State estimation of hybrid dynamic systems with Markov jumps and state constraints

Abstract

In some state estimation applications, the dynamic system can be represented by a hybrid model that is determined by the interaction between analog and digital (mode) states. The hybrid stochastic filtering problem is to provide estimates for both analog and digital states from a sequence of noisy sample measurements and such hybrid model. For these systems, the filter should track an exponentially increasing number of possible trajectories, posing a practical challenge to solve this problem. Therefore, approximate solutions are often pursued, trading off the filter precision for processing time. In this work, we investigate two issues related to the state estimation of hybrid systems. First, we present a modified version of the multiple models and multiple hypotheses (M3H) algorithm to suboptimally solve the problem of state estimation for Markov jump nonlinear systems. We employ Gaussian mixture reduction methods as an alternative for the merging of hypotheses of the original M3H. Thus, information from both the analog and digital states are employed to merge the hypotheses, while only information from the digital state is employed in the original M3H method. As a contribution, the proposed approach provides an effective mechanism for users to explore the tradeoff between filter precision and processing time. Users set their preferences by defining the maximum number of mixture components. Setting this number in our proposed approach is more efficient than choosing the merging depth in M3H when increased precision is demanded. Second, we consider the problem of state estimation for hybrid systems with state ix equality constraints. We investigate special cases of this problem for both linear and nonlinear systems. We categorize such a problem into three groups according to the linear or nonlinear equality constraint as well as to the dependence of the constraint on the operating mode. For the mode-independent equality constraints case, we present the necessary conditions on the initialization and dynamics for the classical interacting multiple models (IMM) algorithm to yield state estimates satisfying a linear equality constraint for linear systems. However, for linear and nonlinear systems, the modedependent equality constraints must be enforced along time. We present a modified version of the IMM filter to enforce the equality constraint on the state estimates.