Dynamic event-triggered observer design for quadcopters with uncertainties

Abstract

This paper investigates the problem of estimating all 12 states and lumped uncertainties for quadcopters with unknown disturbances. A novel dynamic event-triggered mechanism is proposed for the construction of extended state observers regarding quadcopters with nonlinear control gain matrix. To attenuate the undesired uncertainty oscillation in the conventional extended state observer, an improved estimation method is developed to acquire higher estimation accuracy. Apart from validating the uniform ultimate boundedness of estimation errors, the proposed scheme is also free of Zeno behaviour. Comparative experiments in simulation demonstrate the effectiveness of the proposed observer, which shows outstanding performance in state estimation and communication saving. Hardware experiments for the quadcopter are also conducted to prove the practicability of the proposed dynamic event-triggered observer.

Keywords

Quadcopter dynamic event-triggered mechanism extended state observer uncertainty estimation nonlinear system

Introduction

Quadcopter is a kind of rotary wing aircraft which is usually used as the unmanned aerial vehicle. With a simpler mechanical structure and a smaller size than the common aircraft, quadcopters can provide better manoeuvrability including vertical take-off and landing. Owing to the attractive advantages, quadcopters have been widely utilized in applications like search and rescue (Damanik et al., 2022), wildfire monitoring (Zenkin et al., 2020), product delivery and military (Chaudhary et al., 2019). However, obtaining accurate real-time state information in practice is an indispensable issue for controlling quadcopters in the case of external disturbances. In general, some states and disturbances always cannot be measured directly or are too expensive to have access to accurate measurement (Firdaus and Tokhi, 2019). For state estimators, the existence of estimation deviation from the true states may lead to instability of the quadcopter system and even crashes. Several challenges such as complex nonlinearities (Santoso et al., 2018), parameter uncertainty (Xie et al., 2021), un-modelled dynamics (Ali et al., 2020) and external disturbances exist unavoidably in the quadcopter system (Geronel and Bueno, 2021). Hence, it is critical to design a robust state estimator for unmanned quadcopter systems in the flight process.

In order to get real-time state information of the nonlinear quadcopter, robust state estimators have been developed in the past to deal with the problem of keeping the observer valid and system stable in the case of disturbances like extended Kalman filters (EKF), interval observer (IO) and extended state observer (ESO). EKF can linearize the nonlinear system dynamics at sampling time, making it a common approach to filter the noise for complex nonlinear systems like quadcopters (Hajiyev et al., 2018). However, the real parameter of existing external disturbances remains unknown in applications, which means the EKF system may not match the real quadcopter system due to the parameter uncertainty and leads to low estimation accuracy. The approach of the Lipschitz constant is usually used to deal with the nonlinearity (Azid et al., 2023). IO always takes advantage of it to build the boundaries of the real states for nonlinearity (Yin et al., 2023). It can ensure the states fall in the state interval even if uncertainties exist. Nevertheless, the highly complex nonlinearity of quadcopters especially the state-related control gain matrix makes it difficult for IO to find the narrow interval (Khan et al., 2021). It also troubles how to convert two bounds of IO into one which can be fed into the controller. ESO augments the state vector with unknown lumped uncertainty to consider the impact of disturbances on states, allowing it to observe the states and disturbances simultaneously and become a preferred choice in estimation (Li et al., 2023). You et al. (2020) propose a positional control scheme for quadcopters based on ESO. Xiao (2020) combines the sliding mode control with ESO to achieve trajectory planning. However, due to the disturbance estimation structure of traditional ESO, there exists the possibility for states to quiver around the true values, which can finally contribute to the deviation especially when an event-triggered scheme is designed.

Apart from the structure of estimator, communication resources are highly limited in quadcopters due to their small size and lightweight, which underscores the importance of reducing communication resource demands. To reduce the number of communication event numbers, event-triggered schemes based on observers have been recently researched to replace the commonly used time-triggered approach (Ge et al., 2021; Sun et al., 2021, 2022). A static event-triggered scheme compares the real-time state with a predefined threshold value to decide whether to communicate or not (Tian et al., 2019). Huang et al. (2017) employ this scheme into the ESO structure to lower the communication cost of observer and controller. Yang et al. (2021) combine the static event-triggered scheme with controllers to reduce the update rate of controlling signal. Li et al. (2022) further design the dynamic event-triggered mechanism to achieve the adaptive triggering performance. However, the above investigations about the event-triggered mechanism did not consider the 12 states of quadcopter systems which are highly nonlinear and state-space matrix state-related.

Considering the above challenges, this paper combines the dynamic event-triggered mechanism with ESO to propose a new dynamic event-triggered uncertainty stable extended state observer (DEUESO) for the quadcopter system. Thus, the main contributions of this paper are highlighted as follows:

A new dynamic event-triggered ESO is proposed to provide a new solution for systems with highly nonlinear state-related state-space matrices in event-triggered ESO, which can save the majority of communication resources without compromising much state accuracy, especially for quadcopter systems.

An improved uncertainty estimation method in ESO is proposed to obtain more precise state and uncertainty estimation compared to the conventional event-triggered ESO, which leads to less estimation oscillation under the event-triggered mechanism.

Both simulations and hardware experiments validate the effectiveness of the proposed scheme DEUESO for the quadcopter. DEUESO saves 86.69% communication and reduces the uncertainty estimation error by 81.31% in simulations. To prove the practicability, hardware experiments on the quadcopter are conducted resulting in 73.03% communication saved and a 71.72% decrease in uncertainty estimation error.

The remainder of this paper is organized as follows. Section ‘System modelling and problem statement’ presents the dynamics and kinetics model of the quadcopter and problem statement. The design of DEUESO is provided in section ‘Main results’. Section ‘Simulation and hardware results’ illustrates the results of simulations and hardware experiments. Section ‘Conclusion’ summarizes the main findings of this paper.

System modelling and problem statement

Mathematical model of the quadcopter

The inertial frame $e$ and body frame $b$ are usually utilized to present the motion of the quadcopter as Figure 1 shows. Two frames can be transformed into each other by Euler angles in the body frame, which are denoted by roll $ϕ$ , pitch $θ$ and yaw $ψ$ . The rotation matrix is defined as

R_{b}^{e} = [\begin{matrix} c ψ c θ & c ψ s θ s ϕ - s ψ c ϕ & c ψ s θ c ϕ + s ψ s ϕ \\ s ψ c θ & s ψ s θ s ϕ + c ψ c ϕ & s ψ s θ c ϕ - c ψ s ϕ \\ - s θ & c θ s ϕ & c θ c ϕ \end{matrix}]

(1)

where $c * = \cos (*)$ and $s * = \sin (*)$ .

Assumption 1. The quadcopter is a rigid body with a symmetrical structure. The geometric centre of the quadcopter coincides with its centre of gravity. The mass and moment of inertia of the quadcopter remain invariant.

Under Assumption 1, the Newton–Euler equation for the quadcopter kinematics of translational motion and rotational motion can be described as

{\begin{matrix} \begin{matrix} m_{t} {\overset{\cdot\cdot}{P}}_{e} & = G_{e} + F_{e} \\ J_{e} {\overset{\cdot\cdot}{Θ}}_{b} & = - {\overset{\cdot}{Θ}}_{b} \times (J_{e} {\overset{\cdot}{Θ}}_{b}) + τ_{f} \end{matrix} \end{matrix}

(2)

where $P_{e} = [x, y, z]^{T}$ presents the position in the inertial frame and $Θ_{b} = [ϕ, θ, ψ]^{T}$ presents the Euler angles in the inertial frame. $m_{t}$ is the total mass. $G_{e} = [0, 0, - m_{t} g]^{T}$ is the gravity force with gravitational acceleration $g$ . $F_{b}$ is the total thrust in the body frame and can be converted into the inertial frame as $F_{e}$ . $J_{e} = diag {I_{xx}, I_{yy}, I_{zz}}$ is the inertial matrix with the inertial moment $I$ in the body frame. $τ_{f}$ presents the torque produced by rotors. $F_{b}$ and $τ_{f}$ are described as

F_{b} = K_{f} (ω_{1}^{2} + ω_{2}^{2} + ω_{3}^{2} + ω_{4}^{2})

(3)

τ_{f} = (\begin{matrix} τ_{ϕ} \\ τ_{θ} \\ τ_{ψ} \end{matrix}) = (\begin{matrix} l K_{f} (ω_{2}^{2} - ω_{4}^{2}) \\ l K_{f} (ω_{2}^{2} - ω_{4}^{2}) \\ K_{m} (ω_{1}^{2} - ω_{2}^{2} + ω_{3}^{2} - ω_{4}^{2}) \end{matrix})

(4)

where $l$ is the arm length. $ω_{i}$ is the rotation speed of the $i$ th rotor. $K_{f}$ and $K_{m}$ are respectively rotor thrust coefficient and rotor torque coefficient of quadcopters. Given the angular velocities in the frame $b$ and inertial frame $e$ denoted by $[p, q, r]^{T}$ and $[\overset{\cdot}{ϕ}, \overset{\cdot}{θ}, \overset{\cdot}{ψ}]^{T}$ respectively and $t * = \tan (*)$ , they are associated as

(\begin{matrix} \overset{\cdot}{ϕ} \\ \overset{\cdot}{θ} \\ \overset{\cdot}{ψ} \end{matrix}) = (\begin{matrix} 1 & t θ s ϕ & t θ c ϕ \\ 0 & c ϕ & - s ϕ \\ 0 & s ϕ / c θ & c ϕ / c θ \end{matrix}) (\begin{matrix} p \\ q \\ r \end{matrix})

(5)

Then, the mathematical dynamics and kinematics of the nonlinear quadcopter can be described by

{\begin{matrix} \overset{\cdot}{x} = u_{e} \\ \overset{\cdot}{y} = v_{e} \\ \overset{\cdot}{z} = w_{e} \\ \begin{matrix} {\overset{\cdot}{u}}_{e} = - \frac{F_{b}}{m_{t}} (c ψ s θ c ϕ + s ψ s ϕ) \\ {\overset{\cdot}{v}}_{e} = - \frac{F_{b}}{m_{t}} (s ψ s θ c ϕ - c ψ s ϕ) \\ {\overset{\cdot}{w}}_{e} = g - \frac{F_{b}}{m_{t}} (c ϕ c θ) \\ \overset{\cdot}{ϕ} = p + qt θ s ϕ + rt θ c ϕ \\ \overset{\cdot}{θ} = qc ϕ - rs ϕ \\ \overset{\cdot}{ψ} = q \frac{s ϕ}{c θ} + r \frac{c ϕ}{c θ} \\ \overset{\cdot}{p} = \frac{1}{I_{xx}} (τ_{ϕ} + qr (I_{yy} - I_{zz})) \\ \overset{\cdot}{q} = \frac{1}{I_{yy}} (τ_{θ} + pr (I_{zz} - I_{xx})) \\ \overset{\cdot}{r} = \frac{1}{I_{zz}} (τ_{ψ} + pq (I_{xx} - I_{yy})) \end{matrix} \end{matrix}

(6)

where $[u_{e}, v_{e}, w_{e}]^{T}$ are the translational velocities in the inertial frame. Let state vector $ζ$ and control input vector $U$ be described as

ζ = {(x, y, z, u_{e}, v_{e}, w_{e}, ϕ, θ, ψ, p, q, r)}^{T}

(7)

U = (\begin{matrix} u_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{matrix}) = (\begin{matrix} F_{b} \\ τ_{ϕ} \\ τ_{θ} \\ τ_{ψ} \end{matrix})

(8)

Therefore, equation (8) can be rewritten as

{\begin{matrix} \overset{\cdot}{ζ} (t) = A ζ (t) + B (ζ (t)) U (t) + f (ζ (t)) + d (t) \\ y (t) = C ζ (t) \end{matrix}

(9)

by defining

A = (\begin{matrix} 0_{3 \times 3} & I_{3 \times 3} & 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & 0_{3 \times 3} & 0_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & 0_{3 \times 3} & 0_{3 \times 3} & A_{q} \\ 0_{3 \times 3} & 0_{3 \times 3} & 0_{3 \times 3} & 0_{3 \times 3} \end{matrix}), A_{q} = (\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix})

(10)

and

B (ζ (t)) = (\begin{matrix} 0_{3 \times 1} & 0_{3 \times 3} \\ \begin{matrix} (\begin{matrix} - \frac{1}{m_{t}} (c ψ s θ c ϕ + s ψ s ϕ) \\ - \frac{1}{m_{t}} (s ψ s θ c ϕ - c ψ s ϕ) \\ - \frac{1}{m_{t}} (c ϕ c θ) \end{matrix}) \end{matrix} & 0_{3 \times 3} \\ 0_{3 \times 1} & 0_{3 \times 3} \\ 0_{3 \times 1} & B_{τ} \end{matrix})

(11)

B_{τ} = diag {\frac{1}{I_{xx}}, \frac{1}{I_{yy}}, \frac{1}{I_{zz}}}

(12)

where $A$ is the state matrix and $C$ is the output matrix. $B (ζ (t))$ is the state-related time-variant control gain matrix. $d (t)$ is the lumped uncertainties. $f (ζ (t))$ is the nonlinear function of state expressed as

f (ζ (t)) = (\begin{matrix} 0_{5 \times 1} \\ \begin{matrix} - g \\ qt θ s ϕ + rt θ c ϕ \\ qc ϕ - rs ϕ \\ qc ϕ - rs ϕ \\ q \frac{s ϕ}{c θ} + r \frac{c ϕ}{c θ} \\ \frac{I_{yy} - I_{zz}}{I_{xx}} qr \\ \frac{I_{zz} - I_{xx}}{I_{yy}} pr \\ \frac{I_{xx} - I_{yy}}{I_{zz}} pq \end{matrix} \end{matrix})

(13)

Problem statement

Taking into account the input saturation phenomenon of quadcopter rotors in practical application denoted by $M (u_{i})$

M (u_{i}) = {\begin{matrix} u_{i}, & if | u_{i} | \leq u_{Mi} \\ sign (u_{i}) u_{Mi}, & if | u_{i} | > u_{Mi} \end{matrix}, i = 1, 2, 3, 4

(14)

where $U_{M} = [u_{M 1}, u_{M 2}, u_{M 3}, u_{M 4}]^{T}$ is the input maximum limit of the aircraft system. $sign (t)$ refers to the signum function described as

sign (t) = {\begin{matrix} 1, & if t \geq 0 \\ - 1, & if t < 0 \end{matrix}

(15)

Then, the dynamics of quadcopters can be given as

\overset{\cdot}{ζ} = A ζ + B (ζ) M (U) + f (ζ) + d

(16)

This paper is aimed at designing a new extended state estimator under a dynamic event-trigger mechanism to make the estimation errors of state and uncertainties converge to the neighbourhood of zero with less communication for nonlinear systems.

Assumption 2. The lumped disturbances $d$ of the quadcopter dynamics (16) is bounded such that

∥ d ∥ \leq d_{M 1}, ∥ \overset{\cdot}{d} ∥ \leq d_{M 2}

(17)

where $d_{M 1}$ and $d_{M 2}$ are the positive constant vectors.

Assumption 3. The nonlinear function $f (ζ)$ satisfies Lipchitz condition as

∥ f (ζ + Δ ζ) - f (ζ) ∥ \leq α_{f} ∥ Δ ζ ∥

(18)

where $α_{f}$ is a Lipchitz positive constant.

Assumption 4. (Rajamani, 1998) $B (ζ)$ is bounded by constant $B_{M}$ as

∥ B (ζ) ∥ \leq B_{M}

(19)

and simultaneously, there is the Lipchitz constant $α_{B}$ such that $B (ζ)$ satisfies the Lipchitz condition as

∥ B (ζ + Δ ζ) - B (ζ) ∥ \leq α_{B} ∥ Δ ζ ∥

(20)

Definition 1. (Fei et al., 2022) Given states $ζ$ and the associated Lyapunov candidate denoted as $V (ζ)$ . If $V (ζ) = 0$ works only in the case of $∥ ζ ∥ = 0$ and $V (ζ)$ is bounded by constant $T_{ζ}$ as

∥ V (ζ) ∥ \leq T_{ζ}, if t \geq t_{ζ}

(21)

where $t_{ζ}$ is the satisfied time, then $ζ$ is uniformly ultimately bounded (UUB).

Lemma 1. (Fei et al., 2022) Given states $ζ$ and the associated Lyapunov candidate denoted as $V (ζ)$ . $∥ ζ ∥$ is said to be UUB in the neighbourhood [0, $T_{ζ}$ ], if $V (ζ)$ satisfies

\overset{\cdot}{V} (ζ) \leq 0, if ∥ ζ ∥ \geq T_{ζ}

(22)

where $T_{ζ}$ is a positive constant.

Main results

Static event-triggered UESO

Considering the state-related time-variant control gain matrix and complex nonlinearity, a traditional ESO for the quadcopter system (16) is designed as

{\begin{matrix} \dot{\hat{ζ}} = A \hat{ζ} + B (\hat{ζ}) M (U) + f (\hat{ζ}) + \hat{d} + γ_{1} \tilde{ζ} \\ \dot{\hat{d}} = μ_{1} \tilde{ζ} \end{matrix}

(23)

With

\tilde{ζ} = ζ - \hat{ζ}

(24)

where $\hat{ζ}$ is the estimation state of $ζ$ and $\hat{d}$ is the lumped uncertainty estimation. $γ_{1}$ presents a diagonal positive definite matrix of state error and $μ_{1}$ presents a diagonal positive definite error gain matrix for uncertainty.

The static event-triggered condition can be described as

∥ ζ (k_{i}) - ζ ∥ \leq α_{s}

(25)

where $α_{s}$ is a constant.

To design the event-triggered mechanism, this paper defines $k_{i}$ as the $i$ th event-triggered instant with $k_{i + 1} > k_{i}$ for $i \in Z^{+}$ , and $ζ (k_{i})$ is the sampled measurement of state $ζ$ by sensors in the instant $k_{i}$ . For $t \in [k_{i}, k_{i + 1})$ , the measurement information of states from sensors will remain the same. The conventional ESO (Yao et al., 2014) can be transformed as

{\begin{matrix} \dot{\hat{ζ}} = A ζ (k_{i}) + B (ζ (k_{i})) M (U) + f (ζ (k_{i})) + \hat{d} + γ_{1} \tilde{ζ} (k_{i}) \\ \dot{\hat{d}} = μ_{1} \tilde{ζ} (k_{i}) \end{matrix}

(26)

Though ESO structure (26) is effective in most cases. When the communication (25) is not triggered, $\tilde{ζ} (k_{i}) \neq \tilde{ζ}$ . Due to $\dot{\hat{d}}$ is only decided by $\tilde{ζ} (k_{i})$ according to $\dot{\hat{d}} = μ_{1} \tilde{ζ} (k_{i})$ , uncertainty estimation error may be large. Then, the oscillation in uncertainty estimation will further lead to the decline of estimation accuracy in states.

Therefore, the proposed event-triggered UESO adds $- μ_{2} \hat{d}$ to $\dot{\hat{d}}$ to hold the stability of uncertainty $d$ when the sensor measurement communication not triggered and thus improves the accuracy of state estimation. The proposed event-triggered uncertainty stable ESO can be expressed as

{\begin{matrix} \dot{\hat{ζ}} = A ζ (k_{i}) + B (ζ (k_{i})) M (U) + f (ζ (k_{i})) + \hat{d} + γ_{1} \tilde{ζ} (k_{i}) \\ \dot{\hat{d}} = μ_{1} \tilde{ζ} (k_{i}) - μ_{2} \hat{d} \end{matrix}

(27)

where $μ_{2}$ is a diagonal positive definite matrix for uncertainty.

Define $\tilde{d} = d - \hat{d},$ $\tilde{ζ} (k_{i}) = ζ (k_{i}) - \hat{ζ}$ and $E_{ζ} = ζ - ζ (k_{i})$ , then the error systems can be derived as

{\begin{matrix} \dot{\tilde{ζ}} = A (ζ - ζ (k_{i})) + (B (ζ) - B (ζ (k_{i})) M (U) + f (ζ) \\ - f (ζ (k_{i}) + \tilde{d} - γ_{1} \tilde{ζ} + γ_{1} E_{ζ} \\ \dot{\tilde{d}} = \overset{\cdot}{d} - μ_{1} \tilde{ζ} + μ_{1} E_{ζ} - μ_{2} \tilde{d} + μ_{2} d \end{matrix}

(28)

Zeno phenomenon should not happen which indicates events cannot be triggered in a finite time for infinite times. Therefore, the stability of states and the nonexistence of Zeno phenomenon is illustrated in the following theorem.

Theorem 1. If $μ_{2} > 0$ , $μ_{1} > 0$ , and $γ_{1} > 0$ hold, then $\tilde{ζ}$ and $\tilde{d}$ under static event-triggered UESO (27) are all UUB and static event-triggered UESO is free of Zeno phenomenon given system (16) and Assumptions 2, 3 and 4 hold.

Proof. Choosing the Lyapunov function as

V = \frac{{\tilde{ζ}}^{T} μ_{1} \tilde{ζ} + {\tilde{d}}^{T} \tilde{d}}{2}

(29)

then $\overset{\cdot}{V}$ can be derived from $V$ as

\begin{matrix} \overset{\cdot}{V} (t) = \frac{{\dot{\tilde{ζ}}}^{T} μ_{1} \tilde{ζ} + {\tilde{ζ}}^{T} μ_{1} \dot{\tilde{ζ}} + {\dot{\tilde{d}}}^{T} \tilde{d} + {\tilde{d}}^{T} \dot{\tilde{d}}}{2} \\ = {\tilde{ζ}}^{T} μ_{1} [A (ζ - ζ (k_{i})) + (B (ζ) - B (ζ (k_{i})) M (U)] \\ + {\tilde{ζ}}^{T} μ_{1} (f (ζ) - f (ζ (k_{i})) + {\tilde{ζ}}^{T} μ_{1} \tilde{d} + {\tilde{d}}^{T} \overset{\cdot}{d} \\ - {\tilde{d}}^{T} \dot{\hat{d}} - {\tilde{ζ}}^{T} μ_{1} γ_{1} \tilde{ζ} (k_{i}) \\ \leq {\tilde{ζ}}^{T} μ_{1} [(B (ζ) - B (ζ (k_{i})) M (U)] - {\tilde{ζ}}^{T} μ_{1} γ_{1} \tilde{ζ} + {\tilde{d}}^{T} \overset{\cdot}{d} \\ + {\tilde{ζ}}^{T} μ_{1} (γ_{1} + A + α_{f}) E_{ζ} + {\tilde{d}}^{T} μ_{2} d - {\tilde{d}}^{T} μ_{2} \tilde{d} + {\tilde{d}}^{T} μ_{1} E_{ζ} \\ \leq α_{B} α_{s} ∥ μ_{1} M (U) ∥ ∥ \tilde{ζ} ∥ - {\tilde{ζ}}^{T} μ_{1} γ_{1} \tilde{ζ} + d_{M 2} ∥ \tilde{d} ∥ \\ + α_{s} ∥ μ_{1} (γ_{1} + A + α_{f}) ∥ ∥ \tilde{ζ} ∥ + \bar{σ} (μ 2) d_{M 1} ∥ \tilde{d} ∥ \\ - {\tilde{d}}^{T} μ_{2} \tilde{d} + α_{s} ∥ \tilde{d} ∥ ∥ μ_{1} ∥ \\ \leq - {ϒ_{S}}^{T} Λ_{S} ϒ_{S} + Δ_{S} ϒ_{S} \end{matrix}

(30)

where

ϒ_{S} = [∥ \tilde{ζ} ∥, ∥ \tilde{d} ∥]^{T}

(31)

Λ_{S} = diag {μ_{1} γ_{1}, μ_{2}}

(32)

\begin{matrix} Δ_{S} = [α_{B} α_{s} ∥ μ_{1} M (U) ∥ + α_{s} ∥ μ_{1} (γ_{1} + A + α_{f}) ∥, d_{M 2} \\ + α_{s} ∥ μ_{1} ∥ + \bar{σ} (μ 2) d_{M 1}] \end{matrix}

(33)

Hence, from Lemma 1, we have $ϒ_{S}$ is UUB if $∥ ϒ_{S} ∥ \leq \frac{∥ Δ_{S} ∥}{\underline{σ} (Λ_{S})}$ is satisfied when $t \to \infty$ , further leading to the boundedness of $\tilde{ζ}$ and $\tilde{d}$ .

In addition, Zeno phenomenon will not happen equal to

\min {k_{i + 1} - k_{i}} \geq l_{s}

(34)

The static event-triggered condition can be transformed as

∥ ζ (k_{i}) - ζ ∥ = \int_{k_{i}}^{t} \frac{d ∥ E_{ζ} ∥}{d t} d k

(35)

The time derivative of $∥ E_{ζ} ∥$ can be described as

\begin{matrix} \frac{d ∥ E_{ζ} ∥}{dt} \leq \dot{∥ E_{ζ} ∥} \\ \leq ∥ \overset{\cdot}{ζ} - \overset{\cdot}{ζ} (k_{i}) ∥ \\ \leq ∥ B (ζ) M (U) + d ∥ \\ \leq B_{M} U_{M} + d_{M 1} \end{matrix}

(36)

Note that when $∥ ζ (k_{i}) - ζ ∥ > α_{s}$ , the event will be triggered. Therefore,

α_{s} < ∥ ζ (k_{i}) - ζ ∥ \leq (B_{M} U_{M} + d_{M 1}) (k_{i + 1} - k_{i})

(37)

l_{s} > \frac{α_{s}}{(B_{M} U_{M} + d_{M 1})} > 0

(38)

Thus, Zeno phenomenon will not exist. This completes the proof. □

Remark 1. Although the static event-triggered scheme can save communication, it needs the predefined threshold $α_{s}$ , leading to poor adaptation ability. By contrast, the dynamic event-triggered scheme takes advantage of the adaption law of threshold, which can utilize the properties of systems, bringing much more flexibility in the aspect of triggering.

Dynamic event-triggered UESO

Considering the stability of states and uncertainty, the dynamic event-triggered flag of whether to transmit the sensor measurement information to the observer or not is designed as

G_{DET} = {\begin{matrix} 1, & if G (E_{ζ}) \geq λ_{1} ξ_{DET} \\ 0, & if G (E_{ζ}) < λ_{1} ξ_{DET} \end{matrix}

(39)

With

\begin{matrix} G (E_{ζ}) = \frac{∥ μ_{1} (γ_{1} + A + α_{f}) E_{ζ} ∥^{2} + {(α_{B} ∥ μ_{1} M (U) ∥)}^{2} ∥ E_{ζ} ∥^{2}}{4} \\ + ∥ μ_{1} E_{ζ} ∥^{2} - β_{1} {\tilde{ζ}}^{T} Λ_{1} \tilde{ζ} \end{matrix}

and

{\overset{\cdot}{ξ}}_{DET} = - h ξ_{DET} - G (E_{ζ}) + λ_{2}

(40)

where $λ_{1}$ and $λ_{2}$ are both positive definite matrices. $β_{1}$ and $h$ are the two positive constants with $0 < β_{1} < 1$ . $ξ_{DET}$ is the dynamic event-triggered threshold satisfying equation (40). $Λ_{1}$ is defined as

Λ_{1} = μ_{1} γ_{1} - 2 I_{12}

(41)

Based on equation (39), there are two cases about the event-triggered flag function $G_{DET}$ :

Case 1: When $G (E_{ζ}) \geq λ_{1} ξ_{DET}$ , $E_{ζ} = ζ - ζ (k_{i}) = 0$ and ${\overset{\cdot}{ξ}}_{DET} > - h ξ_{DET} + λ_{2}$ , leading to

ξ_{DET} \geq \frac{λ_{2}}{h} > 0

(42)

Case 2: When $G (E_{ζ}) < λ_{1} ξ_{DET}$ , ${\overset{\cdot}{ξ}}_{DET} > - (h + λ_{1}) ξ_{DET} + λ_{2}$ , causing

ξ_{DET} \geq \frac{λ_{2}}{h + λ_{1}} > 0

(43)

In summary of two cases based on equations (42) and (43), $ξ_{DET}$ satisfies $ξ_{DET} \geq \frac{λ_{2}}{h + λ_{1}} > 0$ .

Theorem 2. If $\underline{σ} (μ_{2}) > \frac{1}{2}$ holds and $Λ_{1}$ satisfies positive definite, then the event-triggered mechanism is free of Zeno behaviour and $\tilde{ζ}$ , $\tilde{d}$ , $ξ_{DET}$ of the dynamical event-triggered UESO (27) are all UUB given system (16) and Assumptions 2, 3 and 4 hold.

Proof. Choosing the Lyapunov function as

V = \frac{{\tilde{ζ}}^{T} μ_{1} \tilde{ζ} + {\tilde{d}}^{T} \tilde{d}}{2} + ξ_{DET}

(44)

then $\overset{\cdot}{V}$ can be derived from $V$ as

\begin{matrix} \overset{\cdot}{V} (t) = \frac{{\dot{\tilde{ζ}}}^{T} μ_{1} \tilde{ζ} + {\tilde{ζ}}^{T} μ_{1} \dot{\tilde{ζ}} + {\dot{\tilde{d}}}^{T} \tilde{d} + {\tilde{d}}^{T} \dot{\tilde{d}}}{2} + {\overset{\cdot}{ξ}}_{DET} \\ = {\tilde{ζ}}^{T} μ_{1} [A (ζ - ζ (k_{i})) + (B (ζ) - B (ζ (k_{i})) M (U)] \\ + {\tilde{ζ}}^{T} μ_{1} (f (ζ) - f (ζ (k_{i})) + {\tilde{ζ}}^{T} μ_{1} \tilde{d} + {\tilde{d}}^{T} \overset{\cdot}{d} - {\tilde{d}}^{T} \dot{\hat{d}} \\ - {\tilde{ζ}}^{T} μ_{1} γ_{1} \tilde{ζ} (k_{i}) + {\overset{\cdot}{ξ}}_{DET} \\ \leq {\tilde{ζ}}^{T} μ_{1} [(B (ζ) - B (ζ (k_{i})) M (U)] - {\tilde{ζ}}^{T} μ_{1} γ_{1} \tilde{ζ} + {\tilde{d}}^{T} \overset{\cdot}{d} \\ + {\tilde{ζ}}^{T} μ_{1} (γ_{1} + A + α_{f}) E_{ζ} + {\tilde{d}}^{T} μ_{2} d - {\tilde{d}}^{T} μ_{2} \tilde{d} \\ + {\tilde{d}}^{T} μ_{1} E_{ζ} + {\overset{\cdot}{ξ}}_{DET} \\ \leq α_{B} ∥ μ_{1} M (U) ∥ ∥ E_{ζ} ∥ ∥ \tilde{ζ} ∥ - {\tilde{ζ}}^{T} μ_{1} γ_{1} \tilde{ζ} + d_{M 2} ∥ \tilde{d} ∥ \\ + ∥ μ_{1} (γ_{1} + A + α_{f}) E_{ζ} ∥ ∥ \tilde{ζ} ∥ + \bar{σ} (μ 2) d_{M 1} ∥ \tilde{d} ∥ \\ - \underline{σ} (μ 2) ∥ \tilde{d} ∥^{2} + ∥ \tilde{d} ∥ ∥ μ_{1} E_{ζ} ∥ - h ξ_{DET} - G (E_{ζ}) + λ_{2} \end{matrix}

(45)

According to Young’s inequality, $ab \leq \frac{a^{2}}{2 ε} + \frac{ε b^{2}}{2}$ , where $ε \in ℝ^{+}$ , then we have

\begin{matrix} α_{B} ∥ μ_{1} M (U) ∥ ∥ E_{ζ} ∥ ∥ \tilde{ζ} ∥ \leq ∥ \tilde{ζ} ∥^{2} + \frac{{(α_{B} ∥ μ_{1} M (U) ∥ ∥ E_{ζ} ∥)}^{2}}{4} \\ ∥ μ_{1} (γ_{1} + A + α_{f}) E_{ζ} ∥ ∥ \tilde{ζ} ∥ \leq \frac{∥ μ_{1} (γ_{1} + A + α_{f}) E_{ζ} ∥^{2}}{4} + ∥ \tilde{ζ} ∥^{2} \\ ∥ \tilde{d} ∥ ∥ μ_{1} E_{ζ} ∥ \leq \frac{∥ \tilde{d} ∥^{2}}{4} + ∥ μ_{1} E_{ζ} ∥^{2} \\ d_{M 2} ∥ \tilde{d} ∥ + \bar{σ} (μ 2) d_{M 1} ∥ \tilde{d} ∥ \leq \frac{∥ \tilde{d} ∥^{2}}{4} + {(d_{M 2} + \bar{σ} (μ 2) d_{M 1})}^{2} \end{matrix}

(46)

Therefore, defining $Γ = \underline{σ} (μ_{2}) - \frac{1}{2}$ , we have

\begin{matrix} \overset{\cdot}{V} (t) \leq (β_{1} - 1) {\tilde{ζ}}^{T} Λ_{1} \tilde{ζ} - Γ ∥ \tilde{d} ∥^{2} - h ξ_{DET} + λ_{2} \\ + {(d_{M 2} + \bar{σ} (μ 2) d_{M 1})}^{2} \leq - ϒ^{T} Λ_{2} ϒ + Δ \end{matrix}

(47)

Where

ϒ = [\sqrt{ξ_{DET}}, ∥ \tilde{ζ} ∥, ∥ \tilde{d} ∥]^{T}

(48)

Λ = diag {h, (1 - β_{1}) \underline{σ} (Λ_{1}), Γ}

(49)

and

Δ = {(d_{M 2} + \bar{σ} (μ 2) d_{M 1})}^{2} + λ_{2}

(50)

Hence, from Lemma 1, $\tilde{ζ}$ , $\tilde{d}$ and $ξ_{DET}$ are all UUB if $∥ ϒ ∥ \leq \sqrt{\frac{Δ}{\underline{σ} (Λ_{2})}}$ is satisfied when $t \to \infty$ .

Besides the stability analysis, proving the Zeno phenomenon will not happen is also significant. If the communication works, $E_{ζ} = 0$ . Given the case that $∥ \tilde{ζ} ∥ = 0$ , the event-triggered interval can be presented by time lag between $G (E_{ζ}) = 0$ to $G (E_{ζ}) = λ_{1} ξ_{DET}$ . Therefore, we have

2 \sqrt{\frac{λ_{1} ξ_{D E T}}{Λ_{3}}} = \int_{k_{i}}^{k_{i + 1}} \frac{d ∥ E_{ζ} ∥}{d t} d k

(51)

where $Λ_{3} = \bar{σ} {(μ_{1})}^{2} [4 + \bar{σ} {(γ_{1} + A + α_{f})}^{2} + α_{B}^{2} U_{M}^{2}]$ .

The time derivative of $∥ E_{ζ} ∥$ can be described as

\begin{matrix} \frac{d ∥ E_{ζ} ∥}{dt} \leq B_{M} U_{M} + d_{M 1} \end{matrix}

(52)

Therefore, the dynamic event-triggered interval can be given as

k_{i + 1} - k_{i} \geq \sqrt{\frac{λ_{1} λ_{2}}{Λ_{3} (h + λ_{1}) (B_{M} U_{M} + d_{M 1})}} > 0

(53)

Hence, there is no chance for Zeno behaviour to happen due to the proposed event-triggered scheme. The whole proof of stability and no Zeno phenomenon is accomplished. □

Remark 2. Compared with ESO (26), if $∥ \tilde{ζ} ∥$ is large at the initial time, then $∥ \tilde{ζ} (k_{i}) ∥$ must rise to a high value to satisfy the event-triggered condition (39), leading to large error of uncertainty estimation under the dynamic event-triggered condition. Thus, the proposed UESO can improve the accuracy of state and uncertainty estimation.

Remark 3. The existing works on event-triggered ESO all assume that the state-space matrices of systems especially the control gain matrix is static and state-invariant (Huang et al., 2017; Li et al., 2022; Yang et al., 2021). The proposed DEUESO provides a new solution for systems with nonlinear state-correlated state-space matrices to build a new dynamic event-triggered ESO as shown in Figure 2. This paper utilizes the boundness and Lipchitz property of the nonlinear control gain matrix in Assumption 4 to obtain the uniform ultimate boundness of estimation errors and avoid Zeno phenomenon under the dynamic event-triggered mechanism, which can be specifically applied to the highly nonlinear quadcopter system.

Simulation and hardware results

To justify the effectiveness of the proposed DEUESO, this paper utilizes the Parrot Mambo Minidrone as the quadcopter model for simulation and hardware experiments. The Parrot Mambo is a lightweight indoor drone with arm length $l = 0.108$ m, mass $m_{t} = 0.063$ kg and inertial moment $[I_{xx}, I_{yy}, I_{zz}] = [5.85 \times 10^{- 5}, 7.17 \times 10^{- 5}, 1 \times 10^{- 4}]$ .

To validate the superiority of the proposed DEUESO especially in the accuracy of state observation, stability of uncertainty estimation and consumption of communication resources, these contrast experiments are carried out:

The proposed DEUESO (27) and (39).

The dynamic event-triggered conventional extended state observer (DEESO) (26) and (39).

The static event-triggered uncertainty stable extended state observer (SEUESO) (27) and (25).

The time-triggered uncertainty stable extended state observer (TTUESO) (27).

Simulation results

The proposed scheme is first tested in MATLAB/Simulink. The sampling time is $T_{s} = 5$ ms. $α_{s} = 0.05 .$ The desired position input signal of the quadcopter makes the quadcopter complete two square flight paths. The maximum input limit is set as $U_{M} = [0.8, 0.6, 0.6, 0.6]^{T}$ which is related to the hardware of our quadcopter platform. The key parameters in DEUESO are selected as $h = 1$ , $λ_{1} = 0.6$ , $λ_{2} = 0.5$ , $ξ_{DET} (t_{0}) = 2$

\begin{matrix} γ_{1} & = & diag {0.1, 0.1, 0.1, 0.6, 0.3, 0.3, 5, 5, 5, 0.3, 0.3, 0.3} \\ μ_{1} & = & diag {3, 3, 3, 9, 9, 12, 0.1, 0.1, 0.1, 3, 3, 12} \end{matrix}

\begin{matrix} μ_{2} & = & diag {6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6} \end{matrix}

The initial states are chosen as

\begin{matrix} ζ (t_{0}) = [- 0.05; - 0.05; - 0.1; 0; 0; 0; - 0.05; - 0.05; \\ - 0.0005; 0; 0; 0] \end{matrix}

(54)

Considering the trade-off between the estimation accuracy and hardware computation burden, setting the sampling time $T_{s}$ as 5 ms in simulation is the optimal choice, which is the same as that in the hardware quadcopter platform. The initial states and parameters in DEUESO are chosen according to our hardware quadcopter platform. The initial states reflect the difference between the position and attitude of the quadcopter we set and the original.

Figure 3 gives the boundness of $∥ \tilde{d} ∥$ of the four groups. In simulations, uncertainty estimation error $∥ \tilde{d} ∥$ is bounded under 0.1102 in DEESO, whereas 0.0206 in DEUESO, which means the proposed DEUESO can reduce the uncertainty estimation error to 81.31% of DEESO and thus leads to less state oscillation for trajectory tracking.

Trigger intervals of the proposed static and dynamic event-triggered mechanism are given in Figure 4, where one point in the stem figure means one triggered event and the value in the y-axis indicates the time interval from the last trigger to the current trigger. The total event numbers of DEESO, DEUESO, SEUESO and TTUESO are respectively 882, 699, 1102 and 6000. DEUESO outperforms other groups with raising the trigger interval from 5 to 43 ms and 86.69% trigger numbers saved.

The estimation error of state $ζ$ in simulations is shown in Figure 5. All the states are found to be ultimately bounded for the designed four groups. Compared with the results of DEESO and SEUESO, the proposed DEUESO can achieve obviously better stability and less oscillation in the neighbourhood of zero. Moreover, DEUESO can obtain faster state convergence especially in position estimation which results from the error of initial value.

In summary, the proposed DEUESO can reduce above 85% cost of communication resources and meanwhile achieve much the same performance of state observation and uncertainty estimation as TTUESO. DEUESO outperforms DEESO with a relatively more stable quadcopter system especially in uncertainty estimation and overweighs SEUESO both in communication saving and estimation accuracy.

Hardware results

Similar to the simulations, the hardware experiments verify the practicality of the proposed scheme. Different from simulations, large measurement noise due to sensor capability exists in the hardware experiments just like reality and thus can degrade the performance of the dynamic event-triggered scheme.

The estimation error of uncertainty is shown in Figure 6. The upper boundaries of $∥ \tilde{d} ∥$ are 0.3526 in DEESO and 0.0962 in DEUESO, which means the proposed DEUESO reducing 71.72% disturbance error boundary compared with DEESO.

Event-triggered intervals in hardware experiments are given in Figure 7. The results have a decline compared with simulations due to the hardware especially the sensor noise. The total event-triggered number is respectively 1369, 1618, 6000 and 2352 for the designed groups DEESO, DEUESO, TTUESO and SEUESO. The proposed DEUESO can raise the trigger interval from 5 to 19 ms and save 73.03% event numbers.

The propagation of state estimation error is shown in Figure 8. The results are similar to the simulations. The flight trajectories of the quadcopter are illustrated in Figure 9. Compared with DEESO and SEUESO, DEUESO can guide smoother motions especially around the corner and less swing in altitude.

Given the above experimental results, the proposed scheme DEUESO can still save above 70% communication without compromising much state accuracy in hardware experiments, although due to sensor limits and measurement noise, the performance of dynamic event-triggered mechanism reduces a little compared with simulation. DEUESO can generally lead to less oscillation in quadcopter states and uncertainty estimation than DETESO and SEUESO. Therefore, it can be concluded that DEUESO has competitive robust properties when uncertainty is present.

Conclusion

This paper proposes a novel estimation method of the observer under a dynamic event-triggered scheme for systems with nonlinear control gain matrix and uncertainties, especially for quadcopter systems. The proposed DEUESO is constructed to obtain accurate real-time state and uncertainty estimation at a low communication cost, particularly in the presence of external disturbances. Simulations and hardware experiments both validate the effectiveness of the proposed DEUESO. However, the proposed method may not work well in aggressive trajectories. In the future, we will consider the controller algorithm against the external disturbances based on DEUESO. This work underscores the importance of the dynamic event-triggered scheme applied on the ESO for quadcopters, contributing to less communication resource demand.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was partially supported by the National Natural Science Foundation of China (No. 52220105001, 72322002, 62103295) and the Jiangsu Funding Program for Excellent Postdoctoral Talent (No. 2023ZB246)

ORCID iD

Xincheng Tang

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