DES - Definition of 'State of a System'

Synthesis:

For a differentiable system, think of the State as the condition needed to solve the Cauchy problem (~ex.: $f^{\prime }(0)=3$, $f(0)=5$, → $\left[f^{\prime }(0),f(0)\right]$ is the State of the System).

For a non-differentiable system, and more in generale the State is composed by all the conditions needed to calculate the output of the system given only the inputs.

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State of a System:

The state of a system is the set of variables whose value at time $t_0$ is required in order to determine uniquely the output $y(t)$ for all $t\ge t_0$, given the input $u(t)$ for all $t\ge t_0$

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~Example: state of a capacitor system

$v(t)=v(0)+\frac{1}{C}{\int }_0^{t}i(\tau )d\tau$

So the state is:

• $x=[v]$
• $y=[v]$

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~Example: determine the space needed to stop the car if one applies a constant breaking force $\vec{F}$

The system is: $s(t)=s(0)+\dot{s}(0)t-\frac{F}{M}\frac{t^2}{2}$

So the resulting state is:

• $x=\left[\begin{array}{c}s\\ \dot{s}\end{array}\right]$
• $y=[s]$

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