The Valse team studies the estimation and control problems arising in the analysis and the design of distributed, uncertain, and interconnected dynamical systems:

The Valse team works in the domains of control science: dynamical systems,
stability analysis, estimation, and automatic control. Our developments
are focused on the theoretical and applied aspects related to the control
and estimation of large-scale multi-sensor and multi-actuator systems
based on the use of the theories of finite-time/fixed-time/hyperexponential
convergence and homogeneous systems. The Lyapunov function method
and other methods of analysis of dynamical systems form a basis for
the studies in the Valse team.

The key idea of the research program for the team is that a fast (non-asymptotic) convergence of the regulation and estimation errors increases the reliability of intelligent distributed actuators and sensors in complex scenarios, such as interconnected cyber-physical systems (CPSs).

The expertise of Valse's members in theoretical developments of control and estimation theory (finite-time control and estimation algorithms in centralized context 61, 47, 58, 57, 54, homogeneity framework for differential equations 62, 49, 48, 50, 52, 63, 59, time-delay systems 51, 53, 66, distributed systems 60 and algebraic-based methods for estimation 64, 65) is an essential ingredient to achieve our objective.

The generic chart of different goals and tasks included in the scientific work program of Valse and interrelations between them are presented in Fig. 1. We have selected three main objectives to pursue with the related tasks to fulfill:

All these objectives are interconnected: from a particular problem
in an IoT application, it is planned to design control or estimation
algorithms, leading to the development of theoretical tools; and vice
versa, a new theoretical advance can provide a possibility for the development
of novel tools which can be used in applications.

To explain our motivation: why use finite time? Applying any method for control/estimation
has a price in terms of its advantages and disadvantages. There is
no universal framework that is the best always and everywhere. Finite-time
may appear as a luxurious property for a physical system, requiring
nonlinear tools. Of course, if an asymptotic convergence
and a linear model are enough for solving a given problem, then there
is no reason to develop something else. However, most of the present
problems in CPS and IoT are nonlinear (i.e., they have various
local behaviors that cannot be collected in only one linear model).
Design and analysis of various local linearized models and solutions are luxurious, too. The theory of homogeneity can go beyond linearity, offering many new features while not appearing as severe as other nonlinear tools and having almost all hints of the linear framework. Suppose that, thanks to the homogeneity theory, finite-time/fixed-time can be obtained with little difficulty while adding the bonuses of stronger robustness and faster convergence compared to the linear case? We are convinced that the price of going beyond linear control and estimation can be strongly dropped by maturing the theory of homogeneity and finite/fixed-time convergence. We are also convinced that it will be compensated in terms of robustness and speed, which can be demanded in the new areas of application such as IoT for example.

An objective of the team is the application of the developed control and estimation algorithms for different scenarios in IoT or CPSs. Participation in various potential applications allows the Valse team to better understand the features of CPSs and their required performances, and to properly formulate the control and estimation problems that must be solved. Here is a list of ongoing and potential applications addressed by the team:

It is worth highlighting a widespread distribution of various scientific
domains in the list of applications for the team given above. Such
interdisciplinarity for Valse is unsurprising since control theory is a science of systems whose interest today is, by
nature, to interface with other disciplines and their fields of application.
This is also well aligned with the domain of CPSs, which by its origin
requires multidisciplinary competencies.

In 12, the problem of finite-time and fixed-time stability analysis is considered for a class of nonlinear systems

The paper 20 provides a sufficient condition to ensure output finite-time and fixed-time stability. Comparing with analogous researches, the proposed result is less restrictive and obtained for a wider class of systems. The presented output stability condition is used for adaptive control design, where the state vector of a plant is extended by adjustable control parameters.

The paper 21 is devoted to the problem of finite-time and fixed-time observation of linear multiple input multiple output control systems. The proposed dynamic observers do not require system transformation to a canonical form and guarantee convergence of the observation error to zero in a finite or in a fixed time. It is shown that the observers are robust (in input-to-state sense) against input disturbances and measurement noises. The results are supported with simulation examples.

Razumikhin-like theorems on hyperexponential and fixed-time stability of time-delay systems are proposed for both explicitly and implicitly defined Lyapunov functions in 4. While the former method is useful for stability analysis, the latter approach is more suitable for control synthesis. Examples of systems that can be stabilized hyperexponentially and in fixed time are given. The control parameters tuning algorithm is presented in the form of linear matrix inequalities.

The concept of practical fixed-time input-to-state stability for neutral time-delay systems with exogenous perturbations is introduced in 16. Lyapunov-Krasovskii theorems are formulated in explicit and implicit ways. Further, the problem of static nonlinear output-feedback stabilization of a linear system with parametric uncertainties, external bounded state and output disturbances by using artificial delays is considered. The constructive control design consists in solving linear matrix inequalities with only four tuning parameters to be chosen. It is shown both, theoretically and numerically, that the system governed by the proposed controller converges faster to the given invariant set than in the case of using its linear counterpart.

A discretization of a homogeneous controller for a double integrator is developed in 34, 35. It preserves the finite-time stability property even in the case of the sampled-time implementation of the control law. Theoretical results are supported by numerical simulations.

In the paper 5, the objective is to design a nonlinear controller under time and state constraint for quadrotor. The nonlinear quadrotor model is built by the Euler-Lagrange approach while ignoring the Coriolis terms, hub moment and force. The designed regulator is an implicit PID controller, where the feedback gains are obtained from LMIs (Linear matrix inequalities). LMI system characterizing the system stability and convergence properties is built based on convex embedding approach and implicit Lyapunov function method. To demonstrate the application prospects of implicit PID controller, robustness analysis is provided under external disturbance. The key novelty of this paper is that the implicit PID controller is proven feasible for applying to the quadrotor under time and state constraints, which is also the main outcome.

The article 17 proposes a procedure for upgrading a linear proportional controller to a sliding mode one preventing a degradation of a control quality. Two nonlinear algorithms are studied in this context: Unit and Homogeneous Sliding Mode Controllers (SMCs). The parameters of the nonlinear controllers are defined using the gains of the (already well-tuned) linear controller. The main idea of this upgrading procedure is to split the state-space into two regions, the region of the linear control and the region of the nonlinear control. This is done via a suitable design parameter. The theoretical developments are validated experimentally on a rotary inverted pendulum. Comparisons between the already well-tuned linear controller and the proposed upgraded nonlinear controllers are presented. The experiments in a rotary inverted pendulum demonstrate that the upgraded controller significantly improves the control precision without degradation of the control signal.

In

2, it is demonstrated that the dynamic regressor extension and mixing (DREM) method provides a fixed-time converging parameter estimation for persistently excited regressor under bounded measurement noises. Application of DREM method using delays as extension filters to harmonic signals estimation in power grids is considered in

7,

6. Integrating renewable energy sources into an unbalanced distribution network requires fast and accurate extraction of fundamental frequency of positive- and negative-sequence components from the unbalanced three-phase grid voltage signals, and the proposed fixed-time estimators respond these requirements.

In 11, the problem of adaptive state observation for linear time-varying systems with delayed measurements and unknown parameters is studied. It is shown that the generalized parameter estimation-based observer provides a very simple solution, which being combined with the DREM estimation procedure, guarantees the parameter convergence in fixed-time imposing extremely weak excitation assumptions.

Generalized Persidskii systems represent the dynamics described by the superposition of a linear part with multiple sector nonlinearities and exogenous perturbations. They can be used to model many physical and engineering phenomena.

A state observer is designed in 14 for a class of generalized Persidskii systems with nonlinear measurements, state disturbances, and output noise. We assume that all nonlinearities are diagonal and belong to a sector. The robust stability and convergence conditions for the estimation error are obtained by applying the theory of input-to-output stability. These conditions are established in the form of linear matrix inequalities.

In paper 13, input-to-state stability and stabilization conditions in time-delay generalized Persidskii systems are studied. These conditions are formulated in terms of linear matrix inequalities, which may depend on delay values, and be local or global in state space. Numerical examples of opinion dynamics and the Lotka-Volterra model illustrate the efficiency of the proposed results.

Mechanical systems under strongly nonlinear potential and dissipative forces, with nonlinear nonstationary perturbations having zero mean values, is studied in 8. Proposing a special construction of Lyapunov function, the conditions are found, under which the perturbations do not influence the asymptotic stability of the trivial equilibrium position of the system. These conditions include the requirements on asymptotic stability of the disturbance-free system and the relations of the nonlinearity orders between potential and dissipative forces. The developed theoretical approach is extended to the problem of monoaxial stabilization of a rigid body.

For two canonical models of mechanical systems with disturbances presented by Rayleigh-and Liénard-type equations, several designs of Lyapunov functions are investigated in 9. Under given restrictions, the existence of these functions implies global or local input-to-state stability property for the systems. Extending these results, the conditions of finite-time and fixed-time (integral) input-to-state stability are derived in 1. The efficiency of the proposed designs of Lyapunov functions is demonstrated in several applications.

The problem of delay-independent stability is investigated in 10 for a class of mechanical systems under dissipative, non-conservative and potential forces, which are described by homogeneous terms. The obtained conditions are extended to the case of switched force model under arbitrary and restricted commutation laws. The proof is based on analysis of a complete Lyapunov-Krasovskii functional (common and multiple ones are considered for switched scenario).

In the paper 19, an event-triggered observation scheme is considered for a perturbed nonlinear dynamical system connected to a remote location via a communication channel, which can only transmit a limited amount of data per unit of time. The dynamical system, which is supposed to be globally Lipschitz, is subject to bounded state perturbations. Moreover, at the system’s location, the output is measured with some bounded errors. The objective is to calculate estimates of the state at the remote location in real-time with maximum given error, whilst using the communication channel as little as possible. An event-triggered communication strategy is proposed in order to reduce the average number of communications. An important feature of this strategy is to provide an estimation of the relation between the observation error and the communication rate. The observation scheme’s efficiency is demonstrated through simulations of unicycle-type robots.

The paper 3 proposes a new simplified pointing model as a feedback-based dynamical system, including both human and computer sides of the process. It takes into account the commutation between the correction and ballistic phases in pointing tasks. We use the mouse position increment signal from noisy experimental data to achieve our main objectives: to estimate the model parameters online and predict the task endpoint. Some estimation tools and validation results, applying linear regression techniques on the experimental data are presented. We also compare with a similar prediction algorithm to show the potential of our algorithm's implementation. This work is done in collaboration with the LOKI team of Inria.

In the papers 37, 28, Visible Light Communication (VLC) paradigm has been expedited by the fast evolving and deployment of light emitting diodes (LED), and the possibility to simultaneously exploit the communication and illumination, enabling seamless connectivity based on the lighting infrastructure. Synchronization is a big deal in all the wireless communication systems and the key features of VLC paradigm make the synchronization techniques existing for radio frequencies not suitable for VLC. A novel technique, OSCI-LIGHT based on Andronov-Hopf oscillators, is proposed in this work in order to realize an effective synchronization mechanism in a VLC system. In particular, phase alignment and robustness towards noise have been tested through both numerical simulation and experimental results and a comparison with widely employed synchronization techniques, based on Phase Locked Loop (PLL), has been provided. Experimental results show that our technique outperforms PLL techniques in terms of noise robustness, showing a proper steady state phase delay and a lower Synchronization Error Rate, even in presence of highly noisy environmental conditions.

Leonid Fridman, UNAM, Mexico, 12/2022

The members of the team permanently perform reviews for leading journals and conferences in the domain of the theory of control.

Seminars:

The members of the team participated in many PhD and HDR juries this year in France and abroad.

An article on collaboration with 52 Hertz startup: When mathematical modelling helps divers communicate