A Comparison of Moving Horizon and Bayesian State Estimators with an Application to a pH Process

Recursive Elimination Method in Moving Horizon Estimation for a Class of Nonlinear Systems and Non-Gaussian Noise

“…where d 0 ∈ R is an additional scalar variable, and det D 0 denotes the product of all the diagonal polynomials of D 0 . Note that if (19) has a solution, the value of det D 0 is given as det D 0 = 1/d 0 0, which means any denominator included in (13) does not vanish at the solution; in other words, all the denominators do not vanish as long as (19) holds. Consequently, (13) is equivalent to (n + 1) algebraic equations, namely, (18) and (19).…”

“…Note that if (19) has a solution, the value of det D 0 is given as det D 0 = 1/d 0 0, which means any denominator included in (13) does not vanish at the solution; in other words, all the denominators do not vanish as long as (19) holds. Consequently, (13) is equivalent to (n + 1) algebraic equations, namely, (18) and (19). Note that if det D 0 is strictly positive or negative on its domain, (19) always has a solution and can be omitted.…”

“…Consequently, (13) is equivalent to (n + 1) algebraic equations, namely, (18) and (19). Note that if det D 0 is strictly positive or negative on its domain, (19) always has a solution and can be omitted.…”

“…The process of the conversion from (13) to the algebraic equations (18) and (19) can also be performed for (14) and (15) in the same way. In the following discussion, we denote a vector of polynomials [n T 0 1 − d 0 det D 0 ] T by F 0 and derive the vectors of polynomials F k (k = 1, .…”

“…While the EKF, UKF, and other Kalman-like filters estimate the current state by using the fixed previous estimate, MHE reoptimizes the past estimates on the horizon and estimates the current state by using those reoptimized past estimates. Indeed, it has been reported that this reoptimization property improves the current state estimate in some examples [19]. The estimation problem at each sampling time is formulated as a nonlinear optimization problem, which is usually computationally demanding to solve while the sampling interval is limited.…”