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4.5. Fuzzy controller and fuzzy control

4.5.1. Generic concept of fuzzy control

The concept of the fuzzy controller is nothing but another, non-conventional look at the design of feedback control (Astrom and Wittenmark, 1984). In a nutshell, the problem is formulated as follows. Given is a reference x0 (this could be a fixed value or a certain time trajectory). Design a controller that is a mapping from the observable states of the system under control to control actions that help achieve x0. Given now a point of reference (setpoint) x0, we are interested to apply such control action “u” that the system achieves this setpoint smoothly and as quickly as possible. Imagine that we will assume the role of such a controller, forming a closed loop control structure, Fig. 4.7.


Figure 4.7  Generic structure of a closed loop control system

What knowledge about the behavior of the system is required to make relevant action? Well, we may consider error and change of error as two essential pieces of information that should definitely imply our decisions. The error is defined as a difference between the setpoint and the current output x,

The change of error reflects a trend in the reported values of error and is defined as the difference between two successive values of error

where t′ < t and t′ → t (more precisely we can talk about the derivative of error, ).

When applying control actions in the closed loop structure, the typical behavior of the system would look like the one shown in Fig. 4.8.


Figure 4.8  Typical response of a closed loop control system

As the system exhibits dynamics, one cannot expect it to immediately reach the setpoint. In fact, there is always a compromise between control actions. Especially, too significant control actions lead to a faster response of the system; however, this produces substantial overshoots and oscillations. Similarly, very limited control signals lead to a sluggish response of the system.

Now let us think about this simple strategy in terms of rules. Looking at the previously taken control actions and monitoring the current value of error and its change, we ask ourselves in which way the next control actions should be changed to get the desired performance of the system. It depends. Based on our intuition we can summarize our experience. For instance, if error is around zero and change of error is around zero, it is intuitively justifiable to maintain the previous control. This means that the change in control should be around zero. We have developed the control rule:

if error is ZERO and change of error is ZERO then change of control is ZERO.

This rule looks straightforward. Now what about control if error assumes Negative Big values while the change of error is around zero. This is an example of when the system exhibits overshoot, Fig. 4.8 - a possible control action to alleviate this situation would be to reduce control significantly, that is change of control should be Negative Big or Negative Medium to avoid eventual oscillations.

There are several ways of knowledge elicitation. We elaborate on the two main classes:

utilization of commonsense knowledge: the knowledge of this type is quite readily available and can be easily structured into more formal framework. This is the case in the present approach.

(i)  the rules can be developed for relatively small problems involving very few variables. As their number increases so is our effort to come up with the meaningful and transparent rules. Under these circumstances we usually end up with control protocols that are incomplete or/and inconsistent (including conflicting rules).
(ii)  the rules easily apply to systems that somewhat comply with our intuition on how the control actions should be taken. The protocol we developed pertain to the class of systems, fortunately broad enough to be of a practical relevance. The same qualitative rules will not be completely relevant when considering another class of systems like those with delay in system variables, say

(τ stands for the delay time) or non-minimal phase systems. This is because the systems of these classes do not respond to control in a somewhat “expected” way. For instance, in any system with delay the control affects the system after a certain time (caused by so-called transportation delay) - this somewhat makes the commonsense error - control relationships less evident. The control of non-minimal phase system is even more intricate The reason for that is the common reversed behavior of system output in a time period, under a control action we expect to drive the system output to the desired direction.

acquiring knowledge through interactive computer simulations: this is a powerful and flexible way of knowledge acquisition. The designer (user) is put through a series of highly interactive computer simulations and play with the system simulated via a detailed model and learns or tests the control protocols. After a series of such experiments, after which the control knowledge becomes highly refined, it could be transformed into a formal structure of rules.

An important fact is that the developer becomes fully aware of eventual specificities of control that should be incorporated into the control protocol. As of today, the existing simulation capabilities are unlimited to a great extent. The designer can easily and effectively exercise his/her control skills being enhanced by diverse mechanisms of virtual reality (VR). Taking this into account, it is very likely that the VR approach to knowledge acquisition will be gaining in importance as an essential design vehicle. The VR environment can also take care of collecting and organizing the experimental data.

4.5.2. Design principles of the fuzzy controller

In this section we elaborate on the main components which assemble a fuzzy controller. The characteristic mechanisms involved in its construction and their functioning are described. The basic structure of a fuzzy controller is depicted in Fig. 4.9.


Figure 4.9  Basic structure of a fuzzy controller

The first point to note is that a fuzzy controller resembles the general architecture of a fuzzy model. Recall that, in general terms, a fuzzy model comprises an input interface, a processing module and an output interface. In this view, the encoding and decoding mechanisms embody the input and output interface, whereas the rules and inference are constituents of the processing module.


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