Well-posed Infinite-dimensional Systems Research Articles

Sampled-data output regulation of unstable well-posed infinite-dimensional systems with constant reference and disturbance signals

We study the sample-data control problem of output tracking and disturbance rejection for unstable well-posed linear infinite-dimensional systems with constant reference and disturbance signals. We obtain a sufficient condition for the existence of finite-dimensional sampled-data controllers that are solutions of this control problem. To this end, we study the problem of output tracking and disturbance rejection for infinite-dimensional discrete-time systems and propose a design method of finite-dimensional controllers by using a solution of the Nevanlinna-Pick interpolation problem with both interior and boundary conditions. We apply our results to systems with state and output delays.

Optimal boundary control of coupled parabolic PDE–ODE systems using infinite-dimensional representation

The optimal boundary control problem is studied for coupled parabolic PDE–ODE systems. The linear quadratic method is used and exploits an infinite-dimensional state-space representation of the coupled PDE–ODE system. Linearization of the nonlinear system is established around a steady-state profile. Using appropriate state transformations, the linearized system has been formulated as a well-posed infinite-dimensional system with bounded input and output operators. It has been shown that the resulting system is a Riesz spectral system. The linear quadratic control problem has been solved using the corresponding Riccati equation and the solution of the corresponding eigenvalue problem. The results were applied to the case study of a catalytic cracking reactor with catalyst deactivation. Numerical simulations are performed to illustrate the performance of the proposed controller.

Boundary optimal control of coupled parabolic PDE-ODE systems

This paper deals with boundary optimal control problem for coupled parabolic PDEODE systems. The problem is studied using infinite-dimensional state space representation of the coupled PDE-ODE system. Linearization of the non-linear system is established around a steady state profile. Using some state transformations, the linearized system is formulated as a well-posed infinite-dimensional system with bounded input and output operators. It has been shown that the resulting system is a Riesz Spectral system. The LQ-control problem is studied on the basis of the solution of the corresponding eigenvalues problem. The results have been applied to the case study of catalytic cracking reactor with catalyst deactivation. Numerical simulations are performed to illustrate the performances of the developed controller.

Stabilization of Well-Posed Infinite-Dimensional Systems by Dynamic Sampled-Data Feedback

It is shown that a set of six natural conditions is necessary and sufficient for the existence of a finite-dimensional stabilizing sampled-data controller for a well-posed infinite-dimensional system. The underlying stability concept for the sampled-data system is reminiscent of the notion of input-to-state stability from nonlinear control theory.

The lq-controller synthesis problem for infinite-dimensional systems in factor form

The general lq-problem with infinite time horizon for well-posed infinite-dimensional systems has been investigated by George Weiss and Martin Weiss and by Olof Staffans with a complement by Kalle Mikkola and Olof Staffans. Our aim in this paper is to present a solution of a general lq-optimal controller synthesis problem for infinite-dimensional systems in factor form. The systems in factor form are an alternative to additive models, used in the theory of well-posed systems, which rely on leading the analysis exclusively within the basic state space. As a result of applying the simplified analysis in terms of the factor systems and an another derivation technique, we obtain an equivalent, however, astonishingly not the same formulae expressing the optimal controller in the time-domain and the method of spectral factorization. The results are illustrated by two examples of the construction of both the optimal control and optimal controller for some standard lq-problems met in literature: a control problem for a class of boundary controlled hyperbolic equations initiated by Chapelon and Xu, to which we give full solution and an example of the synthesis of the optimal control/controller for the standard lq-problem with infinite-time horizon met in the problem of improving a river water quality by artificial aeration, proposed by Żolopa and the author.

Wave Equation Stabilization by Delays Equal to Even Multiples of the Wave Propagation Time

For a string equation with time delays in the output feedback loop, we study stability and show that the system is a Riesz spectral system and prove that the spectrum-determined growth condition holds for all delays. When the delay is equal to the even multiples of the wave propagation time, we develop the necessary and sufficient conditions for the feedback gain and time delay which guarantee the exponential stability of the closed-loop system. In particular, we show that as the delay of even multiples is increasing to infinity, the stability bound on the feedback gain decays to zero. We also show that whenever the delay is an odd multiple of the wave propagation time, the closed-loop system is unstable. The lack of robustness to a small perturbation in time delay is specifically discussed for the delay equal to two. A numerical simulation for the case of the delay equal to two is presented to illustrate the convergence. Finally, an alternative stability analysis is conducted within the framework of well-posed infinite-dimensional systems.

A Sampled-Data Servomechanism for Stable Well-Posed Systems

In this technical note, an approximate tracking and disturbance rejection problem is solved for the class of exponentially stable well-posed infinite-dimensional systems by invoking a simple sampled-data low-gain controller (suggested by the internal model principle). The reference signals are finite sums of sinusoids and the disturbance signals are asymptotic to finite sums of sinusoids.

Integral control of infinite-dimensional systems in the presence of hysteresis: an input-output approach

This paper is concerned with integral control of systems with hysteresis. Using an input-output approach, it is shown that application of integral control to the series interconnection of either (a) a hysteretic input nonlinearity, an L2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L2-stable, time-invariant linear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certain constant determined by a positivity condition in the frequency domain. The input-output results are applied in a general state-space setting wherein the linear component of the interconnection is a well-posed infinite-dimensional system.

Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control

We derive absolute stability results for well-posed innite-dimensional systems which, in a sense, extend the well-known circle criterion to the case that the underlying linear system is the series interconnection of an exponentially stable well-posed innite-dimensional system and an integrator and the nonlinearity satises a sector condition of the formh(u);(u) aui0 for some constant a> 0. These results are used to prove convergence and stability properties of low-gain integral feedback control applied to exponentially stable, linear, well-posed systems subject to actuator nonlinearities. The class of actuator nonlinearities under consideration contains standard nonlinearities which are important in control engineering such as saturation and deadzone.

Equivalence of input-output stability and exponential stability

For a large class of well-posed infinite-dimensional systems we show that if their state space realisation is exponentially stabilizable and detectable, then exponential stability of the system operator is equivalent to input-output stability.