Discretization of Continuous Time State Space Systems

The discretization of the continuous system means to convert continuous parameter into individual & separate parameter to analyze the system easily. This can be happened by dividing the continuous-time system into a specific time interval based system. 

Let's learn it from an example:

Suppose we have the continuous-time state-space system 

as per the above-given system, changes in (t) will bring changes in the system as per equation.

To obtain a discrete-time model system, we have to make the system independent of (t). To do that, we will use matrices G & H.

In the above equation, we used (T) as a specific time interval despite of (t) to represent the system. G(T), H(T) are constants that represent changes that can happen due to discrete time intervals. 

We will now determine the values of matrices G(T) and H(T). They are constant for a particular sampling interval. They depend on the value of the sampling interval (T). 

We start by using the solution of (1) to calculate the values of the state x at times kT and (k + 1)T.

above equation represent at specific time interval based value of equation (1) based system.

We want to write x((k+1)T) in terms of x(kT) so we multiply all terms of (6) by e^AT and solve for e^A(K+1)T x(0), obtaining

By multiplying equation (6) with e^AT and we will get the above equation by changing terms to the left-hand side to the right-hand side.

Substituting equation (7) in equation (5) & will obtain

we will get equation in form of e^AT and terms of integration

which, by the linearity of integration, is equivalent to

as per basics of integration, we will get integration interval from kT to (k+1)T

Next, we notice that within the interval from kT to (k+1)T, u(t) = u(kT) is constant, as is the matrix B, so we can take them out of the integral to obtain

where 𝓣 is part of an interval

We can take the e^A(k+1)T inside the integral to obtain

we can take term e^A(k+1)T inside integration as per basic of integration

Now we see that as 𝓣 ranges from kT to (k + 1)T (the lower to the upper limit of integration) the exponent of e ranges from T to 0. Accordingly, let’s define a new variable λ = (k +1)T −𝓣 . Then dλ = −d𝓣, and λ range from T to 0 as 𝓣 ranges from kT to (k + 1)T. Thus we have

the purpose of defining new variable λ to reduce the complexity of the equation

Now, by comparing equation (13) with equation (3), will obtain

so here are the terms, which will give specific value at specific input

Now, if want to remove integral from H(T),

formula of derivation

By implying 

we will get H(T) without integral function

Finally, note that while I restricted the value of λ and 𝓣 to lie within a single sampling interval, k appears nowhere in the expressions for G(T) and H(T). Our solution to (3) is thus

and we can see that at the sampling instants kT, this has exactly the same value as is obtained using (1). Specifically,

and since the input, u(t) is constant on sampling intervals,

As we can see, we converted a continuous-time state-space system equation into a discrete-time system. As per the equation of the discrete-time system, we can now get specific output at specific input with respect to variation in time.

This is it for this one, see ya in the next one.

Thanks & Keep Learning.