The state equation for a linear time-invariant system:

$$ x’(t) = A x(t) + B u(t) $$

Can be solved for $x(t)$. We collect all terms in $x$:

$$ x’(t) - A x(t) = B u(t) $$

Multiply equation by $e^{-At}$

$$ x’(t) e^{-At} - A x(t) e^{-At} = B u(t) e^{-At} $$

Using product rule $d(f;g) = f;dg + g;df$, where:

To give:

In modern control approaches, systems are analyzed in time domain as a set of differential equations. Higher order differential equations are decomposed into sets of first order equations of state variables that represent the system internally. This produces three sets of variables:

• Input variables are stimuli given to the system. Denoted by $u$.
• Output variables are the result of the current system state and inputs. Denoted by $y$.
• State variables represent the internal state of a system which may be obscured in the output variables. Denoted by $x$.

State-space equations

A state-space model is represented by two sets of equations.

A transfer function relates the output of a system to its input when it is represented in the Laplace domain. An assumption is made that initial steady-state response is 0. If $Y(s)$ is the output of a system, $X(s)$ is the input, then the transfer function is:

$$ H(s) = \frac{Y(s)}{X(s)} $$

Example - A Car