A Convex Optimization Framework for Computing Robustness Margins of Kalman Filters

Abstract

This paper proposes a novel convex optimization framework for designing robust Kalman filters that guarantee a user-specified steady-state error while maximizing process and sensor noise. The proposed framework simultaneously determines the Kalman gain and the robustness margin in terms of the process and sensor noise. This is the first paper to present such a joint formulation for Kalman filtering. The proposed methodology is validated through two distinct examples: the Clohessy-Wiltshire-Hill equations for a chaser spacecraft in an elliptical orbit and the longitudinal motion model of an F-16 aircraft.

Figures (6)

• Figure 1: Positional error evolution. Solid lines denote mean error in relative position $(x,y,z)$, whereas shaded regions show $\mu_i \pm \sqrt{\Sigma_{ii}}$ where $i \in (e_x, e_y, e_z)$
• Figure 2: Velocity error evolution. Solid lines denote mean error in relative velocity $(\dot{x},\dot{y},\dot{z})$, whereas shaded regions show $\mu_i \pm \sqrt{\Sigma_{ii}}$ where $i \in (e_{\dot{x}}, e_{\dot{y}}, e_{\dot{z}})$
• Figure 3: Bar plot comparing process(left) and sensor(right) noise variances between case 1(c1) and case 2(c2) for the continuous-time system. The units for $(w_{F_x}, w_{F_y}, w_{F_z})$, $(n_x,n_y,n_z)$, and $(n_{\dot{x}}, n_{\dot{y}}, n_{\dot{z}})$ are $(m/s^2)^2$, $m^2$, and $(m/s)^2$ respectively.
• Figure 4: Comparison of process(left) and sensor(right) noise variances between case 1(c1) and case 2(c2) for the discrete-time system. The units for $(w_{F_x}, w_{F_y}, w_{F_z})$, $(n_x,n_y,n_z)$, and $(n_{\dot{x}}, n_{\dot{y}}, n_{\dot{z}})$ are $(m/s^2)^2$, $m^2$, and $(m/s)^2$ respectively.
• Figure 5: Bar plot comparing process(left) and sensor(right) noise variances between case 1(c1) and case 2(c2) for the longitudinal flight dynamics. The units for $w_V$, $(w_\alpha,n_\alpha)$, $(w_q,n_q)$, $(n_{\dot{u}}, n_{\dot{w}})$ and $n_{\bar{q}}$ are $(ft/s)^2$, $rad^2$, $(rad/s)^2$, $(ft/s^2)^2$, and $(lb/ft^2)^2$ respectively.

Theorems & Definitions (10)

• Theorem 1
• proof
• Corollary 1
• proof
• Remark 1
• Remark 2
• Theorem 2
• proof
• Corollary 2
• proof