Exponential Stability and Transfer Functions of Processes Governed by Symmetric Hyperbolic Systems (original) (raw)

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 ≤ ∞.

Dynamic transmission conditions for linear hyperbolic systems on networks

Journal of Evolution Equations, 2021

We study evolution equations on networks that can be modeled by means of hyperbolic systems. We extend our previous findings in Kramar et al. (Linear hyperbolic systems on networks. arXiv:2003.08281, 2020) by discussing well-posedness under rather general transmission conditions that might be either of stationary or dynamic type—or a combination of both. Our results rely upon semigroup theory and elementary linear algebra. We also discuss qualitative properties of solutions.

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

2018 IEEE Conference on Decision and Control (CDC), 2018

The paper deals with output feedback stabilization of exponentially stable systems by integral controllers. We propose appropriate Lyapunov functionals to prove exponential stability of the closed-loop systems. 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 contribution of the paper is extending the forwarding technique to infinitedimensional systems by elaborating Lyapunov functionals.

Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain

Esaim-control Optimisation and Calculus of Variations, 2009

We study hyperbolic partial differential equations on a one dimensional spatial domain with control and observation at the boundary. Using the idea of feedback we show these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C 0 -semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity. keywords: Infinite-dimensional systems, hyperbolic boundary control systems, C 0 -semigroup, well-posedness. * file:artikel/Dirac/well-posedness 4.tex

An Output Controllability Problem for Semilinear Distributed Hyperbolic Systems

International Journal of Applied Mathematics and Computer Science, 2007

An Output Controllability Problem for Semilinear Distributed Hyperbolic SystemsThe paper aims at extending the notion of regional controllability developed for linear systems to the semilinear hyperbolic case. We begin with an asymptotically linear system and the approach is based on an extension of the Hilbert uniqueness method and Schauder's fixed point theorem. The analytical case is then tackled using generalized inverse techniques and converted to a fixed point problem leading to an algorithm which is successfully implemented numerically and illustrated with examples.

Stability analysis of a degenerate hyperbolic system modelling a heat exchanger

Mathematics and Computers in Simulation, 2007

Mathematical modelling of a heat exchanger in a carbon dioxide heat pump, an evaporator, is considered. A reduced model, called the the zero Mach-number limit, is derived from the Euler equations of compressible fluid flow through elimination of time scales associated with sound waves. The well-posedness of the resulting partial differentialalgebraic equation (PDAE) is investigated by analysis of a frozen coefficient linearisation as well as by numerical experiments. The linear stability analysis is done through transformation to a canonical form with one hyperbolic component, one parabolic component and one algebraic component. Using this canonical form it is seen how to prescribe boundary and initial data and an energy estimate is derived. Numerical experiments on the nonlinear PDAE using a finite difference spatial discretisation support the linear stability analysis.

Steady-state analysis for a class of hyperbolic systems with boundary inputs

Archives of Control Sciences, 2013

Results of a steady-state analysis performed for a class of distributed parameter systems described by hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of the system with two state variables and two boundary inputs, the analytical expressions for the steady-state distribution of the state variables are derived, both in the exponential and in the hyperbolic form. The influence of the location of the boundary inputs on the steady-state response is demonstrated. The considerations are illustrated with a practical example of a shell and tube heat exchanger operating in parallel- and countercurrent-flow modes.

On stability of nonlinear hyperbolic systems with reaction and switching

2013 American Control Conference, 2013

This paper investigates the exponential stability in L 2 norm of scalar nonlinear hyperbolic systems of balance laws with the reaction that may be accumulative or dissipative. Two Lyapunov-based stability criteria that depend on the system parameters and boundary data are proposed with fully considering the reactions' characteristics. The new results can help to construct a common Lyaunov function to stabilize the switched nonlinear hyperbolic systems under arbitrary switching. Several traffic system examples are taken to illustrate the theoretical results.

Feedback control of hyperbolic PDE systems

AIChE Journal, 1996

This article deals with distributed parameter systems described by first-order hyperbolic partial differential equations (PDEs), for which the manipulated input, the controlled output, and the measured output are distributed in space. For these systems, a general output-feedback control methodology is developed employing a combination of theory of PDEs and concepts ffom geometric control. A concept of characteristic index is introduced and used for the synthesis of distributed state-feedback laws that guarantee output tracking in the closed-loop system. Analytical formulas of distributed outputfeedback controllers are derived through combination of appropriate distributed state observers with the developed state-feedback controllers. Theoretical analogies between our approach and available results on stabilization of linear hyperbolic PDEs are also identified. The developed control methodology is implemented on a nonisothermal plug-flow reactor and its performance is evaluated through simulations. P. D. Christofides is presently at the approach limits the controller performance, and may Iead to unacceptable control quality.