Infinite Dimensional Bilinear System Research Articles

Towards a Taylor-Carleman bilinearization approach for the design of nonlinear state-feedback controllers

The Carleman bilinearization is an approach that performs an exact conversion of a finite-dimensional nonlinear system into an infinite-dimensional bilinear system. A finite-dimensional system is later obtained through a truncation for analysis and control purposes. This paper investigates the linear matrix inequality (LMI)-based design of a switched state-feedback control law for the model obtained via Carleman bilinearization of a first-order nonlinear system. It is shown that in order to obtain feasible design conditions, the performance requirements must be relaxed in a neighborhood of the zero equilibrium point, so that problems arising from the uncontrollability of the linear part of the model can be avoided. The effectiveness of the proposed approach is shown using a numerical example and experimental results using a multi-input tank system.

On the feedback stabilization of distributed bilinear systems with time delay

In the present paper, we address the output stabilization for a class of infinite dimensional bilinear system with time delay on a Hilbert space. Then, we show the output weak stabilization under the weak observability condition. Furthermore, a condition like the exact observability allows exponential and strong stabilization of the output of such a system. Illustrative examples and simulations are given to validate established theoretical results.

Regional optimal control problem of a class of infinite-dimensional bi-linear systems

ABSTRACTThe purpose of this paper is to study regional optimal control of an infinite-dimensional bi-linear system, excited by unbounded or bounded controls. Then we prove the existence of an optimal control that minimises a functional cost and we give the characterisation of such a control. Also we give the condition that ensures the uniqueness of the optimal control. The obtained results lead to an algorithm that we illustrate by simulations for a bi-dimensional diffusion system excited by a zone actuator.

Exponential and Weak Stabilization of Constrained Bilinear Systems

This paper considers feedback stabilization for the infinite-dimensional bilinear system $\frac{dz(t)}{dt}=Az(t)+v(t)Bz(t)$, where $A$ is the infinitesimal generator of a linear $C_0-$semigroup of contractions on a Hilbert space $H$ and $B:H\rightarrow H$ is a linear bounded operator. Sufficient conditions for exponential and weak stabilization are given. A minimization problem is also considered. Applications to parabolic and hyperbolic equations are presented.

A LUENBERGER OBSERVER FOR AN INFINITE DIMENSIONAL BILINEAR SYSTEM: A UV DISINFECTION EXAMPLE

Inspired by the UV disinfection processes in food and water treatment industry, we design a Luenberger observer which works at the boundary of the infinite dimensional bilinear system. Existence of a solution, stability and some observer design issues are shown. Simulations of a disinfection process and its observer illustrate the results.

Lyapunov equation for infinite-dimensional discrete bilinear systems

Mean-square stability for discrete systems requires that uniform convergence is preserved between input and state correlation sequences. Such a convergence preserving property holds for an infinite-dimensional bilinear system if and only if the associate Lyapunov equation has a unique strictly positive solution.