Observer design for linear models of multi-rate asynchronous aerospace systems

Abstract

This paper proposes an optimization-based computational approach to design multi-rate observers for linear models of aerospace systems with asynchronous measurements. A case study on the estimation of the state of an autonomous quadrotor UAV around its hovering operating point will be used as an example of the applicability of the proposed method. There are two theoretical contributions of the proposed method. The first is the derivation of a set of Linear Matrix Inequalities (LMIs) for the design of linear observers yielding convergence of the state estimation error given the maximum allowable sampling period (MASP) for each sensor. The second contribution is to propose an optimization problem subject to LMIs for finding the MASPs that guarantee exponential stability of the estimation error. The two contributions enable the analysis and design of multi-rate observers for systems with linear models, including quadrotor UAVs around a hovering operating point. The examples show a very intuitive result: as the sampling frequency of one sensor decreases, another sensor must sample faster to guarantee convergence of the estimated state to the real state. Simulations of a quadrotor with 12 states and 9 measurements show the effectiveness of the approach.

Appendix

Lemma 3

The Lyapunov–Krasovskii functional (18) satisfies conditions (19) and (20).

Proof

First, it is proved that the Lyapunov–Krasovskii functional (18) is non-increasing at the instants $t_n$, $n\in {\mathbb {N}}$, i.e. condition (20) is satisfied. To this end, note that the first component of V, i.e. $V_1$, is a continuous function. The functional $V_2$ is non-negative right before the sampling instants $t_n$, $n\in {\mathbb {N}}$, and vanishes at the sampling instants $t_n$, because based on (9) the lower and upper limits of the integral become equal. Furthermore, the functional $V_3$ is the sum of m non-negative integrals. At each sampling instant $t_n$, one or more of these integrals vanish as the lower and upper limits of the integral become equal (recall that each instant $t_n$, $n\in {\mathbb {N}}$, is equal to at least one instant $s_k^i$, $k\in {\mathbb {N}}$, $i\in \{1,\ldots ,m\}$). The other integrals, however, remain continuous. Finally, the last component $V_4$ is non-negative right before the instants $t_n$ and vanishes at the sampling instants, because $e(t)=e(t_n)$ at $t=t_n$. This finishes the first part of the proof.

Next, we show that the Lyapunov–Krasovskii functional (18) is positive definite and decrescent, i.e. condition (19) is satisfied. The function $V_1$ is a quadratic function and $P>0$. Furthermore, the functionals $V_2$, $V_3$, and $V_4$ are non-negative. Therefore,

$$\begin{aligned} \lambda _\text {min}(P)|e(t)|^2=\lambda _\text {min}(P)|e_t(0)|^2\le V, \end{aligned}$$

which is the left-hand side of condition (19). Considering (12), observe that

$$\begin{aligned}&|e(t)|\le ||e_t||_{\mathcal {W}}, ~|\overline{e}(t)|\le \sqrt{m}||e_t||_{\mathcal {W}}, ~|e(t_n)|\le ||e_t||_{\mathcal {W}}, \\&\int _{t-\rho }^t |\dot{e}(s)|^2 \,\mathrm {d}s = \int _{-\rho }^0 |\dot{e}_t(r)|^2 \,\mathrm {d}r \le ||e_t||_{\mathcal {W}}^2, \\&\int _{t-\rho ^i}^t |\dot{e}(s)|^2 \,\mathrm {d}s = \int _{-\rho ^i}^0 |\dot{e}_t(r)|^2 \,\mathrm {d}r \le ||e_t||_{\mathcal {W}}^2. \end{aligned}$$

Therefore,

$$\begin{aligned}&V_1 \le \lambda _\text {max}(P)||e_t||_{\mathcal {W}}^2, \\&V_2 \le \tau \lambda _\text {max}(R_0)(1+m\tau +\tau )||e_t||_{\mathcal {W}}^2, \\&V_3 \le \sum _{i=1}^m\tau ^i\lambda _\text {max}(R_i)||e_t||_{\mathcal {W}}^2, \\&V_4 \le 4\tau \lambda _\text {max}(X)||e_t||_{\mathcal {W}}^2, \end{aligned}$$

where we used the fact that ${\text {exp}}(\alpha (s-t))\le 1$ for any $s\in [t-\rho ,t]$. Adding these inequalities leads to the upper bound on V in (19). $\square$