Algorithmic Linear Algebra by Herbert Möller

By Herbert Möller

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Letting T go to infinity, G in (61) will approach zero matrix, and hence from (62) we have T (63) 38 2. Optimal Control of Linear Discrete-Time Systems In this computation we need only (w]~, wJ;f, the first n eigenvectors associated with the eigenvalues of matrix H outside the unit circle. A numerical example of the computation is in order here. EXAMPLE. 5, we have the following relevant matrices (cf. 044163 } W 21 ........................................................ Of the above (6 X 3) matrix [k l ,k2 ,k3], the upper half (3 X 3) matrix and the lower half (3 X 3) matrix correspond to WI I and W 21 in (55), respectively.

Ys as a unique solution of ,\S I (9') ,\s Ys s Step 3. Calculate the coefficients ai (i = I, ... , Step 4. Define 1=[J 0 0 0 a=[] D= al I a2 al as-I as - 2 al 56 3. Observers for Linear Discrete-Time Systems Step 5. Specify a row vector h of dimension r, and define s h h h= s- , and X rs matrix ~ as A=== h Step 6. Compute column s-vector g by g Step 7. Define G, s X = [DhsA] -I(y - G). (17') r matrix, as the product of gin (17') and h in Step 5: G ===gh. , w(t) === z(t) + Gy(t), (30) where z(t) = [A22 - GA 12 ]z(t - 1) X y(t - 1) + [B22 - + [A21 - GAil I- (A22 - GAI2)G] GBII]v(t) (25) with a given initial value z(O).

For the infinite-horizon case of minimizing the cost criterion function 00 J' = 2: 1=1 {x T(t - I)Zx(t - 1) + VT(t)V(t)}, (16') where Z and are positive definite matrices, under the system x(t) = Ax(t - 1) + Bv(t), with given initial value x(O) = X t = 1,2, ... 2. Controllers for One-Period Lag Equation Systems (26 t ) on the assumption of state controllability of (17t). v (t) =- (26 t ) Kx (t - 1), where K is the limit of K(t) in (27) as t --700 and is given by K ==[ B TSB + rIB TSA, S = A TS[ A - BK] = [A (gain matrix) +Z (27t) (Riccati equation) - BKfS[ A - BK] + KTK + Z.

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