7 research outputs found
Bilateral boundary control of an input delayed 2-D reaction-diffusion equation
In this paper, a delay compensation design method based on PDE backstepping
is developed for a two-dimensional reaction-diffusion partial differential
equation (PDE) with bilateral input delays. The PDE is defined in a rectangular
domain, and the bilateral control is imposed on a pair of opposite sides of the
rectangle. To represent the delayed bilateral inputs, we introduce two 2-D
transport PDEs that form a cascade system with the original PDE. A novel set of
backstepping transformations is proposed for delay compensator design,
including one Volterra integral transformation and two affine Volterra integral
transformations. Unlike the kernel equation for 1-D PDE systems with delayed
boundary input, the resulting kernel equations for the 2-D system have singular
initial conditions governed by the Dirac Delta function. Consequently, the
kernel solutions are written as a double trigonometric series with
singularities. To address the challenge of stability analysis posed by the
singularities, we prove a set of inequalities by using the Cauchy-Schwarz
inequality, the 2-D Fourier series, and the Parseval's theorem. A numerical
simulation illustrates the effectiveness of the proposed delay-compensation
method.Comment: 11 pages, 3 figures(including 8 sub-figures
Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation
To stabilize PDE models, control laws require space-dependent functional
gains mapped by nonlinear operators from the PDE functional coefficients. When
a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent,
a gain-scheduling (GS) nonlinear design is the simplest approach to the design
of nonlinear feedback. The GS version of PDE backstepping employs gains
obtained by solving a PDE at each value of the state. Performing such PDE
computations in real time may be prohibitive. The recently introduced neural
operators (NO) can be trained to produce the gain functions, rapidly in real
time, for each state value, without requiring a PDE solution. In this paper we
introduce NOs for GS-PDE backstepping. GS controllers act on the premise that
the state change is slow and, as a result, guarantee only local stability, even
for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear
recirculation using both a "full-kernel" approach and the "gain-only" approach
to gain operator approximation. Numerical simulations illustrate stabilization
and demonstrate speedup by three orders of magnitude over traditional PDE
gain-scheduling. Code (Github) for the numerical implementation is published to
enable exploration.Comment: 16 pages, 5 figure
Nonlinear bilateral output-feedback control for a class of viscous Hamilton¿Jacobi PDEs
We tackle the boundary control and estimation problems for a class of viscous Hamilton–Jacobi PDEs, considering bilateral actuation and sensing, i.e., at the two boundaries of a 1-D spatial domain. We first solve the nonlinear trajectory generation problem for this type of PDEs, providing the necessary feedforward actions at both boundaries. We then design an observer-based output-feedback control law, which consists of two main elements—a nonlinear observer that is constructed utilizing measurements from both boundaries and state-feedback laws, which are employed at the two boundary ends. All of our designs are explicit since they are constructed interlacing a feedback linearizing transformation with backstepping. Due to the fact that the linearizing transformation is locally invertible, only a regional stability result is established, combining this transformation with backstepping, suitably formulated to handle the case of bilateral actuation and sensing. We illustrate the developed methodologies via application to traffic flow control and we present consistent simulation results
Nonlinear bilateral output-feedback control for a class of viscous Hamilton–Jacobi PDEs
Nikolaos Bekiaris-Liberis was supported by the funding from the European
Commission’s Horizon 2020 research and innovation programme under the Marie
Sklodowska- Curie Grant Agreement No. 747898, project PADECOT.
Rafael Vazquez acknowledges financial support of the Spanish Ministerio de Economia y Competitividad under grant MTM2015-65608-P.Summarization: We tackle the boundary control and estimation problems for a class of viscous Hamilton–Jacobi PDEs, considering bilateral actuation and sensing, i.e., at the two boundaries of a 1-D spatial domain. We first solve the nonlinear trajectory generation problem for this type of PDEs, providing the necessary feedforward actions at both boundaries. We then design an observer-based output-feedback control law, which consists of two main elements—a nonlinear observer that is constructed utilizing measurements from both boundaries and state-feedback laws, which are employed at the two boundary ends. All of our designs are explicit since they are constructed interlacing a feedback linearizing transformation with backstepping. Due to the fact that the linearizing transformation is locally invertible, only a regional stability result is established, combining this transformation with backstepping, suitably formulated
to handle the case of bilateral actuation and sensing. We illustrate the developed methodologies via application to traffic flow control and we present consistent simulation results.Presented on: Automatic