Mamadou Diagne

Hongpeng Yuan, Ji Wang, Mamadou Diagne

This paper develops a switched event-triggered adaptive boundary control for a class of reaction-diffusion PDE-ODE cascade systems, where the system and input matrices in the ODE as well as the spatially-varying reaction coefficient in the PDE are uncertain. A two-step backstepping transformation is constructed to derive the continuous-time control law. Then a novel dynamic event-triggered control strategy for the PDE-ODE cascade is proposed based on a switched event-triggering mechanism, ensuring global exponential stability of the closed-loop system in place of the exponential convergence commonly achieved with backstepping-based classical dynamic ETC, while inherently excluding Zeno behavior. To address the uncertainties in the PDE-ODE cascade, adaptive update laws are developed, leading to time-varying gain kernels that are adaptively scheduled through the event-triggered control mechanism. Furthermore,to facilitate efficient real-time implementation, deep neural operators (DeepONet) are employed to approximate the backstepping kernels as mappings from the estimated parameters to kernel functions, thereby eliminating the need to repeatedly solve kernel PDEs online. Through a Lyapunov analysis that incorporates the effects of the event-triggering mechanism, parameter adaptation, and kernel approximation errors, we prove the $L^2$ global asymptotic regulation of the resulting closed-loop system. In summary, the key contributions of the paper are threefold: (i) developing an adaptive DeepONet-based framework for reaction-diffusion PDE-ODE cascade systems; (ii) extending the existing adaptive event-triggered control design for reaction-diffusion PDEs to the case with more complex uncertainties; and (iii) generalizing switched dynamic ETC with global exponential stability to PDE-ODE cascades. The effectiveness of the proposed approach is demonstrated through numerical simulations.

Shanshan Wang, Mamadou Diagne, Miroslav Krstić

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.

Shanshan Wang, Mamadou Diagne, Miroslav Krstić

Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has proven capable of approximating PDE backstepping designs in which a single Goursat-form PDE governs a single feedback gain function. In boundary control of coupled PDEs, coupled Goursat-form PDEs govern two or more gain kernels-a PDE structure unaddressed thus far with DeepONet. In this paper, we explore the subject of approximating systems of gain kernel PDEs for hyperbolic PDE plants by considering a simple counter-convecting $2\times 2$ coupled system in whose control a $2\times 2$ kernel PDE system in Goursat form arises. Engineering applications include oil drilling, the Saint-Venant model of shallow water waves, and the Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow. We establish the continuity of the mapping from a total of five plant PDE functional coefficients to the kernel PDE solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and ensure that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. Taking into account anti-collocated boundary actuation and sensing, our $L^2$-Globally-exponentially stabilizing (GES) approximate gain kernel-based output feedback design implies the deep learning of both the controller's and the observer's gains. Moreover, the encoding of the output-feedback law into DeepONet ensures semi-global practical exponential stability (SG-PES). The DeepONet operator speeds up the computation of the controller gains by multiple orders of magnitude. Its theoretically proven stabilizing capability is demonstrated through simulations.