0.3 The concept of state

We are given the following differential equation describing a system. Note that $u(t)=0$ .

Using the Laplace transform techniques described in the module on Linear Systems with Constant Coefficients , we can find a solution for $y(t)$ :

As we need the information contained in $y(t_0)$ for this solution, $y(t)$ defines the state.

The differential equation describing an unforced system is:

Finding the $q(s)$ function, we have

The roots of this function are ${}_1=-1$ and ${}_2=-2$ . These values are used in the solution to the differential equation as the exponents of the exponential functions:

where $c_1$ and $c_2$ are constants. To determine the values of these constants we would need two equations (with two equations and two unknowns, we can find the unknowns). If we knew $y(0)$ and $\frac{d^{1}y(0)}{dt^1}$ we could find two equations, and we could then solve for $y(t)$ . Therefore the system's state, $x(t)$ , is

In fact, the state can also be defined as any two non-trivial (i.e. independent) linear combinations of $y(t)$ and $\frac{d^{1}y(t)}{dt^1}$ .