What is static and dynamic systems

Static and dynamic systems are classifications used to describe the behavior and characteristics of systems in various fields of science, engineering, and mathematics based on how they respond to inputs over time.

Static Systems:

1. Definition: A static system, also known as time-invariant or memoryless system, is characterized by its output depending solely on the current value of the input at the same instant of time.

2. Properties:

   • The output does not change with time or history of the input.
   • No internal state or memory that retains past inputs or outputs.
   • Output is a function of the current input only.

3. Examples:

   • Gain or scaling operations: y(t)=k⋅x(t)y(t) = k \cdot x(t)y(t)=k⋅x(t), where kkk is a constant.
   • Static nonlinear operations: y(t)=x(t)2y(t) = x(t)^2y(t)=x(t)2, y(t)=?x(t)?y(t) = |x(t)|y(t)=?x(t)?.

4. Characteristics:

   • Easily described by algebraic equations or simple functional relationships.
   • Does not involve differential equations or integrators.
   • Analysis often involves direct calculation or inspection of the input-output relationship.

Dynamic Systems:

1. Definition: A dynamic system is characterized by its output depending on both the current input and the history of inputs over time.

2. Properties:

   • Output changes with time due to past and current inputs.
   • Involves differential equations, integrators, or state variables to describe system behavior.
   • Has internal state or memory that affects future outputs.

3. Examples:

   • First-order systems: τdy(t)dt+y(t)=x(t)\tau \frac{dy(t)}{dt} + y(t) = x(t)τdtdy(t)?+y(t)=x(t).
   • Second-order systems: d2y(t)dt2+2ζωndy(t)dt+ωn2y(t)=x(t)\frac{d^2y(t)}{dt^2} + 2\zeta\omega_n \frac{dy(t)}{dt} + \omega_n^2 y(t) = x(t)dt2d2y(t)?+2ζωn?dtdy(t)?+ωn2?y(t)=x(t), where ωn\omega_nωn? is the natural frequency and ζ\zetaζ is the damping ratio.

4. Characteristics:

   • Requires differential equations or difference equations to describe system dynamics.
   • Analysis involves studying stability, transient response, and frequency response.
   • Commonly encountered in control systems, mechanical systems, biological systems, and many physical phenomena.

Practical Considerations:

• Modeling and Analysis: Understanding whether a system is static or dynamic helps in choosing appropriate mathematical models and analysis techniques.

• Control and Design: Dynamic systems require consideration of time-dependent behaviors and stability criteria, whereas static systems are simpler to control and analyze.

• Real-world Application: Many real-world systems exhibit dynamic behavior, necessitating detailed modeling and simulation to predict their response to inputs and environmental changes.

In summary, static systems have no memory and produce an output solely based on the current input, while dynamic systems exhibit time-dependent behavior influenced by both current and past inputs, often described by differential equations or state-space representations.