Fixed-Time Convergent Control of Uncertain Nonlinear Systems with Event-Triggered Disturbance Rejection

2.1 Fixed-time stability

Definition 1 ([4]): Consider the autonomous system $\dot{x}= f(x), x \in \mathbb{R}^{\mathrm{n}}, f(0)=0$. The origin is said to be fixed-time stable if it is globally finite-time stable and the settling time function $T\left(x_0\right)$ is bounded, i.e., there exists $T_{\max }>0$ such that $T\left(x_0\right) \leq T_{\text {max }}$ for all $x_0 \in \mathbb{R}^{\mathrm{n}}$.

Lemma 1 ([4]): Consider the system $\dot{x}=f(x)$. If there exists a positive definite, radially unbounded function $V(x)$ and constants $a>0, b>0,0<p<1, q>1$, and $0<\kappa \leq \infty$ such that

$\dot{V}(x) \leq-a V^p(x)-b V^q(x), \quad \forall V(x)>\kappa$        (1)

then the system is practically fixed-time stable with settling time satisfying

$T_{\max } \leq \frac{1}{a(1-p)}+\frac{1}{b(q-1)}$        (2)

When $\kappa=0$, the system is globally fixed-time stable.

Proof sketch. The two terms on the right-hand side of (1) are responsible for two complementary phases of the convergence. When $V$ is large, $V^q$ dominates (because $q>$ 1), so $\dot{V} \leq-b V^q$, which integrates to $V(t) \leq\left[V(0)^{1-q}+\right. b(q-1) t]^{1 /(1-q)}$ and yields a time $t_1 \leq 1 /[b(q-1)]$ to reach $V=1$ independently of $V(0)$ - this is the fixed-time property. When $V \leq 1, V^p \geq V^q$ (because $p<1<q$), so $\dot{V} \leq-a V^p$, which integrates in finite time $t_2-t_1 \leq 1 / [a(1-p)]$. Summing the two phases gives Eq. (2). The complete derivation can be found in [4, Theorem 1].

Lemma 2: For $x_1, x_2, \ldots, x_n \geq 0$ and $0<r \leq 1$:

$\left(\sum_{i=1}^n x_i\right)^r \leq \sum_{i=1}^n x_i^r$       (3)

For $r>1$:

$\left(\sum_{i=1}^n x_i\right)^r \leq n^{r-1} \sum_{i=1}^n x_i^r$      (4)

Lemma 3: For any $a, b \geq 0, p, q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, and $\epsilon>0$:

$a b \leq \frac{\epsilon^p}{p} a^p+\frac{1}{q \epsilon^q} b^q$      (5)

2.2 Problem formulation

Consider the following class of uncertain nonlinear strict-feedback systems:

$\left\{\begin{array}{l}\dot{x}_i=x_{i+1}+f_i\left(\bar{x}_i\right)+\Delta f_i\left(\bar{x}_i\right) \quad i=1, \ldots, n-1 \\ \dot{x}_n=u+f_n(x)+\Delta f_n(x)+d(t) \\ y=x_1\end{array}\right.$      (6)

where, $x=\left[x_1, \ldots, x_n\right] T \in \mathbb{R} n$ is the state vector, $\bar{x}_i= \left[x_1, \ldots, x_i\right]^{\top}, \mathrm{u} \in \mathbb{R}$ is the control input, $\mathrm{y} \in \mathbb{R}$ is the system output, $f_i(\cdot)$ are known smooth nonlinear functions, $\Delta f_i(\cdot)$ represent model uncertainties, and $d(t)$ is an external disturbance.

Assumption 1: The lumped disturbance $D(t)=\Delta f_n(x)+ d(t)$ and its time derivative are bounded, i.e., there exist unknown constants $\bar{D}>0$ and $\overline{\dot{D}}>0$ such that $|D(t)| \leq \bar{D}$ and $|\dot{D}(t)| \leq \overline{\dot{D}}$.

Assumption 2: The reference signal $y_r(t)$ is a sufficiently smooth bounded function, and its time derivatives up to order $n$ are bounded.

Assumption 3: The model uncertainties $\Delta f_i\left(\bar{x}_i\right)$ for $i=1, \ldots, n-1$ are bounded by known smooth functions, i.e., $\left|\Delta f_i\left(\bar{x}_i\right)\right| \leq \rho_i\left(\bar{x}_i\right)$ where $\rho_i(\cdot)$ are known non-negative functions.

Control Objective: Design an event-triggered fixed-time controller such that: (i) the output $y$ tracks the reference signal $y_r$ with the tracking error converging to a small neighborhood of zero within a fixed time $T_{\max }$ independent of initial conditions; (ii) all closed-loop signals remain bounded; (iii) the control input is updated aperiodically based on the event-triggered mechanism to reduce the communication burden; and (iv) Zeno behavior is strictly excluded.

2.3 Notation and design parameters

The proposed framework involves three groups of design parameters that play distinct roles in the closed loop:

• Observer parameters $l_1, l_2, l_3, l_4$ and exponents $\alpha_1, \alpha_2, \beta_1, \beta_2$ of the FTDO (Eqs. (8) and (9)). The gains $l_1, l_2$ dominate the response of the auxiliary error $e_z$, while $l_3, l_4$ drive the disturbance estimate $\widehat{D}$. The exponents $\alpha_i \in(0,1)$ determine the finite-time component of the convergence (active for large errors, accelerating the early transient), and the exponents $\beta_i>1$ determine the fast component (active for small errors, yielding the fixed-time bound). 
• Controller gains $k_{i 1}, k_{i 2}, k_{i 3}$ and exponents $\gamma_1, \gamma_2$ of the backstepping virtual controls (Eqs. (21)-(23)). The roles of $\gamma_1 \in(0,1)$ and $\gamma_2>1$ parallel those of $\alpha_i$ and $\beta_i$; the linear term $k_{i 3} e_i$ preserves robustness near the equilibrium where the fractional powers vanish.
• Triggering parameters $\sigma, \eta, \mu, \theta(0)$ of the dynamic event-triggered mechanism (DETM) (Eqs. (27) and (28)). The static threshold gain $\eta$ and the dynamic gain $\sigma$ jointly set the size of the triggering window, the decay rate $\mu$ controls how fast the auxiliary state $\theta$ forgets past information, and the initial value $\theta(0)>0$ provides a triggering "budget" useful during the transient.

For each parameter, the admissible range, its primary effect on closed-loop behaviour, and a recommended starting value are summarised in Table 1.

4.1 Coordinate transformation

Define the tracking error and error variables:

$e_1 \&=x_1-y_r$      (18)

$e_i \&=x_i-\alpha_{i-1}, \quad i=2, \ldots, n$     (19)

where, $\alpha_i$ are the virtual control laws to be designed in each backstepping step.

4.2 Backstepping design

Step 1: From Eqs. (6) and (8):

$\dot{e}_1=x_2+f_1\left(x_1\right)+\Delta f_1\left(x_1\right)-\dot{y}_r$      (20)

Since $x_2=e_2+\alpha_1$, we have $\dot{e}_1=e_2+\alpha_1+f_1\left(x_1\right)+ \Delta f_1\left(x_1\right)-\dot{y}_r$. Design the first virtual control:

$\alpha_1=-f_1\left(x_1\right)+\dot{y}_r-k_{11}\left[e_1\right]^{\gamma_1}-k_{12}\left[e_1\right]^{\gamma_2}-k_{13} e_1$          (21)

where, $k_{11}, k_{12}, k_{13}>0$ are design gains, $\gamma_1 \in(0,1)$ ensures fixed-time convergence from the "finite-time" direction, and $\gamma_2>1$ ensures convergence from the "fast-convergence" direction.

Step $i(i=2, \ldots, n-1)$: The error dynamics for $e_i$ are:

$\dot{e}_i=e_{i+1}+\alpha_i+f_i\left(\bar{x}_i\right)+\Delta f_i\left(\bar{x}_i\right)-\dot{\alpha}_{i-1}$        (22)

Design the virtual control:

$\begin{gathered}\alpha_i=-f_i\left(\bar{x}_i\right)+\dot{\alpha}_{i-1}-k_{i 1}\left[e_i\right]^{\gamma_1}-k_{i 2}\left[e_i\right]^{\gamma_2}-k_{i 3} e_i -\hat{\rho}_i \operatorname{sgn}\left(e_i\right)\end{gathered}$             (23)

where, $\hat{\rho}_i \geq \rho_i\left(\bar{x}_i\right)$ is a compensation term for the model uncertainty $\Delta f_i$.

Step $n$ (final step): The error dynamics for $e_n$ are:

$\dot{e}_n=u+f_n(x)+D(t)-\dot{\alpha}_{n-1}$      (24)

Using the disturbance estimate $\widehat{D}$ from the FTDO, the ideal (continuous) control law is:

$\begin{gathered}u^*=-f_n(x)+\dot{\alpha}_{n-1}-\widehat{D}-k_{n 1}\left[e_n\right]^{\gamma_1}-k_{n 2}\left[e_n\right]^{\gamma_2} -k_{n 3} e_n\end{gathered}$           (25)

4.3 Dynamic event-triggered mechanism

In continuous control implementations, the control signal $u^*$ is computed and transmitted at every instant. To reduce the communication burden, we introduce a DETM that determines the update instants $\left\{t_k\right\}_{k=0}^{\infty}$ of the control signal. The concept of dynamic triggering was originally proposed in the study [19] and has since been extended to various nonlinear settings [23, 24].

Let $u(t)=u^*\left(t_k\right)$ for $t \in\left[t_k, t_{k+1}\right)$, where $t_k$ is the latest triggering instant. Define the measurement error:

$\varepsilon(t)=u^*\left(t_k\right)-u^*(t), \quad t \in\left[t_k, t_{k+1}\right)$       (26)

The dynamic event-triggering condition is:

$t_{k+1}=\inf \left\{t>t_k \mid \varepsilon^2(t)>\sigma \theta(t)+\eta e_n^2(t)\right\}$       (27)

where, $\sigma>0$ and $0<\eta<k_{n 3}$ are design parameters, and $\theta(t)$ is an auxiliary dynamic variable governed by:

$\dot{\theta}(t)=-\mu \theta(t)+\varepsilon^2(t)-\eta e_n^2(t), \quad \theta(0)>0$       (28)

with $\mu>0$ being a positive constant.

In plain words, $\theta(t)$ acts as a bandwidth budget: it accumulates whenever the tracking error and the measurement error are simultaneously small (a quiescent phase), and it is spent during transients when frequent re-transmissions are necessary. The bus operator therefore sees a controller that "remembers" how much communication slack it has saved during easy phases and uses that slack to remain silent during slightly harder phases - without ever compromising the fixed-time stability guarantee of Theorem 2. In contrast, a static triggering law has no such memory and forces the controller to re-transmit at the same accuracy floor irrespective of the operating regime.

Remark 1: The dynamic variable $\theta(t)$ acts as an internal "energy reservoir." When $\theta(t)$ is large (accumulated from periods of small measurement errors), the triggering threshold is effectively enlarged, producing longer inter-event intervals. Compared with static event-triggered schemes [19], which use only $\varepsilon^2(t)>\eta e_n^2(t)$, the DETM provides additional flexibility and provably reduces the number of control updates. The parameter $\mu$ controls the decay rate of the dynamic variable; larger $\mu$ makes the mechanism more conservative (closer to static triggering), while smaller $\mu$ allows more aggressive event reduction.

The actual control applied to the plant is therefore:

$u(t)=u^*\left(t_k\right), \quad \forall t \in\left[t_k, t_{k+1}\right)$       (29)

4.4 Comparison with existing triggering mechanisms

The dynamic event-triggering law Eqs. (27)-(28) introduces an auxiliary state $\theta(t)$ that enlarges the triggering threshold beyond the static law $\varepsilon^2>\eta e_n^2 \mathrm{u} $ [19, 40, 41]. The next proposition makes this advantage precise.

Proposition 1 (Inter-event lower bound). Let $\left\{t_k^{\text {DETM}}\right\}$ and $\left\{t_k^{\mathrm{ST}}\right\}$ denote, respectively, the triggering sequences generated by the proposed DETM Eqs. (27)-(28) and by the static law $\varepsilon^2>\eta e_n^2$ when both schemes are applied to the closed-loop system of Section 4 starting from the same initial condition. Suppose $\theta\left(t_k^{\text {DETM}}\right)>0$. Then the next inter-event interval of the DETM satisfies

$t_{k+1}^{\mathrm{DETM}}-t_k^{\mathrm{DETM}} \geq \sqrt{1+\frac{\sigma \theta\left(t_k^{\mathrm{DETM}}\right)}{\eta e_n^2\left(t_k^{\mathrm{DETM}}\right)} \cdot\left(t_{k+1}^{\mathrm{ST}}-t_k^{\mathrm{ST}}\right) .}$

Proof. Between two consecutive events the measurement error satisfies $\varepsilon\left(t_k\right)=0$ and $|\dot{\varepsilon}| \leq L$ for some Lipschitz constant $L>0$ (all closed-loop signals are bounded by Theorem 2, Part 1). Hence $\varepsilon^2(t) \leq L^2\left(t-t_k\right)^2$. The static law triggers when $L^2\left(t-t_k\right)^2>\eta e_n^2\left(t_k\right)$, i.e., after at least $\Delta_{\mathrm{ST}} \triangleq 1 / L \sqrt{\eta e_n^2\left(t_k\right)}$.

The proposed DETM triggers when $L^2\left(t-t_k\right)^2> \sigma \theta\left(t_k\right)+\eta e_n^2\left(t_k\right)$, i.e., after at least $\Delta_{\mathrm{DETM}} \triangleq 1 / L \sqrt{\sigma \theta\left(t_k\right)+\eta e_n^2\left(t_k\right)}$. Taking the ratio yields (P1).

The bound (P1) quantifies the worst-case enlargement of the inter-event interval. In practice $\theta(t)$ accumulates whenever $\varepsilon^2$ stays well below $\eta e_n^2$ (typical of the steady-state phase), so the ratio $\sigma \theta /\left(\eta e_n^2\right)$ can grow significantly above 1 and the inter-event interval can be several times larger than the static one - a phenomenon that we verify experimentally in Section 6.3 (histogram of inter-event intervals). Conceptual diagram of how the auxiliary dynamic state $\theta(t)$ of the DETM "lifts" the triggering threshold above the static level $\eta e_n^2(t)$. Triggering events occur when $\varepsilon^2(t)$ crosses the upper (dynamic) envelope; the static law would have triggered at the lower (static) envelope, producing a denser event sequence as shown in Figure 1.

Figure 1. Conceptual diagram showing how the dynamic event-triggered mechanism (DETM) dynamic state raises the triggering threshold and reduces event occurrences

4.4.1 Design principles of the triggering parameters

This subsection translates the conditions established in Theorem 2 (Eqs. (38)-(39)) into a concrete tuning procedure for the four DETM parameters and supports the claims with a parametric sweep on the manipulator benchmark (Figure 2).

•  $\eta$ (static threshold gain). From Eq. (38), $\eta$ must satisfy $\eta<k_{n 3}$. Increasing $\eta$ widens the triggering window and reduces the number of events, but it enlarges the steady-state ultimate bound proportionally to $\sqrt{\eta}$. Recommended range: $\eta \in[0.2,0.4]$.
• $\sigma$ (dynamic gain). $\sigma>0$ controls how much the dynamic state $\theta$ is allowed to lift the threshold; small $\sigma$ → DETM is close to static; large $\sigma \rightarrow$ very aggressive event-reduction at the price of a larger residual. Recommended: $\sigma=0.1$.
• $\mu$ (decay rate of $\theta$). From Eq. (39), $\mu>1+\sigma$. Larger $\mu \rightarrow \theta$ decays faster, DETM behaves more like static. Smaller $\mu \rightarrow \theta$ accumulates, large savings appear in steady state. Recommended: $\mu=2.0$.
• $\theta(0)>0$ (initial budget). Provides finite slack at $t=$ 0 ; useful when the initial tracking error is small. Recommended: $\theta(0)=1$.

Step-by-step tuning procedure. When porting the controller to a new plant:

1. Pick $\gamma_1=0.6, \gamma_2=1.4, \alpha_2=0.6, \beta_2=1.4;$ the corresponding $\alpha_1, \beta_1$ follow from Theorem 1.
2. Choose $k_{i 1}, k_{i 2}$ so that the desired $T_{\max }$ follows from Eq. (30); aim for $k_{i 1}(1-p) \approx k_{i 2}(q-1)$.
3. Set $k_{i 3}=4$; if oscillations near the equilibrium appear, increase $k_{i 3}$ before increasing $k_{i 1}, k_{i 2}$.
4. Pick $\eta \in[0.2,0.4]$: start at the upper end if the steadystate bound is acceptable, lower it otherwise.
5. Set $\sigma=0.1, \mu=2.0$: they satisfy $\mu>1+\sigma$ and consistently yield the best trade-off.
6. Choose $l_1=8, l_2=5, l_3=10, l_4=6$ initially; if the observer response is too slow, scale all four by $1.5-2.0$.

Effect of the three DETM parameters on the trade-off between number of triggering events $N_{e v}$ (bars, left axis) and RMS tracking error of joint 1 (circles, right axis). (a) Sweep on $\eta \in\{0.1,0.2,0.3,0.4,0.5\}$ with $\sigma=0.1, \mu=2.0$. (b) Sweep on $\sigma \in\{0.02,0.05,0.1,0.2,0.5\}$ with $\eta=0.3, \mu=2.0$. (c) Sweep on $\mu \in\{0.5,1.0,2.0,3.0,5.0\}$ with $\eta=0.3, \sigma=0.1$. The chosen nominal values are marked with arrows, as shown in Figure 2.

Figure 2. Dynamic event-triggered mechanism (DETM) parameter sweep

4.5 Control energy and communication load

4.5.1 Control-energy budget

Define, on a horizon [0, T], the control energy

$E_u(T) \triangleq \int_0^T\|u(t)\|^2 d t$.

Because the controller is held constant between successive events, $\|u(t)\|$ is piecewise constant and $E_u(T)=\sum_{k: t_k<T} \left\|u\left(t_k\right)\right\|^2\left(t_{k+1}-t_k\right)$.

Lemma 2 (energy budget). Under Assumptions 1-3 and the parameters of Table 2, the closed-loop control energy on any horizon $T>T_{\max }$ satisfies

$E_u(T) \leq\left[k_1^2 \bar{e}_n^{2 \gamma_1}+k_2^2 \bar{e}_n^{2 \gamma_2}+k_3^2 \bar{e}_n^2+D_{\max }^2\right] T+C_0$,

Table 2. Comparison of the proposed scheme with closely related fixed-time / event-triggered controllers

with $C_0$ a finite constant independent of T. In particular, the per-unit-time energy is bounded by

$\bar{P}_u \triangleq \limsup _{T \rightarrow \infty} \frac{E_u(T)}{T} \leq k_1^2 \bar{e}_n^{2 \gamma_1}+k_2^2 \bar{e}_n^{2 \gamma_2}+k_3^2 \bar{e}_n^2+D_{\max }^2$.

Proof. Substitute the explicit control law Eqs. (13) into (20), apply Young's inequality $(a+b+c)^2 \leq 3\left(a^2+b^2+c^2\right)$, and use $\left\|e_n(t)\right\| \leq \bar{e}_n$ for $t \geq T_{\max }$ to obtain Eq. (21) with $C_0= \int_0^{T_{\text {max }}}\|u(t)\|^2 d t$, which is finite because all signals are uniformly bounded on $\left[0, T_{\max }\right]$.

4.5.2 Communication-load distribution

Let $\quad \Delta_k \triangleq t_{k+1}-t_k \quad$ and $\quad N_{e v}(T) \triangleq \#\left\{k: t_k \leq T\right\}$. Proposition 1 ensures $\Delta_k \geq \Delta_{\min }>0$, hence $N_{e v}(T) \leq T / \Delta_{\text {min}}$. The experimental histogram (Section 6.3) confirms that the average inter-event interval is several times larger than $\Delta_{\text {min}}$: a designer can size the shared bus capacity to the average rate $\bar{f}_{e v} \approx 168 \mathrm{~Hz}$ (manipulator) and $\approx 164 \mathrm{~Hz}$ (quadrotor) instead of the 1000 Hz periodic-control budget - a $6 \times$ reduction in bus loading.

To validate the proposed method, we apply it to a two-link planar robotic manipulator, which is a standard benchmark for nonlinear control [1, 2, 13].

6.1 System model

The dynamics of a two-link robot are:

$M(q) \ddot{q}+C(q, \dot{q}) \dot{q}+G(q)=\tau+d(t)$       (44)

where, $\mathrm{q}=\left[\mathrm{q}_1, \mathrm{q}_2\right]^{\mathrm{T}} \in \mathbb{R}^2$ is the vector of joint angles, $\tau= \left[\tau_1, \tau_2\right]^{\top}$ is the control torque, $d(t)$ is the external disturbance, and:

$\begin{gathered}M(q) \&=\left[\begin{array}{ll}m_{11} & m_{12} \\ m_{12} & m_{22}\end{array}\right] \\ m_{11}=\left(m_1+m_2\right) l_1^2+m_2 l_2^2+2 m_2 l_1 l_2 \cos \left(q_2\right) \\ m_{12}=m_2 l_2^2+m_2 l_1 l_2 \cos \left(q_2\right), \\ m_{22} \&=m_2 l_2^2\end{gathered}$                  (45)

$\begin{aligned} & C(q, \dot{q}) \&  =\left[\begin{array}{cc}-m_2 l_1 l_2 \sin \left(q_2\right) \dot{q}_2 & -m_2 l_1 l_2 \sin \left(q_2\right)\left(\dot{q}_1+\dot{q}_2\right) \\ m_2 l_1 l_2 \sin \left(q_2\right) \dot{q}_1 & 0\end{array}\right]\end{aligned}$            (46)

$\begin{aligned} & G(q) \& =\left[\begin{array}{c}\left(m_1+m_2\right) g l_1 \sin \left(q_1\right)+m_2 g l_2 \sin \left(q_1+q_2\right) \\ m_2 g l_2 \sin \left(q_1+q_2\right)\end{array}\right]\end{aligned}$        (47)

The physical parameters are: $m_1=1.0 \mathrm{~kg}, m_2=0.8 \mathrm{~kg}$, $l_1=1.0 \mathrm{~m}, l_2=0.8 \mathrm{~m}, g=9.81 \mathrm{~m} / \mathrm{s}^2$.

By defining the state as $x=\left[q_1, \dot{q}_1, q_2, \dot{q}_2\right]^{\top}$, the system can be written in the form Eq. (6) with $n=2$ for each joint channel (the control is designed independently for each joint via the computed-torque pre-compensation).

6.2 Controller parameters

The controller and observer parameters are selected as shown in Table 3.

Table 3. Design parameters

The reference trajectories are $q_{r 1}(t)=\sin (t)$ and $q_{r 2}(t)=\cos (t)$. The external disturbances are:

$d(t)=\left[\begin{array}{l}0.5 \sin (3 t)+0.2 \cos (5 t) \\ 0.3 \cos (2 t)+0.4 \sin (4 t)\end{array}\right]$.         (48)

The initial conditions are $q(0)=[-1.5,2.0]^{\top}$ rad and $\dot{q}(0)=[0,0]^{\top}$ rad/s, chosen to be far from the reference to test the initial-condition independence of fixed-time convergence.

Three controllers are compared:

• Proposed: Fixed-time controller with FTDO and DETM (this work).
• FT-Cont: Fixed-time controller with continuous updates (no event triggering, same FTDO).
• Finite-Time: Finite-time controller using only the $\gamma_1$ terms (no $\gamma_2$-terms, no event triggering).

Energy and communication-load analysis. Applying Lemma 2 to the manipulator parameters gives the predicted asymptotic power $\bar{P}_u \approx 315 (\mathrm{N} \cdot \mathrm{~m})^2 / \mathrm{s}$, which matches the measured $E_u / T=3155 / 10=315.5$. Each avoided update suppresses a high-frequency switching component of $u$ that carries no useful information once $|\varepsilon|$ is below the threshold; consequently the residual control signal is smoother and $\|u(t)\|^2$ is on average $9.9 \%$ lower than that of the periodic FT-Cont and $6 \%$ lower than that of SMC-DO. The histogram of inter-event intervals (Figure 5) shows a heavier tail than SET-FT, with mean $\approx 5.9$ ms vs. 5.2 ms , formally substantiating the bandwidth saving. Histogram of inter-event intervals for the proposed DETM versus the static SET-FT trigger on the manipulator benchmark. The DETM produces a heavier tail with mean $\approx 5.9$ ms (against 5.2 ms for SET-FT), confirming that the dynamic threshold $\sigma \theta_i$ opens additional silence intervals during slow phases of the trajectory, as shown in Figure 5.

Figure 5. Manipulator inter-event histogram

6.4 Trajectory tracking of a 6-DOF quadrotor UAV

To stress-test the proposed scheme on a higher-dimensional, faster, and more uncertain plant, we apply the controller to a 6-DOF quadrotor UAV operating under simultaneous mass uncertainty, wind disturbances, and a tight communication budget.

6.4.1 Plant model and operating conditions

Using the standard time-scale separation between the (fast) attitude inner loop and the (slow) translational outer loop, the translational dynamics in the inertial frame are

$\begin{gathered}m(t) \ddot{p}(t)=u_{\mathrm{pos}}(t)+d_{\mathrm{pos}}(t)-m(t) g e_z, \\ p(t) \in \mathbb{R}^3,\end{gathered}$

with time-varying mass $m(t)=m_0(1+0.30 \sin (0.3 t))$ kg, $m_0=1.5 \mathrm{~kg}$ (a $30 \%$ payload uncertainty representative of a delivery drone with shifting load), and wind/gust disturbance $d_{\text {pos}}(t)=[1.2 \sin (0.8 t),-1.0 \cos (0.6 t), 0.8 \sin (0.5 t)]^{\top} \mathrm{N}$.

The reference is the helical trajectory $p_r(t)= [2 \cos (0.5 t), 2 \sin (0.5 t), 1+0.2 \sin (t)]^{\top}$ m. Thrust is saturated at 35 N ; initial state $p(0)=[2,0,1]^{\top}, \dot{p}(0)=0$. The control law uses the FTDO of Section 3 along each axis, the fixed-time backstepping of Theorem 1, and the DETM of Section 4.2. Gains: $\left(k_1, k_2, k_3\right)=(3.0,1.8,2.5)$, exponents $\left(\gamma_1, \gamma_2\right)=(0.6,1.4)$, DETM $(\sigma, \eta, \mu)=(1.5,3.0,2.0)$.

6.4.2 Results

Figure 6 reports position tracking, Figure 7 the implied tilt commands, Figure 8 the FTDO output $\hat{\mathrm{D}}$, Figure 9 the triggering instants of the proposed DETM versus SET-FT, and Figure 10 the histogram of inter-event intervals. All metrics in Table 4 computed over the post-transient horizon $t \in[3,10]$ s consistent with 6.3, which begins well after the longest settling time observed in Table 4 $\left(\mathrm{T}_5 \leqslant 0.318 \right)$ s.

Figure 9. Triggering instants (DETM vs SET-FT) for quadrotor

Note: dynamic event-triggered mechanism (DETM); static event-triggered fixed-time control (SET-FT)

Figure 10. Quadrotor inter-event histogram

Table 4. Quadrotor UAV — performance comparison under 30% mass uncertainty and unmatched wind disturbance

Note: dynamic event-triggered mechanism (DETM); static event-triggered fixed-time control (SET-FT); fixed-time disturbance observer (FTDO); sliding-mode control with a disturbance observer (SMC-DO)

Update reduction. The proposed scheme issues 1638 updates against 10000 periodic — an 83.6% reduction — and 48% fewer than the static SET-FT trigger. The dynamic threshold $\sigma \theta_i(t)$ adapts the budget to the local disturbance.

Tracking accuracy. The steady-state RMS position error $\left(2.7 \times 10^{-4} \mathrm{~m}\right)$ is statistically indistinguishable from the periodic FT-Cont ($2.4 \times 10^{-4} \mathrm{~m}$); the DETM does not trade accuracy for bandwidth.

Energy efficiency. $E_u=3143.8$ for the proposed scheme vs. 3153.0 for SMC-DO ($0.3 \%$ advantage) with $30 \%$ lower peak thrust (20.7 N vs. 29.5 N).

Robustness. The FTDO tracks the wind disturbance to within 5% of its peak within 0.3 s, consistent with the fixed-time bound of Eq. (12).

Generality. The same gain set of Table 2 transfers verbatim from the manipulator to the 6-DOF UAV with no retuning.

Quadrotor position tracking under 30% mass uncertainty. All five controllers are plotted against the helical reference. Trajectories overlap to plotting accuracy except in the first 0.3 s, where the proposed scheme shows the smallest overshoot, as shown in Figure 6.

Effective tilt commands $\phi_d, \theta_d$ (rad). All controllers respect the ± 0.4 rad envelope; proposed produces visibly smoother commands than SMC-DO, as shown in Figure 10. FTDO output $\widehat{D}$ along the three inertial axes. Convergence achieved within ≤ 0.3 s on all three channels, as shown in Figure 11. Triggering instants for the proposed DETM (top) and SET-FT (bottom). The proposed scheme leaves long quiescent intervals during slow phases of the helix, as shown in Figure 12. Histogram of inter-event intervals. Proposed mean ≈ 6.1 ms vs. SET-FT 3.2 ms, confirming the dynamic threshold opens additional silence intervals, as shown in Figures 7-10.

Figure 11. Joint angle tracking: (a) Joint 1, (b) Joint 2. Solid lines: Actual; dashed lines: Reference

Figure 12. Tracking errors: $e_1=q_1-q_{r 1},$ (b) $e_2=q_2-q_{r 2}$