Proceedings of the 28th IEEE Conference on Decision and Control
A characterization of the structure of uncertainty called the generalized matching condition is p... more A characterization of the structure of uncertainty called the generalized matching condition is proposed. The uncertainty is further decomposed into two parts that are `orthogonal' to each other. On the basis of this decomposition a linear feedback control design is proposed for linear uncertain systems. The control always exists if the generalized matching condition is met. The design is illustrated
Lecture Notes in Computational Science and Engineering, 2004
... Eigenvalues James Louisell 193 The Effect of Approximating Distributed Delay Control Laws on ... more ... Eigenvalues James Louisell 193 The Effect of Approximating Distributed Delay Control Laws on Stability Wim Michiels, Sabine Mondie, Dirk ... Emilia Fridman 233 Robust Stability Analysis of Various Classes of Delay Systems Catherine Bonnet, Jonathan R. Partington 245 On ...
This paper addresses the stability problem of a special class of dynamical systems of coupled dif... more This paper addresses the stability problem of a special class of dynamical systems of coupled differential and difference equations arising from the mathematical description of various engineering systems that contain lossless propagation media (pipes or electrical lines). More explicitly, we shall derive some sufficient stability conditions including delay information using degenerate Liapunov-Krasovskii functionals under appropriate model transformations. Note that the corresponding model transformations induce additional dynamics that will be also characterized.
... AL The significance of Theorems 2 and 3 is seen from the following corollary: Corollary 1. A ... more ... AL The significance of Theorems 2 and 3 is seen from the following corollary: Corollary 1. A system satisfying assumption I is quadratically stable if and only if p = max[n. g(Q)]- min[-ng(Q)] > 0 (10) QET QET ... wf1(Q, E1) + (1 -w)#1(Q, E2) <max{/h(Q,El),fdj(QjE2)} ...
This paper focuses on the closed-loop stability analysis of single-input-single-output (SISO) sys... more This paper focuses on the closed-loop stability analysis of single-input-single-output (SISO) systems subject to input (or output) delays in the presence of PID-controllers. More precisely, using a geometric approach, we present a simple and user-friendly method for the closed-loop stability analysis as well as for the fragility of such PID controllers. The proposed approach is illustrated on several examples encountered in the control literature.
This paper focuses on the stability analysis of the closed-loop single-inputsingle-output (SISO) ... more This paper focuses on the stability analysis of the closed-loop single-inputsingle-output (SISO) system subject to input delays and to PI controllers. A complete characterization of the crossing set, which consists of all frequencies where the number of unstable characteristic roots changes, is given in the parameter space defined by the controller. Next, we explicitly derive the PI controllers corresponding to each frequency in the crossing set. More precisely, we make a partition of the controller parameter space in regions where the number of unstable characteristic roots remains constant. Finally, we also present a methodology to compute the number of unstable roots in each region. Several illustrative examples end the presentation.
The stability problem of discrete-time linear uncertain systems is considered. The uncertainty is... more The stability problem of discrete-time linear uncertain systems is considered. The uncertainty is expressed in terms of uncertain system matrix, and is allowed to be arbitrary time varying. The criterion is based on quadratic stability. The necessary and sufficient condition of the quadratic stability is formulated in a two level optimization problem. The higher level of this optimization is proved
This paper gives an overview of the stability analysis of systems with delay-dependent coefficien... more This paper gives an overview of the stability analysis of systems with delay-dependent coefficients. Such systems are frequently encountered in various scientific and engineering applications. Most such analyses are generalization of those on systems with delay-independent coefficients. Therefore an introduction on systems with delay-independent coefficients is also given, with an emphasis on the τ-decomposition approach. Methods for two key ingredients of this approach are discussed, namely the identification of imaginary characteristic roots with the corresponding delays, and local behavior analysis of these roots as the delay increases through these critical values. For systems with delay-dependent coefficients, we review the methods of analysis for systems with a single delay and commensurate delays, their application to output feedback control and a geometric perspective that establishes a link between systems with and without delay-dependent coefficients. We provide the main ideas of various stability analysis methods and their advantages and limitations. We also present our perspectives on future directions of research on this interesting topic.
2015 54th IEEE Conference on Decision and Control (CDC), 2015
This paper studies the migration of double imaginary roots of the characteristic equation for sys... more This paper studies the migration of double imaginary roots of the characteristic equation for systems with two delays when the delay parameters are subjected to small deviations. As the double roots are not differentiable with respect to the delay parameters, Puiseux series is often used in such a situation in the literature. In this article, we study the "least degenerate" case, and a more traditional analysis was used without Puiseux series. It was found that the local stability crossing curve has a cusp at the point in the parameter space that causes the double root, and it divides the neighborhood of this point into a G-sector and an S-sector. When the parameters move into the G-sector, one of the roots moves to the right half plane, and the other moves to the left half plane. When the parameters move into the S-sector, both roots move either to the left half plane or the right half plane depending on the sign of some value explicitly expressed in terms of derivatives of the characteristic function up to the third order.
For systems with both discrete and distributed delays, the article by Fridman and Tsodik formulat... more For systems with both discrete and distributed delays, the article by Fridman and Tsodik formulated computable conditions for stability, satisfaction of H 1 performance requirement, and design of state feedback control. As is now well known, in order to obtain stability conditions using Lyapunov-Krasovskii functional (LKF) approach, it is necessary to use a complete quadratic LKF, as is done in this article. As a general quadratic functional involve infinite number of parameters, some discretization process is necessary in order to render conditions in a computable form. A piece-wise linear parameterization such as widely used in finite element methods seems to be the most natural. However, there is a fundamental difference between these two discretizations: in finite element analysis, the objective is purely to approximate the solution; in the discretization of LKF, it is also necessary to guarantee the satisfaction of quadratic inequalities. Therefore, the discretization of LKF is a much more sophisticated process.
This paper reviews some subtleties in time-delay systems of neutral type that are believed to be ... more This paper reviews some subtleties in time-delay systems of neutral type that are believed to be of particular relevance in practice. Both traditional formulation and the coupled differential-difference equation formulation are used. The discontinuity of the spectrum as a function of delays is discussed. Conditions to guarantee stability under small parameter variations are given. A number of subjects that have been discussed in the literature, often using different methods, are reviewed to illustrate some fundamental concepts. These include systems with small delays, the sensitivity of Smith predictor to small delay mismatch, and the discrete implementation of distributed-delay feedback control. The framework prsented in this paper makes it possible to provide simpler formulation and strengthen, generalize, or provide alternative interpretation of the existing results.
International Journal of Dynamics and Control, 2014
The characteristic function of a system with three scalar delay channels contains cross terms of ... more The characteristic function of a system with three scalar delay channels contains cross terms of different delays. This article studies the parameterization and geometric structure of the stability crossing set (the set of delay combinations with at least one characteristic root on the imaginary axis) for such systems. Understanding the structure of this set is crucial to the identification of stable regions in the delay parameter space using the D-subdivision method. The presence of the cross terms significantly complicates the analysis, and requires a quite different method than the case without these cross terms, and it involves more numerical computation.
Proceedings of the 28th IEEE Conference on Decision and Control
A characterization of the structure of uncertainty called the generalized matching condition is p... more A characterization of the structure of uncertainty called the generalized matching condition is proposed. The uncertainty is further decomposed into two parts that are `orthogonal' to each other. On the basis of this decomposition a linear feedback control design is proposed for linear uncertain systems. The control always exists if the generalized matching condition is met. The design is illustrated
Lecture Notes in Computational Science and Engineering, 2004
... Eigenvalues James Louisell 193 The Effect of Approximating Distributed Delay Control Laws on ... more ... Eigenvalues James Louisell 193 The Effect of Approximating Distributed Delay Control Laws on Stability Wim Michiels, Sabine Mondie, Dirk ... Emilia Fridman 233 Robust Stability Analysis of Various Classes of Delay Systems Catherine Bonnet, Jonathan R. Partington 245 On ...
This paper addresses the stability problem of a special class of dynamical systems of coupled dif... more This paper addresses the stability problem of a special class of dynamical systems of coupled differential and difference equations arising from the mathematical description of various engineering systems that contain lossless propagation media (pipes or electrical lines). More explicitly, we shall derive some sufficient stability conditions including delay information using degenerate Liapunov-Krasovskii functionals under appropriate model transformations. Note that the corresponding model transformations induce additional dynamics that will be also characterized.
... AL The significance of Theorems 2 and 3 is seen from the following corollary: Corollary 1. A ... more ... AL The significance of Theorems 2 and 3 is seen from the following corollary: Corollary 1. A system satisfying assumption I is quadratically stable if and only if p = max[n. g(Q)]- min[-ng(Q)] > 0 (10) QET QET ... wf1(Q, E1) + (1 -w)#1(Q, E2) <max{/h(Q,El),fdj(QjE2)} ...
This paper focuses on the closed-loop stability analysis of single-input-single-output (SISO) sys... more This paper focuses on the closed-loop stability analysis of single-input-single-output (SISO) systems subject to input (or output) delays in the presence of PID-controllers. More precisely, using a geometric approach, we present a simple and user-friendly method for the closed-loop stability analysis as well as for the fragility of such PID controllers. The proposed approach is illustrated on several examples encountered in the control literature.
This paper focuses on the stability analysis of the closed-loop single-inputsingle-output (SISO) ... more This paper focuses on the stability analysis of the closed-loop single-inputsingle-output (SISO) system subject to input delays and to PI controllers. A complete characterization of the crossing set, which consists of all frequencies where the number of unstable characteristic roots changes, is given in the parameter space defined by the controller. Next, we explicitly derive the PI controllers corresponding to each frequency in the crossing set. More precisely, we make a partition of the controller parameter space in regions where the number of unstable characteristic roots remains constant. Finally, we also present a methodology to compute the number of unstable roots in each region. Several illustrative examples end the presentation.
The stability problem of discrete-time linear uncertain systems is considered. The uncertainty is... more The stability problem of discrete-time linear uncertain systems is considered. The uncertainty is expressed in terms of uncertain system matrix, and is allowed to be arbitrary time varying. The criterion is based on quadratic stability. The necessary and sufficient condition of the quadratic stability is formulated in a two level optimization problem. The higher level of this optimization is proved
This paper gives an overview of the stability analysis of systems with delay-dependent coefficien... more This paper gives an overview of the stability analysis of systems with delay-dependent coefficients. Such systems are frequently encountered in various scientific and engineering applications. Most such analyses are generalization of those on systems with delay-independent coefficients. Therefore an introduction on systems with delay-independent coefficients is also given, with an emphasis on the τ-decomposition approach. Methods for two key ingredients of this approach are discussed, namely the identification of imaginary characteristic roots with the corresponding delays, and local behavior analysis of these roots as the delay increases through these critical values. For systems with delay-dependent coefficients, we review the methods of analysis for systems with a single delay and commensurate delays, their application to output feedback control and a geometric perspective that establishes a link between systems with and without delay-dependent coefficients. We provide the main ideas of various stability analysis methods and their advantages and limitations. We also present our perspectives on future directions of research on this interesting topic.
2015 54th IEEE Conference on Decision and Control (CDC), 2015
This paper studies the migration of double imaginary roots of the characteristic equation for sys... more This paper studies the migration of double imaginary roots of the characteristic equation for systems with two delays when the delay parameters are subjected to small deviations. As the double roots are not differentiable with respect to the delay parameters, Puiseux series is often used in such a situation in the literature. In this article, we study the "least degenerate" case, and a more traditional analysis was used without Puiseux series. It was found that the local stability crossing curve has a cusp at the point in the parameter space that causes the double root, and it divides the neighborhood of this point into a G-sector and an S-sector. When the parameters move into the G-sector, one of the roots moves to the right half plane, and the other moves to the left half plane. When the parameters move into the S-sector, both roots move either to the left half plane or the right half plane depending on the sign of some value explicitly expressed in terms of derivatives of the characteristic function up to the third order.
For systems with both discrete and distributed delays, the article by Fridman and Tsodik formulat... more For systems with both discrete and distributed delays, the article by Fridman and Tsodik formulated computable conditions for stability, satisfaction of H 1 performance requirement, and design of state feedback control. As is now well known, in order to obtain stability conditions using Lyapunov-Krasovskii functional (LKF) approach, it is necessary to use a complete quadratic LKF, as is done in this article. As a general quadratic functional involve infinite number of parameters, some discretization process is necessary in order to render conditions in a computable form. A piece-wise linear parameterization such as widely used in finite element methods seems to be the most natural. However, there is a fundamental difference between these two discretizations: in finite element analysis, the objective is purely to approximate the solution; in the discretization of LKF, it is also necessary to guarantee the satisfaction of quadratic inequalities. Therefore, the discretization of LKF is a much more sophisticated process.
This paper reviews some subtleties in time-delay systems of neutral type that are believed to be ... more This paper reviews some subtleties in time-delay systems of neutral type that are believed to be of particular relevance in practice. Both traditional formulation and the coupled differential-difference equation formulation are used. The discontinuity of the spectrum as a function of delays is discussed. Conditions to guarantee stability under small parameter variations are given. A number of subjects that have been discussed in the literature, often using different methods, are reviewed to illustrate some fundamental concepts. These include systems with small delays, the sensitivity of Smith predictor to small delay mismatch, and the discrete implementation of distributed-delay feedback control. The framework prsented in this paper makes it possible to provide simpler formulation and strengthen, generalize, or provide alternative interpretation of the existing results.
International Journal of Dynamics and Control, 2014
The characteristic function of a system with three scalar delay channels contains cross terms of ... more The characteristic function of a system with three scalar delay channels contains cross terms of different delays. This article studies the parameterization and geometric structure of the stability crossing set (the set of delay combinations with at least one characteristic root on the imaginary axis) for such systems. Understanding the structure of this set is crucial to the identification of stable regions in the delay parameter space using the D-subdivision method. The presence of the cross terms significantly complicates the analysis, and requires a quite different method than the case without these cross terms, and it involves more numerical computation.
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