Time Domain Analysis is crucial for understanding how systems respond to inputs over time. It involves solving differential equations to predict system behavior, analyzing transient and steady-state responses, and evaluating key performance metrics like rise time and overshoot.

This analysis connects to the broader chapter by applying linear differential equations to real-world systems. It provides practical tools for assessing system stability, performance, and response characteristics, bridging theory with practical applications in dynamic systems.

Time-Domain Response of Systems

Analyzing Linear Dynamic Systems

• The time-domain response of a linear dynamic system represents the output of the system as a function of time when subjected to an input signal
• Obtain the time-domain response by solving the differential equation that describes the system's behavior
• The solution to the differential equation consists of two parts:
  • Homogeneous solution (natural response) represents the system's response due to its initial conditions
  • Particular solution (forced response) represents the system's response due to the input signal
• The total response of the system is the sum of the homogeneous and particular solutions
• Use the time-domain response to analyze the system's stability, transient behavior, and steady-state behavior

Solving Differential Equations

• Linear dynamic systems are described by linear differential equations
• Techniques for solving linear differential equations include:
  • Laplace transform method converts the differential equation into an algebraic equation in the s-domain
  • Variation of parameters method finds the particular solution by assuming a solution form and determining the coefficients
  • Undetermined coefficients method assumes a particular solution form based on the input signal and solves for the coefficients
• The initial conditions of the system are used to determine the constants in the homogeneous solution
• The input signal is used to determine the particular solution

System Response to Input Functions

Common Input Functions

• Step input: a sudden change in the input signal from one constant value to another, represented by the unit step function
  • The system's response to a step input is called the step response, which provides information about the system's transient and steady-state behavior
• Ramp input: a linearly increasing function of time, represented by the unit ramp function
  • The system's response to a ramp input is called the ramp response, which helps analyze the system's ability to track a constantly changing input
• Impulse input: an infinitely short duration signal with an infinitely high amplitude, represented by the Dirac delta function
  • The system's response to an impulse input is called the impulse response, which characterizes the system's fundamental behavior

Convolution and Superposition

• Convolution is a mathematical operation that determines the system's response to any arbitrary input signal using the impulse response
• The convolution integral expresses the output of a linear system as the integral of the product of the input signal and the system's impulse response
• The principle of superposition states that the response of a linear system to a sum of inputs is equal to the sum of the responses to each individual input
• Superposition allows for the decomposition of complex input signals into simpler components, making the analysis more manageable

Transient vs Steady-State Behavior

Transient Response

• The transient response is the part of the system's response that decays to zero over time, typically occurring immediately after a change in the input signal
• Determined by the system's poles and initial conditions
• Characteristics of the transient response include:
  • Rise time: the time required for the response to rise from a lower percentage (usually 10%) to an upper percentage (usually 90%) of its final value
  • Settling time: the time required for the response to settle within a specified percentage (usually 2% or 5%) of its final value
  • Overshoot: the percentage by which the response exceeds its final value during the transient response
  • Peak time: the time at which the response reaches its maximum value during the transient response

Steady-State Response

• The steady-state response is the part of the system's response that remains constant or varies periodically after the transient response has decayed to zero
• Determined by the input signal and the system's transfer function
• For stable systems, the steady-state response can be found by applying the final value theorem to the system's transfer function and the Laplace transform of the input signal
• Steady-state error is the difference between the desired output and the actual output of the system in the steady-state condition
• Steady-state error can be classified into:
  • Position error: the error in the system's response to a step input
  • Velocity error: the error in the system's response to a ramp input
  • Acceleration error: the error in the system's response to a parabolic input

Evaluating System Properties from Response

Time-Domain Specifications

• Rise time: a measure of the system's speed of response, affected by the system's bandwidth and damping ratio
  • A shorter rise time indicates a faster response (e.g., in a temperature control system, a shorter rise time means the system reaches the desired temperature more quickly)
• Settling time: a measure of how quickly the system's transient response decays, affected by the system's poles and damping ratio
  • A shorter settling time indicates a more rapidly decaying transient response (e.g., in a positioning system, a shorter settling time means the system reaches and maintains its final position more quickly)
• Overshoot: a measure of the system's damping, affected by the system's poles and damping ratio
  • A lower overshoot indicates a more damped system (e.g., in a servo motor control system, a lower overshoot means less oscillation around the desired position)
• Peak time: the time at which the system's response reaches its maximum value during the transient response
  • A shorter peak time generally indicates a faster response (e.g., in a mechanical system, a shorter peak time means the system reaches its maximum displacement more quickly)

Evaluating Stability and Performance

• Stability refers to a system's ability to return to an equilibrium state after a disturbance or change in input
• A stable system has a bounded output for a bounded input and all its poles in the left half of the complex plane
• Evaluate stability by examining the system's poles and the transient response
  • If the poles have negative real parts, the system is stable (e.g., a mass-spring-damper system with positive damping coefficient)
  • If any pole has a positive real part, the system is unstable (e.g., a chemical reaction with positive feedback)
• Performance refers to how well a system achieves its desired objectives, such as minimizing steady-state error, maximizing speed of response, or maintaining robustness
• Evaluate performance by comparing the system's time-domain specifications to the desired values
  • If the rise time, settling time, and overshoot are within acceptable limits, the system has good performance (e.g., a well-tuned PID controller in a process control system)
  • If the system fails to meet the desired specifications, the performance is considered poor (e.g., an under-damped or over-damped second-order system)