**6.4 The simulation of the cascaded SISO closed-loop system for camera movement adjustment**

In this section, we are going to compare the closed-loop response results of the cascaded control system where the outer-loop controllers are designed using the two aforementioned methods: feedback linearization (Section 6.2) and model linearization (Section 6.3). The simulation results are obtained with the original nonlinear camera model (43). In addition, for the linearized plant approach, we will also illustrate how varying the damping ratio ζ affects the responses. For both methods, we chose the bandwidth of the inner-loop as 100*rad=s* and the bandwidth of the outer-loop as 10*rad=s*.

We compare the simulation responses by choosing six different damping ratios. Two are chosen as the overdamped systems (ζ> 1Þ, one is chosen as a critically damped system ð Þ ζ ¼ 1 , and three as the underdamped systems (ζ<1Þ. We have simulated four cases and compared all six systems for each case. Each case is different due to varying the initial angle *φ* (see Eq. (40)) and the input disturbance *dqv* . Two cases are simulated without the input disturbance while the other two are simulated with the disturbance to compare the robustness of the controlled system.

The step responses of the image coordinate *<sup>u</sup>*c*<sup>R</sup>* is shown in **Figures 16**–**19**. The intrinsic camera parameters are selected to be: *f <sup>u</sup>* ¼ 2*:*8 *mm* (Focal length) and *α* ¼ 120° (Angle of view). In cases 1 and 2, the responses of feedback linearization are displayed in black dashed lines while all other lines are the responses of the controller that is designed with the linearized plant and varying the damping ratio ζ.

**Figure 16.** *Step responses of u*^*<sup>R</sup> for the case 1: φ*< *<sup>α</sup>* <sup>2</sup> ¼ 20°, *dqv* ¼ 0°*.*

*Role of Uncertainty in Model Development and Control Design for a Manufacturing Process DOI: http://dx.doi.org/10.5772/intechopen.104780*

**Figure 17.** *Step responses of <sup>u</sup>*^*<sup>R</sup> for the case 2: <sup>φ</sup>* <sup>¼</sup> *<sup>α</sup>* <sup>2</sup> ¼ 60°, *dqv* ¼ 0°*.*

**Figure 18.** *Step responses of u*^*<sup>R</sup> for the case 3: φ* ¼ 0°, *dqv* ¼ 5°*.*

The case 1 is simulated with the initial angle *φ*< *<sup>α</sup>* 2 , while the case 2 is simulated when *<sup>φ</sup>* <sup>¼</sup> *<sup>α</sup>* <sup>2</sup>, the largest possible initial angle within the angle of view. It can be shown clearly that without any input disturbance, both methods are able to drive the closed-loop responses to the final value. The step response of the feedback linearization has an overshoot*:* In addition, for the second method, the model linearization approach, it can be seen from the two simulation cases that there exists a damping ratio, ζopt, such that.

**Figure 19.** *Step responses of <sup>u</sup>*^*<sup>R</sup> for the case 4: <sup>φ</sup>* <sup>¼</sup> <sup>20</sup>°, *dqv* ¼ � *<sup>α</sup>* <sup>2</sup> ¼ �60°*.*


It can be estimated from **Figures 16** and **17** that ζopt ffi 1 in the case 1 and ζopt ffi 0*:*5 in the case 2. The most desirable system is the one without overshoot and fastest step response. When ζ ¼ ζopt, the system has the fastest response and no (or little) overshoot. Therefore, we can state that the best performance of the controlled system is when setting the damping ratio ζ ¼ ζopt. Clearly, the value of ζopt varies with *φ*, the initial angle of the reference point with respect to the inertial frame.

In the cases 3 and 4, the input disturbance is introduced to the system. In the case 3, a small disturbance (*dqv* ¼ 5°) is added to the actuator input. The case 4 is a combined case where both *φ* and *dqv* are present (*φ* ¼ 20°, *dqv* ¼ �60°). **Figures 18** and **19** do not display the step responses of feedback linearization approach. The step responses of feedback linearization are unstable when input disturbances are introduced. It can be shown that any input disturbance drastically alters the nonlinear interface parameters, used in feedback linearization, and hence, results in an unstable system. On the other hand, the linear controller designed based on the linearized plant model is robust to the input disturbances even with the significantly large disturbances (case 4). Similar to the no disturbance cases, ζopt exists for the cases with disturbances.

From the discussion above, the plant linearization method is the preferred and the recommended method for the camera movement adjustment.

*Role of Uncertainty in Model Development and Control Design for a Manufacturing Process DOI: http://dx.doi.org/10.5772/intechopen.104780*
