State Space Form

Definition

The state of a system is a collection of variables which describe the evolution of the system until the current time.

Definition (operational)

The state $x(t)$ of a system is a collection of variables such that if we know $x(t_0)$ and the input signal $u(t)$, $t\in [t_0,t_1]$, then we can compute $x(t_1),y(t_1)$

State space representation is a mathematical model of a physical system as a set of input, output, and state variables related by first-order differential (or difference) equations. This model is widely used in control systems engineering for designing, analyzing, and simulating dynamic systems. The state space representation is particularly advantageous because it can model systems that are not necessarily easily described by transfer functions, especially when dealing with multiple inputs and outputs or systems that are inherently MIMO (Multiple Input, Multiple Output).