Model Predictive Control for a Linear Parameter Varying Model of an UAV

Abstract

This paper presents a Model Predictive Control (MPC) based autopilot for a fixed-wing Unmanned Aircraft Vehicle (UAV) for meteorological data sampling tasks, named Aerosonde. Aerosonde missions are featured by predetermined operating conditions, allowing the design of ad-hoc controllers for each control task by using the future knowledge of the reference signals driving the aircraft during operations. To develop the controller, the nonlinear dynamics of the vehicle has been described by a Linear Parameter-Varying (LPV) model identified from the plant data by using a subspace identification technique. The LPV model is used to design a MPC to drive the UAV. Two different Linear Parameter-Varying MPC (MPC_LPV) algorithms have been proposed by introducing the previewing technique in the controller due to the a priori knowledge of full reference signals. In the design of the inner Attitude Controller (AC), a future LPV scheduling parameters estimation policy has been introduced (P_F −MPC_LPV) for improving the control results of the standard Previewing MPC_LPV (P-MPC_LPV). Furthermore, an anticipative switching approach (P_S −MPC_S) has been considered for the altitude External Controller (EC) to improve the control performances of the standard previewing switching MPC (P-MPC_S). Both P_F −MPC_LPV and P_S −MPC_S algorithms have been compared to the P-MPC_LPV and P-MPC_S baseline algorithms, showing the effectiveness of proposed methods.

Appendix A: UAV Nonlinear Model and Parameters

$$ \begin{array}{@{}rcl@{}} &&V_{a}=\| V_{b}-W_{b} \|_{2} \end{array} $$

(26)

$$ \begin{array}{@{}rcl@{}} &&\alpha=\arctan(\frac{w}{u}) \end{array} $$

(27)

$$ \begin{array}{@{}rcl@{}} &&\beta=\arcsin(\frac{v}{V_{a}}) \end{array} $$

(28)

$$ \begin{array}{@{}rcl@{}} &&\dot \alpha=\frac{d \alpha}{dt} \end{array} $$

(29)

$$ \begin{array}{@{}rcl@{}} &&\dot \beta=\frac{d \beta}{dt} \end{array} $$

(30)

$$ \begin{array}{@{}rcl@{}} &&p_{cd}=0.5\rho {V_{a}^{2}} \end{array} $$

(31)

$$ \begin{array}{@{}rcl@{}} &&M=\frac{V_{a}}{a} \end{array} $$

(32)

$$ \begin{array}{@{}rcl@{}} &&C_{L}=C_{L0}+C_{L}^{\alpha}\alpha+C_{L}^{\delta_{f} }\delta_{f}+C_{L}^{\delta_{e}}\delta_{e}+\frac{c}{2V_{a}}\left( C_{L}^{\dot\alpha}\dot\alpha+{C_{L}^{q}} q\right)+{C_{L}^{M}} M \end{array} $$

(33)

$$ \begin{array}{@{}rcl@{}} &&C_{D}=C_{D0}+\frac{(C_{L0}-C_{L})^{2}}{\pi e \frac{b^{2}}{S}}+C_{D}^{\delta_{f} }\delta_{f}+C_{D}^{\delta_{e}}\delta_{e}+C_{D}^{\delta_{a}}\delta_{a}+C_{D}^{\delta_{r}}\delta_{r}+{C_{D}^{M}} M \end{array} $$

(34)

$$ \begin{array}{@{}rcl@{}} &&C_{Y}=C_{Y}^{\beta} \beta+C_{Y}^{\delta_{a}}\delta_{a}+C_{Y}^{\delta_{r}}\delta_{r}+\frac{b}{2V_{a}}({C_{Y}^{p}} p+{C_{Y}^{r}} r) \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} &&C_{m}=C_{m0}+C_{m}^{\alpha} \alpha+C_{m}^{\delta_{f} }\delta_{f}+C_{m}^{\delta_{e}}\delta_{e}+\frac{c}{2V_{a}}(C_{m}^{\dot \alpha}\dot \alpha+{C_{m}^{q}} q)+{C_{m}^{M}} M\end{array} $$

(36)

$$ \begin{array}{@{}rcl@{}} &&C_{l}=C_{l}^{\beta} \beta+ C_{l}^{\delta_{a}} \delta_{a}+C_{l}^{\delta_{r}} \delta_{r}+\frac{b}{2V_{a}}({C_{l}^{p}} p+{C_{l}^{r}} r) \end{array} $$

(37)

$$ \begin{array}{@{}rcl@{}} &&C_{n}=C_{n}^{\beta} \beta+ C_{n}^{\delta_{a}} \delta_{a}+C_{n}^{\delta_{r}} \delta_{r}+\frac{b}{2V_{a}}({C_{n}^{p}} p+{C_{n}^{r}} r) \end{array} $$

(38)

$$ \begin{array}{@{}rcl@{}} &&F_{D}=0.5\rho {V_{a}^{2}} S C_{D} \end{array} $$

(39)

$$ \begin{array}{@{}rcl@{}} &&F_{Y}=0.5\rho {V_{a}^{2}} S C_{Y} \end{array} $$

(40)

$$ \begin{array}{@{}rcl@{}} &&F_{L}=0.5\rho {V_{a}^{2}} S C_{L} \end{array} $$

(41)

$$ \begin{array}{@{}rcl@{}} &&M_{l}=0.5\rho {V_{a}^{2}} S C_{l} \end{array} $$

(42)

$$ \begin{array}{@{}rcl@{}} &&M_{m}=0.5\rho {V_{a}^{2}} S C_{m} \end{array} $$

(43)

$$ \begin{array}{@{}rcl@{}} &&M_{n}=0.5\rho {V_{a}^{2}} S C_{n} \end{array} $$

(44)

$$ \begin{array}{@{}rcl@{}} &&J=\frac{\pi V_{a}}{{\varOmega} R_{pr}} \end{array} $$

(45)

$$ \begin{array}{@{}rcl@{}} &&F_{prop}=\frac{4}{\pi^{2}}\rho R_{pr}^{4}{\varOmega}^{2}C_{T} \end{array} $$

(46)

$$ \begin{array}{@{}rcl@{}} &&M_{prop}=-\frac{4}{\pi^{3}}\rho R_{pr}^{5}{\varOmega}^{2}C_{P} \end{array} $$

(47)

$$ \begin{array}{@{}rcl@{}} &&[F_{uelFlow},T_{q},A_{f}]=f(T_{hr},p_{cd},M_{ix},{\varOmega},T) \end{array} $$

(48)

$$ \begin{array}{@{}rcl@{}} &&(I_{pr}+I_{eng})\dot {\varOmega}=M_{prop}+M_{eng} \end{array} $$

(49)

$$ \begin{array}{@{}rcl@{}} &&c_{1}-c_{9} \leftarrow [CG_{pos},M_{ass},J_{x},J_{y},J_{z},J_{xz}]=g(F_{uelFlow}) \end{array} $$

(50)

Table 7 Aircraft model parameters

Table 8 Aircraft model parameters