In previous lectures, we have studied state space representations and have shown how one can obtain a transfer function from such a description. If we are given a transfer function and we are interested in applying the techniques of pole-placement to such a system, we must obtain a state space representation for that transfer function. We will address this issue in this section where we will show how we can generate a state-space model for a given transfer function.


Realization Problem: Given a transfer function T(s) , find a state-space realization A, B, C, D. For single-input single-output systems one solution is as follows.

Suppose we are given the transfer function:

One realization of this transfer function is given by:

where

This type of realization is called the Controllable Canonical Form. Note that this realization is not unique. For each transfer function, there are many different possible realizations.

Example 1: Given the transfer function we want to find a state space realization. We begin rewriting our transfer function in the form given above for the controllable canonical form.

Now, considering the form given above, we can write down the matrices corresponding to one realization.

Let us check if this realization is controllable.

In fact, this result is not surprising. Any realization in controllable canonical form will be controllable. Note that the question of controllability only applies to state space realizations and not to transfer functions. Thus, some realizations of a transfer function will be controllable and some will not.

Example 2: Find a state space representation for the system that has the following transfer function

We begin rewriting our transfer function in the form given above for the controllable canonical form.

Therefore, we have

System Observability

Consider the state space representation

Observability is the ability to estimate the state variables from the output measurement.

In order to test the observability of the system we do the following steps:

- Construct the observability matrix (n is the number of state variables)

- Calculate the determinant of the observability matrix det(Q ).

- If the determinant is not equal to zero then the system is observable. If the determinant is equal to zero then the system is not observable.


Example 3: Is the following system observable

We first construct the observability matrix

Note that

We now calculate the determinant of the observability matrix

We conclude that the system is observable.