Output-feedback stabilization of an unstable wave equation

We consider the problem of stabilization of a one-dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated “boundary damper” feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more complex due to the open-loop instability of the plant. The controller and observer gains are designed using the method of “backstepping,” which results in explicit formulae for the gain functions. We prove exponential stability and the existence and uniqueness of classical solutions for the closed-loop system. We also derive the explicit compensators in frequency domain. The results are illustrated with simulations.     

Stabilization of hybrid neutral stochastic differential delay equations by delay feedback control

This paper is concerned with the problem of exponential mean-square stabilization of hybrid neutral stochastic differential delay equations with Markovian switching by delay feedback control. A delay feedback controller is designed in the drift part so that the controlled system is mean-square exponentially stable. We discussed two types of structure controls; that is, state feedback and output injection. The stabilization criteria are derived in terms of linear matrix inequalities.     

PI stabilization of first-order systems with time delay

This paper considers the problem of stabilizing a first-order plant with dead time using a PI controller. Using a version of the Hermite–Biehler Theorem applicable to quasipolynomials, a complete and constructive characterization of all stabilizing PI gain values is obtained.     

Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation

We consider the dynamic elasticity equations with a locally distributed damping in a bounded domain. The local dissipation of the form a(x)y_t involves coefficients a that vanish on a negligible portion of the subset where the damping is effective. Using multiplier techniques, interpolation inequalities, and a judicious application of Hölder inequality, we prove sharp energy decay estimates. The results obtained generalize and improve on earlier works by Nakao, and the author in the framework of the ordinary wave equation.     

New results on stabilization of Markovian jump systems with time delay

This paper studies the problem of stochastic stabilization for a class of Markovian jump systems with time delay. A new delay-dependent stochastic stability criterion on the stochastic stability of the system is derived based on a novel Lyapunov-Krasovskii functional (LKF) approach. The equivalence and superiority to existing results are demonstrated. Then a state feedback controller, which guarantees the stochastic stability of the closed-loop system, is designed. Illustrative examples are provided to show the reduced conservatism and effectiveness of the proposed techniques.     

Delay dependent stabilization of linear time‐varying system with time delay

This paper investigates both stability and stabilization dependent on the delay of a class of time‐varying linear systems with a constant point time delay. The matrices, describing the state space dynamics, are parameterized by time‐varying function matrices. A numerical example is given in order to verify the theoretical results. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Stability properties of second order delay integro-differential equations

A basic theorem on the behavior of solutions of scalar linear second order delay integro-differential equations is established. As a consequence of this theorem, a stability criterion is obtained.     

A note on spectral realisations for delay systems

In this paper we show that a large class of SISO delay systems has a spectral realisation.     

A note on the controllability for the wave equation in nonsmooth plane domains

We study exact boundary controllability for a two-dimensional wave equation in a region which is an angular sector of a circle or an angular sector of an annular region. The control, of Neumann type, acts on the curved part of the boundary, while in the straight part we impose homogeneous Dirichlet boundary condition. The initial state has finite energy and the control is square integrable.     

Some remarks on stabilization by means of discontinuous feedbacks

By means of Ryan's necessary condition and of some examples, we see how the interpretation of solutions of systems with discontinuous right-hand side influences stabilizability results for systems which do not admit continuous stabilizing feedback laws.     

Stability for first order delay dynamic equations on time scales

In this paper, we establish some sufficient conditions for the uniform stability and the uniformly asymptotical stability of the first order delay dynamic equation

where is a time scale, p(.) is rd-continuous and positive, the delay function . Our results unify the corresponding ones for differential and difference equations. To the best of our knowledge, this is the first time to discuss the asymptotical behavior of delay dynamic equations on time scales.     

An example for the switching delay feedback stabilization of an infinite dimensional system: The boundary stabilization of a string

For vibrating systems, a delay in the application of a feedback control may destroy the stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end.We show that certain delays in the boundary feedback preserve the exponential stability of the system. In particular, we show that the system is exponentially stable with delays freely switching between the values 4L/c and 8L/c, where L is the length of the string and c is the wave speed.     

A note on oscillation criteria of second order nonlinear neutral delay differential equations

In recent years, many results on oscillation criteria of second order nonlinear neutral delay differential equations have been obtained, but some of these criteria are incorrect due to a mistake in a crucial auxiliary result. This note examines several of these results, provides some counterexamples and points out what causes the problems.     

Oscillation criteria for second order quasilinear neutral delay differential equations on time scales

In this paper, we consider the second-order quasilinear neutral dynamic equation

(r(t)|Z^Δ(t)|^α−1Z^Δ(t))^Δ+q(t)|x(δ(t))|^β−1x(δ(t))=0,     

Asymptotic stabilization with locally semiconcave control Lyapunov functions on general manifolds

Asymptotic stabilization on noncontractible manifolds is a difficult control problem. If a configuration space is not a contractible manifold, we need to design a time-varying or discontinuous state feedback control for asymptotic stabilization at the desired equilibrium.     

Robust stabilization of uncertain systems with unknown input delay

This paper is concerned with the robust controller design for uncertain input-delayed systems. The time delay is assumed to be an unknown constant. A controller with delay feedback for the robust stabilization of the system is proposed. The stability criterion of the closed-loop system is derived in terms of linear matrix inequalities (LMIs). Examples show that in many cases our method can give less conservative results than those by the existing methods. Moreover, for the same cases, our controllers have lower feedback gains than the existing ones.     

A note on robust stabilization for systems with parameters

We consider the problem of stabilizing a linear time invariant multivariable system whose transfer function admits a factorization which contains parametric type uncertainties. The parameters are known to take values in some region Ω_a. We first give a sufficient condition for robust stability when a proper feedback compensator is used. Here it is important to note that no assumption is made about the number of unstable plant poles. Following this we focus attention on how to compute an appropriate robustly stabilizing compensator. This is done in a two step procedure. Initially the solutions of a Diophantine equation generate a class of proper compensators, each of which stabilizes the ‘nominal’ closed loop system. This is followed by a Nyquist Theorem inspired minimization which yields the final choice. An illustrative physical example is given. The compensator thus constructed guarantees stability over . A discussion about how to compute the size of the attainable robust stability region is also included.     

Exact controllability of wave equations with variable coefficients coupled in parallel

In this paper, we investigate exact controllability for coupled wave equations which have variable coefficients principal part. Observability inequality is obtained by using Riemannian geometry method and Carleman estimates. Furthermore, the exact controllability result is established with Dirichlet boundary controls when T>T₀, where the lower bound T₀ for the control time T is different from that obtained in the constant coefficient principal part case. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

The existence of time optimal control of semilinear parabolic equations

This work is concerned with the existence of the time optimal control for some semilinear parabolic differential equations with control distributed in a subdomain.     

Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation

An Euler-Bernoulli beam equation under boundary control and delayed observation is considered. An observer-predictor based scheme is developed to stabilize the equation. On a time interval where the observation is available, the state is tracked by the observer; when the observation is not available due to the delay, the state is estimated by the predictor. The estimation is then used in proportional feedback. It is shown that the state of closed-loop system decays exponentially for smooth initial values.