Converting a continuous statespace model to ARMA model

In many situation we need to convert a continuous statespace model in to a linear difference equation model or ARMA/ARMAX discreet time model. This post describes this conversion with some simple examples. 

Example 1: Conversion of First Order Differential Equation to Linear Difference Equation


we will start with a simple example here.  
Problem:
Let us take a simple first order differential equation: 

   dx
  -----  =  - x
   dt

This state space model is with A=-1 and B=0,

we want to discreetize it with T=1s and get a linear differential equation model;

Solution:

First calculate A1 and B1 as following:

1. Using the properties of z=e^-(st), we get that

A1=exp(AT);

A1=1/e;

 or in other words, we can say that

  A1=Linv((sI-A)^-1)  with t=T; 
  A1=Linv(1/(s+1))     =exp(-t) at (t=T)   =e^-1

then similarly

  B1= A^-1 (A1-I)B;= -1(e-1)*0 = 0

 

so the new model is

x[n+1]=A1*x[n]+B1*u[n]

x[n+1]=1/e x[n]   

So the final model in linear differential equation is

  
           x[n+1]-1/e x[n]=0

is the ARM model you wanted

Note :
       exp used is matrix exponential
       exp(A)= I+A+A^2/2!+A^3/3!+... = Linv{(sI-A)^-1}

Example 2: Considering the Inputs

Problem:
Now consider this

dx               
---- =-x+u(t);
dt               

Solution:

So we have

 A1=exp(AT);

 A1=1/e;

 or in other words, we can say that

  Linv((sI-A)^-1)  with t=T; 
 A1=Linv(1/(s+1))     =exp(-t) at (t=T)   = e^-1

then

  B1= inv(A)(A1-I)B;=-1(1/e-1)*1=1-1/e;

so the new model is

x[n+1]=A1 x[n]+B1u[n]

       x[n+1]=1/e x[n]+(1-1/e)u[n];

        x[n+1]-1/e x[n]=(1-1/e)u[n];

x[n+1]-1/e x[n]=(1-1/e)u[n];

Another Method Using Transfer Functions

Let us learn one more method for this.

first calculate the transfer function of this state space model
which is

 G(s)= 1/(s+1);

now discreetize it by using

z=exp(sT)=exp(s) as T=1;

This pretty tricky so we will take the help of MATLAB here to calculate it.

On MATLAB, write the following :

 %make the transfer function using tf
 g=tf(1,[1 1]);

 %convert it to discreet
 g1=c2d(g,1);

you will get

g1=(1-1/e)/(z-1/e);

Now it is very easy to convert it to time domain

X(z)/U(z)=(1-1/e)/(z-1/e);

z X(z)-X(z)/e=(1-1/e)U(z);

The final model is

x[n+1]-1/e x[n]= (1-1/e)u[n]

which is same as earlier one.