Page 384 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

7.5 SOLUTIONS OF STATE EQUATIONS FOR DISCRETE-TIME LTI SYSTEMS
A. Solution in the Time Domain:

Consider an N-dimensional state representation
q[n + 1] = Aq[n] + bx[n] (7.22a)

Y [n] = cq[n] + &[n] (7.226)
where A, b, c, and d are N x N, Nx 1, 1 x N, and 1 X 1 matrices, respectively. One
method of finding q[n], given the initial state q[O], is to solve Eq. (7.22a) iteratively. Thus,
q[l] = Aq[OJ + bx[O]
q[2] = Aq[l] + bx[l] = A{~q[0] + bx[O] ] + bx[l]
+
= A~~[O] Abx[O] + bx[l]
By continuing this process, we obtain

If the initial state is q[n,] and x[n] is defined for n 2 no, then, proceeding in a similar
manner, we obtain

The matrix An is the n-fold product

n
and is known as the state-transition matrix of the discrete-time system. Substituting
Eq. (7.23) into Eq. (7.2231, we obtain
n- I
y[n] =cAnq[O] + CA"-'-~ bx[k] +&[n] n>O (7.25)
k =O
The first term cAnq[O] is the zero-input response, and the second and third terms together
form the zero-state response.

B. Determination of An:
Method 1: Let A be an N X N matrix. The characteristic equation of A is defined to be (App. A)

where JAI - Al means the determinant of Al -A and I is the identiry matrix (or unit
matrir) of Nth order. The roots of c(A) = 0, A, (k = 1,2,. . . , N), are known as the
eigenualues of A. By the Cayley-Hamilton theorem An can be expressed as [App. A, Eq.
(AS7)I

When the eigenvalues A, are all distinct, the coefficients b,, b,, . . . , 6,- , can be found