ss2zp

Converting filter parameters from the state space to a form with zeros, poles, and a gain factor.

Library

EngeeDSP

Syntax

Function call

z,p,k = ss2zp(A,B,C,D) — transforms the representation in the state space

for a given continuous or discrete system, into an equivalent representation of zeros, poles, and gain coefficients

the zeros, poles, and gain coefficients of which represent the transfer function in a factorized form.

z,p,k = ss2zp(A,B,C,D,ni) — indicates that the system has multiple inputs and that ni-the input was excited by a single pulse.

Arguments

Input arguments

# A is a matrix of states

+ the matrix

Details

The matrix of states. If the system has entrances and outputs and is described state variables, then A has the dimension on .

Типы данных

Float32, Float64

# B is the input-state matrix

+ the matrix

Details

The input-state matrix. If the system has entrances and outputs and is described state variables, then B has the dimension on .

Типы данных

Float32, Float64

# C is the output-state matrix

+ the matrix

Details

The output-state matrix. If the system has entrances and outputs and is described state variables, then C has the dimension on .

Типы данных

Float32, Float64

# D is the end—to- end transmission matrix

+ the matrix

Details

The end-to-end transmission matrix. If the system has entrances and outputs and is described state variables, then D has the dimension on .

Типы данных

Float32, Float64

# ni — entry index

+ 1 (by default) | scalar

Details

The index of the input, set as an integer scalar. If the system has inputs, use the function ss2zp with the last argument ni to calculate the response to a single pulse applied to ni- I’m coming in. Specifying this argument causes the function to ss2zp use ni-f columns of matrices B and D.

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. The matrix z it has as many columns as there are outputs (rows in the matrix C).

# 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 discrete system

Details

Consider a discrete system defined by a transfer function

We will determine the zeros, poles, and gain directly from the transfer function. Let’s add zeros to the numerator so that its length is equal to the length of the denominator.

import EngeeDSP.Functions: tf2zp

b = [2 3 0]
a = [1 0.4 1]
z, p, k = tf2zp(b, a)
println("z = ", z, "\np = ", p, "\nk = ", k)

z = [-1.5, 0.0]
p = ComplexF64[-0.20000000000000004 - 0.9797958971132713im, -0.20000000000000004 + 0.9797958971132713im]
k = 2.0

Let’s imagine the system in the form of a state space and define the zeros, poles, and gain using the function ss2zp.

import EngeeDSP.Functions: tf2ss, ss2zp

A, B, C, D = tf2ss(b, a)
z, p, k = ss2zp(A, B, C, D, 1)
println("z = ", z, "\np = ", p, "\nk = ", k)

z = [-1.4999999999999998; -2.1535571616345988e-16;;]
p = ComplexF64[-0.20000000000000004 + 0.9797958971132713im; -0.20000000000000004 - 0.9797958971132713im;;]
k = 2.0

Algorithms

Function ss2zp finds the poles by the eigenvalues of the matrix A. Zeros represent the final solutions to the generalized eigenvalue problem.:

using LinearAlgebra
eigvals([A B; C D], diagm([ones(n); 0]))

In many situations, this algorithm produces false large but finite zeros. Function ss2zp interprets these large zeros as infinite.

Function ss2zp finds the gain factor by solving the equation for the first nonzero Markov parameters.

Literature

Laub, A. J., and B. C. Moore. «Calculation of Transmission Zeros Using QZ Techniques.» Automatica. Vol. 14, 1978, p. 557.