System Realization

This experiment shows that any transfer function can be written in an infinite number of state space forms. A minimal realization is a state space representation of a system that has the minimum possible number of states. A minimal realization is always both controllable and observable. However, there are still an infinite number of minimal realizations for any transfer function. This is illustrated by plotting the impulse response for three systems, motivated by an example in Section 9.1 of John Bay's book.

Observable but not Controllable

The first system has the realization

where c is any scalar nonzero constant. This system is observable but not controllable. Using the well-known formula for computing the transfer function

(where D = 0 in this case) we obtain the transfer function G(s) = 1 / (s + 1). Since the transfer function is first order (that is, the highest power of s in the denominator is 1) we know that we can realize this transfer function with a first order state equation (that is, a state equation with one state).

Controllable but not Observable

The second system has the realization

where, again, c is any scalar nonzero constant. This system is controllable but not observable. We can determine that this system has the same transfer function as the first system; that is, G(s) = 1 / (s + 1). Again, since the transfer function is first order (that is, the highest power of s in the denominator is 1) we know that we can realize this transfer function with a first order state equation (that is, a state equation with one state).

Controllable and Observable

The third system has the realization

where, again, c is any scalar nonzero constant. This system is both controllable and observable. We can determine that this system has the same transfer function as the two systems above; that is, G(s) = 1 / (s + 1). Since the transfer function has the same order as the state equation, we know that this is a minimal realization.

The impulse responses of the three systems described above are simulated below. It can be seen that all three systems have the same impulse response, regardless of the value of the constant c. Since a transfer function is entirely determined in terms of its impulse response, this shows that there are any infinite number of observable and uncontrollable realizations, an infinite number of controllable and unobservable realization, and an infinite number of controllable and observable realizations, all for the same transfer function.