What information is needed to describe the behavior of a system in a control problem?

To describe the behavior of a system in a control problem, the scientific formulation requires two primary categories of information, both specified in a mathematically precise way:1

• System Behavior: The effect of any potential control action on the system must be precisely known under all possible environmental circumstances. This is typically achieved by developing a mathematical model that represents the system's internal dynamical state. This state is often expressed as a state vector—a list of numbers that quantifies the effect of all past external influences, allowing the future evolution of the system to be determined from its present state and future inputs.1
• Purpose and Environment: The purpose of the control (the performance criterion) and the nature of the environment (disturbances) must be clearly defined.1

Because it is often unrealistic to measure every component of the state vector exactly, control problems frequently incorporate state determination, which involves using an optimal filter to estimate the current state. Ultimately, the goal is to use this information to deduce an optimal control law, which determines the necessary action at any given instant.1