Delay-Adaptive Control of First-order Hyperbolic PIDEs

We develop a delay-adaptive controller for a class of first-order hyperbolic partial integro-differential equations (PIDEs) with an unknown input delay. By employing a transport PDE to represent delayed actuator states, the system is transformed into a transport partial differential equation (PDE) with unknown propagation speed cascaded with a PIDE. A parameter update law is designed using a Lyapunov argument and the infinite-dimensional backstepping technique to establish global stability results. Furthermore, the well-posedness of the closed-loop system is analyzed. Finally, the effectiveness of the proposed method was validated through numerical simulation

Backstepping Control of Coupled General Hyperbolic-Parabolic PDE-PDE Systems

This paper considers the backstepping state feedback and observer design for hyperbolic and parabolic PDEs, which are bidirectionally interconnected in a general coupling structure. Both PDE subsystems consist of coupled scalar PDEs with the heterodirectional hyperbolic PDE subsystem subject to actuation and sensing. By making use of a multi-step approach to construct the transformation into a stable target system, it is shown that a backstepping state feedback and observer design only requires to solve the well-known kernel equations for the hyperbolic and parabolic subsystems as well as additional decoupling equations. The latter are standard initial boundary value problems for parabolic PDEs. This significantly facilitates the well-posedness analysis and the numerical computation of the backstepping controller. Exponential stability is verified for the state feedback loop, the observer error dynamics, and the closed-loop system using an observer-based compensator. The proposed backstepping design procedures are demonstrated for numerical examples.Comment: 8 pages, 6 figures, journal paper under revie

Backstepping PDE Design: A Convex Optimization Approach

Abstract\u2014Backstepping design for boundary linear PDE is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to non-strict feedback structures. Based on the compactness of the Volterra and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using Sumof- Squares (SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L2-norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs

Neural Operators for Delay-Compensating Control of Hyperbolic PIDEs

The recently introduced DeepONet operator-learning framework for PDE control is extended from the results for basic hyperbolic and parabolic PDEs to an advanced hyperbolic class that involves delays on both the state and the system output or input. The PDE backstepping design produces gain functions that are outputs of a nonlinear operator, mapping functions on a spatial domain into functions on a spatial domain, and where this gain-generating operator's inputs are the PDE's coefficients. The operator is approximated with a DeepONet neural network to a degree of accuracy that is provably arbitrarily tight. Once we produce this approximation-theoretic result in infinite dimension, with it we establish stability in closed loop under feedback that employs approximate gains. In addition to supplying such results under full-state feedback, we also develop DeepONet-approximated observers and output-feedback laws and prove their own stabilizing properties under neural operator approximations. With numerical simulations we illustrate the theoretical results and quantify the numerical effort savings, which are of two orders of magnitude, thanks to replacing the numerical PDE solving with the DeepONet

Event-triggered gain scheduling of reaction-diffusion PDEs

This paper deals with the problem of boundary stabilization of 1D reaction-diffusion PDEs with a time- and space- varying reaction coefficient. The boundary control design relies on the backstepping approach. The gains of the boundary control are scheduled under two suitable event-triggered mechanisms. More precisely, gains are computed/updated on events according to two state-dependent event-triggering conditions: static-based and dynamic-based conditions, under which, the Zeno behavior is avoided and well-posedness as well as exponential stability of the closed-loop system are guaranteed. Numerical simulations are presented to illustrate the results.Comment: 20 pages, 5 figures, submitted to SICO

Deep Learning of Delay-Compensated Backstepping for Reaction-Diffusion PDEs

Deep neural networks that approximate nonlinear function-to-function mappings, i.e., operators, which are called DeepONet, have been demonstrated in recent articles to be capable of encoding entire PDE control methodologies, such as backstepping, so that, for each new functional coefficient of a PDE plant, the backstepping gains are obtained through a simple function evaluation. These initial results have been limited to single PDEs from a given class, approximating the solutions of only single-PDE operators for the gain kernels. In this paper we expand this framework to the approximation of multiple (cascaded) nonlinear operators. Multiple operators arise in the control of PDE systems from distinct PDE classes, such as the system in this paper: a reaction-diffusion plant, which is a parabolic PDE, with input delay, which is a hyperbolic PDE. The DeepONet-approximated nonlinear operator is a cascade/composition of the operators defined by one hyperbolic PDE of the Goursat form and one parabolic PDE on a rectangle, both of which are bilinear in their input functions and not explicitly solvable. For the delay-compensated PDE backstepping controller, which employs the learned control operator, namely, the approximated gain kernel, we guarantee exponential stability in the $L^2$ norm of the plant state and the $H^1$ norm of the input delay state. Simulations illustrate the contributed theory

Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

This is an open access article under the CC BY-NC-ND license.In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate