Quadcopter attitude control with vibration reduction by additive-state-decomposition dynamic inversion design with a notch filter

Abstract

In practice, the unbalanced mass of the propeller is the leading cause of vibration in a quadcopter. Therefore, this paper proposes an additive-state-decomposition notch dynamic inversion controller to suppress the vibration noise. Firstly, the vibration mechanics model based on unbalanced mass is established and its characteristic frequency is analyzed. Then, the specific form of the notch filter is designed, and this characteristic frequency is taken as its internal parameter. Next, stability analysis shows that the proposed controller guarantees that all attitude signals are globally uniformly ultimately bounded. In particular, the notch filter can effectively reduce the vibration having a specific frequency. Finally, the proposed controller is performed on a real quadcopter to verify its vibration reduction performance.

Appendix A

1.1 Proof of Theorem 1

Under Assumption 4, the closed-loop system composed of (12), (22) and (45) is described as

$$\begin{aligned} {\dot{\textbf{x}}}&=\textbf{Ax}+\textbf{Bv} \nonumber \\ \mathbf {{\dot{v}}}&= \frac{\partial {\textbf{v}}}{\partial {\textbf{u}}}\frac{\partial {\textbf{u}}}{\partial t}+\frac{\partial {\textbf{v}}}{\partial t}=\left( -\frac{1}{\varepsilon }\frac{\partial {\textbf{h}}}{\partial {\textbf{u}}}\left( {\textbf{I}}+{{\varvec{\delta }}_{q}} \right) + \frac{\partial {{\varvec{\sigma }}}}{\partial {\textbf{x}}} {\textbf{B}} \right) {\textbf{v}} \nonumber \\&\quad +\frac{\partial {\textbf{h}}}{\partial t}+\frac{1}{\varepsilon }\frac{\partial {\textbf{h}}}{\partial {\textbf{u}}}{{\varvec{\delta }}_{q}}{\textbf{u}} +\frac{\partial {{\varvec{\sigma }}}}{\partial t}+ \frac{\partial {{\varvec{\sigma }}}}{\partial {\textbf{x}}} \textbf{Ax}. \end{aligned}$$

(52)

Let ${{\gamma }_{1}}$ be the smallest eigenvalue of the positive definite matrix ${\textbf{M}}$, one obtains

$$\begin{aligned} {{{\textbf{x}}}^{\text {T}}}\left( {{\mathbf {{A}}}^{\text {T}}}{\textbf{P}}+{\textbf{P}}\mathbf {{A}} \right) {\textbf{x}}\le -{{\gamma }_{1}}{{\left\| {\textbf{x}} \right\| }^{2}}. \end{aligned}$$

(53)

Define a candidate Lyapunov function $V={{{\textbf{x}}}^{\text {T}}}\textbf{Px}+{{{\textbf{v}}}^{\text {T}}}{\textbf{v}}$. Taking the derivative of V along the solution of (52) yields

$$\begin{aligned} {\dot{V}}={{{\dot{\textbf{x}}}}^{\text {T}}}\textbf{Px}+{{{\textbf{x}}}^{\text {T}}}{\textbf{P}{\dot{\textbf{x}}}}+2{{{\textbf{v}}}^{\text {T}}}{\dot{\textbf{v}}}. \end{aligned}$$

(54)

Substituting (52) and (53) into (54) results in

$$\begin{aligned} {\dot{V}}&\le -{{\gamma }_{1}}{{\left\| {\textbf{x}} \right\| }^{2}}-2{{{\textbf{v}}}^{\text {T}}}\left( \frac{1}{\varepsilon }\frac{\partial {\textbf{h}}}{\partial {\textbf{u}}}{\left( { {\textbf{I}} + {{\varvec{\delta }}_q}} \right) }- \frac{\partial {{\varvec{\sigma }}}}{\partial {\textbf{x}}} {\textbf{B}} \right) {\textbf{v}} \nonumber \\&\quad +2{{{\textbf{v}}}^{\text {T}}}\left( {{{\textbf{B}}}^{\text {T}}}{\textbf{P}}+\frac{\partial \mathbf {\sigma }}{\partial {\textbf{x}}}{\textbf{A}} \right) {\textbf{x}} \nonumber \\&\quad +2{{{\textbf{v}}}^{\text {T}}}\left( \frac{1}{\varepsilon }\frac{\partial {\textbf{h}}}{\partial {\textbf{u}}}{{\varvec{\delta }}_{q}}{\textbf{u}}+\frac{\partial {\textbf{h}}}{\partial t}+\frac{\partial {{\varvec{\sigma }}}}{\partial t} \right) . \end{aligned}$$

(55)

Under Assumptions 1–2, the following results are gained:

$$\begin{aligned}&2{{{\textbf{v}}}^\text {T}}\left( -\frac{1}{\varepsilon }\frac{\partial {\textbf{h}}}{\partial {\textbf{u}}}\left( {\textbf{I}}+{{\varvec{\delta }}_{q}} \right) { + { \frac{{\partial {\mathbf {\sigma }}}}{{\partial {{\textbf{x}}}}}} {{\textbf{B}}}} \right) {\textbf{v}}\le 2\gamma _{2}^{'}{{\left\| {\textbf{v}} \right\| }^{2}} \nonumber \\&\quad -\frac{2}{\varepsilon }(1+{\underline{l}}_{{{\text {J}}_\varDelta }})(1+{\underline{\delta }}_{q}){{\left\| {\textbf{v}} \right\| }^{2}}, \nonumber \\&2{{{\textbf{v}}}^{\text {T}}}\left( {{{\textbf{B}}}^{\text {T}}}{\textbf{P}}+\frac{\partial \mathbf {\sigma }}{\partial {\textbf{x}}}{\textbf{A}} \right) {\textbf{x}}\le 2\gamma _{3}^{'}\left\| {\textbf{v}} \right\| \left\| {\textbf{x}} \right\| \nonumber \\&2{{{\textbf{v}}}^{\text {T}}}\left( \frac{\partial {\textbf{h}}}{\partial t}+\frac{1}{\varepsilon }\frac{\partial {\textbf{h}}}{\partial {\textbf{u}}}{{\varvec{\delta }}_{q}}{\textbf{u}}+\frac{\partial {{\varvec{\sigma }}}}{\partial t} \right) \le 2\left\| {\textbf{v}} \right\| {{l}_{{\text {J}_{t}}}}\left\| {\textbf{u}} \right\| \nonumber \\&\quad +2\left\| {\textbf{v}} \right\| \left( \frac{1}{\varepsilon }(1+{\bar{l}}_{{{\text {J}}_\varDelta }}){{{\bar{\delta }}}_{q}}\left\| {\textbf{u}} \right\| +{{l}_{{{\sigma }_{t}}}}\left\| {\textbf{x}} \right\| +{{{{\bar{d}}}}_{\sigma }} \right) , \end{aligned}$$

(56)

where

$$\begin{aligned} \gamma _{2}^{'}={k_\sigma }\left\| {{\textbf{B}}} \right\| , \gamma _{3}^{'}=\left\| {\textbf{B}} \right\| \left\| {\textbf{P}} \right\| +{k_\sigma }\left\| {\textbf{A}} \right\| . \end{aligned}$$

(57)

Then, substituting (56) into (55) yields

$$\begin{aligned} {\dot{V}}&\le -2\left( \frac{1}{\varepsilon }\left( 1+{{{{\underline{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) \left( 1+{{{{\underline{\delta }}}}_{q}} \right) -\gamma _{2}^{'} \right) {{\left\| {\textbf{v}} \right\| }^{2}} \nonumber \\&\quad +2\left\| {\textbf{v}} \right\| \left( {{l}_{{{\text {J}}_{t}}}}\left\| {\textbf{u}} \right\| +\frac{1}{\varepsilon }\left( 1+{{{{\bar{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) {{{\bar{\delta }}}_{q}}\left\| {\textbf{u}} \right\| +{{l}_{{{\sigma }_{t}}}}\left\| {\textbf{x}} \right\| +{{{{\bar{d}}}}_{\sigma }} \right) \nonumber \\&\quad -{{\gamma }_{1}}{{\left\| {\textbf{x}} \right\| }^{2}}+2\left\| {\textbf{v}} \right\| \gamma _{3}^{'}\left\| {\textbf{x}} \right\| \end{aligned}$$

(58)

Based on Lemma 2 of Ref. [21], ${\textbf{h}}$ is rewritten as

$$\begin{aligned} {\textbf{h}}={\textbf{h}}\left( t,{\textbf{0}} \right) +\left( \int \limits _{0}^{1}{\frac{\partial {\textbf{h}}}{\partial {\textbf{z}}}{{|}_{{\textbf{z}}=s{\textbf{u}}}}\text {d}s} \right) {\textbf{u}}. \end{aligned}$$

(59)

Assumption 1 implies that $\left\| \varvec{\sigma } \right\| \le {k}_{\sigma }\left\| {\textbf{x}} \right\| +{{\bar{\delta }}_{\sigma }}$; then, one has

$$\begin{aligned} \left\| {{\textbf{u}}} \right\| \le \frac{1}{(1+{\underline{l}}_{{{\text {J}}_\varDelta }})}\left( {\left\| {{\textbf{v}}} \right\| + {k_\sigma }\left\| {{\textbf{x}}} \right\| + {{{{\bar{\delta }} }_{\sigma } }} } \right) . \end{aligned}$$

(60)

Substituting (60) into (58) results in

$$\begin{aligned} {\dot{V}}&\le -{{\gamma }_{1}}{{\left\| {\textbf{x}} \right\| }^{2}}-\left( \frac{2}{\varepsilon }(1+{\underline{l}}_{{{\text {J}}_\varDelta }})(1+{\underline{\delta }}_{q})-2{{\gamma }_{2}} \right) {{\left\| {\textbf{v}} \right\| }^{2}} \nonumber \\&\quad +\frac{1}{\left( 1+{{{{\underline{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) }{{\left( {{{\bar{\delta }}}_{\sigma }}{{l}_{{{\text {J}}_{t}}}}+{{{\bar{\delta }}}_{\sigma }}\left( 1+{{{{\bar{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) {{{\bar{\delta }}}_{q}}+\left( 1+{{{{\underline{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) {{{{\bar{d}}}}_{\sigma }} \right) }^{2}} \nonumber \\&\quad +2{{\gamma }_{3}}\left\| {\textbf{v}} \right\| \left\| {\textbf{x}} \right\| \end{aligned}$$

(61)

where

$$\begin{aligned} {{\gamma }_{2}}&=\gamma _2^{'} +\frac{\left( 2{{l}_{{{\text {J}}_{t}}}}+1 \right) \varepsilon +2\left( 1+{{{{\bar{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) {{{\bar{\delta }}}_{q}}}{2\left( 1+{{{{\underline{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) \varepsilon } \nonumber \\ {{\gamma }_{3}}&=\frac{{{k}_{\sigma }}{{l}_{{{\text {J}}_{t}}}}}{\left( 1+{{{{\underline{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) }+\frac{{{k}_{\sigma }}\left( 1+{{{{\bar{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) {{{\bar{\delta }}}_{q}}}{\left( 1+{{{{\underline{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) \varepsilon }+{{l}_{{{\sigma }_{t}}}}+\gamma _{3}^{'}. \end{aligned}$$

(62)

Since the inequality $ab\le {\left( {{a}^{2}}+{{b}^{2}} \right) }/{2}\;$ holds, $2{\gamma _3}\left\| {\textbf{v}} \right\| \left\| {\textbf{x}} \right\|$ satisfies the following inequality:

$$\begin{aligned} 2{{\gamma }_{3}}\left\| {\textbf{v}} \right\| \left\| {\textbf{x}} \right\|&=\left( \sqrt{{{\gamma }_{1}}}\left\| {\textbf{x}} \right\| \right) \frac{{{2\gamma }_{3}}\left\| {\textbf{v}} \right\| }{\sqrt{{{\gamma }_{1}}}}\le \frac{{{\gamma }_{1}}{{\left\| {\textbf{x}} \right\| }^{2}}}{2}\nonumber \\&\quad +\frac{2\gamma _{3}^{2}}{{{\gamma }_{1}}}{{\left\| {\textbf{v}} \right\| }^{2}}. \end{aligned}$$

(63)

Assume that

$$\begin{aligned} \eta \left( \varepsilon \right)&=\min \left( \frac{{{\gamma }_{1}}}{2{{\lambda }_{\max }}\left( {\textbf{P}} \right) },\right. \nonumber \\&\left. \quad \times 2\left( \frac{\left( 1+{{{{\underline{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) \left( 1+{{{{\underline{\delta }}}}_{q}} \right) }{\varepsilon }-{{\gamma }_{2}}-\frac{\gamma _{3}^{2}}{{{\gamma }_{1}}} \right) \right) , \end{aligned}$$

(64)

where ${{\lambda }_{\max }}\left( {\textbf{P}} \right)$ denotes the maximum eigenvalue. Then substituting (63) into (61), one has

$$\begin{aligned} {\dot{V}}\le -\eta \left( \varepsilon \right) V+\frac{1 }{(1+{\underline{l}}_{{{\text {J}}_\varDelta }})}{V_\varepsilon ^2}. \end{aligned}$$

(65)

where ${V_\varepsilon }={{\bar{\delta }}_{\sigma }}{{l}_{{{\text {J}}_{t}}}}+{{\bar{\delta }}_{\sigma }}\left( 1+{{{{\bar{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) {{\bar{\delta }}_{q}}+\left( 1+{{{{\underline{l}}}}_{{{\text {J}}_{\varDelta }}}} \right) {{{\bar{d}}}_{\sigma }}$. If (46) is satisfied, then $\eta \left( \varepsilon \right) >0$. For any initial condition ${\textbf{x}}_0$, the values ${\varvec{\delta }_{\sigma }}\left( t \right)$, ${\varvec{d}_{\sigma }}\left( t \right)$, ${\varvec{\delta }_{q}}(t)$ have upper bounds ${{\bar{\delta }}_{\sigma }}$, ${{\bar{d}}_{\sigma }}$, ${{\bar{\delta }}_{q}}$, respectively, the state vector ${\textbf{x}}$ satisfies

$$\begin{aligned} \left\| {\textbf{x}}\left( t \right) \right\| \rightarrow {\mathcal {B}}\left( V_\varepsilon \sqrt{\frac{1 }{{{\lambda }_{\min }}\left( {\textbf{P}} \right) \eta \left( \varepsilon \right) (1+{{{\underline{l}}}_{{{\text {J}}_\varDelta }}})}} \right) \end{aligned}$$

(66)

where ${\mathcal {B}}\left( \delta \right) \overset{\varDelta }{\mathop {=}}\,\left\{ \xi \in {\mathbb {R}}|\left\| \xi \right\| \le \delta \right\} ,\delta >0$. This implies that the state vector ${\textbf{x}}$ in the closed-loop system (52) is globally uniformly ultimately bounded. This completes the proof of Theorem 1.

1.2 Proof of Corollary 1

Let us consider that the controlled system (12) is simplified into a class of linear system (47) with specific frequency disturbance $\varvec{\sigma } \left( t,\omega _{0} \right)$. Assume that the disturbance signal ${{\varvec{\sigma }}\left( t,\omega _{0} \right) }$ is a cosine signal vector with frequency $\omega _{0}$. Let $\varvec{\sigma }\left( t, \omega _{0} \right) ={{{\textbf{1}}}_3}\cos \left( \omega _{0} t \right)$, where ${{{\textbf{1}}}_3} \in {\Re ^3}$ denotes a column vector whose elements are all ones, without loss of generality. According to the frequency retention characteristics of the linear system, its Laplace transform is expressed as

$$\begin{aligned} {{\varvec{\sigma }}\left( s,\omega _{0} \right) }=\frac{s}{{{s}^{2}}+\omega _{0}^{2}}{{{\textbf{1}}}_3}. \end{aligned}$$

(67)

Each steady-state variable in the dynamic equation (47) must contain a cosine signal with a frequency of $\omega _{0}$. Suppose that these states are expressed as

$$\begin{aligned} {\textbf{x}}_{\text {ss}}\left( t \right) =\left[ \begin{matrix} {{\alpha }_{1}}\cos \left( \omega _{0} t+{\bar{\varphi }_{1}} \right) \\ {{\alpha }_{2}}\cos \left( \omega _{0} t+{\bar{\varphi }_{2}} \right) \\ \cdots \\ {{\alpha }_{6}}\cos \left( \omega _{0} t+{\bar{\varphi }_{6}} \right) \\ \end{matrix} \right] \end{aligned}$$

(68)

where ${\alpha }_{1}, {\alpha }_{2},\ldots , {\alpha }_{6}$ denotes the amplitudes, $\bar{\varphi }_{1}, \bar{\varphi }_{2}, \ldots , \bar{\varphi }_{6}$ are phase. Then, the newly defined steady-state output ${\textbf{y}}_{\text {ss}}$ is expressed as

$$\begin{aligned} {\textbf{y}}_{\text {ss}}={{{\textbf{C}}}^{\text {T}}}{\textbf{x}}_{\text {ss}}=\left[ \begin{matrix} \alpha _{1}^{L}\cos \left( \omega _{0} t+{{\varphi }^{'}_{1}} \right) \\ \alpha _{2}^{L}\cos \left( \omega _{0} t+{{\varphi }^{'}_{2}} \right) \\ \alpha _{3}^{L}\cos \left( \omega _{0}t+{{\varphi }^{'}_{3}} \right) \\ \end{matrix} \right] \end{aligned}$$

(69)

where $\alpha _{i}^{L}$ is a new amplitude, and ${{\varphi }^{'}_{i}}$ is a new phase delay, $i=1, 2, 3$. Substituting (69) into (39) results in

$$\begin{aligned} {\textbf{u}}_{\omega , \text {ss}}\left( s \right)&=-{{\left( {{\left( s{{{\textbf{I}}}_{3}}+\varvec{\varLambda } \right) }^{-1}}{{{\textbf{C}}}^{\text {T}}}{\textbf{B}} \right) }^{-1}}\left( \omega _{0}/q \right) \nonumber \\&\quad \times \frac{s}{{{s}^{2}}+\omega _{0}^{2}}{{{\textbf{C}}}^{\text {T}}}{\textbf{x}}_{ \text {ss}}\left( s \right) . \end{aligned}$$

(70)

Since ${\textbf{x}}_{ \text {ss}}\left( s \right)$ contains some sinusoidal time-varying signals, the control signal will have the following component:

$$\begin{aligned} \left( \omega _{0}/q \right) \frac{s}{{{s}^{2}}+\omega _{0}^{2}}\frac{s}{{{s}^{2}}+\omega _{0}^{2}}. \end{aligned}$$

(71)

According to the conclusion of Ref. [26], if two $\frac{s}{{{s^2} + \omega _{0}^2}}$ are connected in series, the transfer function will be unstable, that is, the system will diverge. However, Theorem 1 implies that the control signal is bounded, so a contradiction appears. Therefore, it is inferred that the state vector ${{\textbf{x}}}\left( t \right)$ cannot contain the disturbance signal with the frequency $\omega _{0}$. Similarly, in a linear system, the controller (38) with ${\textbf{Q}}\left( s \right)$ shown in (37) can completely suppress the disturbance ${{\varvec{\sigma }}\left( t,\omega _{0} \right) }$ having only constant and specific frequency $\omega _{0}$ components, namely, $\underset{{t}\rightarrow 0}{\mathop {\lim }}\,\left\| {\textbf{x}}(t) \right\| = {\textbf{0}}$. This completes the proof of Corollary 1.