Set-based methods in estimation and control


  • Radoslav Paulen
  • Mario Eduardo Villanueva
  • Benoît Chachuat

The workshop will run on 11 July 2020 from 10:00 until 17:00 Berlin time (10am until 5pm CEST/UTC+2h). A recording of the workshop will also be available for streaming from 12 July until 31 August 2020 for registered participants.

Slides of the presentations and papers related to the workshop are available here.


  • Radoslav Paulen, Slovak University of Technology in Bratislava, Slovakia
  • Mario Eduardo Villanueva, ShanghaiTech University, Shanghai, China
  • Benoît Chachuat, Imperial College London, London, United Kingdom


Problems in estimation and control can often be formulated, analyzed and/or solved using methods which exploit the properties of properly chosen or constructed sets. For example, constraints and uncertainties–as well as the effects of the latter on a control system–can naturally be described using sets. Likewise, estimating the response of a system to disturbances, as well as the inherent error of the estimation algorithm can also be described within a set-based formalism. By looking at concepts like invariance, bounded-error estimation, uncertainty propagation, and robust control, the ubiquity of set theoretic methods in the control and estimation community is undeniable.
The goal of this workshop is to provide a tutorial on the use set-based methods in estimation and control. We first present the basic principles of computer representable sets and their arithmetics. This is followed by recent developments in set-theoretic methods for parameter/state estimation and control. The workshop finishes with a discussion of applications and future developments.
The workshop will address topics like:
•  Set arithmetics for outer-approximation of sets described by algebraic constraints
•  Set based methods for uncertainty propagation through dynamic systems
•  Set based methods in set-membership parameter/state estimation
•  Set based methods in robust optimization and predictive control
Due to tutorial nature of the workshop, the prerequisites for attending the workshop cover just basics of convex analysis, optimization and process control.


10:00 Welcome and Introduction

10:15 Introduction to set-based computing – Motivation, models and arithmetics, Benoît Chachuat

This talk gives an overview of methods and tools for bounding functions that can be represented by a directed acyclic graph (DAG), also known as factorable functions. These provide the building blocks of set-theoretic techniques discussed later in the workshop. A central question is how to parameterize an enclosure of the image set of a function. We introduce the formalism of affine set-parameterization, which encompasses interval box, ellipsoid, Taylor models, etc. as special cases and provides a common ground to analysis their complexity and convergence properties. We also briefly discuss existing set-arithmetic software.

11:35 Set-membership parameter/state estimation – classical and regression approaches and Model-based design of experiments for set-based estimation, Radoslav Paulen

The first part of this talk is concerned with set-membership estimation and set-membership regression applied to nonlinear dynamic systems in a context of bounded measurement error. The problem is to determine the subregion in parameter space enclosing all solutions to an estimation/regression problem in the presence of bounded uncertainty on the observed variables. A set-inversion algorithm is applied, whereby the estimation/regression set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a given threshold on the approximation level is met. In the second part of this talk a model-based optimal experiment design (OED) of nonlinear systems is studied. OED represents a methodology for optimizing the geometry of the parametric joint-confidence regions (CRs), which are obtained in an a posteriori analysis of the (least-squares or set-membership) parameter estimates. The optimal design is achieved by using the available (experimental) degrees of freedom such that more informative measurements are obtained. Unlike the commonly used approaches, which base the OED procedure upon the linearized CRs, we explore a path where we explicitly consider the exact CRs in the OED framework.

13:35 Uncertainty propagation through dynamic systems and Set based methods in robust nonlinear model predictive control, Mario Eduardo Villanueva

The first part of this talk gives an overview of methods for propagating sets through dynamic systems described by ordinary differential equationsalso called reachability analysis or set-valued integration. It focuses on continuous-time methods, i.e. methods that rely on a set of auxiliary differential equations that describe the evolution of outer approximations of the reachable sets. We pay particular attention to their theoretical and numerical properties such as convergence and asymptotic stability, as well as briefly discuss their implementation. The second part of the talk is concerned with set-propagation through dynamic systems in the context of robust model predictive controlparticularly tube-based model predictive control. It builds on the formalism presented in the first part, i.e., continuous-time methods for reachability analysis of nonlinear control systems. Besides the tools for set-propagation, we explore the use of set arithmetics for constructing auxiliary optimal control problems whose receding horizon solution provides an MPC feedback law guaranteeing constraint satisfaction.

16:05 Roundtable discussion

(live discussions take place between the talks and in the last session)