H-Infinity Filters Steve Rogers A Kalman filter1 set of equations is given. −¿ y k −H k ̂x ¿k −¿+ F k K k ¿ −¿=F k ̂x ¿k ̂x ¿k +1 T
−¿ H k + R k ¿ H k Pk ¿ ¿ −¿ H Tk ¿ K k =P ¿k + ¿ F Tk −1+Q k−1 −¿=F k−1 P ¿k−1 P ¿k −¿ +¿=( I −K k H k ) P ¿k P ¿k The purpose of the Kalman filter is to estimate the states of a linear dynamic system defined by the equations x k +1=F k x k + w k
y k =H k x k + v k The Kalman filter may also be expressed as6 −¿ y k −H k ̂x ¿k −¿+ F k K k ¿ −¿=F k ̂x ¿k ̂x ¿k +1
−¿ T −1 ¿ I + H k Rk H k P k ¿ ¿ −¿ ¿ K k =P ¿k −¿ ¿ I + H Tk R−1 k H kPk ¿ ¿ −¿ ¿ −¿=F k P ¿k P ¿k +1 The Kalman filter has been used for a variety of industrial applications for many decades. There are a few limitations: 1) we must know the mean and correlation of the noises, 2) we must know the covariance matrices Qk and Rk (however, these matrices may also be used as design tuning parameters), 3) works best with Gaussian noise, and 4) we must know the linear model matrices Fk and Hk. The H∞ filter6, also known as the minimax filter makes no assumptions about noise and it minimizes the worst-case estimation error. The given system equations are x k +1=F k x k + w k y k =H k x k + v k
z k =L k x k The H∞ estimation filter equations are T Ś k =L k S k L k
−1 K k = P k [ I −θ Ś k P k + H Tk R−1 H Tk R−1 k H k Pk ] k
̂x k +1=F k ̂x k + F k K k ( y k −H k ̂x k ) −1 T P k +1=F k P k [ I −θ Ś k P k + H Tk R−1 k H k P k ] F k +Q k
Note that θ is a small, positive, scalar value. If we set θ to 0, the above equations reduce to the standard Kalman filter presented above. The following condition must hold for the estimator to be valid.
T −1 −1 T −1 ́ ́ P−1 k −θ S k + H k R k H k > 0 →θ S k < P k + H k Rk H k , θ= [ 0. .1 ]
The steady-state version6 of the H∞ estimation filter is T Ś = L SL −1
K =P [ I −θ Ś P+ H T R−1 HP ] H T R−1
̂x k +1=F ̂x k + F K k ( y k −H x̂ k ) −1
P=FP [ I −θ Ś P+ H T R−1 HP ] F T +Q
Types of applications1 where the H∞ filter might be preferred over the Kalman filter are: 1) model is not well known, 2) model changes unpredictably, and 3) systems in which stability margins or worst-case estimation performance are important. 1. Simon, D., Optimal State Estimation – Kalman, H∞ and Nonlinear Approaches, Wiley, 2006, ISBN