State Estimation for Asynchronously Switched Sampled-Data Systems

Asynchronously switched sampled-data systems can help model power systems and vehicles that evolve in continuous-time with switching behavior and discrete time measurements. We address the problem of jointly estimating a switching signal, with uncertainty in the exact switching times, as well as the continuous states of the system. We prove stability of the standard Kalman Filter under uncertainty in the switching times, with statistical bounds relating to the sampling period. We then propose a method for estimation of switching times as well as a method for efficient joint estimation of the state and switching signal inspired by the interacting multiple-model extended-Viterbi algorithm. We validate our algorithms in simulation for a power converter and a maneuvering vehicle.

I. Introduction

Real-world systems are often best modeled in continuous time, for example using equations of motion, but with measurements taken at discrete instants [1]. Many systems also vary their behavior between discrete modes either by their construction or to simplify control [2]; for example, geared robot motion, power systems using switched circuits or sources, or an aircraft with several trim conditions including cruising and banked turning. In real-world systems we must also consider noise in our process and measurements, usually represented by random additive noise. A practical formulation for such systems is a stochastic sampled-data switched system [3], given by $$\begin{gather*} \dot x(t) = f(\sigma (t),x(t),u(t)) + w(t) \\ y\left( {{t_k}} \right) = h\left( {\sigma \left( {{t_k}} \right),x\left( {{t_k}} \right)} \right) + v\left( {{t_k}} \right), \end{gather*}$$ where x(t) is the state, u(t) is an input, w(t) is a process noise, y(t_k), usually denoted y_k, is a measured output subject to random measurement noise v(t_k), usually denoted v_k, σ(t) is a “switching signal” taking values in a finite set that tells us the active mode at time t, and t_k are discrete times of measurements indexed by k. The control of such systems is addressed in [4].