# State and Parameter Estimation Architecture for Quad-Rotor Tail-Sitter VTOL UAV 16 STATE AND PARAMETER ESTIMATION ARCHITECTURE FOR QUAD-ROTOR TAIL-SITTER VTOL UAV 1

Kamelia Khammar, Mohamed Mostefa, Fouad Khenfri

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Kamelia Khammar, Mohamed Mostefa, Fouad Khenfri. State and Parameter Estimation Architecture for Quad-Rotor Tail-Sitter VTOL UAV 16 STATE AND PARAMETER ESTIMATION ARCHITECTURE FOR QUAD-ROTOR TAIL-SITTER VTOL UAV 1. International Journal of Electrical, Electronics and Data Communication (IJEEDC), 2025, 13 (2), pp.16-19. hal-05331597

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# STATE AND PARAMETER ESTIMATION ARCHITECTURE FOR QUAD-ROTOR TAIL-SITTER VTOL UAV

¹KAMELIA KHAMMAR, ²MOSTEFA MOHAMED TOUBA, ³FOUAD KHENFRI

¹,²Electrical Engineering, Department Mohamed Kheider University Biskra, Algeria

³ESTACA'LAB, Laval University Park, Laval, France

E-mail: ¹kamelia.khammar@univ-biskra.dz, ²Mostefa.touba@univ-biskra.dz, ³Fouad.khenfri@estaca.fr

**Abstract** - This paper presents a novel architecture for the simultaneous state and parameter estimation of a quad-rotor tail-sitter Vertical Take-Off and Landing (VTOL) Unmanned Aerial Vehicle (UAV). Tail-sitter VTOL UAVs, known for their hybrid capabilities between hovering and forward flight, pose significant challenges in control and navigation due to their complex dynamics. To address these challenges, our framework integrates multiplier matrices, a data-driven concurrent learning strategy, and an adaptive observer. This combination enables improved performance in state and parameter estimation, providing robustness where traditional methods may falter. A critical feature of the proposed architecture is its ability to guarantee the convergence of parameter estimation error under specific excitation conditions. By ensuring these conditions are met during flight, the estimation error converges to a defined neighborhood around the origin, enhancing the precision of the UAV's control system. This characteristic is essential for real-time adaptability in dynamic environments, where rapid adjustments are necessary to maintain stability. Our solution represents a significant advancement for hybrid VTOL UAVs, particularly in ensuring smooth transitions between different flight modes.

**Keywords** - State Estimation, Parameter Estimation, Quad-Rotor Tail-Sitter VTOL UAV, Extended Luenberger, Nonlinear Observers

## I. INTRODUCTION

Unmanned aerial vehicle (UAV) technology known as vertical take-off and landing (VTOL) is expanding quickly due to its capacity to take off and land in confined locations. The tail-sitter is the most efficient type of VTOL UAV. Beyond the efficiency provided, there are still certain obstacles to be addressed, such as the difficulty of fusing the capacity to fly horizontally like a fixed-wing aircraft with the ability to hover like a helicopter [1].

The convergence properties of the expanded Luenberger state observers can be used to generate precise state estimates for parameter estimation, even when partial state measurements are available [2]. Recent research focuses on the development of adaptive observers capable of simultaneously estimating the state and parameters of nonlinear systems using methods of persistent excitation (PE) and finite excitation [3]. This research aims to allow the measurement of all state variables [4]–[5], [6]. Most methods are limited to a specific class of nonlinear systems and are based on assumptions that may be difficult to meet in practice, such as strict PE conditions [4]–[5], [6].

The methods described in [4] and [5] are effective but limited to the dynamic systems of the Brunovsky canonical form. This paper presents a combination of two approaches to estimate the state and parameters of Affine Nonlinear Systems: The matrix multiplier approach to design a Luenberger observer which guarantees the asymptotic convergence of the state estimation errors, and Concurrent learning (CL) uses recorded data to estimate the system parameters with

high accuracy see [7]. We applied to quadrotor tail-sitter VTOL UAV. This work provides practical perspectives to have Adaptive Control of quadrotor Tail-sitter VTOL UAV in actual settings, paving the way for new advancements in the field of autonomous aerial systems. The rest of the article is structured as follows: Section 2 presents design of state observer; Section 3 presents design of parameter estimator, and Sections 4 presents experimental results and finally, concludes the paper.

## II. DESIGN OF STATE OBSERVER

Consider a nonlinear dynamical system of the form [11]:

$$ \dot{x} = Y(x)\theta + g(x)u, \quad y = Cx, \quad (1) $$

The creation of a state observer that uses an extended Luenberger observer to produce estimates of x is presented in this section.

To simplify the observer design, the nonlinear dynamics given in (1) can be written as [11]:

$$ \dot{x} = Ax + F_{\theta}(x, \theta) + G_{u}(x, u), \quad y = Cx, \quad (2) $$

$$ \text{Where } A = (K_{y1} + K_{g1}), \quad F_{\theta}(x, \theta) = -K_{y1}x + Y(x)\theta $$

$$ \text{And } G_u(x, u) = -K_{g1}x + \sum_{i=1}^{N} g_i(x)(u)_i $$

The estimation error dynamics is given by:

$$ \begin{aligned} \dot{x} = (A - LC)\tilde{x} + F_{\theta}(x, \hat{\theta}) + G_u(x, u) - F_{\theta}[\hat{x} + l_1(y - C\hat{x}), \hat{\theta}] \\ -G_u[\hat{x} + l_2(y - C\hat{x}), u] + F_{\theta}(x, \hat{\theta}), \end{aligned} \quad (3) $$

$$ \text{Where } F_{\theta}(x, \hat{\theta}) := F_{\theta}(x, \theta) - F_{\theta}(x, \hat{\theta}). $$

Let the parameter estimation error is defined $\hat{\theta} := \theta - \hat{\theta}$. To facilitate the design of the state observer, the following assumption is made about the set $\Theta$, which contains $\theta$. The observer error dynamics in (3) can then be expressed as:

$$ \tilde{x} = (A - LC)\tilde{x} + \Theta_y(t, \tilde{x}, \hat{\theta}) + \Theta_g(t, \tilde{x}) + F_{\theta}(x, \hat{\theta}), \quad (4) $$

According to the differential mean value theorem (DMVT) the difference functions $\Theta_y$ and $\Theta_g$ are guaranteed to be bounded, which can be expressed in their quadratic forms as:

$$ \begin{bmatrix} \tilde{x} \\ \Theta_y \end{bmatrix} - \begin{bmatrix} l_1 C & 0 \\ 0 & I_n \end{bmatrix}^T M_y \begin{bmatrix} l_1 - l_2 C & 0 \\ 0 & I_n \end{bmatrix} \begin{bmatrix} \tilde{x} \\ \Theta_y \end{bmatrix} \le 0, \quad (5) $$

And
$$ \begin{bmatrix} \bar{x}_g \\ \bar{\theta}_g \end{bmatrix}^T \begin{bmatrix} I_n - l_2 C & 0 \\ 0 & I_n \end{bmatrix} M_g \begin{bmatrix} I_n - l_2 C & 0 \\ 0 & I_n \end{bmatrix} \begin{bmatrix} \bar{x}_g \\ \bar{\theta}_g \end{bmatrix} \le 0, \quad (6) $$

With their corresponding multiplier matrices designed as:

$$ M_y = \begin{bmatrix} 0 & \frac{-K_{y2}^T - K_{y1}^T}{2} \\ \frac{-K_{y2} - K_{y1}}{2} & I_n \end{bmatrix}, \quad (7) $$

$$ M_g = \begin{bmatrix} 0 & \frac{-K_{g2}^T - K_{g1}^T}{2} \\ \frac{-K_{g2} - K_{g1}}{2} & I_n \end{bmatrix}, \quad (8) $$

A Lyapunov-based analysis that uses the above inequalities to establish boundness of the state estimation error for all $t \in R \ge 0$.

**Analysis of State Observer:** The following Theorem establishes local uniform ultimate boundness of the state estimation errors.

**Theorem 1:** Provided Assumption 1 and 2 hold, there exists a constant symmetric positive definite matrix P, and three observer gains, $l_1$, $l_2$ and L, that satisfy the matrix inequality:

$$ \begin{bmatrix} (A-LC)^T P \\ +(P(A-LC)) + 2aP & P - (I-l_1 C)^T (M_g)_{22} & P - (I-l_2 C)^T (M_g)_{22} \\ P - (M_g)_{21}(I-l_1 C) & -(M_g)_{22} & 0 \\ P - (M_g)_{21}(I-l_2 C) & 0 & -(M_g)_{22} \end{bmatrix} < 0, \quad (9) $$

Then observer error in (3) is locally uniformly ultimately bounded.

**Proof:** Consider the continuously differentiable candidate Lyapunov function, $W: D \to R$ defined as:

$$ W(\bar{x}) := \bar{x}^T P \bar{x}, \quad (10) $$

Which satisfies $\lambda_{min}(P)\|\bar{x}\|^2 \le W(\bar{x}) \le \lambda_{max}(P)\|\bar{x}\|^2$. since P is a constant symmetric positive definite matrix, both eigenvalues are positive. On the set D, the orbital derivative of the Lyapunov function along the trajectories of (3) can be expressed as:

$$ \dot{W}(\bar{x}, t) = \begin{bmatrix} \bar{x} \\ \bar{\theta}_g \end{bmatrix}^T \begin{bmatrix} (A-LC)^T P + P(A-LC) & P & P \\ P & 0 & 0 \\ P & 0 & 0 \end{bmatrix} + \bar{x}^T P F_{\bar{\theta}_g}(x, \bar{\theta}) + F_{\bar{\theta}_g}(x, \bar{\theta}) P \bar{x}, \quad (11) $$

Provided the matrix inequalities in (9) is satisfied for some constant $\alpha \in R_+$,

$$ \dot{W}(\bar{x}, t) \le -2aW(\bar{x}) + 2\lambda_{\max}(P)\bar{F}_{\bar{\theta}_g}\|\bar{x}\|, \quad (12) $$

For all $\bar{x} \in D$, where $\max_{x \in C} \|F_{\bar{\theta}_g}(x, \bar{\theta})\| \le \bar{F}_{\bar{\theta}_g}$ for some $\bar{F}_{\bar{\theta}_g} \in R_+$. Hence, the orbital derivative is bounded along the trajectories of (4) as:

$$ \dot{W}(\bar{x}, t) \le -\alpha W(\bar{x}), \quad \forall \bar{x} \in D, \quad \|\bar{x}\| \ge \xi > 0, \quad (13) $$

Where $\xi = \frac{2\lambda_{\max}(P)\bar{F}_{\bar{\theta}_g}}{\lambda_{\max}(P)}$. Invoking [13, Thereom 4.18], the state estimation error is locally uniformly ultimately bounded. The matrix P and the observer gains L, $l_1$ and $l_2$ can be obtained by solving the LMI:

$$ \begin{bmatrix} (A^T P + PA) \\ -C^T R^T - RC \\ P - (M_g)_{21}(I - l_1 C) & -(M_g)_{22} & 0 \\ P - (M_g)_{21}(I - l_2 C) & 0 & -(M_g)_{22} \end{bmatrix} + 2aP \begin{bmatrix} P - (I - l_1 C)^T (M_g)_{22} & P - (I - l_2 C)^T (M_g)_{22} \\ -(M_g)_{22} & 0 & -(M_g)_{22} \end{bmatrix} < 0, \quad (11) $$

For P, R, $l_1$ and $l_2$.

### III. DESIGN OF PARAMETER ESTIMATOR

The parameter estimator to be designed in this section relies the difference between the state estimates at time t and time $(t - \zeta)$, where $\zeta \in R_+$ denotes the time delay, can be expressed as an affine function of the parameters $\theta$ and a residual that reduces with reducing state estimation errors as described in the following Lemma.

**Lemma 1:** If $x, \tilde{x} \in C$ and if Assumption 1 holds for all $\zeta \ge 0$ and for all $t \ge \zeta$, the state estimates satisfy:

$\hat{x}(t) - \hat{x}(t - \zeta) = \hat{y}(t)\theta + \hat{g}_u(t) + E(t)$, Where $\hat{y}(t) := \int_{t-\zeta}^{t} Y(\hat{x}(\tau))d\tau, \hat{g}_u(t) := \int_{t-\zeta}^{t} g(\hat{x}(\tau))u(\tau)d\tau$, And $E(t) = O(\sup_{\sigma \in [t-\zeta, t]} \|\hat{x}(\sigma)\|)$.

**Proof:** Integrating the dynamics in (1) yields. Lemma 1 implies that the parameter estimation error at any time t can be expressed as: $\hat{y}(t)\hat{\theta}(t) = \hat{x}(t) - \hat{x}(t - \zeta) - \hat{g}_u(t) - \hat{y}(t)\hat{\theta}(t) - E(t)$, Witch motivates the update law:

$$ \hat{\theta} = \begin{cases} k_{\theta} \Gamma_{\emptyset}, & \text{if } \hat{\theta}^{\mathrm{T}} \hat{\theta} < \bar{\theta}^2 \text{ or} \\ \hat{\theta}^{\mathrm{T}} \hat{\theta} = \bar{\theta}^2 \text{ and } (k_{\theta} \Gamma_{\emptyset})^{\mathrm{T}} \hat{\theta} \le 0 & \text{if } \hat{\theta}^{\mathrm{T}} \hat{\theta} \le \bar{\theta}^2 \text{ and } (k_{\theta} \Gamma_{\emptyset})^{\mathrm{T}} \hat{\theta} \le 0 \\ I_p - \frac{T \hat{\theta} \hat{\theta}^{\mathrm{T}}}{\hat{\theta}^{\mathrm{T}} \bar{\theta} \hat{\theta}}, & \text{otherwise} \end{cases} \quad (14) $$

Where

$\emptyset(t) := \sum_{i=1}^{N} \left( \frac{\hat{y}(t_i)}{1+k\|\hat{y}(t_i)\|^2} \right)^T (\hat{x}(t_i) - \hat{x}(t_i - \zeta) - \hat{g}_u(t_i) - \hat{y}(t_i)\hat{\theta})$, $k \in R_{>0}$, the normalization is gain, and $k_{\theta} \in R_{>0}$ is the CL gain. The matrix $\Gamma \in R^{pxp}$ is the least-squares gain matrix updated as:

$$ \hat{\Gamma} = \begin{cases} \beta_1 \Gamma - k_{\theta} \Gamma \frac{\hat{y}(t_i)^T \hat{y}(t_i)}{1 + k \|\hat{y}(t_i)\|^2} \Gamma, & \text{if } \hat{\theta}^{\mathrm{T}} \hat{\theta} < \bar{\theta}^2 \text{ or} \\ \hat{\theta}^{\mathrm{T}} \hat{\theta} = \bar{\theta}^2 \text{ and } (k_{\theta} \Gamma_{\emptyset})^{\mathrm{T}} \hat{\theta} \le 0 & \text{if } \hat{\theta}^{\mathrm{T}} \hat{\theta} \le \bar{\theta}^2 \text{ and } (k_{\theta} \Gamma_{\emptyset})^{\mathrm{T}} \hat{\theta} \le 0 \\ 0, & \text{otherwise} \end{cases} \quad (15) $$

Where $\beta_1 \in R > 0$ is a constant adaptation gain. The update law relies on the time delay $\zeta$, and a history stack $\mathcal{H}$. The history stack represents a set of piecewise constant functions that can be expressed as:

$$ \bar{x} := \begin{cases} \hat{x}(t_1) - \hat{x}(t_1 - \zeta), & \hat{y}(t_1) = \hat{y}(t_1) \\ \hat{x}(t_N) - \hat{x}(t_N - \zeta), & \hat{y}(t_N) = \hat{y}(t_N) \end{cases}, \hat{y} := \begin{bmatrix} \hat{y}(t_1) \\ \hat{y}(t_N) \end{bmatrix}, \hat{g}_u := \begin{bmatrix} \hat{g}_u(t_1) \\ \hat{g}_u(t_N) \end{bmatrix}, \quad (16) $$

Where $\hat{X} \in R^{nN}$, $\hat{Y} \in R^{nNxp}$, and $\hat{g}_u \in R^{nN}$. Using Lemma 1, the parameter estimation error dynamics can be expressed as:

$$ \hat{\theta} = -k_{\theta} \Gamma \sum_{i=1}^{N} \frac{\hat{y}(t_i)^T \hat{y}(t_i)}{1 + k \|\hat{y}(t_i)\|^2} \hat{\theta} - k_{\theta} \Gamma \sum_{i=1}^{N} \frac{\hat{y}(t_i)^T E(t_i)}{1 + k \|\hat{y}(t_i)\|^2}, \quad (17) $$

It is clear from (17) that for the parameter estimation error to be bounded, the matrix $\sum_{i=1}^{N} \frac{\hat{y}(t_i)^T \hat{y}(t_i)}{1+k\|\hat{y}(t_i)\|^2}$ needs to be positive definite, which can be ensured if the trajectories are sufficiently informative and the data $(\hat{x}(t_1) - \hat{x}(t_1 - \zeta), \hat{y}(t_i), \hat{g}_u(t_i))_{i=1}^N$ stored in the history stack $\mathcal{H}$ are recorded carefully. Architecture (The algorithm) presented in Algorithm1 [7]. The availability of accurate state estimates is required for precise parameter estimation. However, the initial history stack, recorded during transients, may contain inaccurate data, requiring a purge of the history stack once more accurate state estimates become available. In such cases, newer state estimates are preferred, subject to the conditions of Theorem 1. A greedy purging algorithm based on dwell time is employed to ensure estimator stability while utilizing newer data. This architecture (algorithm) uses two history stacks: a main stack denoted as $\mathcal{H}$ and a transient stack labeled G. The transient stack is filled until a sufficient dwell time T has elapsed. Then, the main stack is purged, and the transient stack is copied

into the main stack. This approach enables the use of newer, more accurate data while maintaining estimator stability.

**Algorithm 1** Algorithm for Adaptive History Stack Observer. At each time instance t, $\tau_1$ stores the last time an event occurred, $\tau_2$ stores the last time instance $\mathcal{H}$ was purged, $\lambda$ stores the highest minimum eigenvalue encountered so far, T denotes the dwell time, $\lambda^*$ denotes some user selected eigenvalue threshold, $l^*$ denotes some user selected sampling rate and $\xi \in (0, 1]$ is a threshold for purging.
**Require:** $t_f \in \mathbb{R}_{t_0}, t^* \in \mathbb{R}_+, T \in \mathbb{R}_{t_0}, \lambda^* \ge 0$
1. $\mathcal{X} \leftarrow 0, \mathcal{Y} \leftarrow 0, \mathcal{F}_u \leftarrow 0, \tau_1 \leftarrow 0, \tau_2 \leftarrow 0 \quad \triangleright \text{Global variables}$
2. $\lambda \leftarrow \min(\text{eig}(\mathcal{F}^T\mathcal{F}))$, $t_0 \leftarrow 0, \hat{x}_0 \leftarrow \hat{x}(t_0), \hat{\theta}_0 = \hat{\theta}(t_0)$
3. **while** $(t_0 < t_f \text{ do}$
4. &nbsp;&nbsp;&nbsp;&nbsp; integrate DDEs in (21), (22) and (24) over $[t_0, t_f]$
5. &nbsp;&nbsp;&nbsp;&nbsp; **if** $(t - \tau_1) \ge t^*$ **then**
6. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **if** $t \ge \zeta$ **then**
7. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; stop integration, an event has occurred
8. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $j \leftarrow \operatorname{argmax}_{i=1:N}\{\min\{\operatorname{eig}(\mathcal{F}^T\mathcal{F} - \hat{\mathcal{Y}}_i^T\hat{\mathcal{Y}}_i + \hat{\mathcal{Y}}_i^T\hat{\mathcal{Y}}_i)\}\}$
9. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **if** $\max_{i=1:N}\{\min\{\operatorname{eig}(\mathcal{F}^T\mathcal{F} - \hat{\mathcal{Y}}_i^T\hat{\mathcal{Y}}_i + \hat{\mathcal{Y}}_i^T\hat{\mathcal{Y}}_i)\} - \lambda \ge \lambda^*$
10. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\lambda \leftarrow \max_{i=1:N}\{\min\{\operatorname{eig}(\mathcal{F}^T\mathcal{F} - \hat{\mathcal{Y}}_i^T\hat{\mathcal{Y}}_i + \hat{\mathcal{Y}}_i^T\hat{\mathcal{Y}}_i)\}\}$
11. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\{\hat{\mathcal{Y}}_i\}_{i=(j-1)}^j \leftarrow \hat{\mathcal{Y}}(t)$
12. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\{\hat{\mathcal{Y}}_u\}_{u=(j-1)}^j \leftarrow \hat{\mathcal{Y}}_u(t)$
13. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\{\mathcal{F}_u\}_{u=(j-1)}^j \leftarrow \hat{x}(t) - \hat{x}(t-\zeta)$
14. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **if** $\mathcal{G}$ is not full then
15. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; add the data points to $\mathcal{G}$
16. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **else**
17. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; add the data points to $\mathcal{G}$ if (26) holds
18. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **end if**
19. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **if** $\min(\operatorname{eig}(\mathcal{F}^T\mathcal{F})) \ge \xi\lambda$ **then**
20. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **if** $(t - \tau_2) \ge T(t)$ **then**
21. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\mathcal{H} \leftarrow \mathcal{G}, \mathcal{G} \leftarrow 0$, and $\tau_2 \leftarrow t$
22. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **if** $\lambda < \min(\operatorname{eig}(\mathcal{F}^T\mathcal{F}))$ **then**
23. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\lambda \leftarrow \min(\operatorname{eig}(\mathcal{F}^T\mathcal{F}))$
24. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **end if**
25. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **end if**
26. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $t_0 \leftarrow t, x_0 \leftarrow x(t), \hat{\theta}_0 \leftarrow \hat{\theta}(t)$
27. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $l_{\gamma,0} \leftarrow l_{\gamma}(t), \hat{l}_{\gamma,0} \leftarrow \hat{l}_{\gamma}(t)$
28. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **end if**
29. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **end if**
30. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **else**
31. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; no event, keep on integrating the DDEs
32. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **end if**
33. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; $\tau_1 \leftarrow t \quad \triangleright \text{ Set this even if a new event is not detected}$
34. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; **end if**
35. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; no event, keep on integrating the DDEs
36. **end while**

**-Analysis of Parameter Estimator:** in order to rigorously analyze the convergence properties of the parameter estimation error, a precise definition of “Finitely Informative” and “Persistently Informative” data in the history stack [12]. The dynamics of the parameter estimation error in (17) and (15) can be expressed as:

$$ \dot{\theta} = -k_{\theta} \Gamma \psi_{\theta} \tilde{\theta} - k_{\theta} Q_{\theta} \text{ and } \dot{\Gamma} = \beta_{1} \Gamma - k_{\theta} \Gamma \psi_{\theta} \Gamma \quad (18) $$

Respectively, it is important to note that the functions $\psi_{\theta}$ and $Q_{\theta}$ are piece-wise continuous. Thus, the trajectories of (18) are defined in the sense of Caratheodory [5]. Using arguments similar to [4, Theorem 1], provided the conditions of Theorem 1 are satisfied, and the states and state estimation errors remain within the compact sets C and D, respectively, over the time interval $I_{s-1}$ in which the history stack was recorded, then using the error bound developed in Lemma 1 the error terms can be bounded as:

$$ \|E_{\epsilon t}\| \le L_e \bar{e}_{\epsilon}, \forall t \in \{1, \dots, N\}, \forall \tilde{x} \in D, \quad (19) $$

Where $\bar{e}_{\epsilon} := \sup_{t \in I_{s-1}} \|\tilde{x}\|$ and $L_e \in \mathbb{R}_+$ is a constant.

**Theorem 2:** If the state and parameters of the system in (1) are estimated using state and parameter estimators that satisfy the conditions of Theorem 1 and Assumption 3, the signal $(\hat{x}, u)$ is FI, $H$ is populated using Architecture (Algorithm1) [5], and if the excitation lasts long enough for two purging events (i.e. $H_3$ is full rank), then the trajectories of the parameter estimation error are ultimately bounded.

**Proof:** Consider the candidate Lyapunov function V:

$$ \Theta \times \mathbb{R} \ge 0 \rightarrow \mathbb{R} \text{ defined as: } V(\tilde{\theta}, t) := \frac{1}{2} \dot{\theta}^T \Gamma^{-1}(t) \tilde{\theta}, \quad (20) $$

The candidate Lyapunov function satisfies the following inequality:

$$ \frac{1}{2\Gamma} \|\tilde{\theta}\|^2 \le V(\tilde{\theta}, t) \le \frac{1}{2\Gamma} \|\tilde{\theta}\|^2, \quad \forall t \in \mathbb{R}_{\ge 0}, \quad (21) $$

Using the completion of squares, the orbital derivative is then bounded for all $\forall t \in \mathbb{R}_{\ge 0}$ as:

$$ \dot{\tilde{v}}_s(\tilde{\theta}, t) \le -\frac{1}{4} \underline{a} \|\tilde{\theta}\|^2, \quad \forall \|\tilde{\theta}\| \ge \rho(\|u\|), \quad (22) $$

Where $\rho(\|u\|) = \sqrt{\frac{\Gamma}{\underline{a}}(4k_{\theta})} \|u\|^2$ and $= \sqrt{\bar{Q}_s}$. Hence, the conditions of [13, Theorem 4.19] are satisfied, and it can be concluded that (18) is input-to-state stable with state $\tilde{\theta}$ and input $u$. If Architecture (Algorithm 1) is implemented and if the signal $(\hat{x}, u)$ is FI, then there exists a time instance $T_s$, such that for all $t \ge T_s$, the history stack remains unchanged. As a result, using [13, Exercise 4.58], and an ultimate bound on $\tilde{\theta}$ can be estimated as:

$$ \lim_{t \to \infty} \sup \|\tilde{\theta}(t)\| \le \tilde{\theta}(T_s) = \sqrt{\frac{\Gamma}{\underline{\Gamma}}} \left( \frac{4k_{\theta} \bar{Q}(T_s)}{\underline{a}} \right) \quad (23) $$

Where $\bar{Q}(T_s)$ denotes a bound on residual error term $Q_s$, in the history stack $H$ for all $t \ge T_s$. The parameter estimation error can be reduced by reducing the estimation errors corresponding to the state estimates stored in the history stack, which reduces $Q_s$. The projection algorithm and Theorem 1 imply the boundedness of all signals in the closed loop for all $t$. Furthermore, Theorem 1 implies that given any $\epsilon \in \mathbb{R}_+$ the gain $\alpha$ can be selected large enough to ensure that $\hat{x}$ has reached the ultimate bound before $t=T_1$ and that the ultimate bound is smaller than $\epsilon$ so that $\bar{\alpha}_2 \le \epsilon$. Since the history stack $H_3$, which is active over the interval $I_3$, is recorded during the interval $I_2$, the bounds in (19) can be used to show

$$ \bar{Q}_3 = \frac{N_1 \epsilon_2}{2\sqrt{\epsilon}} \le \frac{N_1 \epsilon_2}{2\sqrt{\epsilon}} \epsilon. \text{ As such, if } (\hat{x}, u) \text{ is FI with the excitation lasting long enough so that } H_3 \text{ is full rank, then (23) implies that:} $$

$$ \lim_{t \to \infty} \sup \|\tilde{\theta}(t)\| \le \sqrt{\frac{\Gamma}{\underline{\Gamma}}} \left( \frac{2k_{\theta} N \epsilon_e}{\underline{a}} \right) \epsilon $$

## IV. EXPERIMENTAL RESULTS

<table><caption>TABLE I. SIMULATION PARAMETERS</caption><thead><tr><th>Parameters</th><th>Value</th></tr></thead><tbody><tr><td>g</td><td>9.8 m.s<sup>-2</sup></td></tr><tr><td>Mass(m)</td><td>1.2 kg</td></tr><tr><td>Lengh(l)</td><td>1m</td></tr><tr><td>I<sub>xx</sub>, I<sub>yy</sub></td><td>7.5x10<sup>-3</sup>Kg.m<sup>2</sup></td></tr><tr><td>I<sub>zz</sub></td><td>1.3x10<sup>-3</sup>Kg.m<sup>2</sup></td></tr><tr><td>I<sub>r</sub></td><td>7.5x10<sup>-5</sup>Kg.m<sup>2</sup></td></tr><tr><td>Dragcoefficient(d)</td><td>7.5x10<sup>-7</sup></td></tr><tr><td>Liftcoefficient(b)</td><td>7.5x10<sup>-3</sup></td></tr></tbody></table>

**-Sensor data from the quad-rotor tail-sitter VTOL UAV** used for state and parameter estimation:

1. The estimation process involves: Multiplier matrices for efficient calculations,

2. Data-driven learning for adaptation,

3. Adaptive observer for real-time state estimation.

**-Estimated states and parameters** are used by the control and navigation algorithm.

**-The algorithm** calculates control signals for the actuators.

**-These signals** influence the quad-rotor's movement, leading to improved control and navigation.
We use Matlab/Simulink for our simulations. Taking
into account outside disruptions throughout the
takeoff, hovering, and landing phases,

<figure><img src="image_2.png" /><figcaption>Fig.2.Trajectory tracking comparison for a tail-sitter VTOL
UAV performing flight with vertical takeoff, hovering, Level
flight, and vertical landing, under wind disturbance.</figcaption></figure>

The graph shows the trajectory of a tail-sitter VTOL drone performing a flight with vertical takeoff, hovering, horizontal flight, and vertical landing with wind disturbance, using the new approach of online state and parameter estimation: The drone accurately follows the desired trajectory; there are no significant altitude oscillations.

Without a new approach to online state and parameter
estimation, the drone would have difficulty following
the desired trajectory, and its altitude would oscillate
significantly.

The convergence properties of the developed method
demonstrate the ultimately uniform and local
character of the state estimation errors and the input-
to-state stability of the parameter estimation errors
under finite informative conditions.

V. CONCLUSION

We applied an architecture (algorithm1) that uses a
multiplicative matrix observer design and adaptive
simultaneous learning update laws to predict the state
of a quad-rotor tail-sitter VTOL UAV. The algorithm
works in real time and can predict errors in parameter
estimation. Moreover, under certain excitation

conditions, the parameter estimation error is
guaranteed to converge to a given neighborhood of
the origin.

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<figure><img src="image_3.png" /></figure>