Robust State and Output Feedback Control of Launched MAVs with Unknown Varying External Loads

Abstract

The operation of launched micro aerial vehicles (MAVs) with coaxial rotors is usually subject to unknown varying external disturbance. In this paper, a robust controller is designed to reject such uncertainties and track both position and orientation trajectories. A complete dynamic model of coaxial-rotor MAV is firstly established. When all system states are available, a nonlinear state-feedback control law is proposed based on feedback linearization and Lyapunov analysis. Further, to overcome the practical challenge that certain states are not measurable, a high gain observer is introduced to estimate unavailable states and an output feedback controller is developed. Rigid theoretical analysis verifies the stability of the entire closed-loop system. Additionally, extensive simulation studies have been conducted to validate the feasibility of the proposed scheme.

Appendix

Proof of Theorem 1

Considering a candidate Lyapunov function

$$ {V} = V_{p} + V_{\psi} $$

(48)

with

$$ {V_{p}} = \frac{1}{2}{r_{p}^{T}}{r_{p}}, \quad {V_{\psi}} = \frac{1}{2}r_{\psi}^{T}{r_{\psi}} $$

(49)

Recalling Eq. 29, the time derivative of V_pis given by

$$ \begin{array}{rl} \dot V_{p}=&{r_{p}^{T}} \dot {r_{p}^{T}}\\ =&-c_{p} {r^{T}_{p}} r_{p}+{r_{p}^{T}}{\Phi}_{p}-{r^{T}_{p}}\, \text{sgn} \left( r_{p}\right){\Theta}_{p}\\ \le& -c_{p} {r^{T}_{p}} r_{p}+ {\vert{{r^{T}_{p}}}\vert} {{\Theta}_{p}}-{r^{T}_{p}}\, \text{sgn} (r_{p}){\Theta}_{p}\\ \le& -c_{p} {r^{T}_{p}} r_{p} \end{array} $$

(50)

Further, Eq. 32 can be employed to obtain the derivative of V_ψ

$$ \begin{array}{rl} \dot V_{\psi}=&r_{\psi}^{T} \dot r_{\psi}^{T}\\ =&- c_{\psi} r^{T}_{\psi} r_{\psi} + r^{T}_{\psi}{\Phi}_{\psi}- r^{T}_{\psi}\,\text{sgn} \left( r_{\psi} \right) {\Theta}_{\psi}\\ \le & -c_{\psi} r^{T}_{\psi} r_{\psi}+ {\vert r^{T}_{\psi}\vert}{\Theta}_{\psi}- r^{T}_{\psi} \,\text{sgn} \left( r_{\psi} \right){\Theta}_{\psi}\\ \le&-c_{\psi} r^{T}_{\psi} r_{\psi} \end{array} $$

(51)

Thus, the derivative of V is

$$ \begin{array}{l} \dot V=\dot V_{p} +\dot V_{\psi} \le -c_{\psi} r^{T}_{\psi} r_{\psi} -c_{p} {r^{T}_{p}} r_{p} \le c V \end{array} $$

(52)

where c = min{2c_p, 2c_ψ}. Solving Eq. 52 generates

$$ 0\le V \le V(0)e^{-ct} $$

(53)

where $V(0)=\frac {1}{2}({r_{p}^{T}}(0)r_{p}(0)+r_{\psi }^{T}(0)r_{\psi }(0))$is the initial value.

Obviously, V → 0 as t →∞. This implies that r_p, r_ψ → 0ast →∞. Subsequently, the tracking error $\delta _{p_{i}}$and $\delta _{\psi _{i}}$defined in Eqs. 16 and 23 also converge to zero asymptotically by definition. Thus, the closed-loop system can asymptotically track the reference trajectories p_dandψ_d. With the help of Eq. 18 and Assumption 1 − 2,v and T_z are bounded. R_η and Q_η are also bounded since $-\frac {\pi }{2}< \theta , \phi < \frac {\pi }{2}$. Similarly, the signals ω_x, ω_y and $\dot T_{z}$ are all bounded according to Eq. 18. Through Eq. 25, one can reach that ω_z is bounded, and so is $\dot Q_{\eta }$. Thus,the two control inputs τ and $\ddot T_{z}$are also boundedfrom Eq. 33. □

Proof of Theorem 2

Consider the following Lyapunov function candidate

$$ {\overline V} = \overline{V}_{p} + \overline{V}_{\psi} $$

(54)

with

$$ {\overline V_{p}} = \frac{1}{2}{r_{p}^{T}}{r_{p}}+ {e_{p}^{T}}P{e_{p}}, \quad {\overline V_{\psi}} = \frac{1}{2}r_{\psi}^{T}{r_{\psi}} $$

(55)

Define the estimated filtered position tracking error as

$$ {\hat r_{p}} = {k_{{p_{0}}}}{\delta_{p_{0}} + {k_{{p_{1}}}}{\delta_{p_{1}}}{+ }{k_{{p_{2}}}}{\hat\delta_{p_{2}}}{+}\hat \delta_{p_{3}}} $$

(56)

If enforcing the output feedback controller (45), the filtered position tracking error dynamics (27) can be rewritten as

$$ \begin{array}{rl} {\dot r_{p}} =&-\tanh \left( \frac{\hat r_{p}}{\varepsilon_{p} }\right){\Theta}_{p}+{\Phi}_{p} +{k_{{p_{0}}}}{\delta_{p_{1}} + {k_{{p_{1}}}}{\delta_{p_{2}}}}\\ &+{k_{{p_{2}}}}{\delta_{p_{3}}}- {c_{p}}{ \hat r_{p}} - \left( {k_{p_{0}}}{\delta_{p_{1}}} + {k_{p_{1}}}{ \hat \delta_{p_{2}}}{+}{k_{p_{2}}}{\hat \delta_{p_{3}}}\right)\\ =&-\tanh \left( \frac{\hat r_{p}}{\varepsilon_{p} }\right){\Theta}_{p}+{\Phi}_{p}+(k_{p_{1}}+c_{p}k_{p_{2}})e_{p_{2}}\\ &+(k_{p_{2}}+c_{p})e_{p_{3}}-c_{p}r_{p} \end{array} $$

(57)

Thus, the derivative of $\overline V_{p}$can be given by

$$ \begin{array}{rl} {\dot {\overline V}}_{p} =& {r_{p}^{T}}{\dot r_{p}} + \dot {e_{p}^{T}}P{e_{p}} + {e_{p}^{T}}P{\dot e_{p}}\\ =&-{r_{p}^{T}}\tanh\left( \frac{\hat r_{p}}{\varepsilon_{p} }\right) {\Theta}_{p} +{r_{p}^{T}} {\Phi}_{p}+(k_{p_{1}}+c_{p}k_{p_{2}}){r_{p}^{T}}e_{p_{2}}\\ &+(k_{p_{2}}+c_{p}){r_{p}^{T}}e_{p_{3}}-c_{p} {r_{p}^{T}} r_{p}+ \dot {e_{p}^{T}}P{e_{p}} + {e_{p}^{T}}P{\dot e_{p}}\\ \end{array} $$

(58)

Substituting Eqs. 40 and 42 into Eq. 58 generates

$$ \begin{array}{rl} {\dot {\overline V}}_{p} =&-{r_{p}^{T}}\tanh \left( \frac{\hat r_{p}}{\varepsilon_{p} }\right){\Theta}_{p} +{r_{p}^{T}} {\Phi}_{p}-c_{p} {r_{p}^{T}} r_{p} \\ &+(k_{p_{1}}+c_{p}k_{p_{2}}){r_{p}^{T}}e_{p_{2}}+(k_{p_{2}}+c_{p}){r_{p}^{T}}e_{p_{3}}\\ &+ {e_{p}^{T}}({A^{T}}P + PA){e_{p}} + 2 {e_{p}^{T}}PB{\Phi}_{p}\\ =&-{r_{p}^{T}}\tanh \left( \frac{\hat r_{p}}{\varepsilon_{p} }\right){\Theta}_{p}+{r_{p}^{T}} {\Phi}_{p}-c_{p} {r_{p}^{T}} r_{p} \\ &+(k_{p_{1}}+c_{p}k_{p_{2}}){r_{p}^{T}}e_{p_{2}}+(k_{p_{2}}+c_{p}){r_{p}^{T}}e_{p_{3}}\\ &-\lambda {e_{p}^{T}}{e_{p}} + 2 {e_{p}^{T}}PB{\Phi}_{p}\\ \end{array} $$

(59)

Meanwhile, by resorting to Lemma 1, it is apparent that for the vector$r_{p} \in \mathbb {R}^{3}$ and a positive constant ε_p > 0,

$$\begin{array}{@{}rcl@{}} 0 &\le& {r_{p}^{T}}\left( {\text{sgn}} ({r_{p}}) -\tanh \left( \frac{{{r_{p}}}}{\varepsilon_{p}}\right)\right)\\ &=& {| {{r_{p}}} |} - {r_{p}^{T}}\tanh \left( \frac{{{r_{p}}}}{\varepsilon_{p} }\right) \le 3\kappa \varepsilon_{p} \end{array} $$

(60)

Thus, one further has

$$ \begin{array}{rl} {\dot {\overline V}}_{p} =&-{r_{p}^{T}}\tanh \left( \frac{\hat r_{p}}{\varepsilon_{p} }\right){\Theta}_{p} +{r_{p}^{T}} {\tanh} \left( \frac{r_{p}}{\varepsilon_{p} }\right){\Theta}_{p}\\ & -{r_{p}^{T}} {\tanh} \left( \frac{r_{p}}{\varepsilon_{p} }\right){\Theta}_{p} + {r_{p}^{T}} \text{sgn} \left( {{ r}_{p}}\right){\Theta}_{p}\\ &-{r_{p}^{T}} \text{sgn} \left( {{ r}_{p}}\right){\Theta}_{p}+{r_{p}^{T}} {\Phi}_{p}-c_{p} {r_{p}^{T}} r_{p} \\ &+(k_{p_{1}}+c_{p}k_{p_{2}}){r_{p}^{T}}e_{p_{2}}+(k_{p_{2}}+c_{p}){r_{p}^{T}}e_{p_{3}}\\ &-\lambda {e_{p}^{T}}{e_{p}} + 2 {e_{p}^{T}}PB{\Phi}_{p}\\ \le&-{r_{p}^{T}}\tanh \left( \frac{\hat r_{p}}{\varepsilon_{p} }\right){\Theta}_{p}+{r_{p}^{T}} {\tanh} \left( \frac{r_{p}}{\varepsilon_{p} }\right){\Theta}_{p} \\ &+ 3\kappa\varepsilon_{p}{\Theta}_{p}-{r_{p}^{T}} \text{sgn} \left( {{ r}_{p}}\right){\Theta}_{p}+{r_{p}^{T}} {\Phi}_{p}-c_{p} {r_{p}^{T}} r_{p} \\ &+(k_{p_{1}}+c_{p}k_{p_{2}}){r_{p}^{T}}e_{p_{2}}+(k_{p_{2}}+c_{p}){r_{p}^{T}}e_{p_{3}}\\ &-\lambda {e_{p}^{T}}{e_{p}} + 2 {e_{p}^{T}}PB{\Phi}_{p}\\ \le&3\kappa\varepsilon_{p}{\Theta}_{p}+{r_{p}^{T}}\left( {\tanh} \left( \frac{r_{p}}{\varepsilon_{p} }\right)-\tanh \left( \frac{\hat r_{p}}{\varepsilon_{p} }\right)\right){\Theta}_{p} \\ &(k_{p_{1}}+c_{p}k_{p_{2}}){r_{p}^{T}}e_{p_{2}}+(k_{p_{2}}+c_{p}){r_{p}^{T}}e_{p_{3}}\\ &-c_{p} {r_{p}^{T}} r_{p}-\lambda {e_{p}^{T}}{e_{p}} + 2 {e_{p}^{T}}PB{\Phi}_{p}\\ \end{array} $$

(61)

With the help of Young’s inequality and the fact that | tanh(⋅)|≤ 1, Eq. 61 results in

$$ \begin{array}{rl} {\dot {\overline V}}_{p} \le&3\kappa\varepsilon_{p}{\Theta}_{p}+\frac{{{r_{p}^{T}}{r_{p}}}}{{2{\varepsilon_{1}}}} + 6{\varepsilon_{1}}{{\Theta}_{p}^{2}}-c_{p} {r_{p}^{T}} r_{p}\\ & +\left( {k_{{p_{1}}}}+{c_{p}}{k_{{p_{2}}}}\right)\left( \frac{{{r_{p}^{T}}{r_{p}}}}{{2{\varepsilon_{2}}}} + \frac{{\varepsilon_{2}}e_{{p_{2}}}^{T}{e_{{p_{2}}}}}{2}\right)\\ &+ ({k_{{p_{2}}}}+{c_{p}})\left( \frac{{{r_{p}^{T}}{r_{p}}}}{{2{\varepsilon_{3}}}} + \frac{\varepsilon_{3} e_{p_{3}}^{T} e_{p_{3}}}{2}\right)-\lambda {e_{p}^{T}}{e_{p}}\\ &+\varepsilon_{4} {e_{p}^{T}}e_{p}+\frac{{\Vert {PB{\Phi}_{p} } \Vert^{2}}}{\varepsilon_{4}}\\ \le& -\left( c_{p}-\frac{1}{2\varepsilon_{1}}-\frac{{k_{{p_{1}}}}+{c_{p}}{k_{{p_{2}}}}}{2\varepsilon_{2}} - \frac{ k_{p_{2}}+c_{p} } {2\varepsilon_{3}}\right){r_{p}}^{T}{r_{p}}\\ &- \left( \lambda - \frac{(k_{p_{1}}+c_{p}k_{p_{2}})\varepsilon_{2}}{2} -\frac{(k_{p_{2}}+c_{p})\varepsilon_{3}}{2} - {\varepsilon_{4}}\right){e_{p}^{T}}{e_{p}}\\ &+ 3\kappa\varepsilon_{p}{\Theta}_{p}+ 6{\varepsilon_{1}}{{\Theta}_{p}^{2}}+\frac{{\Vert {PB{\Phi}_{p} } \Vert^{2}}}{\varepsilon_{4}} \end{array} $$

(62)

where ∥⋅∥denotes the L₂-norm,and ε, ε₁, ε₂, ε₃, ε₄$\in \mathbb {R}^{+}$.

Similarly, with the controlled designed in Eq. 45, the filtered orientation tracking error dynamics can be rephrased tobe

$$ \begin{array}{ll} {\dot r_{\psi} } = & - {c_{\psi} }{r_{\psi} - {\tanh} \left( \frac{{r_{\psi} }}{\varepsilon_{\psi}} \right){\Theta}_{\psi}+{\Phi}_{\psi} } \end{array} $$

(63)

Hence, the derivative of $\overline V_{\psi }$ can be given by

$$ \begin{array}{rl} {\dot {\overline V}}_{\psi} =& r_{\psi}^{T}{\dot r_{\psi}}=- r_{\psi}^{T}\tanh \left( \frac{{r_{\psi} }}{\varepsilon_{\psi}} \right){\Theta}_{\psi}+r_{\psi}^{T}{\Phi}_{\psi}- {c_{\psi} }r_{\psi}^{T}{r_{\psi} }\\ \le& - {c_{\psi} }r_{\psi}^{T}{r_{\psi} }+\kappa\varepsilon_{\psi}{\Theta}_{\psi} \end{array} $$

(64)

Combining Eq. 62 with Eq. 64 delivers

$$ \begin{array}{rl} \dot {\overline V}\le & -\left( c_{p}-\frac{1}{2\varepsilon_{1}}-\frac{{k_{{p_{1}}}}+{c_{p}}{k_{{p_{2}}}}}{2\varepsilon_{2}} - \frac{ k_{p_{2}}+c_{p} } {2\varepsilon_{3}}\right){r_{p}}^{T}{r_{p}}\\ &- \left( \lambda - \frac{(k_{p_{1}}+c_{p}k_{p_{2}})\varepsilon_{2}}{2} -\frac{(k_{p_{2}}+c_{p})\varepsilon_{3}}{2} - {\varepsilon_{4}}\right){e_{p}^{T}}{e_{p}}\\ &- {c_{\psi} }r_{\psi}^{T}{r_{\psi} }\!+\kappa\left( 3\varepsilon_{p}{\Theta}_{p}+\varepsilon_{\psi}{\Theta}_{\psi}\right)\,+\, 6{\varepsilon_{1}}{{\Theta}_{p}^{2}}\,+\,\frac{{\Vert {PB{\Phi}_{p} } \Vert^{2}}}{\varepsilon_{4}}\\ \le&-c_{p_{1}}{r_{p}}^{T}{r_{p}}-c_{p_{2}}{e_{p}^{T}}{e_{p}}-c_{\psi}r_{\psi}^{T}{r_{\psi} }+ C \end{array} $$

(65)

where $c_{p_{1}}$,$c_{p_{2}}$, C$\in \mathbb {R}^{+}$with

$$ \begin{array}{rl} c_{p_{1}}=&c_{p}-\frac{1}{2\varepsilon_{1}}-\frac{{k_{{p_{1}}}}+{c_{p}}{k_{{p_{2}}}}}{2\varepsilon_{2}} - \frac{ k_{p_{2}}+c_{p} } {2\varepsilon_{3}}\\ c_{p_{2}}=&\lambda - \frac{(k_{p_{1}}+c_{p}k_{p_{2}})\varepsilon_{2}}{2} -\frac{(k_{p_{2}}+c_{p})\varepsilon_{3}}{2} - {\varepsilon_{4}}\\ C=&\kappa\left( 3\varepsilon_{p}{\Theta}_{p}+\varepsilon_{\psi}{\Theta}_{\psi}\right)+ 6{\varepsilon_{1}}{{\Theta}_{p}^{2}}+\frac{{\Vert {PB{\Phi}_{p} } \Vert^{2}}}{\varepsilon_{4}} \end{array} $$

(66)

Hence, $\dot {\overline V}$will becomes negative as long as

$$ r_{p} \notin {\Omega}_{r_{p}}=\left\{ r_{p}\bigg{|}{\Vert r_{p} \Vert} \le \sqrt{\frac{C}{c_{p_{1}}}} \right\} $$

(67)

or

$$ e_{p} \notin {\Omega}_{e_{p}}=\left\{ e_{p}\bigg{|}{\Vert e_{p} \Vert} \le \sqrt{\frac{C}{c_{p_{2}}}} \right\} $$

(68)

or

$$ r_{\psi} \notin {\Omega}_{r_{\psi}}=\left\{ r_{\psi}\bigg{|}{\Vert r_{\psi} \Vert} \le \sqrt{\frac{C}{c_{\psi}}} \right\} $$

(69)

According to the standard Lyapunov analysis, one has that the filtered tracking errorsr_p, r_ψ, and theobserver error e_pare uniformly ultimately bounded (UUB). Furthermore, the tracking errors can be arbitrarily reduced by increasing control gainsc_ψ, c_p, h₁, h₂andh₃. By following the similar analysisin Proof of Theorem 1, the signals T_z, $\dot T_{z}$,ω are bounded as well asall the elements of R_η, Q_ηand$\dot Q_{\eta }$. Furthermore, theestimation signals $\hat \delta _{p_{2}}$,$\hat \delta _{p_{3}}$and$\hat r_{p}$are bounded since the observererror vector e_pis bounded.Thus, the control signals $\ddot T_{z}$and τ are proved to bebounded from Eq. 45. □