265_adv_control_eng

33 the value of the observer gain matrix ke can be

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Unformatted text preview: • 0 0 F F F 0 Àa0 x1 b0 x1 T x2 U T 1 0 F F F 0 Àa1 UT x2 U T b1 U T• U T UT U T U T F UˆTF F UT F U ‡ T F Uu F SR F S R F S R F S RF F F F F F • xn 0 y ˆ [0 0 F F F 1 À an À 1 PQ x1 T x2 U TU 0 F F F 0 1 ]T F U RFS F xn xn bnÀ1 (8:131) Note that the system matrix of the observable canonical form is the transpose of the controllable canonical form given in equation (8.33). The value of the observer gain matrix Ke can be calculated directly using P Q 0 À a0 T 1 À a1 U T U (8:132) K e ˆ QT U F F R S F nÀ1 À anÀ1 Q is a transformation matrix that transforms the system state equation into the observable canonical form Q ˆ (WNT )À1 (8:133) where W is defined in equation (8.102) and N is the observability matrix given in equation (8.89). If the equation is in the observable canonical form then Q ˆ I. (c) Ackermann's formula: As with regulator design, this is only applicable to systems where u(t) and y(t) are scalar quantities. It may be used to calculate the observer gain matrix as follows Ke ˆ (A)NÀ1 [ 0 or alternatively 0 FFF 0 1 ]T QÀ1 P Q C 0 T CA U T 0 U Ke ˆ (A)T F U T F U R F S RFS F F CAnÀ1 1...
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