Creating Digital Transfer Functions Directly
Vector Method, Polynomial Form
A digital transfer function can be expressed as a numerator polynomial divided by a denominator polynomial, that is, F(z) = N(z)/D(z). The numerator, N(z), is represented by a vector, numf, that contains the coefficients of N(z). Similarly, the denominator, D(z), is represented by a vector, denf, that contains the coefficients of D(z). We form F(z) with the command, F=tf(numf,denf,T), where T is the sampling interval. F is called a linear time-invariant (LTI) object. This object, or transfer function, can be used as an entity in other operations, such as addition or multiplication. We demonstrate with F(z)=150(z^{2}+2z+7 )/(z^{2}-0.3z+0.02) . We use an unspecified sampling interval, T=[ ]. Notice after executing the tf command, MATLAB prints the transfer function.
Vector Method, Factored Form
We also can create digital LTI transfer functions if the numerator and denominator are expressed in factored form. We do this by using vectors containing the roots of the numerator and denominator. Thus, G(s) = K *N(z)/D(z) can be expressed as an LTI object using the command, G=zpk(numg,deng,K,T), where numg is a vector containing the
roots of N(z), deng is a vector containing the roots of D(z), K is the gain, and T is the sampling interval. The expression zpk stands for zeros (roots of the numerator), poles (roots of the denominator), and gain, K. We demonstrate with G(z) = 20(z + 2)(z + 4)/[(z – 0.5)(z – 0.7)(z – 0.8)] and an unspecified sampling interval. Notice after executing the zpk command, MATLAB prints the transfer function.
Rational Expression in z Method, Polynomial Form (Requires Control System Toolbox 9.7)
This method allows you to type the transfer function as you normally would write it. The statement z=tf(‘z’) must precede the transfer function if you wish to create a digital LTI transfer function in polynomial form equivalent to using G=tf(numg, deng,T).
Rational Expression in z Method, Factored Form (Requires Control System Toolbox 9.7)
This method allows you to type the transfer function as you normally would write it. The statement z=zpk(‘z’)must precede the transfer function if you wish to create a digital LTI transfer function in factored form equivalent to using G=zpk(numg, -deng ,K ,T). For both rational expression methods the transfer function can be typed in any form
regardless of whether z=tf(‘z’) or z=zpk(‘z’) is used. The difference is in the created digital LTI transfer function. We use the same examples above to demonstrate the rational expression in z methods.
