Define dynamic and static system

In control theory and systems engineering, systems are classified based on their response to inputs. Two primary classifications are dynamic systems and static systems. Here's a detailed explanation of each:

Dynamic System

A dynamic system is one whose output depends not only on the current input but also on past inputs and/or outputs. In other words, the system's behavior is characterized by a memory of past inputs, resulting in time-dependent behavior.

Characteristics of Dynamic Systems:

1. Memory: Dynamic systems have memory, meaning their output at any given time depends on past inputs and states.
2. Differential/Difference Equations: The behavior of dynamic systems is often described using differential equations (for continuous-time systems) or difference equations (for discrete-time systems).
3. State Variables: Dynamic systems are characterized by state variables that capture the system's memory of past behavior.
4. Time-Dependency: The output evolves over time, reflecting changes in the input and the system's state.

Examples of Dynamic Systems:

1. Mass-Spring-Damper System: A mechanical system where the position of the mass depends on the current and past forces applied.
2. RC Circuit: An electrical circuit where the voltage across the capacitor depends on past and current currents and voltages.
3. Population Models: Models in biology where the population at a given time depends on birth and death rates over time.
4. Financial Systems: Economic models where future financial status depends on past investments and returns.

Mathematical Representation:

For continuous-time systems: dy(t)dt+ay(t)=bx(t)\frac{dy(t)}{dt} + a y(t) = b x(t)dtdy(t)?+ay(t)=bx(t) where y(t)y(t)y(t) is the output, x(t)x(t)x(t) is the input, and aaa and bbb are constants.

For discrete-time systems: y[n]=ay[n−1]+bx[n]y[n] = ay[n-1] + bx[n]y[n]=ay[n−1]+bx[n] where y[n]y[n]y[n] is the output, x[n]x[n]x[n] is the input, and aaa and bbb are constants.

Static System

A static system is one whose output depends only on the current input, with no dependence on past inputs or outputs. In other words, the system has no memory, and its output is a direct, instantaneous response to the input.

Characteristics of Static Systems:

1. No Memory: Static systems have no memory of past inputs; the output depends solely on the current input.
2. Algebraic Equations: The behavior of static systems is described using algebraic equations rather than differential or difference equations.
3. Time-Invariant: The output does not evolve over time based on past inputs; it changes only with the current input.

Examples of Static Systems:

1. Resistor: An electrical resistor where the voltage drop across it depends only on the current passing through it at that moment (Ohm's Law: V=IRV = IRV=IR).
2. Linear Amplifier: An amplifier where the output voltage is a constant multiple of the input voltage.
3. Scale: A weighing scale where the displayed weight is directly proportional to the current load.
4. Thermistor: A temperature sensor where the resistance depends only on the current temperature.

Mathematical Representation:

For a static system: y(t)=f(x(t))y(t) = f(x(t))y(t)=f(x(t)) where y(t)y(t)y(t) is the output and x(t)x(t)x(t) is the input, and fff is an algebraic function.

Comparison of Dynamic and Static Systems:

Summary

• Dynamic Systems: Depend on current and past inputs, have memory, and are described by differential or difference equations.
• Static Systems: Depend only on the current input, have no memory, and are described by algebraic equations.

Understanding the distinction between dynamic and static systems is crucial in designing and analyzing systems across various fields, including engineering, physics, economics, and biology.