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By Cornelius Thomas Leondes

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Extra resources for Control and dynamic systems : advances in theory and applications. Vol. 9

Example text

Z. y. 29) where zi is the output of the minimalorder observer and y. is the plant output vector. Applying the control law u. = K. x. with the state estimate x. of Eq. 29) -1 1-1 —1 29 LESLIE M. NOVAK gives the closed-loop state equation c. _ (A. )x. z. 30) Also applying the same input to the observer gives K )P z T. (A. + BiKi)V1Rixi. 31) Zi+1 - Ti+1(Ai + Bi i i i + Combining Eqs. R. 3 ) z —i The stability properties of the overall closed-loop system become apparent when the system is viewed in a different state space.

0 for all "i". This important special -1 case is considered next. Rather loosely stated, in the absence of measurement noise, "m" of the system states are known exactly and it is only necessary to estimate the remaining "n -m" states.

13), = Ki+1 . 15) into the observer system defined by Eq. 9) gives the result Z. Z. 16) LESLIE M. NOVAK structure to the Kalman filter. If K 1+1 is taken to be the Kalman filter gain matrix, the observer obtained is identical to the Kalman filter. If the designer picks the gain matrix K i+1 according to some other criterion, the observer then may be viewed as a suboptimal Kalman filter. ) Therefore, a Kalman filter is an n-dimensional observer for which the weighting matrix Di+l has been chosen to minimize the mean square estimation error.