**Author details**

**Table 3** compiles the results obtained for the double pendulum system. The table compares the settling time and the integral square error (ISE) for the two different controllers. The ISE

*(t)* and *θ<sup>2</sup>*

0 ∞

where e(t) is the error obtained by subtracting the actual response and the desired response

It can be seen from **Table 3** that the settling time and ISE are better in the case of the controller tuned with 5% noise. Optimizing the controller with 5% noise helps with the system response for larger window of uncertainty. As can be seen from **Table 3**, the ISE values for 10% noise

> **Controllers trained without noise Controllers trained with 5% noise No noise 5% Noise 10% Noise No noise 5% Noise 10% Noise**

*)* 1.632 1.636 1.708 0.3895 0.3906 0.3911

*)* 3.871 3.874 5.723 2.1532 2.1545 2.1560

) 0.8825 0.8853 0.8961 0.3875 0.3943 0.4012

) 0.2412 0.2450 0.3323 0.0985 0.0988 0.0990

*)* 4.431 4.487 5.191 2.347 2.368 2.449

*)* 5.208 5.271 13.538 1.475 1.487 1.531

) 3.4389 3.5963 5.9624 2.2963 2.3440 2.4009

) 3.2142 3.4433 11.0016 0.6271 0.6363 0.6674

**Table 3.** Comparison of settling times (Ts in seconds) and the ISE for the scenarios discussed before [13].

*e* <sup>2</sup>(*t*)*dt* (8)

*(t)* under (a) 5% noise and (b) 10% noise with

is given by,

*θ*̇ <sup>1</sup> <sup>=</sup> *<sup>θ</sup>*̇

*θ*̇ <sup>1</sup> <sup>=</sup> *<sup>θ</sup>*̇

which in our case is zero.

<sup>2</sup> = 0 *Ts(θ<sup>1</sup>*

<sup>2</sup> = 2 *Ts(θ<sup>1</sup>*

*Ts(θ<sup>2</sup>*

ISE(*θ<sup>1</sup>*

ISE(*θ<sup>2</sup>*

*Ts(θ<sup>2</sup>*

ISE(*θ<sup>1</sup>*

ISE(*θ<sup>2</sup>*

initial angular velocities of 2 rad/s [13].

*ISE* = ∫

**Figure 11.** Controllers trained with 5% noise: plots showing *θ<sup>1</sup>*

38 Fuzzy Logic Based in Optimization Methods and Control Systems and Its Applications

Anoop Sathyan\* and Kelly Cohen

\*Address all correspondence to: sathyaap@ucmail.uc.edu

Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, USA
