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This article discusses the invariant subspaces that are restricted to be external direct sums. Some existence conditions are presented that facilitate finding such invariant subspaces. This problem is related to the decomposition of coupled differential-difference equations, leading to the possibility of lowering the dimensions of coupled differential-difference equations. As has been well documented, lowering the dimension of coupled differential-difference equations can drastically reduce the computational time needed in stability analysis when a complete quadratic Lyapunov–Krasovskii functional is used. Most known ad hoc methods of reducing the order are special cases of this formulation.

This article presents a geometric description of the set of stabilizing PID controllers. The three-dimensional parameter space is partitioned by the stability crossing surface into regions such that the number of characteristic roots on the right half-plane (RHP) remains constant within each region. The set of stabilizing PID parameters consists those regions with no RHP root. The stability crossing surface is a ruled surface, and it is completely determined by a curve known as the discriminant. The discriminant may be divided into sectors between its cusps, and each sector corresponds to a positive patch and negative patch. The stability crossing surface is composed of these patches.

This article studies the strong stability of scalar difference equations of continuous time in which the delays are sums of a number of independent parameters τii= 1, 2,..., K. The characteristic quasipolynomial of such an equation is a multilinear function of e-Tis. It is known that the characteristic quasipolynomial of any difference equation set in the form of one-delay-per-scalar-channel (ODPSC) model is also in such a multilinear form. However, it is shown in this article that some multilinear forms of quasipolynomials are not characteristic quasipolynomials of any ODPSC difference equation set. The equivalence between local strong stability the exponential stability of a fixed set of rationally independent delays, and the stability for all positive delays is shown.

In engineering practice, delay-difference is often used to approximate the derivatives of output signals for feedback control, leading to a closed-loop system with delay both in the states and in the system’s coefficients. In this context, our objective is to find all the delay values contained in some interval that guarantee the exponential stability of the closed-loop system subject to the delay-difference approximation. A method for stability analysis of systems with delay-dependent coefficients developed in our previous work is further extended and applied to tackle the particular form of systems considered in this paper. The proposed stability analysis procedure is illustrated through the design of a mobile-robot path-following controller.

This article presents the inverse of the kernel operator associated with the complete quadratic Lyapunov-Krasovskii functional for coupled differential-functional equations when the kernel operator is separable. Similar to the case of time-delay systems of retarded type, the inverse operator is instrumental in control synthesis. Unlike the power series expansion approach used in the previous literature, a direct algebraic method is used here. It is shown that the domain of definition of the infinitesimal generator is an invariant subspace of the inverse operator if it is an invariant subspace of the kernel operator. The process of control synthesis using the inverse operator is described, and a numerical example is presented using the sum-of-square formulation.

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.

This article studies the stability crossing set of linear time-invariant systems with two scalar-delay channels. This study is crucial to the complete stability analysis along the idea of D-subdivision. The characteristic quasipolynomial of such systems contains an exponential term with the sum of two delays (cross term) in its exponent. A complete parameterization and geometric characterization of the stability crossing set is conducted. It was found instrumental to relate it to an associated quasipolynomial without such a cross term.

This paper focuses on the stability analysis of the closed-loop single-input-single-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.

Abstract: This paper focuses on the stability of some second-order linear systems with multiple constant and time-varying delays, under the assumption that the corresponding system without delays is an oscillator. Sufficient conditions for delay-dependent stability will be derived using the integral quadratic constraint approach combined with the generalized eigenvalue distribution of some appropriate finite-dimensional matrix pencils. As applications, we shall discuss some fluid approximation models used in congestion control of high-speed networks, under the natural assumptions that the control-time intervals are constant, but the round-trip times are time-varying.

This article revisits some delay-dependent stability criteria previous discussed in the literature. A more systematic approach is taken, and different forms of stability criteria are developed. The form is more convenient to treat related uncertainty in different system matrices. The relationship between the stability criteria based on simple Lyapunov-Krasovskii functional method and on Razumikhin Theorem are more explicitly shown

This paper investigates the robust stability problem of time-delay systems with multiple passive uncertainties. A stability criterion is derived using the refined discretized Lyapunov functional method. The criterion is written in the form of a linear matrix inequality. Numerical examples are presented to illustrate the effectiveness of the method.

This article improves the formulation of discretized Lyapunov functional method for systems with distributed delays. The main idea is to apply Jensen's inequality and variable elimination in matrix inequalities. The resulting stability criterion is much simpler, and converges to the analytical result much faster. The new formulation is also applicable to a wider class of systems.

This paper investigates the stability of linear uncertain systems with time‐varying delay. Stability criteria are derived based on a generalized discretized Lyapunov functional approach. The kernel of the functional, which is a function of two variables, is chosen as piecewise linear. The stability conditions are written in the form of linear matrix inequalities. Numerical examples indicate significant improvements over the existing results.

The excitation strength for the onset of a new resonant-separatrix in the stochastic layer of the Duffing oscillator is predicted through the energy change in minimum and maximum energy spectra. The widths of stochastic layers are estimated through the use of the maximum and minimum energy which can be measured experimentally. The energy spectrum approach, rather than the Poincaré mapping section method, is applied to detect the resonant-separatrix web in the stochastic layer, and it is applicable for the onset of resonant layers in nonlinear dynamic systems. The analytical condition for the onset of a new resonant-separatrix in the stochastic layer is also presented. The analytical and numerical predictions are in good agreement.

The stability problem of linear uncertain time-delay systems is considered using a quadratic Lyapunov functional. The kernel of the functional, which is a function of two variables, is chosen as piecewise linear. As a result, the stability condition can be written as a linear matrix inequality, which improves the estimate of stability limit over the existing approaches. A number of constraints on the parameters can be introduced to reduce the computational effort needed with some compromised accuracy. For a particular choice of constraints, the existing discretization scheme can be recovered. Copyright © 1999 John Wiley & Sons, Ltd.

The stability problem of linear uncertain time‐delay systems is considered using a quadratic Lyapunov functional. The kernel of the functional, which is a function of two variables, is chosen as piecewise linear. As a result, the stability condition can be written as a linear matrix inequality, which improves the estimate of stability limit over the existing approaches. A number of constraints on the parameters can be introduced to reduce the computational effort needed with some compromised accuracy. For a particular choice of constraints, the existing discretization scheme can be recovered. Copyright © 1999 John Wiley & Sons, Ltd.

The Lyapunov function approach is used to derive a new algorithm for model based adaptive motion control as applied to robot manipulators. The technique does not require a knowledge of state accelerations and if the method is implemented through a Newton-Euler recursive formulation then the computational requirements are quite low. High feedback gains are not needed to ensure stability. The parameter ranges allowed for stability are explicitly expressed, making the design process fairly simple. The algorithm is conceptually simple and easily implemented in industrial robots.

The traditional approach of using free vibration modes in the assumed mode method often leads to an extremely slow convergence rate, especially when discete interactive forces are involved. By introducing a number of forced modes, significant improvements can be achieved. These forced modes are intrinsic to the structure and the spatial distribution of forces. The motion of the structure can be described exactly by these forced modes and a few free vibration modes provided that certain conditions are satisfied. The forced modes can be viewed as an extension of static modes. The development of a forced mode formulation is outlined and a numerical example is presented.

The Lyapunov function approach is used to derive a new algorithm for model based adaptive motion control as applied to robot manipulators. The technique does not require a knowledge of state accelerations and if the method is implemented through a Newton-Euler recursive formulation then the computational requirements are quite low. High feedback gains are not needed to ensure stability. The parameter ranges allowed for stability are explicitly expressed, making the design process fairly simple. The algorithm is conceptually simple and easily implemented in industrial robots.

Uncertainty tolerance and H∞ attenuations are two important concerns in feedback control design. It is therefore important to develop effective methods to compute the trade-off between these two objectives. Using a quasiconvex optimization approach, this paper develops algorithms for two problems in this regard: (1) given the uncertainty size of the system, compute the minimum H∞ norm bound of the closed loop system reachable by state feedback control; and (2) given the required maximum acceptable H∞ norm bound of the closed loop system, compute the maximum uncertainty size which can be tolerated when state feedback control is applied. A numerical example is included to show the trade-off between uncertainty tolerance and H∞ attenuations, and the effectiveness of the algorithms.