011 Society for Industrial and Applied Mathematic