Sub-optimal boundary control of semilinear pdes using a dyadic perturbation observer

In this paper, we present a sub-optimal controller for semilinear partial differential equations, with partially known nonlinearities, in the dyadic perturbation observer (DPO) framework. The dyadic perturbation observer uses a two-stage perturbation observer to isolate the control input from the nonlinearities, and to predict the unknown parameters of the nonlinearities. This allows us to apply well established tools from linear optimal control theory to the controlled stage of the DPO. The small gain theorem is used to derive a condition for the robustness of the closed loop system

Boundary second-order sliding-mode control of an uncertain heat process with unbounded matched perturbation

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Robust Adaptive Boundary Control of Semilinear PDE Systems Using a Dyadic Controller

In this paper, we describe a dyadic adaptive control (DAC) framework for output tracking in a class of semilinear systems of partial differential equations with boundary actuation and unknown distributed nonlinearities. The DAC framework uses the linear terms in the system to split the plant into two virtual sub-systems, one of which contains the nonlinearities, while the other contains the control input. Full-plant-state feedback is used to estimate the unmeasured, individual states of the two subsystems as well as the nonlinearities. The control signal is designed to ensure that the controlled sub-system tracks a suitably modified reference signal. We prove well-posedness of the closed-loop system rigorously, and derive conditions for closed-loop stability and robustness using finite-gain L stability theory

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

Online identification and control of PDEs via Reinforcement Learning methods

We focus on the control of unknown Partial Differential Equations (PDEs). The system dynamics is unknown, but we assume we are able to observe its evolution for a given control input, as typical in a Reinforcement Learning framework. We propose an algorithm based on the idea to control and identify on the fly the unknown system configuration. In this work, the control is based on the State-Dependent Riccati approach, whereas the identification of the model on Bayesian linear regression. At each iteration, based on the observed data, we obtain an estimate of the a-priori unknown parameter configuration of the PDE and then we compute the control of the correspondent model. We show by numerical evidence the convergence of the method for infinite horizon control problems

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

Sliding Mode Dirichlet Boundary Stabilization of Uncertain Parabolic PDE Systems With Spatially Varying Coefficients

Abstract-We consider the robust boundary stabilization problem of an unstable parabolic partial differential equation (PDE) system with uncertainties entering from both the spatially-dependent parameters and from the boundary conditions. The parabolic PDE is transformed through the Volterra integral into a damped heat equation with uncertainties, which contains the matched part (the boundary disturbance) and the mismatched part (the parameter variations). In this new coordinates, an infinite-dimensional sliding manifold that ensures system stability is constructed. For the sliding mode boundary control law to satisfy the reaching condition, an adaptive switching gain is used to cope with the above uncertainties, whose bound is unknown

Robust H

This paper addresses the problem of robust H∞ control design via the proportional-spatial derivative (P-sD) control approach for a class of nonlinear distributed parameter systems modeled by semilinear parabolic partial differential equations (PDEs). By using the Lyapunov direct method and the technique of integration by parts, a simple linear matrix inequality (LMI) based design method of the robust H∞ P-sD controller is developed such that the closed-loop PDE system is exponentially stable with a given decay rate and a prescribed H∞ performance of disturbance attenuation. Moreover, a suboptimal H∞ controller is proposed to minimize the attenuation level for a given decay rate. The proposed method is successfully employed to address the control problem of the FitzHugh-Nagumo (FHN) equation, and the achieved simulation results show its effectiveness