Unstructured Perturbations Research Articles

Perturbation theory of structured matrix pencils with no spillover

This paper aims to determine unstructured or structured perturbations of a matrix pencil such that the perturbed matrix pencil reconstructs a given deflating pair without changing the complementary deflating pair of the unperturbed matrix pencil. If the latter is unknown, then it is referred to as no spillover updating. If the complementary deflating pair of the given pencil is known, then all possible unstructured perturbations are obtained. However, if the complementary deflating pair of the unperturbed pencil is unknown, then we obtain a class of structure-preserving perturbations of the structured pencil. Finally, these results are utilized to obtain parametric solutions of the model updating problem with no spillover for various structured matrix pencils.

Observer-Based H∞ Controller Design for High Frequency Stick-Slip Vibrations Mitigation in Drill-String of Rotary Drilling Systems

The drilling process is among the most crucial steps in exploration and production activities in the petroleum industry. It consists of using mechanical mechanisms to crush rocks by the drill bit to pass through the different geological layers. The drill-string continuously transforms the rotational movement from the top drive motor to the drill bit through the drill pipes. Due to the strong interactions with the rocks, aggressive vibrations can arise in the drill-string in its three dimensions, and consequently, this may create three types of synchronous vibrations: axial, lateral, and torsional. The severe status of the latter is known as the stick-slip phenomenon, and is the most common in rotary drilling systems. Based on field observations, it has been inferred that the high frequency stick-slip vibrations may lead to drill-string fatigues and even to premature rupture. In the best case, it reduces the drilling efficiency by decreasing the rate of penetration, due to which the drilling operations become proportionally expensive. The main novelties of this research work are the design of an H∞ observer-based controller to mitigate the high frequency stick-slip vibrations, and the quantitative analysis of the vibrations’ severity for ten degrees of freedom model. The observer is designed to estimate the non-measurable rotational velocity of the drill bit due to the severity of the vibrations, while the controller is dedicated to suppressing the vibrations by using the top drive inputs. Thus, many scenarios have been considered to test and analyze the observer performance and the controller robustness. Furthermore, a comparison with the LQG observer-based controller has been conducted, where H∞ has demonstrated better efficiency in suppressing the stick-slip vibrations under unstructured perturbations.

Updating structured matrix pencils with no spillover effect on unmeasured spectral data and deflating pair

This paper is devoted to the study of perturbations of a matrix pencil, structured or unstructured, such that a perturbed pencil will reproduce a given deflating pair while maintaining the invariance of the complementary deflating pair. If the latter is unknown, it is referred to as no spillover updating. The specific structures considered in this paper include symmetric, Hermitian, ⋆-even, ⋆-odd and ⋆-skew-Hamiltonian/Hamiltonian pencils. This study is motivated by the well-known Finite Element Model Updating Problem in structural dynamics, where the given deflating pair represents a set of given eigenpairs and the complementary deflating pair represents the remaining larger set of eigenpairs. Analytical expressions of structure preserving no spillover updating are determined for deflating pairs of structured matrix pencils. Besides, parametric representations of all possible unstructured perturbations are obtained when the complementary deflating pair of a given unstructured pencil is known. In addition, parametric expressions are obtained for structured updating with certain desirable structures which relate to existing results on structure preservation of a symmetric positive definite or semi definite matrix pencil.

About Total Stability of a Class of Nonlinear Dynamic Systems Eventually Subject to Discrete Internal Delays

This paper studies and investigates total stability results of a class of dynamic systems within a prescribed closed ball of the state space around the origin. The class of systems under study includes unstructured nonlinearities subject to multiple higher-order Lipschitz-type conditions which influence the dynamics and which can be eventually interpreted as unstructured perturbations. The results are also extended to the case of presence of multiple internal (i.e., in the state) point discrete delays. Some stability extensions are also discussed for the case when the systems are subject to forcing efforts by using links between the controllability and stabilizability concepts from control theory and the existence of stabilizing linear controls. The results are based on the ad hoc use of Gronwall’s inequality.

Stability radii for real linear Hamiltonian systems with perturbed dissipation

We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical systems. We analyze when such a system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the real distance to instability (real stability radius) has been a challenging problem. We analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical stability of a DH system is much more robust than for unstructured perturbations.

Subspace Methods for Computing the Pseudospectral Abscissa and the Stability Radius

The pseudospectral abscissa and the stability radius are well-established tools for quantifying the stability of a matrix under unstructured perturbations. Based on first-order eigenvalue expansions, Guglielmi and Overton [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166--1192] recently proposed a linearly converging iterative method for computing the pseudospectral abscissa. In this paper, we propose to combine this method and its variants with subspace acceleration. Each extraction step computes the pseudospectral abscissa of a small rectangular matrix pencil, which is comparably cheap and guarantees monotonicity. We observe local quadratic convergence and prove local superlinear convergence of the resulting subspace methods. Moreover, these methods extend naturally to computing the stability radius. A number of numerical experiments demonstrate the robustness and efficiency of the subspace methods.

Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations

We study the perturbation theory of structured matrices under structured rank one perturbations, with emphasis on matrices that are unitary, orthogonal, or symplectic with respect to an indefinite inner product. The rank one perturbations are not necessarily of arbitrary small size (in the sense of norm). In the case of sesquilinear forms, results on selfadjoint matrices can be applied to unitary matrices by using the Cayley transformation, but in the case of real or complex symmetric or skew-symmetric bilinear forms additional considerations are necessary. For complex symplectic matrices, it turns out that generically (with respect to the perturbations) the behavior of the Jordan form of the perturbed matrix follows the pattern established earlier for unstructured matrices and their unstructured perturbations, provided the specific properties of the Jordan form of complex symplectic matrices are accounted for. For instance, the number of Jordan blocks of fixed odd size corresponding to the eigenvalue 1 or −1 have to be even. For complex orthogonal matrices, it is shown that the behavior of the Jordan structures corresponding to the original eigenvalues that are not moved by perturbations follows again the pattern established earlier for unstructured matrices, taking into account the specifics of Jordan forms of complex orthogonal matrices. The proofs are based on general results developed in the paper concerning Jordan forms of structured matrices (which include in particular the classes of orthogonal and symplectic matrices) under structured rank one perturbations. These results are presented and proved in the framework of real as well as of complex matrices.

Robust decentralized control of perturbed fractional-order linear interconnected systems

The robust decentralized state feedback stabilization problem for a class of perturbed fractional-order linear interconnected systems is investigated. The systems considered are subject to structured or unstructured perturbations. The decentralized state feedback controllers are designed such that the closed-loop perturbed fractional-order linear interconnected systems are asymptotically stable. By using the definition of positive and negative definite matrices and stability theory of fractional-order systems, sufficient conditions for robust stabilization of closed-loop perturbed fractional-order linear interconnected systems are given. Simulation examples are given to demonstrate the effectiveness of the proposed theoretical results.

Iterative design of structured uncertainty models and robust controllers based on closed-loop data

A joint uncertainty modelling and robust control design method is presented for linear time-invariant (LTI) systems. The system is given as a linear fractional transformation (LFT) with structured uncertainty blocks. The goal is to optimise the robust performance of the closed loop by simultaneously creating an optimised uncertainty model consistent with measurement data and designing a robust controller using skew μ synthesis. The proposed modelling/validation and control design algorithm is placed in the iterative identification and control (IIC) framework (design–experiments–analysis loop). In the proposed IIC approach, weighting functions of structured perturbation models are designed, which may lead to less conservative controllers compared to unstructured perturbations and disturbances. No a priori information is required on the disturbances. An advantage of the proposed method in the μ-synthesis context is that the number of fixed design parameters used in robust control (typically frequency-dependent weighting functions) is reduced to those associated with the control performance specification. The weighting functions of the uncertainty model are parameterised in the frequency domain and tuned subject to model validation constraints. It is shown that using skew μ as a robust performance criterion for both uncertainty modelling and control design, the proposed scheme simultaneously improves the robustness and performance of the control system. The efficiency of the method is illustrated on a vehicle steering problem.

Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations

udy the perturbation theory of structured matrices under structured rank one perturbations, and then focus on several classes of complex matrices. Generic Jordan structures of perturbed matrices are identified. It is shown that the perturbation behavior of the Jordan structures in the case of singular J-Hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in [18, 28, 30, 31]. Thus a generic structured perturbation would not be generic if considered as an unstructured perturbation. In other settings of structured matrices, the generic perturbation behavior of the Jordan structures, within the confines imposed by the structure, follows the pattern of that of unstructured perturbations.

On the stability radii of continuous-time infinite Markov jump linear systems

In this paper we introduce the subject of stability radii for continuous-time infinite Markov jump linear systems (MJLS) with respect to unstructured perturbations. By means of the small-gain approach, a lower bound for the complex radius is derived along with a linear matrix inequality (LMI) optimization method which is new in this context. In this regard, we propose an algorithm to solve the optimization problem, based on a bisectional procedure, which is tailored in such a way that avoids the issue of scaling optimization. In addition, an easily computable upper bound for the real and complex stability radii is devised, with the aid of a spectral characterization of the problem. This seems to be a novel approach to the problem of robust stability, even when restricted to the finite case, which in turn allows us to obtain explicit formulas for the stability radii of two-mode scalar MJLS. We also introduce a connection between stability radii and a certain margin of stability with respect to perturbations on the transition rates of the Markov process. The applicability of the main results is illustrated with some numerical examples.

Limitations of nonlinear periodic sampled-data control for robust stabilization

This paper presents a performance analysis of nonlinear periodically time varying sampled-data controllers acting upon a linear time invariant plant. Time invariant controllers are distinguished from strictly periodically time varying controllers. For a given nonlinear strictly periodic controller, a time invariant controller is constructed. Necessary and sufficient conditions are given under which the time invariant controller gives strictly better control performance than the time invariant controller from which it was obtained, for the robust stabilization of Lp unstructured perturbations, for all p ∈ [1,∞].

Introduction to Feedback Control (Morris, K.; 2001) [Book Review

The book is examined in the context of its suitability for a first engineering course in feedback control. The text covers several topics normally included in the most current introductory feedback control texts, including: transfer functions and state-space descriptions; Routh-Hurwitz and Nyquist stability criteria; and state-feedback and linear-quadratic regulator theory. It also includes more contemporary topics, such as: robust stability and performance for unstructured perturbations; strong stabilization and Youla parameterization; generalized plants and H/sub/spl infin// design; and H/sub/spl infin// optimization by model matching and interpolation theory. Short appendices are included on normed linear spaces, matrix and abstract algebra, and state-space system manipulations. The material is well presented, with careful proofs of key results. Each chapter includes numerical examples, together with a good list of homework exercises. Some Matlab functions for computing numerical solution are mentioned, but the discussion of software is limited. Unfortunately, the mathematical level of the test is too high, at least for most programs in the United States. The book also omits several standard topics associated with engineering course on feedback control, such as modeling of physical systems, root-locus analysis, digital control, and an introduction to nonlinear systems. As it is, the book would make a very good test for a graduate engineering course in linear feedback control. It could also be a valuable reference for graduate engineering students or applied mathematics students interested in research in the control area.

Dichotomy property and Lagrange stability for uncertain pendulum-like feedback systems

In this note, property of dichotomy and Lagrange stability of the uncertain pendulum-like systems with additional, multiplicative and H ∞ uncertainty are considered, respectively. By using some results of H ∞ theory the property of dichotomy and Lagrange stability of uncertain pendulum-like systems with multiple equilibria are transformed into L ∞ or H ∞ norm condition of some transfer functions. The convergence problem of all bounded solutions for some uncertain pendulum-like systems can be converted into the internal stability of another system under unstructured perturbations.

Limitations of nonlinear discrete periodic control for disturbance attenuation and robust stabilization

This paper presents a performance analysis of nonlinear periodically time-varying discrete controllers acting upon a linear time-invariant discrete plant. Time-invariant controllers are distinguished from strictly periodically time-varying controllers. For a given nonlinear periodic controller, a time-invariant controller is constructed. Necessary and sufficient conditions are given under which the time-invariant controller gives strictly better control performance than the time-invariant controller from which it was obtained, for the attenuation of l p exogenous disturbances and the robust stabilization of l p unstructured perturbations, for all p ∈ [ 1 , ∞ ] .

Structured condition numbers and backward errors in scalar product spaces

The effect of structure-preserving perturbations on the solution to a linear system, matrix inversion, and distance to singularity is investigated. Particular attention is paid tolinear and nonlinear structures that form Lie algebras, Jordan algebras and automorphism groups of ascalar product. These include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, unitary, complex orthogonal and symplectic matrices. Under reasonable assumptions on the scalar product, it is shown that there is little or no difference between structured and unstructured condition numbers and distance to singularity for matrices in Lie and Jordan algebras. Hence, for these classes of matrices, the usual unstructured perturbation analysis is sufficient. It is shown that this is not true in general for structures in automorphism groups. Bounds and computable expressions for the structured condition numbers for a linear system and matrix inversion are derived for these nonlinear structures.Structured backward errors for the approximate solution of linear systems are also considered. Conditions are given for the structured backward error to be finite. For Lie and Jordan algebras it is proved that, whenever the structured backward error is finite, it is within a small factor of or equal to the unstructured one. The same conclusion holds for orthogonal and unitary structures but cannot easily be extended to other matrix groups. This work extends and unifies earlier analyses.

Global positioning system receiver tracking-loop optimisation via an H∞ approach

The application of H∞ optimal control theory to global positioning system receiver tracking loops is investigated. The design of an H∞ controller for the receiver tracking-loop is presented. The H∞ controller is particularly attractive because it is a robust design in the sense that small disturbances lead to small tracking errors. Furthermore, it easily accommodates the inclusion of plant uncertainties as part of the plant model. By adding unstructured or structured perturbations to the plant model, it is possible to design controllers that ensure stability robustness and performance robustness of the closed-loop system. In order to apply the H∞ optimal design, the GPS receiver tracking loop is rewritten into a two-input two-output generalised plant model. Various levels of disturbances are derived using a software-based GPS L5 signal generator which are then used as test inputs to evaluate the performance of the proposed tracking loop.

A generic approach to the design of decentralized linear output- feedback controllers

A sufficient condition for failure-tolerant performance stabilization in a desirable performance region under decentralized linear output-feedback is established. To exploit the flexibility in decentralized control beyond multivariable pole assignment, and to address the subsystem design objectives along with those of the overall system, a generic problem on decentralized linear output-feedback is then defined. The problem is reformulated in terms of a constrained nonlinear optimization problem. The proposed methodology results in the optimal reconciliation of failure-tolerant robust performance of the overall system, and (maximal) robustness, disturbance rejection, noninteractive performance, reliability and low actuator gains in the isolated subsystems in the face of unstructured perturbations in the controller and plant parameters. The effectiveness of the proposed approach is demonstrated by a numerical example.

Robust eigenstructure assignment with minimum subspace separation

AbstractA computation method for delivering a state feedback controller that assigns closed‐loop eigenstructure with a minimum subspace separation and is robust towards unstructured perturbations is presented. The subspace separation is measured via the difference between two subspace projections, while the eigenstructure robustness is achieved by maximizing an upper bound on the allowable perturbation size such that the closed‐loop system remains stable. By exploiting the Sylvester equation parametrization, the constraint of pole assignment can be satisfied intrinsically and the robust eigenstructure assignment design essentially becomes an unconstrained optimization task. Analytical gradient formulas of the objective functions to be handled are developed. The design approach is demonstrated through numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.

Regional eigenvalue-clustering robustness analysis for singular systems with both structured and unstructured perturbations

In this paper, under assumptions that the nominal linear singular system is regular and impulse-free, and has all its finite eigenvalues lying inside certain specified regions, a sufficient condition is proposed to preserve the assumed properties when both structured (elemental) and unstructured (norm-bounded) parameter perturbations are added into the nominal singular system. No restriction is imposed on the shapes of the specified regions. The proposed method is applicable to both the continuous-time case and the discrete-time case. When all the finite eigenvalues are just required to locate in the stable region, the proposed criterion will become the stability robustness criterion. For the case that the linear singular system only subjects to structured perturbations, by an illustrated example, the presented sufficient condition is shown to be less conservative than the existing one reported recently in the literature.