There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Envisioning Information

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

A tutorial account of variable structure control with sliding mode is presented. The purpose is to introduce in a concise manner the fundamental theory, main results, and practical applications of this powerful control system design approach. This approach is particularly attractive for the control of nonlinear systems. Prominent characteristics such as invariance, robustness, order reduction, and control chattering are discussed in detail. Methods for coping with chattering are presented. Both linear and nonlinear systems are considered. Future research areas are suggested and an extensive list of references is included. >

Variable structure control: a survey

The synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered. Usually, what is called sliding mode is employed in such synthesis. This sliding mode is characterized, in practice, by a high-frequency switching of the control. It turns out that the deviation of the system from its prescribed constraints (sliding accuracy) is proportional to the switching time delay. A new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced. These algorithms feature a bounded control continuously depending on time, with discontinuities only in the control derivative. It is also shown that the sliding accuracy is proportional to the square of the switching time delay.

Sliding order and sliding accuracy in sliding mode control

One of the most important control problems is control under heavy uncertainty conditions. While there are a number of sophisticated methods like adaptation based on identification and observation, or absolute stability methods, the most obvious way to withstand the uncertainty is to keep some constraints by "brutal force". Indeed any strictly kept equality removes one " uncertainty dimension". The most simple way to keep a constraint is to react immediately to any deviation of the system stirring it back to the constraint by a sufficiently energetic effort. Implemented directly, the approach leads to so-called sliding modes, which become main operation modes in the variable structure systems (VSS) [55]. Having proved their high accuracy and robustness with respect to various internal and external disturbances, they also reveal their main drawback: the so-called chattering effect, i.e., dangerous high-frequency vibrations of the controlled system. Such an effect was considered as an obvious intrinsic feature of the very idea of immediate powerful reaction to the minutest deviation from the chosen constraint. Another important feature is proportionality of the maximal deviation from the constraint to the time interval between the measurements (or to the switching delay).

https://onlinelibrary.wiley.com/doi/pdf/10.1002/rnc.1229

Higher order sliding modes

A straightforward modification of the theory of ‘sliding’ variable structure control systems is used to design a control law that is a continuous arbitrarily close approximation, within a predefined subspace, to the discontinuous variable structure control law. Linear combinations of eigenvectors, and certain null spaces, are used to analyse the state trajectory under the influence of the continuous control law. Comparisons are drawn and the inherent robustness properties of variable structure control systems are retained, while the undesirable chatter motion of the sliding mode is eliminated. The accuracy of the approximation to the idealized case can easily be calculated. The general results are illustrated by the simulation of a time-invariant fourth-order multivariable system with two controls

Continuous approximation of variable structure control

Adaptive Control: the Model Reference Approach

An introduction to sliding mode variable structure control.- An algebraic approach to sliding mode control.- Robust tracking with a sliding mode.- Sliding surface design in the frequency domain.- Sliding mode control in discrete-time and difference systems.- Generalized sliding modes for manifold control of distributed parameter systems.- Digital variable structure control with pseudo-sliding modes.- Robust observer-controller design for linear systems.- Robust stability analysis and controller design with quadratic Lyapunov functions.- Universal controllers: Nonlinear feedback and adaptation.- Lyapunov stabilization of a class of uncertain affine control systems.- The role of morse-Lyapunov functions in the design of nonlinear global feedback dynamics.- Polytopic coverings and robust stability analysis via Lyapunov quadratic forms.- Model-following VSC using an input-output approach.- Combined adaptive and Variable Structure Control.- Variable structure control of nonlinear systems: Experimental case studies.- Applications of VSC in motion control systems.- VSC synthesis of industrial robots.

Variable Structure and Lyapunov Control

(1992). Matrices: Methods and Applications. Journal of the Operational Research Society: Vol. 43, No. 12, pp. 1185-1185.

/pdf/matrices-methods-and-applications-3mhi2vv26h.pdf

Alan S. I. Zinober

Papers

Higher order sliding modes

Continuous approximation of variable structure control

Adaptive Control: the Model Reference Approach

Variable Structure and Lyapunov Control

Matrices: Methods and Applications