Guaranteed set-membership state estimation of an octorotor's position for radar applications

Abstract

In the context of state estimation of dynamical systems subject to bounded perturbations and measurement noises, this paper proposes an application of a guaranteed ellipsoidal-based set-membership state estimation technique to estimate the linear position of an octorotor used for radar applications. The size of the ellipsoidal set containing the real state is minimized at each sample time taking into account the measurements performed by the drone's sensors. Three case studies highlight the efficiency of the estimation technique in finding guaranteed bounds for the octorotor's linear position. The computed guaranteed bounds in the linear trajectory are exploited to find the maximum operating frequency of the radar, a necessary information in radar applications.

Keywords:

• Set-membership state estimation
• linear systems
• UAVs
• ellipsoidal set
• radar application

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The reader will notice that the notion of set-membership estimation is employed by different research communities (control systems, signal processing, applied mathematics etc.), in a different sense/meaning. In this paper, we focus on set-membership state estimation techniques from the control systems point of view.

2 There exist techniques leading to non-convex sets (e.g. based on Taylor models Paulen et al., 2016 or polynomial methods Streif et al., 2013) mainly used for parameters estimation, which are beyond the scope of this paper. A brief overview of the set-based parameter estimation methods for nonlinear systems can be found in Chachuat et al. (2015). In addition, stochastic adaptive parameter estimation algorithms were developed (Bhotto & Antoniou, 2011; Lima & Diniz, 2010; Werner et al., 2007; Werner & Diniz, 2001).

3 Decoupling the 12-state octorotor model into 3 subsystems (5(5) $\{\begin{array}{ll}\left[\begin{array}{c}{z}_{k+1}\\ {\psi }_{k+1}\\ V_{z_{k+1}}\\ {\omega }_{z_{k+1}}\end{array}\right]& =\mathbf{A}\left[\begin{array}{c}{z}_k\\ {\psi }_k\\ V_{z_k}\\ {\omega }_{z_k}\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\\ {\displaystyle \frac{T_e}{m}}& 0\\ 0& {\displaystyle \frac{T_e}{I_{zz}}}\end{array}\right]\left[\begin{array}{c}{F}_{z_k}^R\\ {\tau }_{z_k}^R\end{array}\right]+{\mathbf{E}}_1{\mathit{w}}_k,\\ \left[\begin{array}{c}{z}_k\\ {\psi }_k\end{array}\right]& =\mathbf{C}\left[\begin{array}{c}{z}_k\\ {\psi }_k\\ V_{z_k}\\ {\omega }_{z_k}\end{array}\right]+{\mathbf{F}}_1{\mathit{w}}_k.\end{array}$(5) )–(7(7) $\{\begin{array}{ll}\left[\begin{array}{c}{x}_{k+1}\\ y_{k+1}\\ V_{x_{k+1}}\\ V_{y_{k+1}}\end{array}\right]& =\mathbf{A}\left[\begin{array}{c}{x}_k\\ y_k\\ V_{x_k}\\ V_{y_k}\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\\ {\displaystyle \frac{T_e}{m}}& 0\\ 0& {\displaystyle \frac{T_e}{m}}\end{array}\right]\left[\begin{array}{c}{F}_{x_k}^R\\ F_{y_k}^R\end{array}\right]+{\mathbf{E}}_3{\mathit{w}}_k,\\ \left[\begin{array}{c}{x}_k\\ y_k\end{array}\right]& =\mathbf{C}\left[\begin{array}{c}{x}_k\\ y_k\\ V_{x_k}\\ V_{y_k}\end{array}\right]+{\mathbf{F}}_3{\mathit{w}}_k.\end{array}$(7) ) allows us to reduce the number of LMI constraints in (10) from $2^{3(n_x+n_y)}$ to $3\cdot 2^{n_x+n_y}$.

Additional information

Funding

The authors acknowledge MEyC Spain (PID2019-106212RB-C41), the European Research Council and the UE ERDF (Advanced Grant OCONTSOLAR, Project ID: 789051).