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Interactive Feedback Loop Simulator

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Description

This interactive simulator demonstrates how proportional controller gain affects the step response of a first-order closed-loop system. Students can adjust parameters and observe real-time changes in system behavior.

System Model

Plant: First-order system $$G_p(s) = \frac{1}{\tau s + 1}$$

Controller: Proportional gain $$G_c(s) = K$$

Closed-Loop Transfer Function: $$G_{CL}(s) = \frac{K}{\tau s + 1 + K} = \frac{K/(1+K)}{(\tau/(1+K))s + 1}$$

Key Relationships

Parameter Formula Effect of Increasing K
Steady-State Output $y_{ss} = \frac{K}{1+K}$ Approaches 1 (reference)
Steady-State Error $e_{ss} = \frac{1}{1+K}$ Decreases toward 0
Closed-Loop Time Constant $\tau_{CL} = \frac{\tau}{1+K}$ Faster response
Settling Time (2%) $t_s \approx 4\tau_{CL}$ Faster settling

Interactive Controls

  • Gain K slider (0.5 to 10): Adjusts controller gain
  • Tau τ slider (0.5 to 3): Adjusts plant time constant
  • Run/Pause button: Animates the step response
  • Reset button: Returns to default parameters
  • Show error region: Highlights the area between reference and output

What to Observe

  1. Effect of increasing K:
  2. Output gets closer to reference (less steady-state error)
  3. Response becomes faster (smaller time constant)

  4. But: Some error always remains with proportional control alone

  5. Effect of increasing τ:

  6. Response becomes slower
  7. Same steady-state error (depends only on K)

Learning Objectives

After using this MicroSim, students will be able to:

  1. Demonstrate how changing K affects closed-loop response speed
  2. Calculate steady-state error for a proportional control system
  3. Predict settling time from system parameters
  4. Explain why proportional control alone cannot eliminate steady-state error

Lesson Plan

Warm-Up Exploration (5 minutes)

  1. Set K=1, τ=1. Click "Run" and observe the step response.
  2. What is the final value of the output? Is it equal to the reference?

Guided Investigation (15 minutes)

Part 1: Effect of Controller Gain 1. Keep τ=1 fixed 2. Try K = 0.5, 1, 2, 5, 10 3. For each K, record: - Steady-state output y_ss - Approximate settling time 4. Question: What happens to the error as K increases?

Part 2: Effect of Plant Time Constant 1. Keep K=2 fixed 2. Try τ = 0.5, 1, 2, 3 3. Observe how response speed changes 4. Question: Does τ affect the final steady-state error?

Mathematical Verification (10 minutes)

  1. For K=4, τ=1, predict:
  2. y_ss = K/(1+K) = ?
  3. e_ss = 1/(1+K) = ?
  4. τ_CL = τ/(1+K) = ?
  5. Run the simulation and verify your calculations

Discussion Questions

  1. Why can't proportional control eliminate all steady-state error?
  2. What type of controller action might help reduce steady-state error to zero?
  3. If you need faster response, would you increase K or decrease τ?

Key Insight

The Proportional Control Limitation

With proportional control alone, the steady-state error $e_{ss} = \frac{1}{1+K}$ can never reach zero, no matter how large K becomes. This fundamental limitation motivates the introduction of integral action in PID controllers.

References