Section: New Results

Non-linear, Sampled and Time-delay systems

Participants : Jean-Pierre Richard, Lotfi Belkoura, Gang Zheng, Denis Efimov, Wilfrid Perruquetti.

Nonlinearities, sampling, quantization and time-delays cause serious obstructions for control and observer design in many fields of techniques and engineering (e.g. networked and internet systems, distributed systems etc.). The proposed by the team algebraic approach suits well for estimation and regulation in such a type of systems. The recent results are listed below:

• A new type of stability is introduced and its equivalent Lyapunov characterization is presented in [16] . The problem of global stability for the compact set composed of all invariant solutions of a nonlinear system (several equilibriums, for instance) is studied. It is shown that several well-known multi-stable systems satisfy this new stability property.

• A new state-dependent sampling control is proposed in [23] , [65] , which enlarges the sampling intervals of state feedback control. The case of linear time invariant systems with time delays is considered that guarantees the exponential stability of the system origin for a chosen decay rate. The approach is based on LMIs obtained from the sufficient Lyapunov-Razumikhin stability conditions.

• Nonlinear feedback design for fixed-time stabilization of linear control systems is studied in [31] . Nonlinear control algorithms of two types are presented for uncertain linear plants. Controllers of the first type are stabilizing polynomial feedbacks that allow to adjust a guaranteed convergence time of system trajectories into selected neighborhood of the origin independently on initial conditions. Controllers of the second type are modifications of the second order sliding mode control algorithms. They provide global finite-time stability of the closed-loop system and allow to adjust a guaranteed settling time independently on initial conditions. Control algorithms are presented for both single-input and multi-input systems.

• The problem of natural wave control is addressed in [17] , which involves steering a lattice of oscillators towards a desired natural (i.e. zero-input) assignment of energy and phase across the lattice. This problem is formulated and solved for lattices of linear oscillators via a passivity-based approach.

• The verification problems for transition systems enriched with a metric structure is analysed in [27] . The main novelty compared to an algorithm presented recently by Lerda et al. [2008] consists in introducing a tuning parameter, which improves the performance drastically. A procedure that allows one to prove unbounded safety from the result of the bounded safety algorithm via a refinement step is also established. The algorithm to handle bounded liveness verification is adapted.

• The problem of finite-time output stabilization of the double integrator is addressed in [52] applying the homogeneity approach. A homogeneous controller and a homogeneous observer are designed (for different degree of homogeneity) ensuring the finite-time stabilization. Their combination under mild conditions is shown to stay homogeneous and finite-time stable as well.

• The notes [76] , [77] are dedicated to the stability analysis of bilinear sampled-data systems, controlled via a linear state feedback static controller. A zero order hold device is used. The purpose is to find a constructive way to calculate the maximum allowable sampling period (MASP) that guarantees the local stability of the system. The proposed stability conditions are formulated as linear matrix inequalities (LMI).

• The works [75] , [74] concern the adaptation of sampling times for linear time invariant systems controlled by state feedback. Complementary to various works that guarantee stabilization independently of changes in the sampling rate, there the conditions to design stabilizing sequences of sampling instants is provided. In order to reduce the number of these sampling instants, a dynamic scheduling algorithm optimizes, over a given sampling horizon, a sampling sequence depending on the system state value. The proofs are inspired on switched system techniques combining Lyapunov functions and LMI optimization.

• The mechanism of entrainment to natural oscillations in a class of (bio)mechanical systems described by linear models is investigated in [61] . A nonlinear control strategy (based on the speed gradient control algorithm) is analyzed providing the system oscillation in resonance mode with a natural frequency. It ensures an energy-optimal entrainment performance robustly against perturbations in system parameters in a finite time.

• The paper [29] considers a networked control loop, where the plant is a "slave" part, and the remote controller and observer constitute the "master". Since the performance of Networked Control Systems (NCS) depends on the Quality of Service (QoS) available from the network, a controller is designed that takes into account qualitative information on the QoS in real time.

• In the paper [50] , the theory of non-commutative rings allows determining whether or not there exists an equation called algebraically essential in order to estimate the delay on a nonlinear system. From this equation, it is shown that this equation is generally not enough to guarantee the delay estimation, thus the notion of persistent signal with respect to delay estimation is introduced. Furthermore, based on the definitions of algebraically essential equation and of persistent signal, a delay estimation algorithm is proposed. Some simulation results have been presented in order to highlight the robustness (with respect to measurement noise) of the proposed algorithm.

• The problem of algebraic identifiability for linear and nonlinear dynamical systems is considered in [88] .