Locally positive stabilization of infinite-dimensional linear systems by state feedback

Positivity and Stability of Fractional-Order Linear Time-Delay Systems

Abstract In this paper, for the first time, a linear Lyapunov–Razumikhin function is introduced to find an $\alpha$-exponential state estimate for positive systems with time delays. A new set of conditions for the existence of exponential state boundedness of positive systems with bounded time-varying delays is established. Two procedures are then proposed to find the factor vector of $\alpha$-exponential state estimate, in which some techniques of optimization are applied to find the smallest possible factor vector. Numerical experiments are presented to demonstrate the effectiveness of the linear Lyapunov–Razumikhin function as well as of the optimization techniques. 

Exponential state estimate of positive systems with time-varying delays: a Lyapunov-Razumikhin approach


 This article addresses the stochastically exponential stability and mean stability of positive time-varying systems with stochastic impulses. The term ‘stochastic impulse’ means the randomness of impulsive densities or intensities. More specifically, the impulsive maps are not unique and the impulsive intensities are independent random variables with different distributions. The occurrence instants of impulses are restricted by several different processes, e.g. a mode-dependent average impulsive interval, a Markov chain, a Poisson process and a renewal process. Using a time-varying copositive Lyapunov function and stochastic analysis theory, several stochastic stability conditions are given. Finally, an example with four cases is presented to show the effectiveness of the proposed results.

Stability analysis for time-varying positive systems with stochastic impulses

\n In this work, we investigate the question of designing a positive observer for a class of infinite dimensional linear positive systems. We present a new observer design based on a classical Luenberger-like observer. The proposed observer is positive. That is, it ensures that the state estimates are nonnegative at any time. The existence of such positive observers is proven by a specific choice of the observer gain and using positive bounded perturbation results. We show in particular that the error of the state estimation converges exponentially to zero. Finally, the main result is applied to an isothermal tubular (bio) reactor model, namely the plug-flow (bio) reactor model. The approach is illustrated by some numerical simulations.

Positive observers for linear positive systems in a Hilbert lattice space

The strong stabilizability of linear second order equation is investigated in this paper. It is supposed that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. Then this control is subject to a cone-bounded nonlinearity. Well-posedness and stability results of the closed-loop system under such (nonlinear) control are stated. The first result is proven by using nonlinear semigroups techniques and the Schauder fixed-point theorem and the second one is based on the result of Marx et al. [12]. Our results are then applied to the particular damped and undamped wave equation. Simulation results are presented to validate the theoretical results. Note that some of the results of this paper apply for a large class systems. 

Cone-bounded feedback laws for linear second order systems

In this work, we present a three nonlinear constrained control schemes, with an adjustable parameter, based on the Lyapunov design approach to regulate the different moments for a class of cell population balance model. The model consists of a partial integro-differential equation, describing cell growth, and an ordinary integro-differential equation, accounting for substrate consumption. A numerical simulation is presented to illustrate the performance of the proposed approach.

Lyapunov Design Controllers for Different Moments of Structured Cell Population Balance Model With Input Constraints

Abstract The problem of designing a positive Luenberger observer for a class of infinite-dimensional semilinear positive systems is addressed. The proposed observer is positive. That is, it ensures that the state estimates are non-negative at any time. The positive observer is designed based on positive bounded perturbation results. In particular it is shown that, under suitable conditions the state estimation error converges exponentially to zero. The results are illustrated by numerical simulations based on an exothermic Plug-Flow Tubular reactor model and show the effectiveness of the proposed positive observer. 

Abdellaziz Binid

Papers

Positive observers for linear positive systems in a Hilbert lattice space

Cone-bounded feedback laws for linear second order systems

Lyapunov Design Controllers for Different Moments of Structured Cell Population Balance Model With Input Constraints

Positive observers for semilinear positive systems