sos2ss

Convert digital filter second-order section parameters to state-space form

Syntax

[A,B,C,D] = sos2ss(sos)

[A,B,C,D] = sos2ss(sos,g)

Description

[A,B,C,D] = sos2ss(sos) converts a second-order section representation of a digital filter sos to its equivalent state-space form.

[A,B,C,D] = sos2ss(sos,g) converts a second-order section representation of a digital filter sos to its equivalent state-space form with gain g.

Examples

State-Space Representation of Second-Order Section System

Compute the state-space representation of a simple second-order section system with a gain of 2.

sos = [1  1  1  1   0  -1 ;
      -2  3  1  1  10   1];
[A,B,C,D] = sos2ss(sos,2)

A = 4×4

   -10     0    10     1
     1     0     0     0
     0     1     0     0
     0     0     1     0

B = 4×1

     1
     0
     0
     0

C = 1×4

    42     4   -32    -2

D = 
-4

Input Arguments

sos — Second-order section representation

matrix

Second-order section representation, specified as a matrix. sos is an L-by- 6 matrix of the form

$$\text{sos}=\left[\begin{array}{cccccc}{b}_{01}& b_{11}& b_{21}& 1& a_{11}& a_{21}\\ b_{02}& b_{12}& b_{22}& 1& a_{12}& a_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ b_{0L}& b_{1L}& b_{2L}& 1& a_{1L}& a_{2L}\end{array}\right].$$

The entries of sos must be real for proper conversion to state space.

g — Overall system gain

real-valued scalar

Overall system gain, specified as a real-valued scalar. The function applies the gain to the system as

$$H(z)=g{\displaystyle \prod _{k=1}^L{H}_k}(z).$$

Output Arguments

A — State matrix

2L-by-2L matrix

State matrix, returned as a 2L-by-2L matrix.

B — Input-to-state vector

2L-by-1 vector

Input-to-state vector, returned as a 2L-by-1 vector.

C — Output-to-state vector

1-by-2L vector

Output-to-state vector, returned as a 1-by-2L vector.

D — Feedthrough matrix

scalar

Feedthrough matrix, returned as a scalar.

More About

Transfer Function

The discrete transfer function in second-order section form is given by

$$H(z)={\displaystyle \prod _{k=1}^L{H}_k(z)={\displaystyle \prod _{k=1}^L\frac{b_{0k}+b_{1k}{z}^{-1}+b_{2k}{z}^{-2}}{1+a_{1k}{z}^{-1}+a_{2k}{z}^{-2}}.}}$$

Algorithms

The sos2ss function first converts from second-order sections to transfer function using the sos2tf function, and then from transfer function to state-space form using the tf2ss function.

The single-input, single-output state-space representation is given by

$$\begin{array}{ll}x(n+1)\hfill & =Ax(n)+Bu(n),\hfill \\ y(n)\hfill & =Cx(n)+Du(n).\hfill \end{array}$$

Extended Capabilities

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

Version History

Introduced before R2006a