Stochastic linear hybrid systems: Modeling, estimation, and application

Hybrid systems are dynamical systems which have interacting continuous state and discrete state (or mode). Accurate modeling and state estimation of hybrid systems are important in many applications. We propose a hybrid system model, known as the Stochastic Linear Hybrid System (SLHS), to describe hybrid systems with stochastic linear system dynamics in each mode and stochastic continuous-state-dependent mode transitions. We then develop a hybrid estimation algorithm, called the State-Dependent-Transition Hybrid Estimation (SDTHE) algorithm, to estimate the continuous state and discrete state of the SLHS from noisy measurements. It is shown that the SDTHE algorithm is more accurate or more computationally efficient than existing hybrid estimation algorithms. Next, we develop a performance analysis algorithm to evaluate the performance of the SDTHE algorithm in a given operating scenario. We also investigate sufficient conditions for the stability of the SDTHE algorithm. The proposed SLHS model and SDTHE algorithm are illustrated to be useful in several applications. In Air Traffic Control (ATC), to facilitate implementations of new efficient operational concepts, accurate modeling and estimation of aircraft trajectories are needed. In ATC, an aircraft\u27s trajectory can be divided into a number of flight modes. Furthermore, as the aircraft is required to follow a given flight plan or clearance, its flight mode transitions are dependent of its continuous state. However, the flight mode transitions are also stochastic due to navigation uncertainties or unknown pilot intents. Thus, we develop an aircraft dynamics model in ATC based on the SLHS. The SDTHE algorithm is then used in aircraft tracking applications to estimate the positions/velocities of aircraft and their flight modes accurately. Next, we develop an aircraft conformance monitoring algorithm to detect any deviations of aircraft trajectories in ATC that might compromise safety. In this application, the SLHS model is used to describe the trajectory of a conforming aircraft and the SDTHE algorithm is used to design a filter which generates a residual vector. We show that the residual has approximately a Gaussian distribution with zero mean and a known covariance if the aircraft is conforming to its flight plan. Conformance monitoring is then carried out by statistical tests of the residual. Finally, we consider applications of the SDTHE algorithm in fault detection and isolation. Here, we consider a plant which is modeled as a SLHS. We also consider a set of possible faults in the plant and describe the plant\u27s dynamics in the presence of each fault as a different SLHS. We use a bank of filters designed based on the SDTHE algorithm to generate a set of residuals. Each of these residuals has a known approximate probability distribution corresponds to each fault of the plant. The fault detection and isolation problem is then formulated as a multiple hypothesis test and solved with existing decision making algorithms. The proposed fault detection and isolation algorithm is validated in two illustrative applications