We have studied systems of the form:

where u is the input, y is the output and is the initial condition. Pictorially, the system can be represented as follows:

From the variation of parameters formula, the state x(t) satisfies:

and therefore the output y(t) can be written as:

The first term, , is the unforced response resulting only from the initial conditions and the term, , is the forced response due to the input u(t). A state space description of a system is an internal description because it contains all the information that is available about the internal operation of the system. This information is contained in the state vector. In an external or input/output representation, this internal system information is lost and the effect of the input only on the output is considered. In other words, the output represents only the forced response and the initial conditions are assumed to be equal to zero.

Given the following internal description of a dynamical system

we want to find an input/output relationship in the Laplace domain. We first take the Laplace transform of the above equations with the initial condition set to zero, we have

Solving for X(s) we obtain:

Taking the Laplace transform of the output equation y = Cx+Du , we find that

An internal or state space system representation describes the evolution of the system in the time domain. However, an external or input/output system description is developed in the Laplace domain. We now consider how an external representation may be obtained from an internal one.

The term is called the transfer function of the system and it determines the output Y(s) for any given input U(s). Notice that the equation does not contain any information about the system state or the initial conditions.

Summary: Given the internal representation of a system:

it straightforward to obtain the transfer function

A transfer function can be represented by the following block diagram:

where Y(s) = T(s)U(s).

For a given set of system matrices A, B, C, D, there is only one corresponding transfer function T(s). However, one transfer function may correspond to many different state space representations. A state space description obtained from a transfer function is known as a realization and can take on many different forms. We will study a few of these forms such as
the controllable canonical form and the observable canonical form later on in the course.

Remember that in computing a transfer function, the initial conditions are set to zero. Therefore, a piece of information is lost in the transformation from a state space description to a transfer function.


Example: Consider the following dynamical system in state space form:

(a) Find the transfer function from U to Y.
(b) Find y(t) if u(t) =1 and the initial conditions are equal to zero.


Solution

(a) Transfer function

(b) Using the transfer function T(s) calculated in (a) and the equation Y(s) =T(s)U(s), we can calculate the output for a given input signal.

Note that in the example above, the denominator of the transfer function T(s) is of degree two, which is the same as the number of state variables in our state space system representation. However, this is not always the case. A state space system with n states may have a transfer function whose denominator has degree less than n. The order of a state space system as the number of its state variables. This tells us that the degree of the transfer function (i.e. the degree of the denominator polynomial) is less than or equal to the order of the system. In the special case where the order of the system is equal to the degree of the transfer function, we say that the system has minimal order.