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.

Linear Systems Experiments Home Page

Last Revised: August 5, 2002