Yuri Shtessel Christopher Edwards Leonid Fridman - Sliding Mode Control and Observation

Springer Science+Business Media New York 2014

Yuri Shtessel 1, Christopher Edwards 2, Leonid Fridman 3 and Arie Levant 4

In the formulation of any practical control problem, there will always be a discrepancy between the actual plant and its mathematical model used for the controller design. These discrepancies (or mismatches) arise from unknown external disturbances, plant parameters, and parasitic/unmodeled dynamics. Designing control laws that provide the desired performance to the closed-loop system in the presence of these disturbances/uncertainties is a very challenging task for a control engineer. This has led to intense interest in the development of the so-called robust control methods which are supposed to solve this problem. One particular approach to robust controller design is the so-called sliding mode control technique.

In , the main concepts of sliding mode control will be introduced in an intuitive fashion, requiring only a basic knowledge of control systems. The sliding mode control design techniques are demonstrated on tutorial examples and via graphical exposition. Advanced sliding mode concepts, including sliding mode observers/differentiators and second-order sliding mode control, are studied at a tutorial level. The main advantages of sliding mode control, including robustness, finite-time convergence, and reduced-order compensated dynamics, are demonstrated on numerous examples and simulation plots.

For illustration purposes, the single-dimensional motion of a unit mass (Fig. ) is considered. A state-variable description is easily obtained by introducing variables for the position and the velocity so that where u is the control force, and the disturbance term f ( x 1, x 2, t ), which may comprise dry and viscous friction as well as any other unknown resistance forces, is assumed to be bounded, i.e., f ( x 1, x 2, t ) L > 0. The problem is to design a feedback control law u = u ( x 1, x 2) that drives the mass to the origin asymptotically. In other words, the control u = u ( x 1, x 2) is supposed to drive the state variables to zero: i.e., . This apparently simple control problem is a challenging one, since asymptotic convergence is to be achieved in the presence of the unknown bounded disturbance f ( x 1, x 2, t ). For instance, a linear state-feedback control law provides asymptotic stability of the origin only for f ( x 1, x 2, t ) 0 and typically only drives the states to a bounded domain ( k 1, k 2, L ) for f ( x 1, x 2, t ) L > 0.

The question is whether the formulated control problem can be addressed using only knowledge of the bounds on the unknown disturbance.