zp2tf

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

Syntax

[b,a] = zp2tf(z,p,k)

Description

[b,a] = zp2tf(z,p,k) converts a factored transfer function representation

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

of a single-input/multi-output (SIMO) system to a polynomial transfer function representation

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

Examples

Transfer Function of Mass-Spring System

Compute the transfer function of a damped mass-spring system that obeys the differential equation

$$\underset{}{\overset{¨}{w}}+0.01\underset{}{\overset{\dot{}}{w}}+w=u(t).$$

The measurable quantity is the acceleration, $$y=\underset{}{\overset{¨}{w}}$$, and $$u(t)$$ is the driving force. In Laplace space, the system is represented by

$$Y(s)=\frac{s^{2}U(s)}{s^2+0.01s+1}.$$

The system has unit gain, a double zero at $$s=0$$, and two complex-conjugate poles.

k = 1;
z = [0 0]';
p = roots([1 0.01 1])

p = 2×1 complex

  -0.0050 + 1.0000i
  -0.0050 - 1.0000i

Use zp2tf to find the transfer function.

[b,a] = zp2tf(z,p,k)

b = 1×3

     1     0     0

a = 1×3

    1.0000    0.0100    1.0000

Input Arguments

z — Zeros

column vector | matrix

Zeros of the system, specified as a column vector or a matrix. z has as many columns as there are outputs. The zeros must be real or come in complex conjugate pairs. Use Inf values as placeholders in z if some columns have fewer zeros than others.

Example: [1 (1+1j)/2 (1-1j)/2]'

Data Types: single | double
Complex Number Support: Yes

p — Poles

column vector

Poles of the system, specified as a column vector. The poles must be real or come in complex conjugate pairs.

Example: [1 (1+1j)/2 (1-1j)/2]'

Data Types: single | double
Complex Number Support: Yes

k — Gains

column vector

Gains of the system, specified as a column vector.

Example: [1 2 3]'

Data Types: single | double

Output Arguments

b — Transfer function numerator coefficients

row vector | matrix

Transfer function numerator coefficients, returned as a row vector or a matrix. If b is a matrix, then it has a number of rows equal to the number of columns of z.

a — Transfer function denominator coefficients

row vector

Transfer function denominator coefficients, returned as a row vector.

Algorithms

The system is converted to transfer function form using poly with p and the columns of z.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced before R2006a