# Inria Associate team

## 2016 – 2019

# Overview

Estimation (of the state variables, parameters or inputs) and control in finite time is the principal direction of research of Non-A team applying algebraic tools and the homogeneity approach. Sliding mode algorithms are ones of the most popular solutions providing finite-time convergence. The main advantages of sliding mode control and estimation solutions include convergence in finite time, compensation of matched disturbances and robustness with respect to measurement noises. The basic disadvantage is a chattering phenomenon that occurs due to discontinuity of the right-hand side in such systems. During the last decades a big effort has been applied to overcome the chattering, and new generations of sliding-mode algorithms have been developed avoiding discontinuity as much as possible and having even fixed-time convergence. The last fifth generation of sliding- mode was founded just recently and waits for its theoretical development and application for design of control and estimators.

The goal of the project is development of control and estimation algorithms converging in fixed or in finite time by applying the last generation sliding mode techniques and the homogeneity theory.

# People involved in the project

### Non-A team, Inria Lille – Nord Europe:

The Non-A team (Inria) is developing an estimation theory, built around differential algebra and operational calculation on the one hand, and high gain algorithms (such as sliding mode, *etc*.) on the other hand. Both approaches allow derivatives of noisy signals to be estimated in finite or fixed time, which opens a lot of prospects in control and signal processing. It has resulted in relatively simple, rapid algorithms. Unlike traditional methods, the majority of which pertain to asymptotic statistics, the Non-A estimators are ”non-asymptotic”. These estimators found their applications in electrical engineering, robotics, aerospace and bioscience.

**Denis Efimov** (project leader), CR Inria, is a recognised expert in automatic control theory. His main research interests include nonlinear oscillation analysis, observation and control, nonlinear system stability. Andrey Polyakov, CR Inria, is working with different fields of mathematical control theory. Rosane Ushirobira, CR Inria, is working in algebraic methods for signal and parameter estimation, based on differential algebra and Weyl algebra. Wilfrid Perruquetti, Full Professor at École Centrale de Lille specializes in estimation and control theory, non-linear systems with applications to robotics. Francisco Lopez, PhD student 2015-2018, financially supported by Inria.

### Sliding mode control laboratory:

The Universidad Nacional Autonoma de Mexico (UNAM) team comes from the “Sliding Mode Control” laboratory. A detailed survey on advantages of the variable structure systems can be briefly summarized as follows:

- Insensitivity/uniformity (more than robustness) with respect to so-called matched uncertainties/disturbances acting in the same channels as the control
- Finite-time, exact and uniform convergence.
- The possibility to use the equivalent output injection in order to reveal additional information

UNAM team :

**Leonid Fridman**, Full Professor- Jaime Alberto Moreno Pérez, Full Professor
- Rafael Iriarte, Professor
- Andrea Aparicio, PhD student 2014−2017

# Scientific progress

**First year**- Linear system stabilization by n-sign feedback. The problem of a linear system stabilization using a control dependent on the signs of the state variables is revisited [4,9]. It is shown that depending on the properties of the linear part of the system, the stabilization by n-sign feedback is possible (the previous results show that for the chain of integrators of order 3 and higher it is impossible). The conditions of stability are expressed in the form of linear matrix inequalities, and the domain of stability is evaluated. Next, the control law is augmented by a linear feedback, then an input-to-state stability property of the closed-loop system is established. The problem of finite-time convergence in the obtained system is discussed.
- On converse design of homogeneous Lyapunov functions. It is a well-known fact that (weighted) homogeneous and asymptotically stable dynamical systems possess homogeneous Lyapunov functions [Zubov (1964); Rosier (1992)]. However, the existing methods [Zubov (1964); Rosier (1992)] do not provide an explicit expression for the design of such Lyapunov functions, since they assume that a non-homogeneous Lyapunov function is obtained by the converse Lyapunov theorem, and next its nonlinear transformation is performed. In [1], several analytic expressions of homogeneous Lyapunov functions are proposed. For this purpose, modifications of converse Lyapunov approaches are carried out to directly obtain the desired homogeneous Lyapunov functions from converse arguments.
- Oscillatory global output synchronization of nonidentical nonlinear systems. A global output synchronization problem for nonidentical nonlinear systems having relative degree two or higher is studied in [3,12]. The synchronization is based on a partial projection of the dynamics of individual subsystems into Brockett oscillators. The approach uses a higher order sliding mode observer to estimate the states and perturbations of the synchronized nonlinear systems.

**Second year**__Numeric algorithms for building homogeneous Lyapunov functions__. Using previously obtained extensions of the converse Lyapunov function constructions for homogeneous systems, numeric algorithm has been developed in [7], which quantitatively derives a homogeneous Lyapunov function for a system starting from numeric discretization of its solutions initiated on a sphere, and next by a suitable approximation. The proposed algorithm can be used for verification stability of homogeneous systems.__Homogeneity framework for discrete-time systems__. Homogeneity demonstrated its utility for continuous-time systems granting the dynamics by a superior rate of convergence or robustness, but for discrete-time systems the existing notions were not so useful. In [5,11] we have proposed a new concept which allows to establish (local) stability and convergence rate from degree for a wide class of discrete-time systems.__Fixed-time converging estimators under delayed measurements__. Estimation in linear and nonlinear systems is an important topic of research nowadays [2, 15], especially under delayed measurements due to omnipresence of lags and samplings in networked systems. Usually, appearance of a delay in measurements degrades performance (rate of convergence, stability or robustness) of an observer, and quality of estimation in such a case can be guaranteed for a known bounded delay only. In our collaborative work [6,10] an observer is designed, which use a new structure proposed by our colleagues from UNAM ensuring a uniform convergence for any delay value.

**Third year**__Local homogeneity of discrete-time systems__. Developing the new concept of homogeneity introduced in [3] for discrete-time systems, a local homogeneous approximation for these class of systems is introduced and investigated in [16], and also the robustness abilities with respect to exogenous perturbations for discrete-time systems are established. Study of these properties is useful in a design of estimation and control algorithms in discrete time, which is a direction of future research.__Numeric design of homogeneous Lyapunov functions for homogeneous differential inclusions__. During the previous two years of cooperation, we proposed analytical and numeric tools for construction of homogeneous Lyapunov functions for homogeneous systems using the converse arguments [1, 7]. A restriction we imposed for all these analysis and design is the positiveness of homogeneity degree for a system under consideration (in this case the systems are also locally Lipschitz, which simplifies a lot the application of many methods in [1, 7]). This year these results are developed in [13] to the systems with negative degree (non-Lipschitz case) and to discontinuous systems, since these dynamics demonstrate finite-time convergence and have a special importance in applications (sliding-mode control and estimation systems often have negative homogeneity degree).__Development of the implicit Lyapunov function methodology__. The Lyapunov function method is one of the main tools for analysis of stability of nonlinear systems. Since there is no common routine for such a function construction for an application, there are many methods oriented on special classes of systems or using special forms of Lyapunov functions (see*e.g*. [7]). One of such an approach is implicit Lyapunov function (ILF) method, which is well-developed in Non-A team, and it is applicable for the chains of integrators, where all other terms are treated as perturbations. Placing a lot of items in disturbances introduces conservatism in the method and in the obtained estimates on the time of convergence. Another solution is adaptation of ILF approach to linear -parameter-varying (LPV) systems, which has been performed in [8], or design of adding integrator extension for the ILF method as in [14]. Both results, in [8] and [14], reduces the conservatism of the conventional ILF approach by improving the accuracy of settling-time and robustness margin evaluation.

# Record of activities

Visits of Mexican team to France:

- First year:
- Leonid Fridman, 10/07/2016 − 22/07/2016, “Stability analysis of a sliding-mode control algorithm of second order with time delays”
- Jaime Alberto Moreno Pérez, 27/06/2016 − 08/07/2016, “Recursive design of Lyapunov functions for finite-time stable systems”
- Tonámetl Sánchez Ramírez, 24/10/2016 − 18/11/2016, “Homogeneity for discrete- time systems”
- Juan Gustavo Rueda Escobedo, 24/10/2016 − 18/11/2016, “Finite-time and fixed- time identification of parameters”

- Second year:
- Jaime Alberto Moreno Pérez, 27/06/2017 − 08/07/2017, “Numeric design of Lyapunov functions for homogeneous systems”
- Leonid Fridman, 10/07/2017 − 22/07/2017, “Stability analysis of a sliding-mode control algorithm of high order with time delays”
- Willy Alejandro Apaza Perez, 24/10/2017 − 18/11/2017, “Control and estimation in finite time of mechanical systems”
- Alan Tapia, 24/10/2017 − 18/11/2017, “Development of implicit Lyapunov function method for polytypical systems”

- Third year:
- Jaime Alberto Moreno Pérez, 22/11/2018 − 06/12/2018, “Numeric design of Lyapunov functions for discontinuous homogeneous systems”
- Willy Alejandro Apaza Perez, 16/07/2018 − 19/07/2018, “Control and estimation in finite time of mechanical systems”
- Jose Angel Mercado Uribe, 03/06/2018 − 27/07/2018, “An extension of integral homogeneous controllers for finite-time regulation”
- Jesús Mendoza Avila, 03/06/2018 − 27/07/2018, “Application of numeric tools for Lyapunov function design of non-Lipschitz systems”

Visits of French team to Mexico (also partially supported by ANR projects):

- First year:
- Rosane Ushirobira, 03/12/2016 − 10/12/2016, “Numerical design of homogeneous Lyapunov functions for homogeneous systems with positive degree”
- Denis Efimov, 03/12/2016 − 10/12/2016, “Stability analysis of a sliding-mode control algorithm of second order with time delays”

# Events

*Special issue on differentiators*, editors: Reichhartinger M., Efimov D., Fridman L. International Journal of Control, 91(9), 2018, pp. 1980−1982- L. Fridman obtained an Inria International Chair 2017-2021

# Production

- Efimov D., Ushirobira R., Moreno J.A., Perruquetti W.
*On the numerical construction of homogeneous Lyapunov functions*. Proc. 56th IEEE Conference on Decision and Control (CDC), Melbourne, Australia, 2017. - Rios H.,Efimov D., Moreno J.A., Perruquetti W., Rueda-Escobedo J.G.
*Time-Varying Parameter Identification Algorithms: Finite and Fixed-Time Convergence.*IEEE TAC, issue 99, 2017. - Ahmed H., Ushirobira R., Efimov D., Fridman L.
*Oscillatory global output synchronization of nonidentical nonlinear systems*. Proc. IFAC World Congress, Toulouse, 2017. - Aparicio Martínez A., Efimov D., Fridman L.
*Stabilization of a triple integrator by a 3- sign feedback*. Proc. 55th IEEE Conference on Decision and Control (CDC), Las Vegas, 2016. - Sanchez Ramirez T., Efimov D., Polyakov A., Moreno J.A., Perruquetti W.
*A homogeneity property of a class of discrete-time systems*. Proc. 56th IEEE Conference on Decision and Control (CDC), Melbourne, 2017. - Rueda-Escobedo J-G., Ushirobira R., Efimov D., Moreno J.,
*A Gramian-based observer with uniform convergence rate for delayed measurements*. Proc. European Control Conference, Limassol, Cyprus, 2018. - Efimov D., Ushirobira R., Moreno J.A., Perruquetti W.
*Homogeneous Lyapunov functions: from converse design to numerical implementation*. SIAM Journal on Optimization and Control, ?(?), 2018, pp. ?−?. - Reichhartinger M., Efimov D., Fridman L.
*Special issue on differentiators*. International Journal of Control, 91(9), 2018, pp. 1980−1982 - Tapia A., Efimov D., Bernal M., Fridman L., Polyakov A.
*An Implicit Lyapunov Function Approach for Non-Asymptotic Robust Stabilization of MIMO Nonlinear Systems via Convex Models.*Submitted to Automatica - Aparicio A., Fridman L., Efimov D.
*Stabilization of switched systems by linear feedbacks and its ISS properties*. Submitted to IET Journal on Control Theory & Applications - Rueda-Escobedo J-G., Ushirobira R., Efimov D., Moreno J.
*Gramian-based uniform convergent observer for stable LTV systems with delayed measurements.*Submitted to International Journal of Control - Sanchez Ramirez T., Efimov D., Polyakov A., Moreno J.A., Perruquetti W.
*A homogeneity property of a class of discrete-time systems*. Submitted to Systems & Control Letters - Ahmed H., Ushirobira R., Efimov D., Fridman L.
*Output Global Oscillatory Synchronization of Heterogeneous Systems.*Submitted to International Journal of Control - Uribe J.A.M., Efimov D., Fridman L., Moreno J.A.
*Extension of numerical design of homogeneous Lyapunov function to non-smooth homogeneous systems.*In preparation. - Avila J.M., Efimov D., Fridman L., Polyakov A., Moreno J.A.
*Design of integral homogeneous extensions for continuous sliding-mode controls.*In preparation. - Perez W.A.A., Efimov D., Fridman L., Moreno J.A.
*Design of a finite-time convergent observer for a planar mechanical system.*In preparation. - Sanchez Ramirez T., Efimov D., Polyakov A., Moreno J.A.
*Local homogeneity and robustness against perturbations for discrete-time systems*. In preparation.