Stability tests for two common classes of linear time-delay and hybrid systems

EXPLICIT HYBRID OPTIMAL CONTROLLER FOR DISTURBANCE ATTENUATION IN LINEAR HYBRID SYSTEMS

In this paper, we deal with the problem of output feedback stabilization for a class of linear hybrid systems. This problem is addressed by characterizing the structural properties of such a class of systems. Namely, reachability, controllability, stabilizability, observability, constructibility, and detectability are framed in terms of algebraic and geometric conditions on the data of the system. Two canonical forms, recalling the classical Kalman decompositions with respect to reachability and observability, are given. By taking advantage of this characterization, duality between control and observation structural properties is established and necessary and sufficient conditions for the existence of a linear time-invariant output feedback compensator are stated. Compared with previous results, no assumption is needed on the plant about minimum phaseness, relative degree, or squareness.

This paper is dedicated to the stability of linear time-delay system for a non small delay h. This is motivated by the fact that in some cases introducing a delay in the loop may stabilize a system. Compared to previously derived results the methodology is totally new and the resulting LMI formulas are original contributions. The derived criteria are based on Taylor series of the delay operator for which the Lagrange remainder is treated as a bounded uncertainty. A sequence of results with decreasing conservatism are obtained applying quadratic separation. An extension to robust analysis with respect to uncertain delay is also derived. A numerical example illustrates the efficiency

The published literature for stability analysis of linear time-delay systems has attracted much interest among the researchers over the last five decades. In general, the stability for linear time-delay systems can be checked exactly only by eigenvalue considerations. Two types of stability conditions are delay-independent type, mainly used for arbitrarily large delays, and delay-dependent type, which is mostly useful in applications that consider maximum delay. In this paper, we consider a linear time-invariant system with delay differential equation form for quiet-standing case study as a derivative feedback control. When it is reasonably chosen with intentional delays, the effects of time delay for ankle torque can be used to improve and stabilize the response of the close-loop systems.

This research examines new methods for stabilizing linear time-delay systems that are subject to denial-of-service (DoS) attacks. The study takes into account the different effects that a DoS attack can have on the system, specifically delay-independent and -dependent behaviour. The traditional proportional-integral-derivative (PID) acts on the error signal, which is the difference between the reference input and the measured output. The approach in this paper uses what we call the PID state feedback strategy, where the controller acts on the state signal. Our proposed strategy uses the Lyapunov-Krasovskii functional (LKF) to develop new linear matrix inequalities (LMIs). The study considers two scenarios where the time delay is either a continuous bounded function or a differentiable and time-varying function that falls within certain bounds. In both cases, new LMIs are derived to find the PID-like state feedback gains that will ensure robust stabilization. The findings are illustrated with numerical examples.

Considers the filtering problem for a class of linear hybrid systems with nonlinear uncertainties and Markovian jump parameters. The unknown nonlinearities in the system are time-varying and norm-bounded. First, we show the equivalence of the norm bounded linear and nonlinear uncertainty sets. Then, instead of the original hybrid linear system with nonlinear uncertainties, we consider the same system with linear uncertainties. By using a Riccati equation approach for this new system, a robust filter is designed using two sets of coupled Riccati-like equations such that the estimation error is guaranteed to have an upper bound.

A problem of great interest in the control of hybrid systems is the design of least restrictive controllers for reachability specifications. Controller design typically uses game theoretic methods to compute the region of the state space for which there exists a control such that for all disturbances, an unsafe set is not reached. In general, the computation of the controllers requires the steady state solution of a Hamilton-Jacobi partial differential equation which is very difficult to compute, if it exists. In this paper, we show that for special classes of hybrid systems where the continuous vec-tor fields are linear, the controller synthesis problem is semi-decidable: There exists a computational algo-rithm which, if it terminates in a finite number of steps, will exactly compute the least restrictive con-troller. This result is achieved by a very interesting interaction of results from mathematical logic and o p timal control. 1

Delay-independent stability for linear time-delay systems is analyzed using the structured singular value concept. We show that the delays can be viewed as repeated scalar uncertainty blocks for a 'nominal' plant, and the stability analysis converts to checking the structured singular value of the nominal plant with respect to a specially constructed uncertainty structure. When the system is not stable and independent of delays, we propose a method to compute the maximum tolerable delay for which the system remains stable. Finally, the H/sub /spl infin// control problem for time-delay systems is discussed and shown to be equivalent to a /spl mu/-synthesis problem. The results show that we can use the /spl mu/-approach to analyze and design for time-delay systems.

Some simple ideas on stabilization of time delay systems are exploited to find realistic robust stabilizing controllers. A discrete time controller is applied to stabilize a time delay system with a single input. Some simulation results are given and the relationship between delay duration and sampling period is discussed. >

The stability problem of linear time-delay systems with block-diagonal uncertainty is considered using a quadratic Lyapunov functional. The guaranteed stability condition is written as a linear matrix inequality for any given number of divisions N in discretization. The result can also be interpreted as guaranteed H/sub /spl infin// performance under uncertainty.

In this article, delay-dependent stability criteria for linear retarded and neutral systems with multiple delays are proposed by employing the Lyapunov-Krasovskii functional approach and integral inequality. By the N-segmentation of delay length, we obtain more relaxed results on the delay bounds which guarantee the asymptotic stability of the linear retarded and neutral systems with multiple delays. Simulation results show that the stability boundary calculated by the proposed stability criteria are very near to the real stability boundary of the example systems. Moreover, it is shown that the proposed stability criteria are less conservative than several other existing criteria.

Semi-global stabilization of linear time-delay systems with control energy constraint

This paper extends some of the basic results in the area of Lyapunov (asymptotic) and finite time and practical stability to linear, discrete, time invariant time-delay systems. New definitions have been established for the latter concept of stability. Sufficient conditions for this type of stability, concerning the particular class of linear discrete time-delay systems are derived. To the best knowledge of the authors, such results have not yet been reported. Moreover, the generalization of some previous results, in the area of Lyapunov stability for the same class of systems has been also established and proven.

This paper provides sufficient conditions for both practical stability and finite time stability of linear singular continuous time delay systems which can be mathematically described as Eẋ(t) = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> x(t) + A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> x(t - τ). Considering a finite time stability concept, new delay independent conditions have been derived using the approach based on the Lyapunov-like functions and their properties on the subspace of consistent initial conditions. These functions do not need to have the properties of positivity in the whole state space and negative derivatives along the system trajectories. When the practical stability has been analyzed the above mentioned approach was combined and supported by the classical Lyapunov technique to guarantee the attractivity property of the system behavior.

In this paper, both problems of the stability and stabilization analysis for linear neutral time-delay systems are respectively investigated. The time-delays considered here are assumed bounded but no information to be available. Using Lyapunov functional method, delay-dependent conditions for the stability and stabilization of the systems are presented in terms of linear matrix inequality (LMI). A numerical example is given to illustrate the main results of this paper and make a comparison with the results presented in literatures