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4.5.3. Numerical experiments

This section is devoted to a number of simulation experiments with the fuzzy controller. Let us first discuss a detailed topology of the fuzzy controller used in these experiments. The rules come in the previously discussed form

- if error is A and change of error is B then change of control is c

Here A and B are fuzzy sets defined in the respective universe of error and change of error; c is a certain numeric value of control increment associated with A and B. The control protocol is shown in Fig. 4.10.


Figure 4.10  Linguistic rules of the fuzzy controller NB - negative big, NM - negative medium, NS - negative small, ZE - zero, PS - positive small, PM - positive medium, PB - positive big

All the fuzzy sets of control and change of control are defined in the [-1, 1] interval. The detailed computations of control are performed with the use of t- and s-norms. More specifically, the activation levels of the subconditions (error and change of error) are combined AND-wise with the use of the product operation. The activation levels of the fuzzy sets of control are computed using the probabilistic sum.

To assure that the same linguistic protocol applies to various dynamic systems despite different ranges of their input and output variables, proposed are scaling coefficients, see Fig. 4.11


Figure 4.11  Fuzzy controller and its scaling coefficients

The relationship between e′ and e (likewise de′ and de as well as u′ and u) is a linear function with the scaling coefficient ke (kde, ku, respectively)

While this scaling mechanism becomes common, it should be mentioned that it exhibits some drawback as the value of this scaling factor needs to be defined in advance. To alleviate this drawback, we introduce a mechanism of dynamic adjustment of the scaling factors as shown in Fig. 4.12. This procedure heavily relies upon the computations of a moving average of the input signal; this average is involved in the normalization of the original variable, Fig. 4.12.


Figure 4.12  Normalization of variable with dynamic range

In more detail:

  first, a dynamic range D is determined as

where <x+> is a moving average of the positive samples of e′ (similarly, <x-> results from the averaging of the negative samples),

  second, the normalization follows the expression

where β is a fixed parameter; ∈ is set up to a small positive number and assures us that even in cases when the dynamic range reduces to zero, the division operation remains valid. As heavily relying on averaging, the calculations of the normalization factor help filter out some noise in the collected data.

In essence, the rules of the fuzzy controller as constituting the control protocol, represent a nonlinear PI controller. Observe that an integration effect comes from the implemented incremental form of control (Δ u). All the membership functions are Gaussian-like with the same spread equal 0.1 and the modal values distributed uniformly across [-1, 1] equal to -1.0, -0.66, -0.33, 0, 0.33, 0.66, 1.0, respectively. The use of several nonlinear mechanisms in the fuzzy controller yields the nonlinear numeric characteristics of the fuzzy controller; for illustrative purposes some of them are included in Fig. 4.13.


Figure 4.13  Characteristics of the fuzzy controller u = u(e) for selected values of change of error

The control problem is discussed in the setting of the linear system with the transfer function equal to

Even fairly simple, this function models a broad class of dynamic systems. For detailed experiments we assume β = 0.2 and ω = 5.0. Note that the system is significantly underdamped - this effect is clearly visible in Fig. 4.14.


Figure 4.14  Response of the system to a step signal of amplitude 5


Figure 4.15  Response of the fuzzy controller to a unit step function

The fuzzy controller can also tolerate some transportation delay in the system; the response of the system with a transfer function

where τ = 0.35 is shown in Fig. 4.16.


Figure 4.16  Response of the fuzzy controller to a unit step function - a system with delay

The closed loop system exhibits good tracking performance - its response to a cosine reference is shown in Fig. 4.17. Similarly, the system performs well to a linearly increasing reference function, Fig. 4.18.


Figure 4.17  Response of the fuzzy controller to cosine reference signal


Figure 4.18  Response of the fuzzy controller to linearly increasing reference signal

In the ensuing series of experiments we investigate the role of the scaling factors and their impact on the performance of the system’s response. We experiment with the control scaling coefficients (ku) equal to 1.3, 1.1 and 0.8. The lower the value, the slower the response of the system, Fig. 4.19.

Now consider the scaling factor ku = 0.8 and discuss the same system but with the larger delay times such as τ =0.5, 1.0 and 1.5. As expected, higher delays result in more oscillations, Fig. 4. 20.

A reduction of ku further to ku = 0.4 helps us cope with even higher delays such as τ = 1.5, 2.0, 2.5, and 3.5, Fig. 4.21.

It is interesting to observe the changes of the scaling factor of the input variable of the controller (error) over time as the system approaches a setpoint. This factor changes continuously over time. The factor decreases once the system gets close to the setpoint and zooms up the control rules. This makes the entire protocol active yet at a different scale - an intriguing fractal-like behavior of the controller.


Figure 4.19  System response for several values of ku


Figure 4.20  System response for the fuzzy controller with ku =0.8 and different delays


Figure 4.21  System response for the fuzzy controller with ku =0.4 and different delay times


Figure 4.22  System response for the fuzzy controller with ku = 1.1 along with changes of the scaling factor over time (β = 0.2, ω = 5.0).


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