Interval constrained state estimation of nonlinear dynamical systems using unscented Gaussian sum filter

Equality‐constrained state estimation for hybrid systems

“…This approach is called CIHE. (9). The proposed approach CIMM 1 projects the combined estimate x^k of the IMM filter given by (54) onto some of the hyperplanes defined by the linear equality constraint (9) by means of a projection step.…”

“…Thus, it is necessary to enforce (9) after the data-assimilation step (in both x^k p, s and P k xxp, s (51)). Likewise, we consider that it is necessary to use the projecting the estimates step to obtain combined estimates, x^k (54) and P k xx (55), that satisfy the equality constraint (9). The CIMM 1 employs the prediction and correction of estimates step and the projecting the estimates step to enforce the equality constraint (9) on the state vector; see marker ◼ in Fig.…”

“…This problem consists of scenarios in which the dynamics and disturbances are such that the state vector of the system satisfies an inequality constraint [6] or an equality constraint [7]. Non-linear Kalman filtering algorithms [8,9] and particle filtering methods [10,11] provide approximate solutions for constrained state estimation.…”

“…The next proposition shows that the constrained state estimate x^k p given by (62) minimises the mean-squared error satisfying at least one of the equality constraints (9). Proposition 3: Consider Problem 2 and let p(x k ) denote the a posteriori probability density for x k provided by the IMM filter as the Gaussian mixture:…”

“…In this case, this first proposal does not look very appealing. Alternatively, the second proposal projects the combined estimate, x^k (54) and P kxx (55), in each one of the equality constraints(9). Next, we choose the final constrained estimate, x^k…”