State Estimation of Constrained Nonlinear Discrete-time Dynamical Systems

In this paper, a recursive state estimator is developed to handle the problem of state estimation of nonlinear discrete-time dynamical systems when some of the states of these systems are subject to equality or inequality constraints which are due to physical or practical considerations. The system model and the measurements are assumed to be corrupted by zero mean white Gaussian noise. The proposed algorithm, which is based on the multiple projection approach, differentiates between two cases: the case of strictly satisfied constraints and the case when some of the constraints are violated. In the first case, the obtained estimator is the standard extended Kalman filter. However, in the second case, we use the active set method to formulate a set of equality constraints; the number of these constraints equals the number of violated constraints. This set of equality constraints is treated as a new set of noise free measurements. Then, instead of formulating a quadratic optimization problem to be solved in order to satisfy the violated constraints, we use the multiple projection approach to extend the estimator recursively to handle the assumed new set of measurements. The proposed technique is different from other techniques in the sense that it is recursive in addition to the fact that the obtained estimates propagate with the algorithm to the next sampling instant. To illustrate the effectiveness and the simplicity of the developed estimation technique, two examples are presented. Simulation results indicate that the proposed technique is simple and effective in estimating the states of nonlinear discrete-time systems while satisfying equality and/or inequality constraints imposed on the states of these systems.