Page 436 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

7.58. A discrete-time LTI system is specified by the difference equation
y[n] +y[n - 11 - 6y[n - 21 = 2x[n - 11 +x[n - 21

Write the two canonical forms of state representation for the system.

7.59. Find A" for

(a) Using the Cayley-Hamilton theorem method.
(b) Using the diagonalization method.

- 2(+)" + 30)" 6(f )" - 6(f )"
Am. A" = 3($)" - 2(;)" I
- (i)" + (4)"

7.60. Find A" for

(a) Using the spectral decomposition method.
(b) Using the z-transform method.
[(3)" 0 0
I ;(2)" - ;(-3)" j(2)" - j(-3)"
Am. An= 0 f(2)"+ ;(-3)" f(2)" - +(-3)"
0
7.61. Given a matrix

(a) Find the minimal polynomial m(A) of A.
(b) Using the result from part (a), find An.
Ans. (a) m(A) = (A - 3XA + 3) = A' - 9