INVARIANT SET BASED ANALYSIS OF DYNAMICAL SYSTEMS

1. Overview

This postodoctoral position will be dedicated to the research on the properties and geometry of invariant sets with respect to the trajectories of dynamic systems. Invariant sets have application in fault detection [7] and constrained and robust control, [1]. In particular, the current postdoc will focus on establishing geometric and analytic properties of the invariant sets of time-delay and switched dynamic systems.

A set-theoretic link between the minimal Robust Postively Invariant (mRPI) sets of time-delay and switched systems was established in [6]. It was proven that the mRPI set of a time-delay system is

contained in the mRPI set of the corresponding switched system. This was refined to strict containment by [8]. The following objectives will be pursued in the current work:

(1) measure the distance between the minimal robust positive invariant set of a switched system and

the smallest convex robust positively invariant set.

(2) investigate the dynamics of switched and linear parameter varying systems by extending current

results on convex invariant sets to star-convex sets.

(3) characterise the existence and structure of p-invariant sets which are a relaxation of the traditional

rigid invariant sets

(4) introduce delay and dwell time dependence for invariant sets of time-delay and switched systems.

The mathematics used and the results developed for each topic will be applicable to all of the proposed topics. In particular, topics 1, 2 and 3 are closely related as all these topics concern approximations of the mRPI set of a dynamic system.

2. Research plan (roughly 3 months per topic)

2.1. Distance between mRPI set of a switched system and its convex hull. It is known that the minimal convex RPI set for a switched linear system with a non-convex mRPI set is the convex hull

of this mRPI set. However, the distance between these sets is not fully characterized. This distance has implications in fault detection due to the switching nature of the dynamics.

The distance between the iterative (monotonic) approximations of the mRPI and their convex hull will be exploited. The rate at and lower bound to which this distance decreases would allow estimates of the accuracy and error of finite approximations of the (convex approximations of) mRPI set.

2.2. Star-convex mRPI. Previous work considered the convexity of mRPI sets for time-delay and switched systems subject to disturbances from a symmetric convex set. Convexity of the mRPI of the time-delay system follows immediately from the convexity preserving property of the Minkowski sum,

[5]. However, in general, the mRPI set of a switched system is not convex, regardless of the convexity of the disturbance set.

For switched linear systems, the mRPI set is always star-convex and symmetric. Similar properties are expected when relaxing the convexity of the disturbance set to star-convex topologies.

It is of further interest to find pairs of convex sets which bound the star-convex mRPI set. The lower bound would be the kernel of the star-convex set i.e. the largest convex set which is strictly contained in the star-convex mRPI set. Measuring the degree of star-convexity of a mRPI set will assist in finding the distance between the mRPI set of a switched system and convex approximations of this mRPI set.

2.3. Relaxing rigid invariance requirements. In general, invariant sets have complex geometry.

Finding such sets is infeasible, [3], and approximations of these sets is computationally intensive. Therefore they are difficult to use in applications. A weakening of the rigid requirement for strict invariance of a set is p-invariant sets, [4]. Given a discrete-time system a set is p-invariant if, when a point in the p-invariant set leaves the set it remains outside the set for at most piterations.

So points in a p-invariant set are allowed to leave the set so long as they return after at most p steps. Trivially any invariant set is 0-invariant. In addition to establishing existence and uniqueness conditions, there are three questions of interest:

(1) Given a set what is the minimum p such that the set is p-invariant

(2) Given a p what sets are p-invariant and what are their properties?

(3) Relating to the first topic, how far is a p-invariant set from the mRPI set of a dynamic system?

2.4. Delay and dwell-time dependent invariant sets. In [6] the mRPI set of a time-delay system was independent of the delay. This is as there was no constraint on the integral of the system. Many systems are subject to such constraints. The time-delay system was refactored in [2] by introducing a kinematic matrix which related the state of the system at subsequent iterations to be directly constrained.

When subject to this additional constraint the invariant set of the system was delay-dependent. It is proposed to extend the results of [2] in particular to the mRPI of delay-dependent systems. Given the focus on switched systems it is also of interest to consider the invariant sets of switched systems for which the switching frequency is constrained.

References

[1] Franco Blanchini and Stefano Miani. Set-theoretic methods in control. Springer, 2008.

[2] Carlos E.T. D ́orea, Sorin Olaru, and Silviu-Iulian Niculescu. “Delay-Dependent Polyhedral Invari-

ant Sets for Continuous-Time Linear Systems”. In: IFAC-PapersOnLine 55.34 (2022). 8th IFAC

Symposium on System Structure and Control SSSC 2022, pp. 108–113. issn: 2405-8963.

[3] Ilya Kolmanovsky and Elmer G Gilbert. “Theory and computation of disturbance invariant sets for

discrete-time linear systems”. In: Mathematical problems in engineering 4 (1998).

[4] Sorin Olaru et al. “Low Complexity Invariant Sets for Time-Delay Systems: A Set Factorization

Approach”. In: Low-Complexity Controllers for Time-Delay Systems. Ed. by Alexandre Seuret et

al. Cham: Springer International Publishing, 2014, pp. 127–139. isbn: 978-3-319-05576-3.

[5] Rolf Schneider. Convex bodies: the Brunn–Minkowski theory. 151. Cambridge university press, 2014.

[6] Nikola Stankovic, Sorin Olaru, and Silviu-Iulian Niculescu. “Further Remarks on Invariance Prop-

erties of Time-delay and Switching Systems.” In: ICINCO (1). 2011, pp. 357–362.

[7] J. Tan et al. “Invariant Set-Based Analysis of Minimal Detectable Fault for Discrete-Time LPV

Systems With Bounded Uncertainties”. In: IEEE Access 7 (2019), pp. 152564–152575.

[8] Christopher Townsend et al. “Switched mRPI Sets are Strictly Contained in Time-Delay mRPI

Sets”. In: 2020 Australian and New Zealand Control Conference (ANZCC). 2020, pp. 7–11. doi:

10.1109/ANZCC50923.2020.9318370.

3. For applications

The postdoc will take place in the Laboratory of Signals and Systems (L2S) of CentraleSupelec (school of engineering within Paris Saclay University). The school of engineering located in Gif sur Yvette will be the employer and the funding for the postdoctoral project will be granted on the basis of an internal selection process by the L2S scientific committee in May-June 2023.

Successful candidates for this position will have their PhD defended at the time of the application. Ideally, the topic of the PhD should be related to the control theory or related fields with a solid experience in set-theoretic methods.

Interested candidates should set a CV and motivation letter to:

• Prof. Sorin Olaru sorin.olaru@centralesupelec.fr – supervisor

• Ms. Stephanie Douesnard stephanie.douesnard@centralesupelec.fr – HR representative.