Being a motion on a discontinuity set of a dynamic system, sliding mode is used to keep accurately a given constraint and features theoretically-infinite-frequency switching. Standard sliding modes provide for finite-time convergence, precise keeping of the constraint and robustness with respect to internal and external disturbances. Yet the relative degree of the constraint has to be 1 and a dangerous chattering effect is possible. Higher-order sliding modes preserve or generalize the main properties of the standard sliding mode and remove the above restrictions. r-Sliding mode realization provides for up to the rth order of sliding precision with respect to the sampling interval compared with the first order of the standard sliding mode. Such controllers require higher-order real-time derivatives of the outputs to be available. The lacking information is achieved by means of proposed arbitrary-order robust exact differentiators with finite-time convergence. These differentiators feature optimal asymptot...

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Higher-order sliding modes, differentiation and output-feedback control

This book, written by two major figures in adaptive control, provides a wealth of material for researchers, practitioners, and students. While some researchers in adaptive control may note the absence of a particular topic, the book‘s scope represents a high-gain instrument. It can be used by designers of control systems to enhance their work through the information on many new theoretical developments, and can be used by mathematical control theory specialists to adapt their research to practical needs. The book is strongly recommended to anyone interested in adaptive control.

Adaptive Control

Homogeneity approach to high-order sliding mode design

http://www.tau.ac.il/~levant/levant_2sl_aut2007_pre.pdf

Principles of 2-sliding mode design

A universal finite-time-convergent controller is developed capable to control the output of any uncertain single-input-single-output system with a known permanent relative degree r. The tracking error /spl sigma/ is steered to zero by means of a control dependent only on /spl sigma/, /spl sigma//spl dot/, ..., /spl sigma//sup (r-1)/ and continuous everywhere except the set /spl sigma/=/spl sigma//spl dot/=/spl middot//spl middot//spl middot/=/spl sigma//sup (r-1)/=0. A robust output-feedback controller version provides for the tracking accuracy proportional to the sampling noise magnitude.

Quasi-continuous high-order sliding-mode controllers

Variable structure control with sliding sector

In general, a Variable Structure (VS) system is designed with a sliding mode. Recently a sliding sector, designed by an algebraic Riccati equation, has been proposed to replace the sliding mode for chattering-free VS controllers. In this paper we extend the design algorithm for the sliding sector to a time-varying sliding sector. The time-varying sliding sector is defined by functions dependent on both state and time, hence time-varying uncertainty can be considered. The VS controller is designed to stabilize an uncertain system, quadratically. The design procedure for real systems is introduced via an implementation to the control of “Furuta pendulum”. To enhance the stability it is necessary to compensate the time-varying nonlinear static friction of the actuator adequately, hence this problem is a good example to demonstrate the performance of the proposed VS control method. In the experiment, it will be shown that the VS control with the time-varying sliding sector is superior to an orthodox chattering-free VS control.

Vs-control with time-varying sliding sector — design and application to pendulum —

This chapter presents some discrete-time variable structure (VS) control design algorithms with sliding sectors and then proposes a discrete-time VS controller with an invariant sliding sector. The invariant sliding sector is an invariant subset of the state space determined by a linear and a quadratic functions on the state variable. To ensure the invariance, a VS control law is implemented. Inside the sector with the VS control law, a Lyapunov function keeps decreasing. If the state is inside the subset in some time instance, then the VS control input will let the state remain inside. A discrete-time VS controller with the invariant sliding sector for discrete-time systems is designed such that the state is moved into the sector in finite steps and stays there from then on. The resultant VS control system is quadratically stable as there exists a Lyapunov function which decreases in the state space. Simulation results are given to show the efficiency of the proposed design algorithm of the discrete-time VS controller.

Discrete-time Variable Structure Control

A sliding sector, inside which a norm of the state decreases, is defined for both a continuous-time system and a discrete-time system. A continuous-time variable structure system (VSS) controller and a discrete-time VSS controller are designed to transfer the state from the outside to the inside of the sliding sector. Simulation results are given.

Yaodong Pan

Papers

Variable structure control with sliding sector

Vs-control with time-varying sliding sector — design and application to pendulum —

Discrete-time Variable Structure Control

A new approach to design a sliding sector for VSS controller

Discrete-time vss controller design