Backstepping-Based Nonlinear RISE Feedback Control for an Underactuated Quadrotor UAV Without Linear Velocity Measurements

Abstract

In this paper, a robust three dimensional output feedback control problem is proposed for a 6-degrees-of-freedom model of a quadrotor unmanned aerial vehicle (UAV) to track a bounded and sufficiently smooth reference trajectory in the presence of slowly varying force disturbances. Due to the underactuation structure of the UAV, a nonlinear output feedback controller based on the robust integral of the sign error signal (RISE) mechanism is first designed for the translational dynamics to ensure position reference tracking without velocity measurement. The angular velocity is then regarded as intermediate control signal for the rotational dynamics to fulfill the task of attitude angle reference tracking. The torque input is designed taking full advantage of the smooth exact differentiator that circumvents derivatives computation of virtual controls, the backstepping technique is then judiciously modified to allow the use of the RISE control technique to compensate for the external disturbances. The proposed controller yields semi-global asymptotic stability tracking despite the added disturbances in the dynamics. Simulation results are shown to demonstrate the proposed approach.

Appendix

For the clarity of the stability proof, we first write the closed loop-system by substituting the control input (15.35) into the dynamics (15.34), yields

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left[ \begin{array}{c} \dot{\boldsymbol{\eta}} \\ \dot{\mathbf{e}}_\psi \\ \end{array} \right]&=&\left[ \begin{array}{c} (k_2+1)({\mathbf{r}}_f-\boldsymbol{\eta})-\mathbf{S}(\omega)\boldsymbol{\eta}+\widetilde{\mathbf{M}}+{\mathbf{M}}_d-K_1\text{Sgn}({\mathbf{e}}_p+{\mathbf{e}}_f)-{\mathbf{e}}_p \\ 0 \\ \end{array} \right]\notag\\ &&+\overline{\mathbf{B}}_C\boldsymbol{z} {} \end{array} \end{aligned} $$

(15.40)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{I}}_f\dot{\mathbf{r}}_a&=&\frac{1}{k_3}\Big(\mathbf{D}-K_2\text{Sng}(\boldsymbol{z})-k_3(k_3{\mathbf{I}}_f+{\mathbf{I}}_3){\mathbf{r}}_a\Big)+{\mathbf{I}}_f({\mathbf{r}}_a-\boldsymbol{z}){} \end{array} \end{aligned} $$

(15.41)

where we have used the following relationship:

$$\displaystyle \begin{aligned} {\mathbf{B}}_C\left[ \begin{array}{c} \boldsymbol{z} \\ 0\\ \end{array} \right]=\left[ \begin{array}{c} -\mathbf{S}(\boldsymbol{\delta}) \boldsymbol{z} \\ T_{\boldsymbol{z}}(\boldsymbol{\varTheta})\boldsymbol{z}\\ \end{array} \right]=\underbrace{\left[ \begin{array}{c} -\mathbf{S}(\boldsymbol{\delta}) \\ T_{\boldsymbol{z}}(\boldsymbol{\varTheta})\\ \end{array} \right]}_{\overline{\mathbf{B}}_C}\boldsymbol{z} \end{aligned}$$

For the closed-loop system (15.41) and (15.40), we consider the Lyapunov function candidate function V  which is Lipschitz positive definite function defined as:

$$\displaystyle \begin{aligned} V(\boldsymbol{y},t)=V_1+V_2+P+Q \end{aligned} $$

(15.42)

where we define the functions V ₁ and V ₂ as follows

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_1 &\displaystyle =&\displaystyle \frac{m}{2}{\mathbf{e}}_p^\top {\mathbf{e}}_p+\frac{1}{2}{\mathbf{e}}_f^\top {\mathbf{e}}_f+\frac{1}{2} {\mathbf{r}}_f^\top {\mathbf{r}}_f+\frac{1}{2}\boldsymbol{\chi}^\top \boldsymbol{\chi}{} \end{array} \end{aligned} $$

(15.43)

$$\displaystyle \begin{aligned} \begin{array}{rcl} V_2 &\displaystyle =&\displaystyle \frac{1}{2}\boldsymbol{z}^\top {\mathbf{I}}_f \boldsymbol{z}+\frac{1}{2}{\mathbf{r}}_a^\top {\mathbf{I}}_f {\mathbf{r}}_a{}\vspace{-3pt} \end{array} \end{aligned} $$

(15.44)

Let y ∈ R ²⁰ be defined as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \boldsymbol{y}&\displaystyle =&\displaystyle [\boldsymbol{X}^\top,\sqrt{P(t)},\sqrt{Q(t)}]^\top{}\\ \boldsymbol{X}&\displaystyle =&\displaystyle [{\mathbf{e}}_p^\top,{\mathbf{e}}_f^\top,{\mathbf{r}}_f^\top,\boldsymbol{\eta}^\top,e_\psi,\boldsymbol{z}^\top,{\mathbf{r}}_a^\top]\notag \end{array} \end{aligned} $$

(15.45)

and the auxiliary function $P\in \mathbb {R}$ and $Q \in \mathbb {R}$ are defined as the Filippov solutions (Filippov 1964) to the following differential equations respectively:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \dot{P} &\displaystyle =&\displaystyle -\boldsymbol{\eta}^\top \Big({\mathbf{M}}_d-K_1\text{Sng}({\mathbf{e}}_p+{\mathbf{e}}_f)\Big) \end{array} \end{aligned} $$

(15.46)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \dot{Q} &\displaystyle =&\displaystyle \frac{{\mathbf{r}}_a^\top}{k_3}\Big(\mathbf{D}-K_2\text{Sgn}(\boldsymbol{z})\Big) \end{array} \end{aligned} $$

(15.47)

with initial conditions P(t ₀) and Q(t ₀) are given as follows

$$\displaystyle \begin{aligned} \begin{array}{rcl} P(t_0) &\displaystyle =&\displaystyle \sum_{i=1}^3 K_1|e_{pi}(0)| -{\mathbf{e}}_p^\top(0)M_d(0) {} \end{array} \end{aligned} $$

(15.48)

$$\displaystyle \begin{aligned} \begin{array}{rcl} Q(t_0) &\displaystyle =&\displaystyle \frac{1}{k_3}\Big(\sum_{i=1}^3 K_2|z_{i}(0)|-\boldsymbol{z}(0)^\top \mathbf{D}(0) \Big){} \end{array} \end{aligned} $$

(15.49)

with the subscript i = 1, …, n denotes the i-th element of the vectors e _p(t) and z(t). The gain control K ₁ and K ₂ are chosen according to the sufficient condition (15.3) underlined in Lemma 2.

It can be verified that the Lyapunov function candidate defined in (15.42) can be bounded as :

$$\displaystyle \begin{aligned} U_1(\boldsymbol{y})\leq V(\boldsymbol{y},t) \leq U_2(\boldsymbol{y}) \end{aligned} $$

(15.50)

where $U_1 : \mathbb {R}^{20} \rightarrow \mathbb {R}$ and $U_2 : \mathbb {R}^{20} \rightarrow R$ are positive definite functions defined as $U_1(\boldsymbol {y})=\min \{\frac {1}{2},\frac {m}{2}\}\| \boldsymbol {y}\|{ }^2$ and $U_2(\boldsymbol {y})=\max \{1,\frac {m}{2}\}\| \boldsymbol {y}\|{ }^2$. Under Filippov’s framework, strong stability of the closed-loop system will be established in the following.

The time derivative of (15.42) along the Filippov trajectories exists almost everywhere (a.e), for almost all t ∈ [t ₀, t], one can get:

(15.51)

where ∂V  is the generalized gradient of V  (Clarke 1990). Since V  is continuously differentiable with respect to y, then one has

$$\displaystyle \begin{aligned} \dot{\widetilde{V}}\subset \nabla V^\top \varPsi[\dot{\mathbf{e}}_p^\top, \dot{\mathbf{e}}_f^\top, \dot{\mathbf{r}}_f^\top, \dot{\boldsymbol{\eta}}^\top, \dot{e}_\psi, \frac{1}{2}P^{\frac{1}{2}} \dot{P}, \frac{1}{2}Q^{\frac{1}{2}} \dot{Q}, 1] \end{aligned} $$

(15.52)

where $\nabla V=\varPsi [{\mathbf {e}}_p^\top , {\mathbf {e}}_f^\top , {\mathbf {r}}_f^\top , \boldsymbol {\eta }^\top , e_\psi ,\boldsymbol {z}^{\top {\mathbf {I}}}_f,{\mathbf {r}}_a^\top {\mathbf {I}}_f, 2P^{-\frac {1}{2}},2Q^{-\frac {1}{2}}]^\top$. Substituting (15.9), (15.15), (15.24), (15.40) and (15.41) into (15.52) and using the calculus for Ψ[.] from Paden and Sastry (1987), yields:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \dot{\widetilde{V}} &\displaystyle \subset&\displaystyle {\mathbf{e}}_p^{\top} (\boldsymbol{\eta}-{\mathbf{e}}_p-{\mathbf{r}}_f-\boldsymbol{\delta})+(1-m){\mathbf{e}}_p^{\top}{\mathbf{R}}^{\top} \dot{\mathbf{p}}_d-{\mathbf{e}}_f^{\top} {\mathbf{e}}_f+{\mathbf{e}}_f^{\top} {\mathbf{r}}_f-{\mathbf{r}}_f^{\top} {\mathbf{r}}_f \\ &\displaystyle &\displaystyle -(k_2+1){\mathbf{r}}^{\top} \boldsymbol{\eta} +{\mathbf{r}}_f^{\top} {\mathbf{e}}_p-{\mathbf{r}}_f^{\top} {\mathbf{e}}_f-(k_2+1)\boldsymbol{\eta}^{\top} \boldsymbol{\eta}+\boldsymbol{\eta}^{\top} \widetilde{\mathbf{M}}\\ &\displaystyle &\displaystyle +\boldsymbol{\eta}^{\top} \Big({\mathbf{M}}_d-K_1\varPsi[\text{Sng}({\mathbf{e}}_p+{\mathbf{e}}_f)]\Big)-\boldsymbol{\eta}^{\top} {\mathbf{e}}_p +(k_2+1)\boldsymbol{\eta}^{\top} {\mathbf{r}}_f\\ &\displaystyle &\displaystyle +\boldsymbol{\eta}^{\top} \overline{\mathbf{B}}_C\boldsymbol{z}-\Big({\mathbf{M}}_d-K_1\varPsi[\text{Sng}({\mathbf{e}}_p+{\mathbf{e}}_f)]\Big)-k_3\boldsymbol{z}^{\top} {\mathbf{I}}_f \boldsymbol{z}-{\mathbf{r}}_a^{\top} {\mathbf{r}}_a\\ &\displaystyle &\displaystyle +\frac{1}{k_3}{\mathbf{r}}_a^{\top}\Big(\mathbf{D}-K_2\varPsi[\text{Sng}(\boldsymbol{z})]\Big)-\frac{1}{k_3}{\mathbf{r}}_a^{\top}\Big(\mathbf{D}-K_2\varPsi[\text{Sng}(\boldsymbol{z})]\Big){} \end{array} \end{aligned} $$

(15.53)

where Ψ[Sng(x)] = SGN(x) such that ∀x = [x ₁, x ₂, x ₃]^⊤, the set valued map SNG(x _i) = 1 if x _i > 0, [−1, 1] if x _i = 0 and − 1 if x _i < 0 for i = 1, 2, 3. Using the fact that the set in (15.53) reduces to a scalar equality since the right hand side is continuous a.e, the following upper bound can be obtained on $\dot {V}$:

(15.54)

Denote by $\gamma =\| \boldsymbol {\delta } \|+(m+1)\| \dot {\mathbf {p}}_d\|$, using the Mean Value Theorem (De Queiroz et al. 1997), the function $\widetilde {\mathbf {M}}$ can be upper bounded as $\| \widetilde {\mathbf {M}} \|\leq \| \varphi (\| \boldsymbol {X} \|)\| \| \boldsymbol {X} \|$, then using the young’s inequality yields:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \| -{\mathbf{e}}_p^\top\big(\boldsymbol{\delta}+(m-1){\mathbf{R}}^\top \dot{\mathbf{p}}_d\big)\| &\displaystyle \leq&\displaystyle \gamma \| {\mathbf{e}}_p\|\leq \frac{\gamma}{\| \boldsymbol{X} \|}\| \boldsymbol{X} \|{}^2 {} \end{array} \end{aligned} $$

(15.55)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \| \widetilde{\mathbf{M}} \| \| \boldsymbol{\eta} \| &\displaystyle \leq&\displaystyle \frac{\| \varphi (\| \boldsymbol{X} \|)\|{}^2}{4k_2}+k_2\| \boldsymbol{\eta}\|{}^2 {}\vspace{-3pt} \end{array} \end{aligned} $$

(15.56)

substituting (15.55) and (15.56) into (15.54), we obtain

(15.57)

It follows through the choice of ς as any given positive constant such that ∥X∥² ≥ ς ², the expression in (15.57) can be upper bounded as:

(15.58)

From (15.58), it can be concluded that for some positive constant C ₀, we have:

(15.59)

It can be concluded from inequalities (15.50) and (15.59) that V (y, t) ∈ L _∞, therefore e _p, e _f, r _f, η, z, r _a, P and $Q \in \mathcal {L}_\infty$. From (15.13), (15.14) and (15.15), a straightforward conclusion can be drawn to show that $\dot {\mathbf {e}}_f, \mathbf {q}, \dot {\mathbf {q}} \in \mathcal {L}_\infty$. From the fact that the desired trajectory is sufficiently smooth then using (15.8) we can conclude that p and $\dot {\mathbf {p}} \in \mathcal {L}_\infty$. From (15.28) and (15.36) we know that $\overline {\mathbf {u}}_1$ and ${\mathbf {u}}_2 \in \mathcal {L}_\infty$. Given these boundedness statements, it is clear from that the time derivative of U(X) = γ∥X∥² is such that $\dot {U}(\boldsymbol {X})\in \mathcal {L}_\infty$, which also implies that U(X) is uniformly continuous. Define the region $\mathcal {S}_{\mathcal {D}}$ as follows:

$$\displaystyle \begin{aligned} \mathcal{S}_{\mathcal{D}}=\left\{\boldsymbol{y} \in \mathcal{D}~|\quad U_2(\boldsymbol{y})< \min\{\frac{1}{2},\frac{m}{2}\} \varphi^{-1}\Bigg(2\sqrt{\frac{k_2(\varsigma-\gamma)}{\varsigma}}\Bigg)^2 \right\} \end{aligned} $$

(15.60)

The region of attraction in $\mathcal {S}_{\mathcal {D}}$ can be made arbitrary large by appropriately tuning the control gain k ₂. As a matter of fact, the region of attraction is calculated based on the set given in (15.60) as follows:

$$\displaystyle \begin{aligned} \| \boldsymbol{y}(t_0) \| \leq \sqrt{\frac{\min\{\frac{1}{2},\frac{m}{2}\}}{\max\{1,\frac{m}{2}\}}}\varphi^{-1}\Bigg(2\sqrt{\frac{k_2(\varsigma-\gamma)}{\varsigma}}\Bigg) \end{aligned} $$

(15.61)

which can be rearranged as in (15.38). Based on the definition of the vector state y in (15.45) and the initial conditions defined in (15.16) and (15.17) and (15.48) and (15.49), an explicit expression for ∥y(t ₀)∥ can be derived as in (15.39).

From (15.59) it can be concluded that C ₀∥X∥² → 0 as time goes to infinity, $\forall \boldsymbol {y}\in \mathcal {S}_{\mathcal {D}}$, it is then obvious to show form the definition of X that e _p, e _f, r _f, e _ψ, η, z, r _a → 0 as t →∞, $\forall \boldsymbol {y}\in \mathcal {S}_{\mathcal {D}}$. This further implies that e _v → [0, 0, −δ]^⊤ as t →∞, consequently the velocity tracking error $m\boldsymbol {\upsilon }-{\mathbf {R}}^\top \dot {\boldsymbol {p}}_d \rightarrow 0$ as t →∞. This completes the proof.