Design & Discretization of Adaptive Sliding-Mode Controllers

It is the purpose of a controller to modify the so-called plant system in such a way that the overall system shows a desired behavior. At the same time, the controlled system shall be insensitive against disturbances that act on the plant. This goal can be achieved by using ideas from variable structure control systems. It is characteristic for these ideas that specific manipulation of the controller structure renders the control-loop resistant against certain kinds of disturbances. A major drawback of this methodology is that disposable knowledge about the system is often disregarded. This may lead to excessive wear in the actuators, e.g. servo drives, hydraulic valves. The underlying project DeDiaSMC Design and Discretization of Adaptive Sliding-Mode Controllers addresses the above-mentioned issue. Structural information that was disregarded since then shall now be incorporated within the design of the control system. The control law to be devised then needs to adapt online to parametric uncertainties, as for example friction coefficients, spring rate or resistor values. In course of the project, rigorous proofs of stability and attainable performance shall be provided. A powerful tool for this purpose is the direct method of Lyapunov. In recent years, a series of novel Lyapunov functions has been found which shows to be promising in particular for designing adaptive, variable structure control systems. The implementation of the continuous-time control laws is achieved via a so-called discretization. Digital realizations of these control laws require a dedicated numerical treatment. It is a goal oft he research activities to find out which control loop properties prevail after discretization. Special attention is paid on stability properties and control accuracy in the closed loop. Methods derived in the project will be evaluated by means of simulations and experimentally on a nano-positioning machine. The machines model structure is well-suited for testing the advocated approach.

The aim of a control system is to impose a predefined desired behavior on a system, the so-called plant. The overall feedback loop should be as insensitive as possible to disturbances. This goal can be achieved using methods from the field of so-called structure-variable systems, for example. The underlying ideas are characterized by the fact that the special structure of the control law makes it possible to achieve almost complete immunity of the control loop to certain types of interference. However, available system knowledge is often not fully utilized. This can, for example, lead to excessive stress on the actuators. This project addresses the problem described above. Previously unconsidered structural information about the controlled system can now be taken into account when designing the control system. The control laws can independently adapt to existing parametric uncertainties, such as friction coefficients, spring constants and resistance values, at runtime. The combination of adaptive and structure-variable methods enables the design of extremely powerful control laws. Correct implementation on digital hardware is essential for the practical application of the control laws developed. This requires the development of so-called discretization methods as the computation on digital hardware is performed at discrete time instants. As part of the project, new adaptive control algorithms were developed and discretized. The stability of the resulting discrete-time control laws was proven using Lyapunov's Direct method. The methods are characterized by high control quality and prevent the extremely undesirable "discretization chattering" due to the new discretization methods developed. This phenomenon, which significantly deteriorates the behaviour of the feedback loop, is not caused by the control itself, but by the discretization method used. The research activities have not only led to the development of new methods, but have also provided new theoretical insights into existing approaches. The research results have been published at important conferences, and several journal publications are currently being finalized or are already in the review process.