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We prove this by contradiction. Suppose that the rank of M, is less than N. Then there exists an.initia1 state q[O] = q, f 0 such that
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Moq, = 0
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CHAP. 71
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STATE SPACE ANALYSIS
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Now from Eq. (7.65), for x(t ) = 0 and to = 0,
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However, by the Cayley-Hamilton theorem eA'can be expressed as
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Substituting Eq. (7.136) into Eq.(7.135),we get
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in view of Eq. (7.134).Thus, qo is indistinguishable from the zero state and hence the system is not observable. Therefore, if the system is to be observable, then Mo must have rank N.
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7.53. Consider the system in Prob. 7.50.
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( a ) Is the system controllable
( 6 ) Is the system observable
From the result from Prob. 7.50 we have
Now and by Eq. (7.128) the controllability matrix is
M, = [b Ab] =
[-1 -11
and IM,I = 0. Thus, it has a rank less than 2 and hence the system is not controllable. ( b ) Similarly,
and by Eq. (7.133) the observability matrix is
and J M J = - 2 # 0. Thus, its rank is 2 and hence the system is observable. Note from the result from Prob. 7.50(b) that the system function H(s) has pole-zero cancellation. As in the discrete-time case, if H(s) has pole-zero cancellation, then the system cannot be both controllable and observable.
7.54. Consider the system shown in Fig. 7-22.
Is the system controllable ( b ) Is the system observable ( c ) Find the system function H(s).
STATE SPACE ANALYSIS
[CHAP. 7
Fig. 7-22
In matrix form
where Now and by Eq. (7.128) the controllability matrix is
M,= [b Ab] =
[: :]
and lM,I = 0. Thus, its rank is less than 2 and hence the system is not controllable. ( b ) Similarly,
and by Eq. (7.133) the observability matrix is
and lMoI= 0. Thus, its rank is less than 2 and hence the system is not observable. By Eq. (7.52) the system function H(s) is given by
H ( S )= c ( s ~ - A ) - ' ~
Note that the system is both uncontrollable and unobservable.
CHAP. 71
STATE SPACE ANALYSIS
Supplementary Problems
Consider the discrete-time LTI system shown in Fig. 7-23. Find the state space representation of the system with the state variables q , [ n ] and q , [ n ] as shown.
Fig. 7-23
Consider the discrete-time LTI system shown in Fig. 7-24. Find the state space representation of the system with the state variables q , [ n ] and q , [ n ] as shown.
Fig. 7-24
STATE SPACE ANALYSIS
[CHAP. 7
7.57. Consider the discrete-time LTI system shown in Fig. 7-25.
( a ) Find the state space representation of the system with the state variables q , [ n ] and q 2 [ n ] as shown.
( b ) Find the system function H ( z ) .
Find the difference equation relating x [ n ] and y [ n ] .
Fig. 7-25
CHAP. 71
STATE SPACE ANALYSIS
7.58. A discrete-time LTI system is specified by the difference equation
y [ n ] + y [ n - 11 - 6 y [ n - 21
= 2x[n
- 1 1 + 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.
Am. A"
- 2(+)" + 3 0 ) "
6 ( f )" - 6 ( f)" 3($)" - 2(;)"
(i)" (4)" +
7.60. Find A for "
( a ) Using the spectral decomposition method.
( b ) Using the z-transform method.
Am. An=
Given a matrix
[(3)" 0
0 f(2)"+ ;(-3)"
;(2)" - ; ( - 3 ) "
f(2)" - + ( - 3 ) " j(2)" - j ( - 3 ) "
( 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'
STATE SPACE ANALYSIS
[CHAP. 7
Consider the discrete-time LTI system with the following state space representation:
(a) Find the system function H(z). (b) Is the system controllable (c) IS the system observable
Ans.
( 2 - 1)* (b) The system is controllable. (c) The system is not observable.
(a) H ( z ) =
Consider the discrete-time LTI system in Prob. 7.55. (a) (b) (c) (dl Is the Is the IS the Is the system system system system The The The The asymptotically stable BIBO stable controllable observable
Am. (a) (b) (c) (dl
system is asymptotically stable. system is BIBO stable. system is controllable. system is not observable.
The controllability and observability of an LTI system may be investigated by diagonalizing the system matrix A. A system with a state space representation
v [ n + 1 1 = Av[n] + *bx[n]
y [ n ]= b [ n ]
(where A is a diagonal matrix) is controllable if the vector Ib has no zero elements, and it is observable if the vector e has no zero elements. Consider the discrete-time LTI system in Prob. 7.55.
Let d n ] = Tq[n].Find the matrix T such that the new state space representation will have a diagonal system matrix. Write the new state space representation of the system. Using the result from part ( b ) , investigate the controllability and observability of the system.
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