Section: Research Program
Axis 3: Automatic Control
Participants: B. Brogliato, C. Prieur, V. Acary
This last axis is dedicated to the automatic control of nonsmooth dynamical systems, or the nonsmooth control of smooth systems. The first item concerns the discretetime sliding mode control for which significant results on the implicit implementation have been obtained in the BIPOP team. The idea is to pursue this research towards state observers and differentiators (Sect 3.4.1). The second direction concerns the optimal control which brings of nonsmoothness in their solution and their formulation. After the preliminary work in BIPOP on the quadratic optimal control of Linear Complementarity systems(LCS), we propose to go further to the minimal time problem, to impacting systems and optimal control with state constraints (Sect. 3.4.2). In Sect 3.4.3, the objective is to study the control of nonsmooth systems that contain unilateral constraint, impact and friction. The targeted systems are cable–driven systems, multibody systems with clearances and granular materials. In Sect 3.4.4, we will continue our work on the higher order Moreau sweeping process. Up to now, the work of BIPOP was restricted to finitedimensional systems. In Sect 3.4.5, we propose to extend our approach to the control of elastic structures subjected to contact unilateral constraints.
It is noteworthy that most of the problems listed below, will make strong use of the numerical tools analyzed in Axis 2, and of the Modeling analysis of Axis 1. For instance all optimal control problems yield BVPs. Control of granular materials will undoubtedly use models and numerical simulation developed in Axis 1 and 2. And so on. It has to be stressed that the type of nonsmooth models we are working with, deserve specific numerical algorithms which cannot be found in commercial software packages. One of the goals is to continue to extend our software package Siconos, and in particular the siconos/control toolbox with these developments.
Discretetime SlidingMode Control (SMC) and State Observers (SMSO)

SMSO, exact differentiators: we have introduced and obtained significant results on the implicit discretization of various classes of slidingmode controllers [32], [34], [69], [80], [49], with successful experimental validations [70], [69], [71], [93]. Our objective is to prove that the implicit discretization can also bring advantages for slidingmode state observers and Levant's exact differentiators, compared with the usual explicit digital implementation that generates chattering. In particular the implicit discretization guarantees Lyapunov stability and finitetime convergence properties which are absent in explicit methods.

HighOrder SMC (HOSMC): this family of controllers has become quite popular in the slidingmode scientific community since its introduction by Levant in the nineties. We want here to continue the study of implicit discretization of HOSMC (twisting, supertwisting algorithms) and especially we would like to investigate the comparisons between classical (first order) SMC and HOSMC, when both are implicitly discretized, in terms of performance, accuracy, chattering suppression. Another topic of interest is stabilization in finitetime of systems with impacts and unilateral constraints, in a discretetime setting.
Optimal Control

Linear Complementarity Systems (LCS) : With the PhD thesis of A. Vieira, we have started to study the quadratic optimal control of LCS. Our objective is to go further with minimumtime problems. Applications of LCS are mainly in electrical circuits with setvalued components such as ideal diodes, transistors, etc. Such problems naturally yield MPEC when numerical solvers are sought. It is therefore intimately linked with Axis 2 objectives.

Impacting systems : the optimal control of mechanical systems with unilateral constraints and impacts, largely remains an open issue. The problem can be tackled from various approaches: vibroimpact systems (no persistent contact modes) that may be transformed into discretetime mappings via the impact Poincaré map; or the classical integral action minimization (Bolza problem) subjected to the complementarity Lagrangian dynamics including impacts.

State constraints, generalized control : this problem differs from the previous two, since it yields Pontryagin's first order necessary conditions that take the form of an LCS with higher relative degree between the complementarity variables. This is related to the numerical techniques for the higher order sweeping process [33].
Control of nonsmooth discrete Lagrangian systems

Cable–driven systems: these systems are typically different from the cablecar systems, and are closer in their mechanical structure to socalled tensegrity structures. The objective is to actuate a system via cables supposed in a first instance to be flexible (slack mode) but nonextensible in their longitudinal direction. This gives rise to complementarity conditions, one big difference with usual complementarity Lagrangian systems being that the control actions operate directly in one of the complementary variables (and not in the smooth dynamics as in cablecar systems). Therefore both the cable models and the control properties are expected to differ a lot from what we may use for cableway systems (for which guaranteeing a positive cable tension is usually not an issue, hence avoiding slack modes, but the deformation of the cables due to the nacelles and cables weights, is an important factor). Tethered systems are a close topic.

Multibody systems with clearances: our approach is to use models of clearances with dynamical impact effects, i.e. within Lagrangian complementarity systems. Such systems are strongly underactuated due to mechanical play at the joints. However their structure, as underactuated systems, is quite different from what has been usually considered in the Robotics and Control literature. In the recent past we have proposed a thorough numerical robustness analysis of various feedback collocated and noncollocated controllers (PD, linearization, passivitybased). We propose here to investigate specific control strategies tailored to such underactuated systems [48].

Granular systems: the context is the feedback control of granular materials. To fix the ideas, one may think of a “juggling” system whose “object” (uncontrolled) part consists of a chain of aligned beads. Once the modeling step has been fixed (choice of a suitable multiple impact law), one has to determine the output to be controlled: all the beads, some of the beads, the chain's center of mass (position, velocity, vibrational magnitude and frequency), etc. Then we aim at investigating which type of controller may be used (output or state feedback, “classical” or sinusoidal input with feedback through the magnitude and frequency) and especially which variables may be measured/observed (positions and/or velocities of all or some of the beads, position and/or velocity of the chain's center of gravity). This topic follows previous results we obtained on the control of juggling systems [50], with increasing complexity of the “object”'s dynamics. The next step would be to extend to 2D and then 3D granular materials. Applications concern vibrators, screening, transport in mining and manufacturing processes.

Stability of structures: our objective here is to study the stability of stacked blocks in 2D or 3D, and the influence on the observed behavior (numerically and/or analytically) of the contact/impact model.
Switching LCS and DAEs, higherorder sweeping process (HOSwP)

We have gained a strong experience in the field of complementarity systems and distribution differential inclusions [33], [51], that may be seen as some kind of switching DAEs. We plan to go further with nonautonomous HOSwP with switching feedback inputs and nonuniform vector relative degrees. Switching linear complementarity systems can also be studied, though the exact relationships between both point of views remain unclear at the present time. This axis of research is closely related to cyberphysical systems in section 3.2.
Control of Elastic (Viscoplastic) systems with contact, impact and friction

Stabilization, trajectory tracking: until now we have focused on the stability and the feedback control of systems of rigid bodies. The proposal here is to study the stabilization of flexible systems (for instance, a “simple” beam) subjected to unilateral contacts with or without setvalued friction (contacts with obstacles, or impacts with external objects line particle/beam impacts). This gives rise to varying (in time and space) boundary conditions. The best choice of a good contact law is a hard topic discussed in the literature.

Cableway systems (STRMTG, POMA): cablecar systems present challenging control problems because they usually are underactuated systems, with large flexibilities and deformations. Simplified models of cables should be used (RitzGalerkin approach), and two main classes of systems may be considered: those with moving cable and only actuator at the station, and those with fixed cable but actuated nacelles. It is expected that they possess quite different control properties and thus deserve separate studies. The nonsmoothness arises mainly from the passage of the nacelles on the pylons, which induces frictional effects and impacts. It may certainly be considered as a nonsmooth setvalued disturbance within the overall control problem.