Lyapunov Functionals for Output Regulation of Exponentially Stable Semigroups via Integral Action and Application to Hyperbolic Systems (original) (raw)

Adding Integral Action for Open-Loop Exponentially Stable Semigroups and Application to Boundary Control of PDE Systems

IEEE Transactions on Automatic Control, 2019

The paper deals with output feedback stabilization of exponentially stable systems by an integral controller. We propose appropriate Lyapunov functionals to prove exponential stability of the closed-loop system. An example of parabolic PDE (partial differential equation) systems and an example of hyperbolic systems are worked out to show how exponentially stabilizing integral controllers are designed. The proof is based on a novel Lyapunov functional construction which employs the forwarding techniques.

Output stabilization of semilinear parabolic systems with bounded feedback

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2018

In this paper, we will study the output feedback stabilization of in…nite-semilinear parabolic systems evolving on a spatial domain and in a subregion ! of (interior to or on its boundary @). We consider the condition of admissibility and the decomposition methods technique of the state space via the spectral properties of the system. Then we apply this approach to a regional exponential stabilization problem using bounded feedback. Applications are presented.

Design of a Proportional Integral Control Using Operator Theory for Infinite Dimensional Hyperbolic Systems

IEEE Transactions on Control Systems Technology, 2014

This paper considers the control design of a nonlinear distributed parameter system in infinite dimension, described by the hyperbolic Partial Differential Equations (PDEs) of de Saint-Venant. The nonlinear system dynamic is formulated by a Multi-Models approach over a wide operating range, where each local model is defined around a set of operating regimes. A new Proportional Integral (PI) feedback is designed and performed through Bilinear Operator Inequality (BOI) and Linear Operator Inequality (LOI) techniques for infinite dimensional systems. The new results have been simulated and also compared to previous results in finite and infinite dimension, in order to illustrate the new theoretical contribution.

Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov's second method

ESAIM: Control, Optimisation and Calculus of Variations, 2008

In this paper we study asymptotic behaviour of distributed parameter systems governed by partial differential equations (abbreviated to PDE). We first review some recently developed results on the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionals we prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDE systems. Then we apply the result to establish exponential stability of various chemical engineering processes and, in particular, exponential stability of heat exchangers. Through concrete examples we show how Lyapunov's second method may be extended to stability analysis of nonlinear hyperbolic PDE. Meanwhile we explain how the method is adapted to the framework of Banach spaces L p , 1 < p ≤ ∞.

Internal stabilization of semilinear parabolic systems

Journal of Mathematical Analysis and Applications, 2003

This work is concerned with the internal stabilization of the steady-state solutions to semilinear parabolic systems via finite-dimensional feedback controllers. The internal controller is active on a nonempty open subset and in one equation only.

Exponential Stability and Transfer Functions of Processes Governed by Symmetric Hyperbolic Systems

ESAIM: Control, Optimisation and Calculus of Variations, 2002

In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial differential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of well-posed infinite-dimensional linear systems, we prove that the system is regular and derive various properties of its transfer functions, which are potentially useful for controller design. Our results remain valid for a wide class of processes governed by symmetric hyperbolic systems.

On the Stabilization of Infinite Dimensional Semilinear Systems

Ima Journal of Mathematical Control and Information, 2019

This chapter considers the question of the output stabilization for a class of infinite dimensional semilinear system evolving on a spatial domain Ω by controls depending on the output operator. First we study the case of bilinear systems, so we give sufficient conditions for exponential, strong and weak stabilization of the output of such systems. Then, we extend the obtained results for bilinear systems to the semilinear ones. Under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given.

On continuity of solutions for parabolic control systems and input-to-state stability

Journal of Differential Equations, 2018

We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.