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The lower scaling factor ku =0.5 leads to the slower response of the system, see Fig. 4.23. Observe a similar pattern in the changes of the scaling factor.


Figure 4.23  System response for the fuzzy controller with ku = 0.5 along with changes of the scaling factor over time (β = 0.2, ω = 5.0).

The same experiment is repeated for ku = 1.5; Fig. 4. 24.


Figure 4.24  System response for the fuzzy controller with ku =1.5 along with changes of the scaling factor over time (β =0.2, ω =5.0).

Now we investigate tracking properties of the fuzzy controller by assuming a general and highly irregular form of the reference signal, Fig. 4.25.


Figure 4.25  System response for the fuzzy controller with ku = 4.0 along with changes of the scaling factor over time (β =0.2, ω = 5.0).

4.5.4. Fuzzy scheduler

A concept of combining fuzzy controllers and some “standard” control algorithms has been around for a number of years (Pedrycz, 1995). Realizations of such hybrid structures are quite different. In some sense, the very concept of such a hierarchy of control has been around in control engineering (Khalil, 1996); however, fuzzy sets added to it an important conceptual enhancement. In general, we consider standard numeric controllers to be distributed at the lower level and situated close to the system under control. The fuzzy controller plays a role of a task supervisor (scheduler) whose role is to dispatch control actions coming from the individual controllers at the lower (execution) level. One of the options we would like to discuss in detail concerns a fuzzy controller - scheduler switching between control actions of a series of PI or PID controllers. The parameters of each of these controllers are adjusted to some local characteristics of the system under control. Two architectures governing the switching mechanism are envisioned, Fig. 4.26:

  switching based on the state of the system. These rules assume the form

  if error is E and change of error is DE then activation level of PIDi

where E and DE are fuzzy sets in the respective universes of discourse while PIDi describes a control action of the i-th controller.

  switching between the parameters of a single PI or PID controller that leads to a so-called variable structure of the controller. Here the rules assume the format

  if error is E and change of error is DE then parameters of the PID


Figure 4.26  Two topologies of hierarchical control: fuzzy scheduler and local PID controllers (i) and PID controller with variable parameters scheduled by the fuzzy scheduler (ii)

In the following numerical experiment we discuss the first version of the fuzzy scheduler aimed at switching between two local PID controller. The system to be controlled is almost the same as discussed before with one significant modification: its parameters are state dependent. More specifically, we set them up in the form

where x is the state variable of the system. We also utilize two local PID controllers at the lower level of control. The first one, PID fast provides with a fast yet quite oscillatory response. The other one delivers a smooth, overshoot-free yet very sluggish response, refer to Fig. 4.28. The scheduling rules are summarized in Fig. 4.27; the calculations of the activation level of the fuzzy controllers are completed in the same way as in the case of the fuzzy controller. The calculations of the control are worked out in the form

where PID fast and PID slow are the control outputs produced by these two controllers while act_slow and act_fast are the relevant activation levels describing a level of contribution coming from each of these two controllers.


Figure 4.27  Rules of the fuzzy scheduler

The stratified control architecture results in a significant improvement in control. Fig. 4.28 summarizes the responses of the two PID controllers along with the response of the system with the gain scheduler. Evidently, the response becomes fast yet without any meaningful overshoot - the main benefit provided by the smooth switching between the individual controllers.


Figure 4.28  Response of the system under different control algorithms (i) slow PID controller (ii) fast PID controller (iii) fuzzy scheduler

The similar improvement can be witnessed in the case of the system of the second order with the transfer function

Here the lower (execution) layer of the controller is composed of two PD controllers; the fast PD controller is described by the parameters k and kd equal to 1.0 and 1.5, respectively. The slow PD controller uses k = 1.0 and kd = 21.0. The rules of the fuzzy scheduler are the same as in the previous example. The time responses of the controlled system are included in Fig. 4.30. Fig. 4.29 includes a comparison of the fast PD controller with the fuzzy scheduler in a tracking task when the system has to follow a sinusoidal reference function. Again, there is some improvement of the tracking properties when using the fuzzy scheduler.


Figure 4.29  Tracking properties of the fast PD controller (i) and the fuzzy scheduler (ii)


Figure 4.30  Response of the system under different control algorithms (i) slow PID controller (ii) fast PID controller (iii) fuzzy scheduler

4.6. Rule-based systems with nonmonotonic operations

Nonmonotonic reasoning (Ginsberg, 1987) has become an area of vigorous research in symbolic computation of Artificial Intelligence. The study by Yager (1987) has expanded the concept of nonmonotonic operations to linguistic variables and fuzzy sets, in particular. In this note we propose a certain computational generalization of the nonmonotonic operations on fuzzy sets. In what follows, we first elaborate on the basic definitions of these operators and discuss their extensions. Secondly, the section addresses several selected applications including an estimation of defaults (default fuzzy sets) and looks into nonmonotonic approximate reasoning.


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