Cascaded Model-Free Fuzzy Control: an Application to the Coupled Tanks System
Abstract
This report proposes a design methodology for cascaded model-free fuzzy control systems. The ordinary Mamdani approach is modified in order to use expert knowledge for variable set-point control without any need of the system model. The methodology is successfully tested in a sub-actuated, naturally delayed setup, known as the coupled tanks system, where the water level is maintained at different set points, both in simulation and in real time.
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Introduction
From a control point of view, cascade systems are challenging, since (a) they are sub-actuated, (b) they naturally incorporate delays in the control action, and (c) they usually pursue several set points. Since the common approach within the control community is to consider the mathematical model of the system, obtained from first principles, in order to design an appropriate control law which stabilizes around the origin, very simple systems such as the coupled tanks may be hard to control due to the changing reference level. The usual solution is to adjust the controller, most commonly a PID, to different set points, as has been done with aircraft altitude control [1]. This, of course, requires a proper tuning which is indeed more complex for cascade systems.
Within model-based methodologies, cascade systems have been widely studied through input-output stability methods [2], input-to-state approaches [3, 4], gain scheduling [5], passivity stabilization [6, 7], sliding mode control [8, 9], backstepping[10], and synchronization [11]. Model-free methodologies, relying on expert knowledge and artificial intelligence, have also proposed a variety of solutions to the problem of analyzing and controlling cascade systems: via nonlinear scheduling [12], fuzzy exact representations [13], fuzzy sliding control [14], genetic algorithms [15], and evolutionary laws [16].
This work proposes a design methodology based on model-free fuzzy control, i.e., a Mamdani representation of expert knowledge about the plant for control purposes. In contrast with the ordinary approach which mixes all the controlled variables in a single fuzzy system to calculate the appropriate control law, it is shown that a cascade system is better (and simpler) dealt with if the fuzzy controller “decouples” the cascaded variables as to allow them to vary according to the desired set points.
The following sections are organized as follows: Section II presents the required preliminaries on model-free fuzzy control and cascade systems this work is concerned about; Section III introduces the novel methodology and explains the heuristic as well as numerical advantages of the proposal; Section IV introduces a typical cascade system: the coupled tanks, which is employed in Section V to illustrate how the proposed technique can be successfully implemented for level control of coupled tanks, both in simulation as well as in real-time, with clear advantages over the PID-based controller of the provider and the traditional fuzzy control; Section VI closes the paper by providing some conclusions.
Conclusion
A novel cascaded model-free fuzzy control has been presented: it allows dealing with cascaded systems by introducing adaptable shape membership functions in order to deal with variable set point goals. By subsequently transmitting the reference from one block to next one, the cascaded fuzzy methodology is able to pick the right shape and “center” for the fuzzy sets as to track the desired set point at the last block. The methodology has been successfully implemented in the coupled tanks system, where it has performed better than the provider PID control and the traditional fuzzy control structure. Time delays have also been appropriately dealt with by simply expanding the error intervals.