4.1 Backstepping controller design

To simplify the controller design, an additional set of assumptions is proposed:

• • Assumption 3: The time derivatives of speed V, altitude h and heading $\psi$ can be neglected as they have a slower rate of change compared with the controlled variables $\alpha, \beta$ and $p_s$ .

• • Assumption 4: Actuators have rapid enough dynamics, thus they can be ignored in the design process.

Assumptions 3 is mainly valid for cruise flight and progressive manoeuvres, when a controlled change in the aircraft equilibrium has a primary effect on the faster attitude dynamics and a secondary one on the navigation variables. Finally, assumption 4 is very common and generally reasonable provided that assumptions 2 and 3 are respected.

Equation (9) is not suitable for the application of a total backstepping controller because the cascade form is not respected, in particular due to the presence of $\beta$ in the $\alpha$ dynamics, and vice versa. However by separating its dynamics as follows:

(10) \begin{align}\dot{p}_s=u_1\end{align}

(11) \begin{align}\begin{cases}\dot{\alpha}=q_s-p_s\tan\beta+\frac{\displaystyle-Lift-T\sin\alpha+mg_2}{\displaystyle mV\cos\beta}\\[3pt] \dot{q}_s=u_2\end{cases}\end{align}

(12) \begin{align}\begin{cases}\dot{\beta}=-r_s+\frac{\displaystyle Y-T\cos\alpha\sin\beta+mg_3}{\displaystyle mV}\quad\\[3pt] \dot{r}_s=u_3\quad\end{cases}\end{align}

three sub-controllers stabilising the desired states $\alpha$ , $\beta$ and $p_s$ can be defined. Cross-coupling exists due to the presence of $p_s$ and $\beta$ in the $\alpha$ dynamics and, at the same time, to the presence of $\alpha$ in the $\beta$ dynamics. During the individual sub-controls design, $\beta$ and $p_s$ are imposed constant in the $\alpha$ controller, and $\alpha$ constant in the $\beta$ controller. On the contrary, during the simultaneous control of the three variables, this assumption is disregarded because it is not physically realistic and not necessary, as shown below. Because of this coupling, the computation of a control action considers, at each moment, the value of the state controlled by another control action. For instance, the control law defining $u_2$ is evaluated with the instantaneous value of $\beta$ controlled by $u_3$ . This solution is beneficial when dealing with manoeuvres where strong coupling exists between longitudinal and lateral–directional planes.

A simple proportional controller is chosen for $p_s$ , Equation (10), while the cascade form of Equations (11) and (12) allows the application of a backstepping controller for $\alpha$ and $\beta$ . Note that Equations (11) and (12) have similar structure:

(13) \begin{align}\begin{cases}\dot{\omega}_1=f(\omega_1,y)+\omega_2\\ \dot{\omega}_2=u_s\\ \end{cases}\end{align}

A single backstepping controller designed for Equation (13) is therefore also suitable for Equations (11) and (12). As it is preferable to have the origin as the desired equilibrium point, a change of variables is defined as

\begin{align*}x_1&=\omega_1-H\\[2pt] x_2&=\omega_2+f(H,y)\\[2pt] \Omega(x_1)&=f(x_1+H,y)-f(H,y)\end{align*}

where H is the reference value for the controlled variable. The resulting dynamics is

(14) \begin{align}\begin{cases}\dot{x}_1=\Omega(x_1)+x_2\\ \dot{x}_2=u_s\\ \end{cases}\end{align}

The external control input $u_s$ controls $x_2$ that, in cascade, acts as virtual control to stabilise $x_1$ . Table 1 summarises the relationships between the variables used in the new and original systems. The functions $f_{\alpha}(\alpha,y_{\alpha})$ and $f_{\beta}(\beta,y_{\beta})$ are

\begin{align*}&f_{\alpha}(\alpha,y_{\alpha})=-p_s\tan\beta+\frac{\displaystyle-Lift-T\sin\alpha+mg_2}{\displaystyle mV\cos\beta}\\[3pt] &f_{\beta}(\beta,y_{\beta})=\frac{\displaystyle Y-T\cos\alpha\sin\beta+mg_3}{\displaystyle mV}\end{align*}

Table 1 Change of variable relationships

As shown in Ref. (Reference HÄrkegrd2), a linear globally stabilising control law for the system of Equation (14) is

\begin{align*}u_s=-k_2(x_2+k_1x_1)\end{align*}

with $k_2>2k_1>\max\{0,k_u\}$ . Using the relationships in Table 1, the control laws for Equations (11) and (12) become

(15) \begin{align}&u_2=-k_{\alpha,2}\left(q_s+k_{\alpha,1}\left(\alpha-\alpha^{ref}\right)+f_{\alpha}(\alpha^{ref},y_{\alpha})\right)\nonumber\\[3pt] &u_3=k_{\beta,2}\left(-r_s+k_{\beta,1}\beta+ f_{\beta}(0,y_{\beta})\right)\end{align}

with

(16) \begin{align}&k_{\alpha,2}>2k_{\alpha,1}, \quad k_{\alpha,1}>\max\{0,k_{\alpha}\}, \quad k_{\alpha}=\underset{\alpha,y_{\alpha}}{\operatorname{\max}}\frac{\partial f_{\alpha}(\alpha,y_{\alpha})}{\partial\alpha}\nonumber\\[2pt] &k_{\beta,2}>2k_{\beta,1}, \quad k_{\beta,1}>\max\{0,k_{\beta}\}, \quad k_{\beta}=\underset{\beta,y_{\beta}}{\operatorname{\max}}\frac{f_{\beta}(\beta,y_{\beta})-f_{\beta}(0,y_{\beta})}{\beta}\end{align}

Finally, the proportional control adopted for $p_s$ is

(17) \begin{align}u_1=k_{p_s}\left(p_s^{ref}-p_s\right), \quad k_{ps}>0. \end{align}

The relation between the control inputs and the stability-axes angular accelerations is defined by $\textbf{u}_c=(u_1, u_2, u_3)^T=\dot{\boldsymbol{\omega}}_s$ . The angular accelerations are the result of the variation in moments originated primarily by the deflection of the aircraft control surfaces. The vector of deflections $\boldsymbol{\delta}$ is obtained from the moment equation thus

(18) \begin{align}\textbf{M}(\boldsymbol{\delta})=I\left(R_{sb}^T\textbf{u}_{\textbf{c}} + \dot{R}_{sb}^T \boldsymbol{\omega}_s \right)+\boldsymbol{\omega}\times I\boldsymbol{\omega}\end{align}

To calculate $\boldsymbol{\delta}$ , a control strategy matching the controlled variables with the aircraft control surfaces must be defined.

4.2 Control strategy

The controller described above stabilises three variables related to the aircraft attitude. A global autopilot configuration capable of controlling speed V, altitude h and heading $\psi$ is required. In a real-life implementation, these variables could be easily measured with, respectively, a pitot tube, a barometric pressure sensor and a magnetometer. The control strategy is defined as follows: the backstepping controller acts on $\alpha$ , $\beta$ and $p_s$ in the inner loop, while three PID controllers act on V, h and $\psi$ in the outer loop. This approach separates the fast dynamics, characterising attitude, from the slower dynamics, characterising navigation. A prompt response of the backstepping controller is necessary when dealing with attitude variables, which are of prime importance for aircraft safety. For instance, immediate control of $\alpha$ in presence of vertical gusts could prevent stall or dangerous flight regimes. Consistent with assumption 3, the slower navigation variables can be successfully handled using traditional PIDs whose gains are tuned manually using an heuristic approach. The goal is to optimise the response in terms of overshoot, rise time, settling time and ringing.

The PID controlling the speed feeds the backstepping controller with the desired angle-of-attack, while the PID controlling the heading defines the desired roll rate. These values are limited in magnitude to avoid the request for motion that is incompatible with the aircraft dynamics during sudden manoeuvres. Saturation is imposed at the stall angle-of-attack for $\alpha^{ref}$ and at a typical roll rate for $p^{ref}$ . Note that the desired roll rate $p^{ref}$ is expressed in body axes, while the conversion to stability axes $p_s^{ref}$ is performed using Equation (5). The control surfaces employed are the elevator $\delta_e$ , the aileron $\delta_a$ and the rudder $\delta_r$ . According to assumption 1, these only generate a variation in moments but not in forces. The deflection vector $\boldsymbol{\delta}=(\delta_e, \delta_a, \delta_r)^T$ is obtained by substituting into Equation (18) the most general expressions for the moments,

(19) \begin{align}&L(\delta_a, \delta_e, \delta_r)=\frac{1}{2} \rho V^2 S b \left( C_{l\beta} \beta + C_{l\dot{\beta}} \dot{\beta} + C_{lp}\hat{p} + C_{lr}\hat{r} + C_{l \delta_a}\delta_a + C_{l \delta_e}\delta_e + C_{l \delta_r}\delta_r \right)\nonumber\\[3pt] &M(\delta_a, \delta_e, \delta_r)=\frac{1}{2} \rho V^2 S c \left( C_{m0} + C_{m\alpha}\alpha + C_{m\dot{\alpha}} \dot{\alpha} + C_{mq}\hat{q} + C_{m \delta_a}\delta_a + C_{m \delta_e}\delta_e + C_{m \delta_r}\delta_r \right)\nonumber\\[3pt] &N(\delta_a, \delta_e, \delta_r)=\frac{1}{2} \rho V^2 S b \left( C_{n\beta} \beta + C_{n\dot{\beta}} \dot{\beta} + C_{np}\hat{p} + C_{nr}\hat{r} + C_{n \delta_a}\delta_a + C_{n \delta_e}\delta_e + C_{n \delta_r}\delta_r \right) \end{align}

and solving the resulting linear system with three equations and three unknowns. The non-dimensional angular rates $\hat{p}$ , $\hat{q}$ and $\hat{r}$ are typically defined as

\begin{align*}\hat{p}=\frac{pb}{2V}, \hat{q}=\frac{qc}{2V}, \hat{r}=\frac{rb}{2V}\end{align*}

and $\rho$ is the air density, b is the aircraft wingspan, c the mean aerodynamic chord and S the wing area. The aerodynamic derivatives are $C_{m0}, C_{m\alpha}, C_{m\dot{\alpha}}, C_{mq}, C_{l\beta}, C_{l\dot{\beta}}, C_{lp}$ , $C_{lr}$ , $C_{n\beta}, C_{n\dot{\beta}}, C_{np}$ and $C_{nr}$ , while the control derivatives are $C_{m \delta_a}$ , $C_{m \delta_e}$ , $C_{m \delta_r}$ , $C_{l\delta_a}$ , $C_{l\delta_e}$ , $C_{l\delta_r}$ , $C_{n\delta_a}$ , $C_{n\delta_e}$ and $C_{n\delta_r}$ . Note that, commonly, the contribution of $C_{m \delta_a}$ , $C_{m \delta_r}$ , $C_{l\delta_e}$ and $C_{n\delta_e}$ is very small or zero. In this case, the calculation of the commands is simpler: $\delta_e$ is found from the $M(\delta_e)$ equation, while $\delta_a$ and $\delta_r$ are found by solving the linear system with $L(\delta_a, \delta_r)$ and $N(\delta_a, \delta_r)$ . The engine thrust vector is considered to be aligned with the aircraft $X_B$ axis and so does not generate moments.

The third PID controls the altitude by defining the required throttle value $\delta_{th}$ independently of the backstepping controller which acts through angular rates. The outer-loop strategy, where control surfaces, in practice the elevator, control the airspeed and the throttle controls the altitude, is a standard autopilot mode. As explained in Ref. (Reference Guglieri30), this approach guarantees better tracking of the airspeed, which is a key flight safety parameter. Table 2 summarises the controlled variables, their commands and the control method.

Table 2 Relationship between variables and commands

Figure 2. Backstepping control strategy for fixed-wing aircraft.

The proposed control scheme is shown in Fig. 2. The computed control inputs act on the aircraft, while its measured controlled states, viz. total speed, altitude and heading angle, are the feedback variables. Their differences from the corresponding reference values, $V^{ref}$ , $h^{ref}$ and $\psi^{ref}$ , define the error inputs for the PIDs. The throttle command and the measured speed are given as inputs to the backstepping controller as required by the control law definition and for the estimation of the inner-loop states. Note that, in fact, the variables $\alpha$ , $\beta$ and $p_s$ , used for the definition of the inner-loop error, are estimated with a good degree of accuracy inside the backstepping controller by integrating Equations (10)–(12), as shown below in Fig. 4(b). A support for the accurate estimation of $\alpha$ and $\beta$ can be provided by the feedback of $\phi$ and $\theta$ , easily measurable with an Inertial Measurement Unit (IMU). These values appear in Equation (7) for the calculation of $g_2$ and $g_3$ . The reason for this unconventional solution lies in the intention of implementing and testing the backstepping controller on a real aircraft. The possibility to effectively estimate these variables greatly simplifies the structure of the autopilot system and significantly reduces the development time and cost. The need for a measure of $\alpha$ and $\beta$ would be undermined by the lack of affordable, reliable and compact aerodynamic angles sensors suitable for small UAVs.

Figure 3. Aircraft employed for the numerical simulations.

5.1 Simulink simulations

Simulink simulations are performed by integrating the equations through a second-order Heun method with 0.01s time step; continuous time blocks are employed. The block scheme follows the structure of Fig. 2. The non-linear equations of motion of Ref. (Reference Etkin and Reid29) are adopted in the aircraft block. Here, actuator transfer functions as from Ref. (Reference Marguerettaz, Sartori, Guglieri and Quagliotti32) and a simplified linear motor model are included. Standard continuous-time Simulink PID blocks are employed. Note that these blocks contain a low-pass filter in the derivative action, $D\frac{C_{PID}}{1+C_{PID}/s}$ , with D derivative gain and s complex variable. The default coefficient value of $C_{PID}=100$ is maintained.

The controller is applied to two self-developed non-linear models representing the MH850 UAV and the Cessna 172P aircraft (Fig. 3). The MH850 is characterised by a tailless configuration, electric propulsion and non-movable vertical fins at the wingtips^{(Reference Marguerettaz, Sartori, Guglieri and Quagliotti32)}. The wingspan is 85cm, the mass 1kg and the cruise speed 15m/s. Aerodynamic control is achieved with elevons, which control the longitudinal motion when symmetrically deflected and lateral–directional motion when antisymmetrically deflected. A numerically derived database including of all aerodynamic derivatives is available to build the non-linear aircraft model^{(Reference Capello, Guglieri, Marguerettaz and Quagliotti33,Reference Capello, Guglieri, Quagliotti and Sartori34)} . The Cessna 172P is a single-combustion-engine aircraft with a standard configuration including high wing and fixed, tricycle landing gear. Take-off weight is around 880kg, and wingspan 11m. The aircraft is powered by a Lycoming O-320-D2J engine able to produce 160hp and guaranteeing a cruise speed of 60m/s. The control surfaces are aileron, elevator and rudder. Its choice is motivated by two reasons: it is a popular aircraft with much technical data available; a detailed aircraft model is available in FlightGear. The two aircraft differ considerably, not only in terms of absolute weight, dimension and power. Relative characteristics of the C172P, such as power-to-weight ratio and wing loading, are poorer than those of the large majority of small UAVs (Table 3). Testing the controller on a lower-performance platform allows to prove its universality and identify its limits.

Table 3 Aircraft specific properties

It is interesting to explain how the calculation of the commands for the MH850 rudderless configuration is performed. As pointed out above, the $\delta_e$ command is found from the $M(\delta_e)$ equation as $C_{m \delta_a}=0$ . Both L and N moments are functions of the remaining $\delta_a$ command, which generates an overdetermined system of two equations with one unknown. It is chosen to disregard the $N(\delta_a)$ equation and to obtain $\delta_a$ from $L(\delta_a)$ . This is motivated by the strong predominance of the rolling moment over the yawing moment in case of aileron deflection, for the MH850 being $C_{l\delta_a}\approx 10 \cdot C_{n\delta_a}$ .

Initially, the MH850 response to contemporary step inputs is tested. Reference values are arbitrarily defined as $V^{ref}=17$ m/s, $h^{ref}=120$ m and $\psi^{ref}=30^{\circ}$ , typical figures expected in standard operations. Longitudinal and lateral–directional commands are applied at the same time. The outer-loop responses are presented in Fig. 4(a) and demonstrate the capability of the controller to effectively achieve good tracking and short settling time. Although no rudder is used, the response on $\psi$ is still satisfactory with just aileron control. Figure 4(b) and (c) respectively show the inner-loop responses and the commands. Each of the inner-loop plots includes the reference value, the state estimated within the backstepping controller and the aircraft state. Accurate velocity tracking is achieved thanks to excellent control of the angle-of-attack in the inner loop. In this case, $\alpha$ is bounded to $\pm 12^{\circ}$ in order to avoid near-stall conditions. The sideslip angle $\beta$ shows some oscillations originating during the step transition. The limited directional damping provided by the vertical fins at the wingtips might be responsible for this. In any case, the magnitude of the oscillations is minimal, with a peak smaller than $0.4^{\circ}$ , barely noticeable in flight. The deflection of the elevons always remains within the $20^{\circ}$ maximum value, and throttle saturation is measured only for few seconds after the step input start.

Figure 4. Simulink responses for MH850.

On the same plots, the comparison with a well-tuned PID controller is proposed. The outer-loop PID gains remain unchanged, while the backstepping controller is replaced by two inner-loop PIDs. The first one determines $\delta_e$ according to the pitch angle error, the reference $\theta$ being the output of the outer PID on V; The second one defines $\delta_a$ based on the roll angle error, the reference $\phi$ being obtained from the outer PID on $\psi$ . A similar configuration, commonly employed in commercial autopilots, was illustrated in Ref. (Reference Capello35). In the outer loop, the PID performance is almost comparable to backstepping, the V response being slightly more oscillatory while the $\psi$ response is slower and has larger overshoot. Similarly, no significant difference is observable in the inner loop. Instead, the commands $\delta_e$ and $\delta_a$ from the PID control show a higher oscillatory behaviour, while the altitude and throttle remain basically unchanged.

A validation of the controller robustness to aircraft parametric uncertainties is performed with two test cases. In both of them, significant variations in aircraft mass, inertia and static margin are introduced. These parameters are altered in the non-linear aircraft model, while the controller remains unchanged. Case 1 considers a heavier aircraft, with higher inertia and a reduced static margin, so that the derivatives $C_{m\alpha}$ and $C_{m \delta_e}$ are weaker. In case 2, the aircraft is lighter and has lower inertia, and its centre of gravity is moved forward, so that the magnitude of the derivatives is higher. The variation in m, I, $C_{m\alpha}$ and $C_{m \delta_e}$ is $\pm$ 30 $\%$ from the nominal values. Figure 5 shows the obtained results in comparison with the nominal case. In the outer-loop response of Fig. 5(a), V and $\psi$ , which are indirectly controlled via backstepping, remain almost unchanged. Slightly higher oscillations in V are observable for case 1 due to a lower pitch damping. The altitude response, controlled with throttle solely through PID, suffers stronger variations from the nominal case. As expected, the aircraft with higher mass and inertia has a slower response to step input, and higher overshoot and settling time. In the inner loop, see Fig. 5(b), lateral oscillations are increased in amplitude in case 1. It is interesting to observe how the $\alpha$ trim values change in the two cases. The plot of commands in Fig. 5(c) confirms that full throttle command is required for longer for the heavier aircraft of case 1. Meaningful step response parameters such as overshoot os, rise time $t_r$ and settling time $t_s$ are listed in Table 4. As explained in Section 4.2, the goal of the heuristic approach used to tune the controllers, both PID and backstepping, is to minimise their values.

Figure 5. Simulink responses for MH850 in presence of uncertainties.

Table 4 Step response parameters for nominal backstepping, PID and backstepping in presence of uncertainties; $t_r$ and $t_s$ are in seconds, os has the unit of the considered variable

The PID controller introduced above is tested for the same perturbed aircraft configurations, and the results are presented in Fig. 6. It is evident that the nominal PID controller is not able to withstand the uncertainties introduced in both cases. The aircraft loses directional control and accelerates while quickly losing altitude, crashing into the ground in less than 10s. Despite the backstepping and PID controllers being almost equivalent in the nominal case, it is clear that a traditional PID configuration is not able to deal with substantial changes in the aircraft parameters. On the contrary, the backstepping controller is proved to be robust as it guarantees satisfactory performance in all cases.

Figure 6. Simulink responses for MH850 with PID controller in presence of uncertainties.

To demonstrate the ability of the backstepping controller to withstand sensor noise, the same simulation is performed but including this disturbance. White Gaussian noise is introduced on velocity, altitude and heading angle measurements. Noise characteristics are based on real data from available sensors: a pitot tube with standard deviation $\sigma_V=0.3$ m/s, a barometric pressure sensor with $\sigma_h=0.5$ m and a magnetometer with $\sigma_{\psi}=1$ deg. A Kalman filter is applied to each noisy feedback variable to mitigate the effect of the disturbance. The simulation results appear in Fig. 7. A comparison with Fig. 4 shows that the aircraft response is equivalent, in particular for the outer-loop variables. Here the velocity is the state most influenced by noise, but it still shows a satisfactory response. In fact, when steady state is achieved, the standard deviation is just 0.074m/s. The inner loop is more affected by noise because of the derivative operation in the PID controller. This causes $\alpha^{ref}$ and $p_s^{ref}$ to sustain high-frequency oscillations which, on the contrary, are not present in the values of $\alpha$ and $p_s$ estimated within the backstepping controller. It is necessary to point out that the PID gains for the noisy example have been slightly adjusted compared with the noise-free case. The major change is the reduction of the derivative gains for the V and h loops. Note that, without any adjustment, the controller would still guarantee satisfactory reference tracking, albeit with a more disturbed response. The calculated commands shown in Fig. 7(c), in particular for the elevator, are also affected by noise but remain compatible with the dynamic response of the actuators.

Figure 7. Simulink responses for MH850 in presence of noise.

The C172P responses to ramp inputs are shown in Fig. 8(a). Excellent tracking performance is achieved, in particular for the speed, as previously observed for the MH850. The different nature of the reference signal is motivated by the different responses expected from the two aircrafts: aggressive for the UAV and progressive for the Cessna. A more aggressive request to the C172P, for instance a higher climbing rate, would still result in zero altitude steady-state error, but with a larger deviation in the climbing phase. This is not due to a problem with the controller but to the lack of power of the C172P. In the C172P case, the presence of the rudder command guarantees excellent heading angle tracking. The less demanding references generate a smoother behaviour of the inner-loop variables, as shown in Fig. 8(b). Note that the steady $0.4^{\circ}$ error in the $\alpha$ tracking is motivated by the effect that the propeller-generated induced velocity has on the elevator. The different flow velocity on the elevator changes its moment contribution to the aircraft equilibrium and thus the trim angle-of-attack. This phenomenon is included in the simulated aircraft model but not in the simplified backstepping controller aircraft scheme. Saturation is observed for the elevator and the throttle in Fig. 8(c).

Figure 8. Simulink responses for C172P.

5.2 Software-in-the-loop simulations

As a first step towards achieving real-time implementation on a microcontroller board, the control law is implemented in C code and applied to the C172P model existing within FlightGear simulator, the aircraft having the same features as described in Section 5.1. The adopted architecture is presented in Fig. 9.

Figure 9. SIL layout.

Figure 10. FlightGear SIL simulated manoeuvre for C172P.

FlightGear is a freeware open-source flight simulator developed by volunteers around the world that offers academic developers a well-used tool to test their aircraft models and control laws; see for instance Refs (Reference Park, Kim, Chang, Kim, Kim and Kim36), (Reference Jung and Tsiotras37). FlightGear version 2.6.0 is used, and the JSBSim flight dynamics library is employed. JSBSim is an open-source flight dynamics model defining the six-degree-of-freedom equations that characterise aircraft motion. Data transfer between the C application and FlightGear is performed via User Datagram Protocol (UDP). FlightGear provides the value of the feedback variables V, h and $\psi$ , the backstepping controller returns the commands $\delta_e$ , $\delta_a$ , $\delta_r$ and $\delta_{th}$ . A 25Hz frequency is chosen in order to guarantee a consistent data rate compatible with real sensors, integration of the equations is performed with a second order Heun method running at 100Hz.

Figure 10 shows the results of a complex manoeuvre. The aircraft is requested first to climb and turn while accelerating, then to maintain the speed while climbing and turning more aggressively, finally to decelerate while performing another turn and rapidly losing altitude. All variables are tracked with good accuracy in every phase of the manoeuvre. Speed control performs the best, where the quick response is guaranteed by the choice of using the elevator instead of the throttle. Similarly, the heading angle shows good results despite some mild overshoot. The altitude response is penalised by some overshoot/undershoot and some mild oscillations in the settling phase. The difference in slope between reference and actual values is caused by the slower engine response and the low power-to-weight ratio. In this paper, priority is given to speed tracking, which is crucial to avoid stall. Finally, it is interesting to observe how the changes in altitude affect the speed. The commands are plotted in Fig. 11. The surface deflections always remain well within the saturation limits, viz. $20^{\circ}$ for elevator and aileron or $16^{\circ}$ for rudder, while a rate limiter is imposed. The motor instead goes full throttle during the climbing phases.

Figure 11. FlightGear SIL simulated manoeuvre commands for C172P.