Delay-Adaptive Boundary Control of Coupled Hyperbolic PDE-ODE Cascade Systems

This paper presents a delay-adaptive boundary control scheme for a $2\times 2$ coupled linear hyperbolic PDE-ODE cascade system with an unknown and arbitrarily long input delay. To construct a nominal delay-compensated control law, assuming a known input delay, a three-step backstepping design is used. Based on the certainty equivalence principle, the nominal control action is fed with the estimate of the unknown delay, which is generated from a batch least-squares identifier that is updated by an event-triggering mechanism that evaluates the growth of the norm of the system states. As a result of the closed-loop system, the actuator and plant states can be regulated exponentially while avoiding Zeno occurrences. A finite-time exact identification of the unknown delay is also achieved except for the case that all initial states of the plant are zero. As far as we know, this is the first delay-adaptive control result for systems governed by heterodirectional hyperbolic PDEs. The effectiveness of the proposed design is demonstrated in the control application of a deep-sea construction vessel with cable-payload oscillations and subject to input delay

Adaptive Control By Regulation-Triggered Batch Least-Squares Estimation of Non-Observable Parameters

The paper extends a recently proposed indirect, certainty-equivalence, event-triggered adaptive control scheme to the case of non-observable parameters. The extension is achieved by using a novel Batch Least-Squares Identifier (BaLSI), which is activated at the times of the events. The BaLSI guarantees the finite-time asymptotic constancy of the parameter estimates and the fact that the trajectories of the closed-loop system follow the trajectories of the nominal closed-loop system ("nominal" in the sense of the asymptotic parameter estimate, not in the sense of the true unknown parameter). Thus, if the nominal feedback guarantees global asymptotic stability and local exponential stability, then unlike conventional adaptive control, the newly proposed event-triggered adaptive scheme guarantees global asymptotic regulation with a uniform exponential convergence rate. The developed adaptive scheme is tested to a well-known control problem: the state regulation of the wing-rock model. Comparisons with other adaptive schemes are provided for this particular problem.Comment: 29 pages, 12 figure

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

Robust piecewise adaptive control for an uncertain semilinear parabolic distributed parameter systems

In this study, we focus on designing a robust piecewise adaptive controller to globally asymptotically stabilize a semilinear parabolic distributed parameter systems (DPSs) with external disturbance, whose nonlinearities are bounded by unknown functions. Firstly, a robust piecewise adaptive control is designed against the unknown nonlinearity and the external disturbance. Then, by constructing an appropriate Lyapunov–Krasovskii functional candidate (LKFC) and using the Wiritinger’s inequality and a variant of the Agmon’s inequality, it is shown that the proposed robust piecewise adaptive controller not only ensures the globally asymptotic stability of the closed-loop system, but also guarantees a given performance. Finally, two simulation examples are given to verify the validity of the design method

Event-triggered boundary control of constant-parameter reaction-diffusion PDEs: a small-gain approach

This paper deals with an event-triggered boundary control of constant-parameters reaction-diffusion PDE systems. The approach relies on the emulation of backstepping control along with a suitable triggering condition which establishes the time instants at which the control value needs to be sampled/updated. In this paper, it is shown that under the proposed event-triggered boundary control, there exists a minimal dwell-time (independent of the initial condition) between two triggering times and furthermore the well-posedness and global exponential stability are guaranteed. The analysis follows small-gain arguments and builds on recent papers on sampled-data control for this kind of PDE. A simulation example is presented to validate the theoretical results.Comment: 10 pages, to be submitted to Automatic

Safe Adaptive Control of Hyperbolic PDE-ODE Cascades

Adaptive safe control employing conventional continuous infinite-time adaptation requires that the initial conditions be restricted to a subset of the safe set due to parametric uncertainty, where the safe set is shrunk in inverse proportion to the adaptation gain. The recent regulation-triggered adaptive control approach with batch least-squares identification (BaLSI, pronounced ``ballsy'') completes perfect parameter identification in finite time and offers a previously unforeseen advantage in adaptive safe control, which we elucidate in this paper. Since the true challenge of safe control is exhibited for CBF of a high relative degree, we undertake a safe BaLSI design in this paper for a class of systems that possess a particularly extreme relative degree: ODE-PDE-ODE sandwich systems. Such sandwich systems arise in various applications, including delivery UAV with a cable-suspended load. Collision avoidance of the payload with the surrounding environment is required. The considered class of plants is $2\times2$ hyperbolic PDEs sandwiched by a strict-feedback nonlinear ODE and a linear ODE, where the unknown coefficients, whose bounds are known and arbitrary, are associated with the PDE in-domain coupling terms that can cause instability and with the input signal of the distal ODE. This is the first safe adaptive control design for PDEs, where we introduce the concept of PDE CBF whose non-negativity as well as the ODE CBF's non-negativity are ensured with a backstepping-based safety filter. Our safe adaptive controller is explicit and operates in the entire original safe set

Mathematical models and numerical simulation of mechanochemical pattern formation in biological tissues

Mechanical and chemical pattern formation in the development of biological tissue is a fundamental and fascinating process of self-complexation and self-organization. Yet, the understanding of the underlying mechanisms and their mathematical description still lacks in many interesting cases such as embryogenesis. In this thesis, we combine recent experimental and theoretical insights and numerically investigate the capacity of mechano-chemical processes to spontaneously generate patterns in biological tissue. Firstly, we develop and numerically analyze a prototypical system of partial differential equations (PDEs) leading to mechanochemical pattern formation in evolving tissues. Based on recent experimental data, we propose a novel coupling by tensor invariants describing stretch, stress or strain of tissue mechanics on the production of signaling molecules (morphogens). In turn, morphogen leads to piecewise-defined active deformations of individual biological cells. The presented approach is flexible and applied to two prominent examples of evolving tissue: We show how these simple interaction rules (“feedback loops”) lead to spontaneous, robust mechanochemical patterns in the applications to embryogenesis and to symmetry breaking in the sweet water polyp Hydra. Our results reveal that the full 3D model geometry is essential to obtain realistic results such as gastrulation events. Also, we highlight predictive numerical experiments that assess the sensitivity of biological tissue with regard to mechanical stimuli, namely to micropipette aspiration. These numerical experiments allow for a cross-validation with experimental observations. Besides, we apply our modeling approach to growing tips in colonial hydroids and investigate the role of rotational and shearing active deformations by comparison to experimental data. Secondly, we develop an efficient, numerical method to reliably solve these strongly coupled, prototypical systems of PDEs that model mechanochemical long-term problems. We employ state-of-the-art finite element methods, parallel geometric multigrid solvers and present a simple, local mesh refinement strategy to obtain an efficient solution approach. Parallel solvers are essential to deal with the huge problem size in 3D and were modified to keep track of biological cells. Further, we propose a stabilization of the structural equation to deal with the strongly coupled system of equations and the challenges of the different timescales of growth (days) and nonlinear elasticity (seconds). Also, this addresses the instabilities which result form the description of homogeneous Neumann values on the entire boundary that is necessary since the locations of patterns is a priori unknown