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Concept# Sliding mode control

Summary

In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamics of a nonlinear system by applying a discontinuous control signal (or more rigorously, a set-valued control signal) that forces the system to "slide" along a cross-section of the system's normal behavior. The state-feedback control law is not a continuous function of time. Instead, it can switch from one continuous structure to another based on the current position in the state space. Hence, sliding mode control is a variable structure control method. The multiple control structures are designed so that trajectories always move toward an adjacent region with a different control structure, and so the ultimate trajectory will not exist entirely within one control structure. Instead, it will slide along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a sliding mode and the geometrical locus consisting of the boundaries is c

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Related lectures (7)

Romain Christophe Rémy Fleury, Xinxin Guo, Hervé Lissek

Absorbing airborne noise at frequencies below 300 Hz is a particularly vexing problem due to the absence of natural sound absorbing materials at these frequencies. The prevailing solution for low-frequency sound absorption is the use of passive narrow-band resonators, whose absorption level and bandwidth can be further enhanced using nonlinear effects. However, these effects are typically triggered at high intensity levels, without much control over the form of the nonlinear absorption mechanism. In this study, we propose, implement, and experimentally demonstrate a nonlinear active control framework on an existing experimental (linear) electroacoustic resonator prototype, allowing for unprecedented control over the form of non-linearity, and arbitrarily low absorption intensity thresholds. More specifically, the proposed architecture combines a linear feedforward control on the front pressure through a first microphone located at the front face of the loudspeaker, and a nonlinear feedback on the membrane displacement estimated through the measurement of the pressure inside the back cavity with a second microphone located in the enclosure. It is experimentally shown that even at a weak excitation level, it is possible to observe and control the nonlinear behaviour of the system. Taking the cubic nonlinearity as an example, we demonstrate numerically and experimentally that in the low frequency range ($[\SI{50}{Hz}, \SI{600}{Hz}]$), the nonlinear control law allows improving the sound absorption performance, i.e. enlarging the bandwidth of optimal sound absorption while increasing the maximal absorption coefficient value. The reported experimental methodology can be extended to implement various types of hybrid linear and/or nonlinear controls, thus opening new avenues for managing wave nonlinearity and achieving non-trivial wave phenomena.

20202004

We address the free boundary problem that consists in finding the shape of a three dimensional glacier over a given period and under given climatic conditions. Glacier surface moves by sliding, internal shear and external exchange of mass. Ice is modelled as a non Newtonian fluid. Given the shape of the glacier, the velocity of ice is obtained by solving a stationary non-linear Stokes problem with a sliding law along the bedrock-ice interface. The shape of the glacier is updated by computing a Volume Of Fluid (VOF) function, which satisfies a transport equation. Climatic effects (accumulation and ablation of ice) are taken into account in the source term of this equation. A decoupling algorithm with a two-grid method allows the velocity of ice and the VOF to be computed using different numerical techniques, such that a Finite Element Method (FEM) and a characteristics method. On a theoretical level, we prove the well-posedness of the non-linear Stokes problem. A priori estimates for the convergence of the FEM are established by using a quasi-norm technique. Eventually, convergence of the linearisation schemes, such that a fixed point method and a Newton method, is proved. Several applications demonstrate the potential of the numerical method to simulate the motion of a glacier during a long period. The first one consists in the simulation of Rhone et Aletsch glacier from 1880 to 2100 by using climatic data provided by glaciologists. The glacier reconstructions over the last 120 years are validated against measurements. Afterwards, several different climatic scenarios are investigated in order to predict the shape the glaciers until 2100. A dramatic retreat during the 21st century is anticipated for both glaciers. The second application is an inverse problem. It aims to find a climate parametrization allowing a glacier to fit some of its moraines. Two other aspects of glaciology are also addressed in this thesis. The first one consists in modeling and in simulating ice collapse during the calving process. The previous ice flow model is supplemented by a Damage variable which describes the presence of micro crack in ice. An additional numerical scheme allows the Damage field to be solved and a two dimensional simulation of calving to be performed. The second problem aims to prove the existence of stationary ice sheet when considering shallow ice model and a simplified geometry. Numerical investigation confirms the theoretical result and shows physical properties of the solution.