## 5 Experimental Tests
### Experimental Tests and Discussions  
Figure 5 shows the experimental results of the proposed nonlinear control scheme during take-off and hovering flight. It is illustrated from Figure 5a-d that the resulting controlled system can achieve a good tracking performance within about 10 s even without any reliable and accurate _a priori_ information about the behavior of the rotational speed system, where \(\varepsilon\) denotes the tracking error between the command signal \(\omega_{c}\) and the actual rotational speed \(\omega\). Thereafter, the tracking error typically remains in the range of 0.75 rad/s, which is sufficient to meet the application requirements of a high-performance MAV. Meanwhile, the adjustable parameters of the nonlinear controller also have good convergence properties as shown in Figure 5e, although they finally fluctuate more or less around their mean values. It is, however, to note that the control signal from the nonlinear controller changes rapidly with time, as shown in Figure 5f.  
Figure 4: The hardware-in-loop (HIL) experimental setup.  
Figure 6 shows the learning process of adjustable parameters with different \(\mu\) or without soft limiting elements. The increase of \(\mu\) can obviously contribute to the convergence of adjustable parameters; however, this results in the significant fluctuations of adjustable parameters in the later stage of the learning process mainly because of measurement noises. It is even worse that the adjustable parameters exhibit sharp fluctuations for a longer period of time if removing the soft limiting elements from the nonlinear adaptive control scheme, as shown in Figure 6b, which can easily render the over-learning problem of adjustable parameters.  
Figure 7 presents the experimental results of the optimized PID controller, with the nonlinear controller removed from the control scheme, where \(\omega_{p}\) is the rotational speed, \(u_{p}\) the controller output, \(\varepsilon_{p}\) the tracking error between \(\omega_{c}\) and \(\omega_{p}\) in this test case. In contrast to what has been shown in Figure 5, the tracking error increases significantly from 0.75 rad/s to 8.0 rad/s, although the control performance is basically acceptable in practice, which also comparatively demonstrates that the proposed nonlinear control scheme can contribute to the significant improvement in the control performance, as shown in Figure 8.  
Figure 9 shows the comparisons of some experimental results of two types of control schemes in a wide range of rotational speeds, where the initial values of the adj  
Figure 8: Comparisons of some experimental results of two types of control schemes. (**a**) \(\varepsilon\) and \(\varepsilon_{p}\); (**b**) \(u\) and \(u_{p}\).  
Figure 6: Comparisons of the learning process of adjustable parameters. (**a**) Adjustable parameters with soft limiting elements: _Case 1: \(\mu=0\), Case 2: \(\mu=1000\left(a_{m1}\dot{e}+a_{m0}e\right)\); (**b**) Adjustable parameters without soft limiting elements.  
Figure 7: Experimental results of the optimized proportional-integral-differential (PID) controller. (**a**) \(\omega_{c}\) and \(\omega_{p}\); (**b**) \(\varepsilon_{p}\).  
nonlinear controller are set to the final values obtained in the first experimental test shown in Figure 5e. The results have once again verified that the dynamic tracking accuracy of the proposed nonlinear adaptive control scheme is much more satisfactory than that of the optimized PID controller, as shown in Figure 9a-c. It is of further note that the achievements largely depend on the fast control capability of the proposed nonlinear adaptive controller, as shown in Figure 9d.