A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

Abstract

In this paper, a strict Lyapunov function is developed in order to show the exponential stability and input-to-state stability (ISS) properties of a diffusion equation for nonhomogeneous media. Such media can involve rapidly time-varying distributed diffusivity coefficients. Based on this Lyapunov function, a control law is derived to preserve the ISS properties of the system and improve its performance. A robustness analysis with respect to disturbances and estimation errors in the distributed parameters is performed on the system, precisely showing the impact of the controller on the rate of convergence and ISS gains. This is important in light of a possible implementation of the control since, in most cases, diffusion coefficient estimates involve a high degree of uncertainty. An application to the safety factor profile control for the Tore Supra tokamak illustrates and motivates the theoretical results. A constrained control law (incorporating nonlinear shape constraints in the actuation profiles) is designed to behave as closely as possible to the unconstrained version, albeit with the equivalent of a variable gain. Finally, the proposed control laws are tested under simulation, first in the nominal case and then using a model of Tore Supra dynamics, where they show adequate performance and robustness with respect to disturbances.

Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

“…In order to achieve an optimal current evolution in the plasma discharges of the ramp-up phase, various dynamic optimization techniques are applied to obtain numerical solutions in an open loop fashion, such as [7], [8], [19], [28]. For online implementations of optimized actuation commands, feedback strategies (e.g., [5,6,20,29]) are necessary to attenuate external perturbations and system uncertainties. However, many challenging research problems still remain open.…”

Lyapunov-based distributed control of the safety-factor profile in a tokamak plasma

“…In particular actuation errors, estimation/measurement errors and state disturbances have been considered in [18].…”

“…An example is [14] and related works, where an infinite-dimensional model, described by partial differential equations (PDEs), is used to construct an extremum-seeking controller for the current profile, considering fixed shape profiles for the current deposited by the RF antennas and for the diffusivity coefficients. Other PDE-control approaches, related to Tore Supra, can also be mentioned: [15], where sum-of-square polynomials are used to construct a Lyapunov function considering constant diffusivity coefficients; [16], where a slidingmode controller was designed for the infinite-dimensional system, considering timeinvariant diffusivity coefficients; [17], where a polytopic linear parameter-varying approach is used to build a common Lyapunov function guaranteeing stability of the discretized system with time-varying diffusivity profiles and finally [18], where an infinite dimensional Lyapunov function is constructed to guarantee the stability and robustness of the controlled system, considering distributed time-varying diffusivity coefficients as well as non-linear shape constraints in the actuation profiles due to the use of two engineering parameters (the power and the parallel refraction index of the lower-hybrid antennas) to control the safety factor profile.…”

“…• building upon [18] and [25], the consideration of time varying diffusivity coefficient profiles in the control design, guaranteeing the stability of the system and its robustness with respect to several common sources of errors and unmodeled dynamics;…”

“…• the extensive robustness analysis presented in [18] provides a clear idea of the impact of the tuning parameters and the profile estimation errors on the behaviour and stability of the system; • the resulting control law is easy to implement and requires little online computing ressources (ideal for real-time implementation); • the resulting control law can be easily extended to include other non-inductive actuators (e.g. Electron Cyclotron Current Drive antennas).…”