# Lecture 10: Perspective

Transfer functions provide only an input/output perspective of what is going on in a system. There may be things going on physically that do not appear in a transfer function, due to cancellations, etc. On the other hand, state-space analysis provides a more complete representation. Furthermore, it can be generalized to time-varying systems, multi- input or output systems, and in some applications leads to very explicit design formulations. There is also much that can be done with nonlinear systems in state variable form.

We have seen that we can describe an LTIC system using a single differential
equation. In state-space analysis, we deal with
*
systems
*
of
equations, but make it so that all equations are
*
first order
*
.
Sometimes this requires introducing some extra variables. The
variables appearing in these equations (with respect to which we
differentiate) are called the
*
state
*
variables. The idea behind
the name is this: for a first order differential equation, if we know
where we are initially (the initial condition), then this provides all
of the information we need to determine where to go.

In circuits, it is common to choose the voltage across the capacitors
and the current through the inductors as state variables. This
provides our first example.

Note: we have

- First order differential equations.
- Each equation is expressed in terms of the state variables and the input.

*linear*equations, we can put the equations in matrix form. Let

Let denote taking the derivatives individually:

In the example above, let

Then we can write

Note: For nonlinear systems, we can still put them in state variable
form, even when we cannot use a matrix for the representation.

In general, a set of state-variable equations can be written

Note that this could be

- Nonlinear
- Multiple inputs
- Multiple outputs

A general
-input,
-output
*
linear
*
system with
state
variables can be written as

where is , is , is and is . (Write out the matrices.)

Have a student work the circuit on the board.