tf2zp

Conversion of the transfer function filter parameters into the form of zeros, poles, and gain.

Library

EngeeDSP

Syntax

Function call

z,p,k = tf2zp(b,a) — finds a matrix of zeros z, the vector of poles p and the corresponding gain vector k from the parameters of the transfer function b and a. The function transforms the polynomial representation of the transfer function

a continuous system with one input and several outputs (Single-Input/Multi-Output, SIMO) into a multiplier form of the transfer function

Function tf2zp it should be used when working with degrees for example, in continuous transfer functions. A similar function tf2zpk it is more useful when working with transfer functions expressed in degrees. .

Arguments

Input arguments

# b — coefficients of the numerator of the transfer function

+ vector | the matrix

Details

The coefficients of the numerator of the transfer function, specified as a vector or matrix. If b — the matrix, then each row b corresponds to the output signal of the system. Argument b contains coefficients in descending order of degrees . Number of columns b must be less than or equal to the length of the argument. a.

Типы данных

Float32, Float64

# a — coefficients of the denominator of the transfer function

+ vector

Details

The coefficients of the denominator of the transfer function, specified as a vector. Argument a contains coefficients in descending order of degrees .

Типы данных

Float32, Float64

Output arguments

# z — zeros

+ the matrix

Details

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

# p — poles

+ vector

Details

The poles of the system, returned as a column vector. Argument p contains the coordinates of the poles of the coefficients of the denominator of the transfer function.

# k — gain factors

+ vector

Details

The system’s gain coefficients, returned as a column vector. Argument k contains the gain coefficients for each transfer function of the numerator.

Examples

Zeros, poles, and the gain of a continuous system

Details

Let’s create a system with the following transfer function

Let’s find the zeros, poles, and gain of the system. Using the function eqtflength to make sure that the numerator and denominator have the same length.

import EngeeDSP.Functions: eqtflength, tf2zp

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

b, a = eqtflength(b, a)
z, p, k = tf2zp(b, a)
println("z = ", z, "\np = ", p, "\nk = ", k)

z = [-1.5, 0.0]
p = ComplexF64[-0.35355339059327373 - 0.35355339059327384im, -0.35355339059327373 + 0.35355339059327384im]
k = 2.0

Let’s plot the poles and zeros to make sure they are in the expected locations. Let’s add a unit circle to the graph for clarity.

import Pkg
Pkg.add("ControlSystems")
using ControlSystems

H = tf(b, a)
z, p = tzeros(H), poles(H)

pzmap(H)
annotate!([(real(zi)+0.1, imag(zi), text("Zero", 8)) for zi in z])
annotate!([(real(pi)+0.1, imag(pi), text("Pole", 8)) for pi in p])

θ = range(0, 2π, length=200)
plot!(cos.(θ), sin.(θ),
      ls=:dash, color=:black,
      label="Unit Circle", aspect_ratio=:equal)

display(plot!())

tf2zp