In this work, we present a scalable Linear Matrix Inequality (LMI) based framework to verify the stability of a set of linear Partial Differential Equations (PDEs) in one spatial dimension coupled with a set of Ordinary Differential Equations (ODEs) via input-output based interconnection. Our approach extends the newly developed state space representation and stability analysis of coupled PDEs that allows parametrizing the Lyapunov function on $L_2$ with multipliers and integral operators using polynomial kernels of semi-separable class. In particular, under arbitrary well-posed boundary conditions, we define the linear operator inequalities on $\mathbb{R}^n \times L_2$ and cast the stability condition as a feasibility problem constrained by LMIs. In this framework, no discretization or approximation is required to verify the stability conditions of PDE-ODE coupled systems. The developed algorithm has been implemented in MATLAB where the stability of example PDE-ODE coupled systems are verified.
We present designs for exponential stabilization of an ODE-heat PDE-ODE coupled system where the control actuation only acts in one ODE. The combination of PDE backstepping and ODE backstepping is employed in a state-feedback control law and in an observer that estimates PDE and two ODE states only using one PDE boundary measurement. Based on the state-feedback control law and the observer, the output-feedback control law is then proposed. The exponential stability of the closed-loop system and the boundedness and exponential convergence of the control law are proved via Lyapunov analysis. Fin…
This paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients. Thereby, the coupling in the PDE is in-domain and at the uncontrolled boundary, whereby the ODE is coupled with the latter boundary. For the state feedback design a two-step backstepping approach is developed, that yields the conventional kernel equations and additional decoupling equations of simple form. The latter can be traced back to simple Volterra integral equations of the second kind, which are directly solvable …
We consider an interlinked production model consisting of conservation laws (PDE) coupled to ordinary differential equations (ODE). Our focus is the analysis of control laws for the coupled system and corresponding stabilization questions of equilibrium dynamics in the presence of disturbances. These investigations are carried out using an appropriate Lyapunov function on the theoretical and numerical level. The discrete $L^2-$stabilization technique allows to derive a mixed feedback law that is able to ensure exponential stability also in bottleneck situations. All results are accompanied by …
In this paper, we present a convex formulation of $H_{\infty}$-optimal control problem for coupled linear ODE-PDE systems with one spatial dimension. First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation (PIE) system and show that stability and $H_{\infty}$ performance of the PIE system implies that of the ODE-PDE system. We then construct a dual PIE system and show that asymptotic stability and $H_{\infty}$ performance of the dual system is equivalent to that of the primal PIE system. Next, we pose a convex dual formulation of the stability and $H_{\infty}$-performan…
We present an adaptive event-triggered boundary control scheme for a parabolic PDE-ODE system, where the reaction coefficient of the parabolic PDE, and the system parameter of a scalar ODE, are unknown. In the proposed controller, the parameter estimates, which are built by batch least-square identification, are recomputed and the plant states are resampled simultaneously. As a result, both the parameter estimates and the control input employ piecewise-constant values. In the closed-loop system, the following results are proved: 1) the absence of a Zeno phenomenon; 2) finite-time exact identif…
We present a control design for semilinear and quasilinear 2x2 hyperbolic partial differential equations with the control input at one boundary and a nonlinear ordinary differential equation coupled to the other. The controller can be designed to asymptotically stabilize the system at an equilibrium or relative to a reference signal. Two related but different controllers for semilinear and general quasilinear systems are presented and the additional challenges in quasilinear systems are discussed. Moreover, we present an observer that estimates the distributed PDE state and the unmeasured ODE …
Motivated by engineering applications of subsea installation by deepwater construction vessels in oil drilling, and of aid delivery by unmanned aerial vehicles in disaster relief, we develop output-feedback boundary control of heterodirectional coupled hyperbolic PDEs sandwiched between two ODEs, where the measurement is the output state of one ODE and suffers a time delay. After rewriting the time-delay dynamics as a transport PDE of which the left boundary connects with the sandwiched system, a state observer is built to estimate the states of the overall system of ODE-heterodirectional coup…
In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of a simplified semilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with a non-smooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the uncontrolled equation, we prove linear convergence …
Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, the DDE representation is limited. In network models with delay, the delayed channels are low-dimensional and accounting for this heterogeneity is not possible in the DDE framework. In addition, DDEs cannot be used to model difference equations. Furthermore, estimation and control of systems in DDE format has proven challenging, despite decades of study. In this paper, we examine alternative representations for systems with delay and provide formulae for conversion between representations. F…
This paper addresses the derivation of generic and tractable sufficient conditions ensuring the stability of a coupled system composed of a reaction-diffusion partial differential equation (PDE) and a finite-dimensional linear time invariant ordinary differential equation (ODE). The coupling of the PDE with the ODE is located either at the boundaries or in the domain of the reaction-diffusion equation and takes the form of the input and output of the ODE. We investigate boundary Dirichlet/Neumann/Robin couplings, as well as in-domain Dirichlet/Neumann couplings. The adopted approach relies on …
Observer Design for a Class of ODE -- Continuum-PDE Cascade Systems Inspired by a Control-Theoretic Model of Large-Scale Arterial Networks of Blood Flow
H$_1$-ISS Analysis and Boundary Control Design for Coupled Linear ODEs and Hyperbolic PDEs
Universidade Federal de Santa Catarina（圣卡塔琳娜联邦大学）；Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，格勒诺布尔国立理工学院，GIPSA实验室）
In this paper we propose two numerical algorithms to solve a coupled PDE-ODE system which models a slow vehicle (bottleneck) moving on a road together with other cars. The resulting system is fully coupled because the dynamics of the slow vehicle depends on the density of cars and, at the same time, it causes a capacity drop in the road, thus limiting the car flux. The first algorithm, based on the Wave Front Tracking method, is suitable for theoretical investigations and convergence results. The second one, based on the Godunov scheme, is used for numerical simulations. The case of multiple b…
In this paper we present a continuation method which transforms spatially distributed ODE systems into continuous PDE. We show that this continuation can be performed both for linear and nonlinear systems, including multidimensional, space- and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing evolution of continuous state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples including transport equations, Kuramoto equations and heat diffusion equations. As a main example, we p…
In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with an ODE that has to hold true in almost all points in space. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the discretization of th…
While for coupled hyperbolic PDEs of first order there now exist numerous PDE backstepping designs, systems with zero speed, i.e., without convection but involving infinite-dimensional ODEs, which arise in many applications, from environmental engineering to lasers to manufacturing, have received virtually no attention. In this paper, we introduce single-input boundary feedback designs for a linear 1-D hyperbolic system with two counterconvecting PDEs and $n$ equations (infinite-dimensional ODEs) with zero characteristic speed. The inclusion of zero-speed states, which we refer to as {\em atac…
A deep-sea construction vessel is used to install underwater parts of an off-shore oil drilling platform at the designated locations on the seafloor. By using extended Hamilton's principle, a nonlinear PDE system governing the lateral-longitudinal coupled vibration dynamics of the deep-sea construction vessel consisting of a time-varying-length cable with an attached item is derived, and it is linearized at the steady state generating a linear PDE model, which is extended to a more general system including two coupled wave PDEs connected with two interacting ODEs at the uncontrolled boundaries…
This paper studies the problem of designing sampled-data observers and observer-based, sampled-data, output feedback stabilizers for systems with both discrete and distributed, state and output time-delays. The obtained results can be applied to time delay systems of strict-feedback structure, transport Partial Differential Equations (PDEs) with nonlocal terms, and feedback interconnections of Ordinary Differential Equations with a transport PDE. The proposed design approach consists in exploiting an existing observer, which features robust exponential convergence of the error when continuous-…
School of Mathematics, Jilin University, 2699 Qianjin Street, Changchun, 130012, Jilin, China. Chair for Dynamics, Control, Machine Learning；Numerics (Alexander von Humboldt Professorship), Department of Mathematics, Friedrich–Alexander-Universit\"at Erlangen–N\"urnberg, 91058 Erlangen, Germany.；Institut de Mathematiques de Bourgogne, Université Bourgogne Europe, CNRS, 21000 Dijon, France.；Division of Mathematical Sciences, School of Physical；Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371, Singapore.；School of Mathematical Sciences, Nankai University, Tianjin 300071, China.；Numerics (Alexander von Humboldt Professorship), Department of Mathematics, Friedrich–Alexander-Universit\"at Erlangen–N\"urnberg, 91058 Erlangen, Germany. Departamento de Matem\'aticas, Universidad Aut\'onoma de Madrid, 28049 Madrid, Spain. Chair of Computational Mathematics, Fundaci\'on Deusto. Av. de las Universidades, 24, 48007 Bilbao, Basque Country, Spain.
In this work, stabilization of an axonal growth in a neuron associated with the dynamics of tubulin concentration is proposed by designing a boundary control. The dynamics are given by a parabolic Partial Differential Equation (PDE) of the tubulin concentration, with a spatial domain of the axon's length governed by an Ordinary Differential Equation (ODE) coupled with the tubulin concentration in the growth cone. We propose a novel backstepping method for the coupled PDE-ODE dynamics with a moving boundary, and design a control law for the tubulin concentration flux in the soma. Through employ…
