High-gain extended Kalman filter for continuous-discrete systems with asynchronous measurements

ABSTRACT

This paper investigates an adaptation of the high-gain Kalman filter for nonlinear continuous-discrete system with multirate sampled outputs under an observability normal form. The contribution of this article is twofold. First, we prove the global exponential convergence of this observer through the existence of bounds for the Riccati matrix. Second, we show that, under certain conditions on the sampling procedure, the observer's asynchronous continuous-discrete Riccati equation is stable and also, that its solution is bounded from above and below. An example, inspired by mobile robotics, with three outputs available is given for illustration purposes.

KEYWORDS:

• Nonlinear sampled-data systems
• multirate continuous-discrete systems
• high-gain observers
• extended Kalman filter
• Riccati stability

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Although restrictive, this condition is necessary in order to apply Lemma A.9 to a time varying matrix C.

2 This time dependency is not explicitly written in the observer's equations to make the notations less cluttered.

3 Note that, matrices $R_{{\sigma }_k}$ and $I_{{\sigma }_k}^{-1}$ do commute. Indeed, by definition, each blocks of $R_{{\sigma }_k}$ correspond to a block of $I_{{\sigma }_k}^{-1}$ made of an identity matrix times some constant parameter.

4 Or a wheeled mobile robot.

5 Boat-beacon line: the line that passes through $(x_1,x_2{)}^{\mathrm{\prime }}$ and the centre of the concerned beacon.

6 As a consequence, the initial guess lacks consistency w.r.t. the problem's physics which makes the task harder for the observer.

7 Having its standard deviation equals to 0.1 for the angle measurements, and 1 for the distance measurements.

8 ρ is defined w.r.t. k – since we need our relations to remain valid for any ${\overline{\tau }}_k$ large enough – and should be understood as ${\rho }_k$. This latter notation is however not used for readability reasons.