Technical Program

Opening remarks

Constructive methods for robust control of distributed parameter systems

Many important plants (e.g. flexible manipulators or heat transfer processes) are governed by partial differential equations (PDEs) and are often described by models with a significant degree of uncertainty. Some PDEs may not be robust with respect to arbitrary small time-delays in the feedback. Robust finite-dimensional controller design for PDEs is a challenging problem. 

In this talk two constructive methods for finite-dimensional control will be presented: 

1. Spatial decomposition (or sampling in space) method, where the spatial domain is divided into N subdomains with N sensors and actuators located in each subdomain; 

2. Modal decomposition method, where the controller is designed on the basis of a finite-dimensional system that captures the dominant dynamics of the infinite-dimensional one. Sufficient conditions ensuring the stability and performance of the closed-loop system are established in terms of simple linear matrix inequalities that are always feasible for appropriate choice of controllers. 

We will discuss delayed and sampled-data implementations as well as application to network-based deployment of multi-agents. 

LMI-based stability analysis of coupled multi-dimensional parabolic PDEs

Coupled PDEs appear in diverse fields, including chemical processes, viral infection and immune responses, and information diffusion in online social networks. Within the control community, considerable research has been dedicated to studying the controllability and stabilization of coupled PDEs. This talk will focus on an LMI-based stability analysis for coupled multi-dimensional parabolic PDEs. I will show how the gridding technique,

originally developed for networked control systems with aperiodic sampling and time-varying delays, can be also useful for stability analysis of PDEs.

LMI-based Feedback Compensator Design for Parabolic MIMO PDEs

Modern industrial applications propose the problem of control system design for many complex processes with spatiotemporal dynamics, e.g., rapid thermal processing processes in materials, energy conversion, and material transport in chemical reactors, and vibration suppression of flexible mechanical structures, etc. The control system of these complex processes is generally modeled by partial differential equations (PDE) with a finite number of control inputs and measurement outputs. The conflict between infinite-dimensional system dynamics and finite-dimensional control inputs/measurement outputs makes such control system design and analysis challenging. On the other hand, control system model parameters are generally time-space-varying and their accurate values are unknown for practical applications. Due to the time-space coupling feature of the system state, control system design of PDEs should be focused on performance enhancement in both time and space domains. In this talk, we will present some our recent work on LMI-based feedback compensator design for parabolic MIMO PDEs, which includes two variants of Poincaré-Wirtinger inequality, parameter-dependent feedback compensator design, cooperative control and centralized state estimation, and spatiotemporal adaptive state feedback control. 

Necessary and sufficient stability conditions for ODE-PDE interconnected systems

This talk deals with the stability analysis of linear ODE-PDE interconnected systems. In the literature, as an extension of the finite-dimensional case, theoretical tools exist to assess stability of these infinite-dimensional systems. It is based on an operator algebra instead of a matrix algebra. However, for implementaion purposes, an approximation need to be performed and the results are released. The design of numerical tools leading to stability guarantees for which the degree of conservatism is evaluated and controlled is then a major issue.

From the perspective of approximation, a finite-dimensional approximated system is isolated and the infinite-dimensional error part is preserved and modeled. Then, the small-gain theorem or Lyapunov arguments lead to various numerical tests of finite dimension which provide under or over regions for stability of the ODE-PDE interconnections. More interestingly, by quantifying the error part, we can even prove the convergence toward the exact region of stability of these numerical criteria and give an idea of their rate of convergence.

Two case studies are treated in particular: ODE-transport and ODE-reaction-diffusion interconnected systems. For the case of ODE-transport couplings, we show that an approximation of Legendre polynomials and the application of Lyapunov methods allow to obtain necessary and sufficient conditions of stability in terms of matrices conditions. The size of these matrices can be calculated explicitly and does not fly away thanks to a supergeometric convergence rate of the approximation.

Verification methods for the Lyapunov-Krasovskii functional inequalities

We will discuss parameterizations of Lyapunov-Krasovskii Functionals (LKF) to analyze the stability of ODE/Transport PDE systems. We discuss the solution to the delay Lyapunov matrix, which constructs an LKF associated with a prescribed time derivative, and relate it to the approaches commonly used in the numerical computation of LKFs. We compare two approaches for the stability analysis of time-delay systems based on semidefinite programming, namely the method based on integral inequalities and the method based on sum-of-squares programming, which have recently emerged as optimization-based methods to compute LKFs. We discuss their main assumptions and establish connections between both methods. Finally, we formulate a projection-based method allowing to use general sets of functions to parameterize LKFs, thus encompassing the sets of polynomial functions in the literature. The solutions of the proposed stability conditions and the construction of the corresponding LKFs as stability certificates are illustrated with numerical examples.

PIETOOLS and PIEs: An Extension of the LMI Framework to PDEs

We explain the recently proposed partial integral equation representation and show how it enables us to solve many problems in analysis, control, and simulation of delayed and partial differential equations. We start by defining the linear algebra of partial integral (PI) operators. Next, we show that through a similarity transformation, the solution of a broad class of delayed and partial differential equations may be equivalently represented using a partial integral equation (PIE) - an equation parameterized by PI operators. We then show that many analysis and control problems for systems represented as a PIE may be solved through convex optimization of PI operators (LPIs or Linear PI Inequalities). Finally, we discuss the PIETOOLS software suite which automates the process of conversion to PIE, analysis, optimal controller synthesis, implementation, and simulation.

Robust State Estimation for ODE-PDEs: Bringing Theory to Industry with PIETOOLS

This talk presents a computational framework for synthesising state estimator of uncertain Partial Differential Equations (PDEs) when they are coupled with uncertain Ordinary Differential Equations (ODEs). To analyze the behavior of the interconnected ODE-PDE systems under uncertainty, we introduce a class of multipliers of Partial Integral (PI) operator type and consider various classes of uncertainties by enforcing constraints on these multipliers. Since the ODE-PDE models are equivalent to Partial Integral Equations (PIEs), we show that the robust stability and performance can be formulated as Linear PI Inequalities (LPIs) and LPIs can be solved by LMIs using PIETOOLS. The method is applied and experimentally tested in an industrial-scale experimental setup from inkject printing manufacturer. 

Studying fluid flows with auxiliary functions and LMIs

The famous problem of turbulence consists in estimating time-averaged quantities at a small cost. Finding bounds for time averages without solving the governing PDEs is a way forward in this problem.

Auxiliary function is a Lyapunov-like function central to our approach. If a differentiable V(x(t)) is bounded then the infinite time average of dV/dt is zero: <dV(x(t))/dt>=0. Hence, if F(x(t))+dV(x(t))/dt<U  then U is an upper bound for <F(x(t))>. For a dynamical system dx/dt=f(x) the condition becomes F[x]+LV[x]<U for all x, where LV=fdV/dx is the Lie derivative of V. The bound can be optimised over V. This idea applies both to ODE and PDE.

For polynomial ODEs this reduces to LMI-constrained optimisation with the help of SOS relaxation, similar to Lyapunov stability. The talk will describe the uncertain system method, which allows to rigorously reduce the auxiliary functional optimisation to LMI optimisation for PDEs of the type typically occurring in fluid dynamics. Examples will be given, and the challenges outlined.

Boundary feedback stabilization of freeway traffic networks: A showcase of numerically tractable conditions for hyperbolic systems

Boundary feedback control of networks of freeway traffic is considered in this talk by means of Partial Differential Equations based techniques and Lyapunov theory. The model includes boundary control and unknown perturbations. Some stability conditions will be given and relaxations will be introduced to derive numerically tractable conditions written in terms of a finite number of LMIs. Main results will be illustrated using a traffic simulation software at the Fourth Ring road of Beijing, China.