Introduction to State Variables and State Space

Summary

• State Variables

  The state of a system is a number, or pair of numbers, or triple of numbers, or..., that characterizes the system’s behavior at at a given time. We write the state as \( (x_1,\dots,x_n) \). So for example, the state of the Glucose/Insulin system might be a pair of numbers \( (G, I) \) representing the glucose and insulin levels of the person at a given time. \( G \) and \( I \) are called state variables for the system.

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• State Space

  We think of the state at a given time as a point in a space, namely, the space of all possible states. This space is called state space. If there is only one state variable, then state space is a 1-dimensional space, that is, a line. If there are 2 state variables, then state space is the 2-dimensional plane. If the 2 state variables are Glucose and Insulin, we call this space “Glucose-Insulin space”, if it’s Sharks and Tuna we call it “Shark-Tuna space”.

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• examples in 1D and 2D
  • 1D : “Temperature space”
  • 2D : “Shark-Tuna space”, “Glucose-Insulin space”