⚙️

CLOSED-LOOP CONTROL SYSTEMS

Interactive Process Control Laboratory  |  PID Dynamics & Nonlinear Response

MVCC · CSSIA · v3.0
01 The Closed-Loop Control Architecture
System Block Diagram
Reference
Setpoint (SP)
→
Σ
error
→
PID
Controller
Kp · Ki · Kd
→
Actuator
Control Output
→
Process
Plant / System
→
Output (PV)
Process Variable
↑  ← ← ← ←  FEEDBACK (Sensor Measurement)  ← ← ← ← ← ↑

Error = Setpoint − Measured Output  ·  The controller continuously minimizes error through feedback

🔄 What Is Closed-Loop Control?

A closed-loop (feedback) control system continuously compares the actual output against the desired setpoint. The difference—called the error signal—drives corrective action. Unlike open-loop systems, it automatically compensates for disturbances and nonlinearities.

⚡ Proportional (P) Action

Produces an output proportional to current error. A higher gain Kp speeds response but risks overshoot. P-only control always leaves a steady-state error (offset).

u_P = Kp × e(t)

∫ Integral (I) Action

Integrates error over time, eliminating steady-state offset. Too much Ki causes integral windup—accumulated error that drives excessive overshoot. The I-term captures the history of the process.

u_I = Ki × ∫e(t) dt

∂ Derivative (D) Action

Responds to the rate of change of error—anticipating future error. Acts like a damper, reducing overshoot and settling time. Sensitive to measurement noise.

u_D = Kd × de(t)/dt

🌊 Overshoot & Nonlinearity

Overshoot occurs when the process variable exceeds the setpoint before settling. Nonlinear systems (e.g., valves with dead bands, temperature asymmetry) worsen this. The derivative term and gain scheduling are key countermeasures.

📊 Key Performance Metrics

Rise Time — Time to reach 10%→90% of setpoint
% Overshoot — Peak above setpoint / setpoint × 100
Settling Time — Time to stay within ±5% of SP
Steady-State Error — Final residual offset

02 PID Tuning Effects Reference
Increase Kp
Rise Time▼ Faster
Overshoot▲ More
Steady-State Error▼ Less
Increase Ki
Rise Time▼ Faster
Overshoot▲ More
Steady-State Error▼ Eliminates
Increase Kd
Rise Time– Slight effect
Overshoot▼ Less
Settling Time▼ Faster
⚠️ Nonlinear Systems: PID parameters tuned at one operating point may not perform well at another. Techniques such as gain scheduling, anti-windup, and setpoint prefiltering are used to handle nonlinear behavior and prevent destructive overshoot.
System Configuration
Setpoint (SP) ?The target value you want the process variable to reach and maintain. 50
Proportional Gain (Kp) ?Controls how strongly the controller responds to current error. Higher = faster but more overshoot risk. 1.5
Integral Gain (Ki) ?Eliminates steady-state error by integrating past errors. Too high causes windup and overshoot. 0.3
Derivative Gain (Kd) ?Anticipates future error using rate of change. Reduces overshoot — the damper of the system. 0.8
Process Delay (τ) ?Dead time in the process — delay between control action and measured effect. Makes control harder. 2
Process Nonlinearity ?Simulates valve dead-band, saturation, or asymmetric plant behavior that makes PID tuning harder. 0.3
Disturbance Magnitude ?External upsets to the process (e.g., load changes, supply variations). Tests disturbance rejection. 5
Anti-Windup Protection ?Prevents the integrator from accumulating error when the output is saturated, avoiding excessive overshoot.
Setpoint Pre-Filter ?Smooths setpoint step changes to reduce sudden large errors that cause windup and overshoot.
PRESETS
Process Variable
0.0
units
Setpoint
50.0
units
Error
+50.0
units
Controller Output
0.0
%
Ready — Configure parameters and press RUN
Response Curve — Process Variable vs. Time
Error Signal & Controller Output
Performance Analysis
Rise Time
—
seconds
% Overshoot
—
%
Settling Time
—
seconds
Thermostat Control Lab
Mode
Room Setpoint Temperature (°F) 72°F
Initial Room Temperature (°F) 55°F
HVAC Power (% capacity) 80%
Thermal Mass (inertia) 0.5
Nonlinear Mode (asymmetric heating/cooling)
Closed-Loop Concept Walkthrough

🌡️ Sensor reads current room temperature (Process Variable)

🎯 Comparator calculates: Error = Setpoint − Measured Temp

🧠 Controller processes error signal using on/off (bang-bang) logic with hysteresis

⚡ Actuator energizes furnace/AC based on control output

♻️ Feedback Loop continuously re-evaluates every cycle

Real-Time Thermostat Visualization
Room Temp
55°F
Setpoint
72°F
Error
+17°F
HVAC State
OFF
55°F
HVAC Activity
💤
SYSTEM IDLE
Heating Load0%
Cooling Load0%
Temperature Response — Showing Overshoot Prevention
■ Room Temp    - - Setpoint    ■ Overshoot Zone    ■ Hysteresis Band
Knowledge Check — Closed-Loop Control
Quick Reference

Open-Loop vs Closed-Loop

Open-loop: no feedback — blind execution. Closed-loop: sensors measure output and feed back to correct errors automatically.

Error Signal

e(t) = SP − PV. If positive: output is below setpoint. If negative: above setpoint. Controller drives error toward zero.

Integral Windup

When controller output saturates (maxes out), the integrator keeps accumulating error. Anti-windup prevents the integrator from charging beyond the saturation limit.

Bang-Bang Control

Simplest closed-loop: output is either fully ON or OFF. Thermostats use hysteresis (deadband) to prevent rapid switching and reduce overshoot oscillation.

Gain Scheduling

Using different PID gain sets at different operating points to handle nonlinear process behavior across the entire operating range.