tf2zp

Convert transfer function filter parameters to zero-pole-gain form

Syntax

[z,p,k] = tf2zp(b,a)

Description

[z,p,k] = tf2zp(b,a) finds the matrix of zeros z, the vector of poles p, and the associated vector of gains k from the transfer function parameters b and a. The function converts a polynomial transfer-function representation

$$H(s)=\frac{B(s)}{A(s)}=\frac{b_1{s}^{n-1}+\cdots +b_{n-1}s+b_n}{a_1{s}^{m-1}+\cdots +a_{m-1}s+a_m}$$

of a single-input/multi-output (SIMO) continuous-time system to a factored transfer function form

$$H(s)=\frac{Z(s)}{P(s)}=k\frac{(s-z_1)(s-z_2)\cdots (s-z_m)}{(s-p_1)(s-p_2)\cdots (s-p_n)}.$$

Note

Use tf2zp when working with powers of s, such as in continuous-time transfer functions. A similar function, tf2zpk, is more useful when working with transfer functions expressed in powers of z^–1.

Examples

Zeros, Poles, and Gain of Continuous-Time System

Generate a system with the following transfer function.

$$H(s)=\frac{2s^2+3s}{s^2+\frac{1}{\sqrt{2}}s+\frac{1}{4}}=\frac{2(s-0)(s-(-\frac{3}{2}))}{(s-\frac{-1}{2\sqrt{2}}(1-j))(s-\frac{-1}{2\sqrt{2}}(1+j))}$$

Find the zeros, poles, and gain of the system. Use eqtflength to ensure the numerator and denominator have the same length.

b = [2 3];
a = [1 1/sqrt(2) 1/4];

[b,a] = eqtflength(b,a);
[z,p,k] = tf2zp(b,a)

z = 2×1

         0
   -1.5000

p = 2×1 complex

  -0.3536 + 0.3536i
  -0.3536 - 0.3536i

k = 
2

Plot the poles and zeros to verify that they are in the expected locations.

zplane(b,a)
text(real(z)+0.1,imag(z),"Zero")
text(real(p)+0.1,imag(p),"Pole")

Input Arguments

b — Transfer function numerator coefficients

vector | matrix

Transfer function numerator coefficients, specified as a vector or matrix. If b is a matrix, then each row of b corresponds to an output of the system. b contains the coefficients in descending powers of s. The number of columns of b must be less than or equal to the length of a.

Data Types: single | double

a — Transfer function denominator coefficients

vector

Transfer function denominator coefficients, specified as a vector. a contains the coefficients in descending powers of s.

Data Types: single | double

Output Arguments

z — Zeros

matrix

Zeros of the system, returned as a matrix. z contains the numerator zeros in its columns. z has as many columns as there are outputs.

p — Poles

column vector

Poles of the system, returned as a column vector. p contains the pole locations of the denominator coefficients of the transfer function.

k — Gains

column vector

Gains of the system, returned as a column vector. k contains the gains for each numerator transfer function.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The complexity of outputs, z and k, might be different in MATLAB^® and the generated code.

• The order of outputs, z and p, might be different in MATLAB and the generated code.

Version History

Introduced before R2006a