Time Response Specifications Lecture Notes

Overview

• Focus on Time Response Specifications for control systems
• Previous topics included response of first and second order systems to signals, steady state error, and damping effects

Key Concepts

System Response Types

• First-order Systems: Analyze steady-state error
• Second-order Systems: Examine response variations with damping (under-damped, critically damped, over-damped)

Design Specifications

• Desired Response Example:
  • Target a value (e.g., temperature set to 24 degrees)
• System Response Expectations:
  • Instantaneous response may not be feasible; acceptable to overshoot or oscillate
  • Analyzing speed to reach desired value and potential overshoot

Step Response Analysis

• Focus on step response for simplicity
• Key metrics to quantify:
  • Delay Time (td): Time to reach 50% of final value the first time
  • Rise Time (tr): Time to rise from 10% to 90% of final value (or 0 to peak for under-damped systems)
  • Peak Time (tp): Time to reach peak value
  • Peak Overshoot (Mp): Percentage deviation from steady-state value at peak
  • Settling Time (ts): Time to remain within a specified tolerance band (2% or 5%) of final value

Second Order Underdamped System

• Transfer function: ( G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} )
• Rise Time (tr): Calculated using the damped natural frequency ( \omega_d )
• Peak Time (tp): Derived from calculus by setting the derivative to 0
• Peak Overshoot (Mp): Exponential function of damping ratio ( \zeta )

Settling Time

• Determined based on time constant ( \tau )
• For 2% tolerance: ( ts = \frac{4}{\zeta \omega_n} )
• For 5% tolerance: ( ts = \frac{3}{\zeta \omega_n} )

System Behavior

• Damping Ratio (( \zeta )) Effects:
  • ( \zeta = 0 ): Oscillatory response, infinite settling time
  • Higher ( \zeta ): Faster settling time and reduced peak overshoot

Applications of Damping

• Over-damped Systems:
  • Example: Tap shut-off valves, automatic door closers
• Critically Damped Systems:
  • Example: Elevator mechanisms, firearms triggers
• Under-damped Systems:
  • Example: String instruments, voltmeters

Steady State Error Analysis

• Basics: Difference between desired and actual output over time
• Error Constants:
  • Position Error Constant (Kp): Defined for step inputs
  • Velocity Error Constant (Kv): Defined for ramp inputs
  • Acceleration Error Constant (Ka): Defined for parabolic inputs

System Type and Steady State Error

• Type 0 System: Constant position error, infinite velocity and acceleration error
• Type 1 System: Zero position error, constant velocity error, infinite acceleration error
• Type 2 System: Zero position and velocity errors, finite acceleration error

Summary

• Discussed system types, performance metrics, and steady state errors
• Next lecture: Stability definitions and analysis using transfer functions and Routh-Hurwitz criteria.