# 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

- 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 [1]. 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 [2], 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]. 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.

# Next year’s work program

New estimation algorithms have been proposed recently in UNAM’s team, which ensure finite-time or fixed-time convergence and based on solution in real time of an adjoint system. These algorithms can be applied for estimation of the state or the parameters, or both. An interesting problem consists in robustness analysis of these algorithms with respect to external disturbances, measurement noises and time delays in measurement channel, which is planned to be investigated in the next year.

In order to apply or implement the control or estimation algorithms in the modern digital controllers it is necessary to discretize them, then the properties of the algorithms can be different from those obtained in the continuous time. An important method to analyse stability and robustness of sliding mode algorithms in continuous time is based on homogeneity, this notion currently exists only for continuous-time systems, and its development for discrete-time case is very challenging and useful for analysis of discretized versions of control and estimation algorithms. This problem is also planned to be considered next year.

During the first year of cooperation, we proposed an analytical tool for construction of homogeneous Lyapunov functions for homogeneous systems using the converse arguments [2]. It is planned to use this new theory to develop numerical algorithms and the corresponding theoretical basements for design of Lyapunov functions in applications.

Despite their many advantages, sliding mode solutions have also several drawbacks mainly originated by the impossibility of a perfect practical realization of sliding motion. Among these shortages it is necessary to mention the chattering phenomenon, which is a high frequency oscillation of the control signal when trajectories stay around the sliding surface. The appearance of chattering may physically destroy the actuator and/or degrade the performance of transients. That is why frequently for practical realization of sliding mode algorithms different approximations of discontinuities (e.g. sign functions) are used. The main drawback of existent approximations is that a chattering reduction is achieved by a price of quality loss (appearance of static error in the presence of matched disturbances

and, consequently, practical stability with an exponential rate of convergence). In a recent collaborative work of the team, a development of the sliding mode control is presented that is based on a sign approximation using the time-delay framework, which guarantees the quality preservation (there is no static error in the presence of a matched disturbance and locally the speed of convergence is faster than any exponential). Further investigations, dealing with discrete-time realization and asymptotic behavior analysis (that is a hard issue for nonlinear discontinuous and time-delayed systems), are necessary.

# Record of activities

Visits of Mexican team to France:

- Jaime Alberto Moreno Pérez, 27/06/2016 − 08/07/2016, “Recursive design of Lyapunov functions for finite-time stable systems”
- Leonid Fridman, 10/07/2016 − 22/07/2016, “Stability analysis of a sliding-mode control algorithm of second order with time delays”
- 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”

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

- Andrey Polyakov, 03/12/2016 − 10/12/2016, “Application of non-differentiable Lyapunov functions for analysis of sliding mode control and estimation algorithms”
- 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”

# Production

- 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. - Efimov D., Ushirobira R., Moreno J.A., Perruquetti W.
*On converse design of homogeneous Lyapunov functions*. Proc. IFAC World Congress, Toulouse, 2017. (submitted) - Ahmed H., Ushirobira R., Efimov D., Fridman L.
*Oscillatory global output synchronization of nonidentical nonlinear systems*. Proc. IFAC World Congress, Toulouse, 2017. (submitted)