*4.3. Model Validation*

In Section 4, a comparison is provided between simulated and measured variables regarding the static characterization of hovering conditions. In what follows, predicted dynamic properties about the same equilibrium are validated through identification methods [33]. To this end, flight data are collected and eventually filtered after performing frequency sweep maneuvers about the hover, according to the approach described in [34]. The frequency response for each selected input–output pair is then identified during an optimization process driven by the difference between the computed and the predicted frequency responses. The fidelity of the model is finally established using time domain verification, according to which time response predicted by the identified model is compared with the response recorded during flight tests.

Different maneuvers and data pairs are considered for the identification of transfer functions, such as *p*(*s*)/*A*1*s*(*s*) and *q*(*s*)/*B*1*s*(*s*), with the aim of validating the predicted dynamic information. For the sake of brevity, the adopted procedure is detailed for the characterization of the heave subsidence mode, whose dominant derivative *Zw* is discussed above. In particular, the first input-output data pair describes the effect of MR collective pitch angle *θ*<sup>0</sup> on vertical acceleration, *az* = *w*˙ , expressed in a body-fixed frame. A detail of the data taken into account for such an identification procedure is reported in Figure 5. The predicted transfer function as obtained from the state-space representation in Equation (22) is:

$$\frac{a\_z(s)}{\theta\_0(s)}\Big|\_{mdl} = \frac{Z\_{\theta\_0}s(s-z\_1)(s-z\_2)(s-z\_3)}{(s-p\_{lv})(s-p\_{ps})\left(s^2 - 2/\tau\_{ph}s + \omega\_{ph}^2\right)}\tag{24}$$

where a set of 4 zeros is determined. The first one is located at the origin, *z*<sup>1</sup> = 0.9983 *pps* is real negative, and *z*2, *z*<sup>3</sup> are a complex–conjugate pair such that *z*<sup>2</sup> *z*<sup>3</sup> = 1.008 *ω*<sup>2</sup> *ph* and *<sup>z</sup>*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>3</sup> <sup>=</sup> 1.059 · 2/*τph*. Heave subsidence mode evidently dominates the motion along *<sup>z</sup>B*, provided that almost perfect pole-zero cancellation characterizes the terms depicted in gray color. It follows:

$$\frac{a\_z(s)}{\theta\_0(s)}\Big|\_{mdl} \approx \frac{Z\_{\theta\_0}s}{s - p\_{hv}}\tag{25}$$

Numerical identification is performed by using a Prediction Error Minimization (PEM) method focused on simulation [35], provided the transfer function in Equation (24) is assumed as the initial guess model. The identified transfer function is:

$$\begin{split} \frac{a\_z(s)}{\theta\_0(s)} \Big|\_{id} &= \frac{0.9964 \, Z\_{\theta\_0}(s - \pi\_1)(s - \pi\_2)}{(s - 1.0421 \, p\_{lv})(s - 0.9912 \, \pi\_2)} \\ &\quad \cdot \frac{(s^2 + \pi\_3 s + \pi\_4)}{(s^2 + 1.0090 \, \pi\_3 s + 0.9964 \, \pi\_4)} \\ &\approx \frac{0.9964 \, Z\_{\theta\_0}s}{(s - 1.0421 \, p\_{lv})} \end{split} \tag{26}$$

where pole–zero cancellation can evidently be performed for the gray terms. It must be noted that *π*<sup>1</sup> ≈ 0, such that the zero at the origin is also recovered. The estimation error between model-predicted and identified parameters is provided in the second line of Equation (26), where the updated values of *Zθ*<sup>0</sup> and <sup>|</sup>*phv*|, respectively, result in being 0.36% smaller and 4.21% bigger than the model-predicted ones in Equation (25). A sample comparison between measured and refined-simulation data after heave subsidence mode characterization is finally provided in Figure 6 for the acceleration.

**Figure 5.** The input–output measured data used for heave subsidence mode characterization near hover (detail).

**Figure 6.** Measured and simulated data after heave subsidence mode characterization at hover (detail).

Encouraging results are indeed obtained for other input-output pairs, thus validating the modeling approach. In all cases, in fact, very good agreement is found between the dynamic properties obtained through numerical simulation and identification techniques.

#### **5. Control System Design and Test**

In what follows, the control system design phase is described based on the mathematical model in Section 3 and the analysis performed in Section 4. The closed-loop system is

first analyzed by Model-In-the-Loop (MIL) simulations, where linear controllers are directly designed and validated in the nonlinear framework by means of an extensive campaign of simulations performed in collaboration with the candidate pilot. HIL tests are then performed to refine the control gains and validate the software/hardware setup [36].
