Signals and Systems

10.6 TRANSFER FUNCTION FROM STATE MODEL

From the given state variable model of SISO it is possible to find its transfer function. Let the state variable model of the SISO be

 

x(k + 1) = Ax(k) + Bu(k)     (10.12)

 

y(k) = Cx(k) = Du(k)     (10.13)

 

The input vector for a SISO system has only one row and hence U (k) = u(k). Let us determine H(z). Taking z transform of Eq. (10.12) we get

 

zIX (z) − zX (0) = AX(z) + BU (z)

 

i.e.,        (zI − A) X(z) = zX (0) BU(z)

 

i.e.,       X (z) = (zI − A)⁻¹ zX (0) + (zI − A)⁻¹ BU(z)     (10.14)

 

Taking X (0) as null matrix, we get from Eq. (10.14)

 

X (z) = (zI − A)⁻¹ BU(z)     (10.15)

 

Taking z–transform of Eq. (10.13) we get

 

Y (z) = CX (z) + DU (z)

 

Y(z) = C[(zI−A)⁻¹ BU (z)] + DU (z)