68  PID: Control Theory

In 60 Seconds

The PID equation Output = Kpe(t) + Kiintegral(e) + Kd*de/dt combines proportional (current error), integral (accumulated error), and derivative (error rate) terms. Select P-only for simple on/off systems, PI for most IoT applications requiring zero steady-state error, and full PID only when overshoot or oscillation must be minimized.

68.1 Learning Objectives

By the end of this chapter, you will be able to:

• Derive the PID Equation: Formulate the combined Proportional-Integral-Derivative output for a given control scenario
• Calculate PID Term Contributions: Compute P, I, and D term outputs from error signals, accumulated error, and error rate
• Evaluate System Response: Assess system behavior metrics including overshoot, settling time, and steady-state error
• Select PID Configurations: Justify the choice of P, PI, PD, or full PID control for specific IoT applications

MVU: The Three PID Terms

Core Concept: PID control combines three complementary strategies - P (Proportional) reacts to current error, I (Integral) accumulates past errors to eliminate offset, and D (Derivative) predicts future error to prevent overshoot. Why It Matters: Each term solves a specific problem that the others cannot - P alone leaves steady-state error, adding I eliminates that error but may overshoot, and D provides the damping needed for smooth, stable control. Key Takeaway: Think of PID as past-present-future: I remembers the past, P responds to the present, D anticipates the future - together they achieve what no single term can accomplish alone.

For Beginners: PID: Control Theory

Process control in IoT is about automatically adjusting systems to maintain desired conditions. Think of cruise control in a car: it continuously measures your speed, compares it to your target, and adjusts the throttle to keep you on track. IoT systems use similar feedback loops to control everything from room temperature to industrial manufacturing processes.

Sensor Squad: Sammy Explains PID Control

Hey there, future engineers! I’m Sammy the Sensor, and today I’m going to tell you about something SUPER cool called PID control!

Imagine you’re riding a bike and trying to stay in the center of a path. Here’s what each letter means:

P is for “Present” - It’s like looking at where you ARE right now. If you’re far from the center, you turn the handlebars a lot. If you’re close, just a tiny turn!

I is for “In the past” - It’s like remembering if you’ve been drifting to one side for a while. Even if you’re almost centered now, if you kept going left before, you need a little extra push to the right!

D is for “Direction” - It’s like noticing how FAST you’re moving toward or away from the center. If you’re zooming toward the edge super fast, you need to turn harder BEFORE you go off the path!

Put them all together and you get a super smart helper that: - Sees where you are NOW (P) - Remembers where you’ve BEEN (I) - Predicts where you’re GOING (D)

Real-world example: Your house thermostat uses PID! When it’s cold, it turns the heater on strong. As it gets warmer, it slows down so it doesn’t make your house TOO hot!

Try this at home: Fill a glass exactly to a line using water. Notice how you start fast, then slow down as you get close? That’s YOU doing P and D control!

68.2 Prerequisites

Before diving into this chapter, you should be familiar with:

• Feedback Fundamentals: Understanding feedback concepts and negative feedback
• Open-Loop and Closed-Loop Systems: Comparing control architectures
• Sensor Fundamentals: Sensors for feedback measurement

The Challenge: Simple On/Off Control Wastes Energy and Causes Wear

The Problem: Bang-bang (on/off) control is fundamentally inefficient:

• Overshoot: System blows past the setpoint, then must reverse direction
• Oscillation: Continuous cycling between “too hot” and “too cold” (or fast/slow)
• Actuator Wear: Frequent switching damages relays, compressors, and motors
• Energy Waste: Heating then immediately cooling wastes significant energy

What We Need:

• Smooth approach to setpoint without overshoot
• Fast recovery from disturbances
• Stable operation with no oscillations
• A general solution that works across different system types

The Solution: PID control uses three complementary strategies to achieve optimal control.

68.3 PID Controller Overview

Figure 68.1: PID control system block diagram showing the three parallel control actions: Proportional (responds to current error), Integral (eliminates steady-state error), and Derivative (anticipates future error)

PID Control System Flow - The feedback loop that enables precise control

Figure 68.2: PID Control System Flow - The feedback loop that enables precise control

While simple closed-loop systems provide basic feedback, achieving optimal performance often requires more sophisticated control algorithms. The PID controller is the most widely used feedback control algorithm in industrial and IoT applications.

A PID controller uses three different control strategies simultaneously, each addressing different aspects of error correction:

1. P (Proportional): Reacts to current error magnitude
2. I (Integral): Reacts to accumulated error over time
3. D (Derivative): Reacts to rate of error change

Figure 68.3: PID Controller Block Diagram: Proportional, Integral, and Derivative Paths

PID Controller Block Diagram: Error signal splits into three parallel paths. P term responds to current error, I term accumulates past errors, D term predicts future based on rate of change. All three combine to produce optimal control signal.

68.4 The PID Equation

\[ u(t) = K_p \cdot e(t) + K_i \cdot \int_{0}^{t} e(\tau) \, d\tau + K_d \cdot \frac{de(t)}{dt} \]

Where:

• \(u(t)\) = Controller output at time \(t\)
• \(e(t)\) = Error (Set Point - Process Variable)
• \(K_p\) = Proportional gain
• \(K_i\) = Integral gain
• \(K_d\) = Derivative gain

Putting Numbers to It

Consider a thermostat targeting 22°C with \(K_p = 5\), \(K_i = 0.2\), \(K_d = 1\). At \(t = 10\)s, temperature is 20°C and falling at 0.1°C/s. The PID output is:

\[ \begin{align} e(t) &= 22 - 20 = 2\text{°C} \\ u_P &= 5 \times 2 = 10 \text{ (proportional)} \\ u_I &= 0.2 \times \int_0^{10} e \, dt \approx 0.2 \times 20 = 4 \text{ (accumulated error)} \\ u_D &= 1 \times 0.1 = 0.1 \text{ (temperature falling)} \\ u(t) &= 10 + 4 + 0.1 = 14.1 \end{align} \]

The P term (10) responds to current 2°C error. The I term (4) compensates for past accumulated offset. The D term (0.1) is small because temperature is changing slowly. Total output = 14.1 units drives the heater.

68.5 Understanding Error in PID Control

The foundation of PID control is the error signal:

Process Variable (PV)

The actual measured value of what we’re controlling (e.g., current temperature)

Set Point (SP)

The desired target value (e.g., target temperature)

Error (e)

The difference between desired and actual values

\[ e(t) = SP - PV(t) \]

Example: Home Temperature Control

Figure 68.4: Practical example of PID control applied to home temperature control system

Figure 68.5: PID temperature control sequence from initial 4C error to stable setpoint

PID Temperature Control Sequence: Initially large error (18 to 22C) produces full heating. As temperature rises, P term decreases (smaller error), D term provides braking (rapid approach), and system smoothly reaches setpoint without overshoot.

Scenario Analysis:

Time	SP	PV	Error	Action
0 min	22C	18C	+4C	Full heating
5 min	22C	20C	+2C	Reduce heating
10 min	22C	21.5C	+0.5C	Minimal heating
15 min	22C	22C	0C	Heating OFF
20 min	22C	21.8C	+0.2C	Small heating pulse

Key Observations:

• As error decreases, control action decreases
• Zero error means no action needed (maintain state)
• Small variations around set point are normal

68.6 Proportional Control (P)

Figure 68.6: Proportional control system diagram showing output proportional to current error magnitude

The Proportional term provides control action directly proportional to the current error. Larger errors produce stronger corrections.

\[ u_p(t) = K_p \cdot e(t) \]

Figure 68.7: Proportional control mechanism with error-proportional output correction

Characteristics:

• Fast response: Immediate reaction to errors
• Simple implementation: Single multiplication
• Proportional output: Big errors produce big corrections; small errors produce small corrections

Analogy: Driving and Lane Changes

Imagine changing lanes on a highway (the “process”) by steering (the “control action”):

Set Point (SP): Center of target lane Process Variable (PV): Current car position Error: Distance from center of target lane

Figure 68.8: Lane change steering analogy comparing low, moderate, and high Kp values

Kp Tuning Impact on Lane Change: Low Kp produces slow, gentle response that may never reach target. Moderate Kp balances speed and stability. High Kp causes fast but unstable response with overshoot and oscillation.

Problems with P-Only Control:

1. Steady-State Error: System may never reach exact set point
2. Overshoot: High Kp causes system to go past target
3. Oscillation: High Kp can cause continuous cycling around set point

P-Only Control Limitation: Steady-State Error

Figure 68.9: Graph comparing proportional control responses with different Kp values

With proportional control alone, there’s often a permanent offset from the set point called steady-state error.

Why?

• As error decreases, control action decreases
• Eventually, control action becomes too weak to eliminate remaining error
• System settles with small persistent error

Example:

• Set point: 22C
• Steady state: 21.7C
• Error: 0.3C persists indefinitely

Solution: Add Integral control to eliminate steady-state error

68.7 Integral Control (I)

Figure 68.10: Diagram showing addition of integral control action to eliminate steady-state error

The Integral term accumulates error over time and provides control action based on the total accumulated error. This eliminates steady-state error that P-only control cannot handle.

\[ u_i(t) = K_i \cdot \int_{0}^{t} e(\tau) \, d\tau \]

Figure 68.11: Integral control mechanism accumulating error over time to eliminate steady-state offset

How Integral Control Works:

• Accumulates error over time
• If error persists (even small error), integral grows
• Integral keeps increasing until error becomes zero
• Also called “automatic reset” - resets bias until error eliminated

Mathematical Intuition:

The integral accumulates (sums) all past errors:

Time | Error | Proportional | Integral Accumulated | Total Output
-----|-------|-------------|---------------------|-------------
0s   | +3C   | Kp×3 = 1.5  | 0                  | 1.5
1s   | +2C   | Kp×2 = 1.0  | Ki×(3+2) = 0.5     | 1.5
2s   | +1C   | Kp×1 = 0.5  | Ki×(3+2+1) = 0.6   | 1.1
3s   | +1C   | Kp×1 = 0.5  | Ki×(3+2+1+1) = 0.7 | 1.2
4s   | 0C    | Kp×0 = 0    | Ki×(...) = 0.7     | 0.7
5s   | 0C    | Kp×0 = 0    | Ki×(...) = 0.7     | 0.7

Kp = 0.5, Ki = 0.1

Key Observation: Even when P term drops to zero (error = 0), I term maintains output, compensating for disturbances.

Benefits of Adding Integral:

1. Eliminates steady-state error: System reaches exact set point
2. Compensates for disturbances: Overcomes constant external forces
3. Automatic adjustment: No manual reset needed

Integral Windup Problem

If a large error persists for a long time (e.g., system startup, actuator saturation), the integral can accumulate to very large values. This is called integral windup.

Consequences:

• When error finally reverses, huge accumulated integral causes massive overshoot
• System may oscillate wildly or become unstable

Solutions:

• Limit integral accumulation (clamping)
• Reset integral when actuator saturates
• Anti-windup algorithms in modern controllers

68.8 Derivative Control (D)

Figure 68.12: Diagram showing addition of derivative control action to reduce overshoot

The Derivative term provides control action based on the rate of change of error. It anticipates future error trends and provides damping to prevent overshoot.

\[ u_d(t) = K_d \cdot \frac{de(t)}{dt} \]

Figure 68.13: Derivative control mechanism using rate of change to prevent overshoot

How Derivative Control Works:

• Measures how fast error is changing: de/dt
• If error decreasing rapidly, derivative is negative
• Negative derivative reduces control action (anticipates reaching target)
• Provides damping effect to prevent overshoot
• Acts as predictive brake

Analogy: High-Speed Lane Change (Need for Derivative)

Now consider changing lanes at high speed where momentum causes overshoot:

Figure 68.14: PI versus PID control comparison showing derivative term preventing overshoot

Derivative Term Prevents Overshoot: PI control without D term overshoots at high speed due to momentum. PID control’s D term detects rapid approach and reduces control action preemptively.

Benefits of Adding Derivative:

1. Reduces overshoot: Dampens aggressive P and I responses
2. Improves stability: Prevents oscillations
3. Faster settling: Reaches steady state more quickly
4. Better response to changing conditions: Anticipates trends

Cautions with Derivative:

1. Noise sensitivity: Amplifies high-frequency noise in sensor readings
2. Difficult to tune: Kd selection is challenging
3. Rarely used alone: Almost always combined with P and/or I

68.9 PID Configurations Comparison

Figure 68.15: P-only, PI, and full PID control configuration performance comparison

PID Configuration Selection Decision Tree

Figure 68.16: PID Configuration Selection Decision Tree

PID Configuration Performance Comparison: P-only provides fast response but leaves steady-state error. PI eliminates error but may overshoot. Full PID combines fast response, zero error, and minimal overshoot.

Mode	Usage	Application Examples
P	Sometimes	Simple temperature control, LED brightness
PI	Most common	HVAC systems, level control, pressure regulation
PID	Sometimes	Motor speed control, precision positioning
PD	Rare	Servo motors, fast response systems

PID Term Summary

Term	Reacts To	Purpose	Issue to Watch
P	Current error magnitude	Provide proportional response	Steady-state error, overshoot
I	Accumulated error over time	Eliminate steady-state error	Integral windup, slow response
D	Rate of error change	Dampen overshoot, predict future	Noise amplification, hard to tune

Simplified Summary:

• P corrects present error
• I corrects past accumulated error
• D corrects future predicted error

Alternative View: PID Terms as Time Perspectives

Figure 68.17: Thinking of PID in temporal terms helps with tuning intuition. If your system overshoots badly (future not anticipated), increase D. If it never quite reaches setpoint (past errors not addressed), increase I.

PID Terms as Time Perspectives - Past, Present, and Future

Figure 68.18: PID Terms as Time Perspectives - Past, Present, and Future

68.10 Knowledge Check

Quiz 1: Proportional Control

Quiz 2: Integral Windup

Quiz 3: Derivative Term Purpose

Quiz 4: Choosing PID Configuration

Quiz 5: PID Time Perspective

68.11 Common Mistakes and Pitfalls

Pitfall Cards: PID Implementation Mistakes

Mistake 1: Starting with Full PID

• What happens: Beginners often implement full PID immediately, making tuning extremely difficult
• Better approach: Start with P-only, add I if steady-state error persists, add D only if overshoot is problematic
• Why it matters: Each term interacts with others - tuning three parameters simultaneously is much harder than one at a time

Mistake 2: Ignoring Sensor Noise

• What happens: The D term amplifies high-frequency noise, causing erratic actuator behavior
• Better approach: Filter sensor readings before PID calculation, or reduce/eliminate Kd
• Real example: Temperature sensor with 0.5C noise + high Kd = heater rapidly cycling on/off

Mistake 3: No Anti-Windup Protection

• What happens: During large setpoint changes or actuator saturation, integral accumulates to extreme values
• Better approach: Implement integral clamping, stop integrating when output saturates
• Consequence: Massive overshoot when system finally responds

Mistake 4: Using Same Sample Rate for All Terms

• What happens: Fast sample rates make D term unstable; slow rates make P term sluggish
• Better approach: Consider separate update rates or digital filtering for D term
• Typical values: 10-100 Hz for temperature; 1-10 kHz for motor control

PID Common Mistakes Decision Tree - Diagnosing Control Problems

Figure 68.19: PID Common Mistakes Decision Tree - Diagnosing Control Problems

68.12 Practical IoT Example: Smart Greenhouse Temperature Control

Real-World Implementation: ESP32 Greenhouse Controller

This example demonstrates PID control for maintaining optimal plant growth temperature.

System Components:

Component	Role	Specifications
ESP32	Controller	240 MHz, WiFi, 10-bit ADC
DHT22	Temperature sensor	+/-0.5C accuracy, 0.5 Hz sample rate
SSR Relay	Actuator interface	25A capacity for heater
Heating element	Process	500W ceramic heater

PID Implementation (Pseudocode):

# PID Constants (tuned for this system)
Kp = 2.0    # Proportional gain
Ki = 0.1    # Integral gain
Kd = 0.5    # Derivative gain

# State variables
integral = 0
last_error = 0
setpoint = 25.0  # Target: 25C for tomatoes

def calculate_pid(current_temp):
    global integral, last_error

    # Calculate error
    error = setpoint - current_temp

    # Proportional term
    P = Kp * error

    # Integral term with anti-windup
    integral += error * dt
    integral = clamp(integral, -100, 100)  # Prevent windup
    I = Ki * integral

    # Derivative term
    derivative = (error - last_error) / dt
    D = Kd * derivative
    last_error = error

    # Combined output (0-100% duty cycle)
    output = P + I + D
    return clamp(output, 0, 100)

Tuning Process Used:

1. Started with Kp=1, Ki=0, Kd=0
2. Increased Kp until slight oscillation (Kp=3), then backed off to Kp=2
3. Added Ki=0.1 to eliminate 0.5C steady-state error
4. Added Kd=0.5 to reduce morning sun disturbance overshoot

Results:

• Steady-state accuracy: +/-0.3C (improved from +/-1.5C with on/off control)
• Settling time: 8 minutes (vs 20+ minutes with on/off oscillation)
• Energy savings: 23% reduction from elimination of overshoot cycles

Quiz 6: Practical PID Application

68.13 Worked Example: PID Tuning for Smart Greenhouse Temperature Control

Worked Example: Tuning P, PI, and PID for a 50m2 Greenhouse With Solar Disturbance

Scenario: A smart greenhouse grows tomatoes requiring 25 degrees C (+/-1 degree C). The greenhouse has a 2 kW heater and a motorized vent. Morning sun causes a +3 degree C/hour disturbance starting at 9 AM. Sensor reads temperature every 5 seconds. The control output ranges from -100% (full vent) to +100% (full heat). Compare P-only, PI, and PID controllers.

Step 1: P-Only Controller (Kp = 4)

Time	Setpoint	Actual	Error	P Output	What Happens
8:00 AM	25.0	22.0	+3.0	12.0% heat	Heater warms greenhouse
8:15	25.0	23.8	+1.2	4.8% heat	Warming slows as error shrinks
8:30	25.0	24.5	+0.5	2.0% heat	Steady state: 0.5 degree C below setpoint
9:00	25.0	24.5	+0.5	2.0% heat	Sun starts – 3 degree C/hr disturbance
9:30	25.0	25.8	-0.8	-3.2% vent	P reacts, but only partially compensates
10:00	25.0	26.2	-1.2	-4.8% vent	Outside +/-1 degree C tolerance

P-only result: Permanent 0.5 degree C offset (steady-state error). Cannot fully reject solar disturbance. Fails the +/-1 degree C requirement.

Step 2: PI Controller (Kp = 4, Ki = 0.1)

Time	Error	P Term	I Term (accumulated)	Total Output	What Happens
8:00	+3.0	12.0	0.3	12.3% heat	Similar to P-only initially
8:15	+1.2	4.8	3.6 (accumulated)	8.4% heat	I term keeps pushing even as error shrinks
8:30	+0.1	0.4	5.1	5.5% heat	Error nearly zero (I eliminated steady-state error)
9:00	0.0	0.0	5.2	5.2% heat	Stable at setpoint when sun starts
9:15	-0.8	-3.2	4.4	1.2% heat	I term slow to discharge – overshoot!
9:30	-1.5	-6.0	2.9	-3.1% vent	Overshoots to 26.5 degree C before I catches up
10:00	-0.3	-1.2	1.8	0.6% heat	Eventually settles, but took 45 min

PI result: Eliminates steady-state error (good). But during solar disturbance, I term causes overshoot because it accumulated positive values during heating phase and takes time to discharge. Marginally meets +/-1 degree C but with slow disturbance rejection.

Step 3: Full PID Controller (Kp = 4, Ki = 0.1, Kd = 2.0)

Time	Error	P	I	D (rate of change)	Total	What Happens
8:30	+0.1	0.4	5.1	0.0	5.5% heat	Same as PI (steady state, no change)
9:00	0.0	0.0	5.2	0.0	5.2% heat	Stable
9:05	-0.2	-0.8	5.0	-2.4 (temp rising fast!)	1.8% heat → opens vent	D detects rate of change BEFORE error grows
9:15	-0.5	-2.0	4.5	-1.2	1.3% vent	D already opened vent 10 min earlier than PI
9:30	-0.6	-2.4	3.9	-0.2	-1.3% vent	Max overshoot: only 0.6 degree C (vs 1.5 for PI)
10:00	-0.1	-0.4	3.5	0.0	3.1% heat	Settled within 30 min (vs 45 for PI)

PID result: D term detects the temperature rising at 9:05 AM (before error exceeds tolerance) and pre-emptively opens the vent. Overshoot reduced from 1.5 degrees C to 0.6 degrees C. Settling time improved from 45 min to 30 min.

Comparison Summary:

Metric	P-Only	PI	PID
Steady-state error	0.5 degree C	0 degree C	0 degree C
Solar disturbance overshoot	1.2 degree C	1.5 degree C	0.6 degree C
Settling time after disturbance	Never (oscillates)	45 min	30 min
Meets +/-1 degree C tolerance?	No	Marginal	Yes
Energy savings vs on/off control	15%	20%	23%

68.14 Interactive: PID Term Calculator

Calculate the P, I, and D term contributions for a given system state.

viewof pid_Kp = Inputs.range([0, 20], {value: 5, step: 0.5, label: "Kp (Proportional gain)"})

viewof pid_Ki = Inputs.range([0, 2], {value: 0.2, step: 0.05, label: "Ki (Integral gain)"})

viewof pid_Kd = Inputs.range([0, 5], {value: 1.0, step: 0.1, label: "Kd (Derivative gain)"})

viewof pid_error = Inputs.range([-10, 10], {value: 2, step: 0.1, label: "Current error e(t) (°C)"})

viewof pid_acc_error = Inputs.range([-100, 100], {value: 20, step: 1, label: "Accumulated error ∫e dt (°C·s)"})

viewof pid_dedt = Inputs.range([-2, 2], {value: 0.1, step: 0.05, label: "Rate of error change de/dt (°C/s)"})

{
  const uP = pid_Kp * pid_error;
  const uI = pid_Ki * pid_acc_error;
  const uD = pid_Kd * pid_dedt;
  const uTotal = uP + uI + uD;
  const pPct = Math.abs(uTotal) > 0.001 ? (uP / uTotal * 100).toFixed(1) : 0;
  const color = Math.abs(uTotal) < 0.5 ? '#7F8C8D' : uTotal > 0 ? '#16A085' : '#E74C3C';
  const action = uTotal > 0.5 ? 'heating / increasing output' : uTotal < -0.5 ? 'cooling / decreasing output' : 'near setpoint - minimal action';

  return html`<div style="background: var(--bs-light, #f8f9fa); padding: 1rem; border-radius: 8px; border-left: 4px solid ${color}; margin-top: 0.5rem;">
  <p><strong>P term (Kp × e):</strong> ${uP.toFixed(2)}</p>
  <p><strong>I term (Ki × ∫e dt):</strong> ${uI.toFixed(2)}</p>
  <p><strong>D term (Kd × de/dt):</strong> ${uD.toFixed(2)}</p>
  <hr style="margin: 0.5rem 0;">
  <p><strong>Total output u(t):</strong> <span style="color: ${color}; font-size: 1.2em; font-weight: bold;">${uTotal.toFixed(2)}</span></p>
  <p><em>Action: ${action}</em></p>
  </div>`;
}

Match the PID Term to Its Role

Order the PID Tuning Steps

Key Concepts

• PID Controller: A feedback control algorithm combining proportional (P), integral (I), and derivative (D) terms to drive a process variable (measured output) toward a setpoint by computing a corrective control output
• Transfer Function: A mathematical representation of the input-output relationship of a linear time-invariant system in the Laplace domain, used to analyze stability, frequency response, and design PID controller gains
• Stability: The property of a control system that returns to equilibrium after a disturbance — a stable PID system’s error converges to zero or a bounded steady-state value rather than growing unbounded
• Frequency Response: The analysis of a control system’s behavior across a range of input frequencies, characterizing bandwidth, gain margin, and phase margin that determine stability and performance limits
• Root Locus: A graphical method showing how closed-loop pole locations change as PID gain varies, used to select gain values that achieve desired transient response (overshoot, settling time) while maintaining stability
• Ziegler-Nichols Method: A heuristic PID tuning procedure that determines initial gain values from the system’s ultimate gain (Ku) and ultimate period (Tu) at the stability boundary, providing a starting point for fine-tuning
• Bode Plot: A frequency domain representation showing gain (dB) and phase (degrees) vs. frequency, used to assess gain margin and phase margin of a PID control loop

Common Pitfalls

1. Applying PID to Nonlinear Systems Without Linearization

PID controllers are designed for linear time-invariant systems. Directly applying PID to highly nonlinear systems (pH control, biological processes, aerodynamics) causes performance that varies with operating point. Linearize around the operating point or use gain scheduling for nonlinear plants.

2. Confusing Control Bandwidth with Sampling Rate

Setting the control loop sampling rate equal to the desired control bandwidth. The Nyquist criterion requires sampling at least 2× the control bandwidth — in practice, 5–20× is needed to avoid aliasing and phase lag from sampling. Undersample a control loop and instability results.

3. Not Accounting for Actuator Saturation in Theory

Designing PID gains based on linear theory without considering that real actuators saturate (valves fully open/closed, motors at max speed). During saturation the integral term continues to accumulate (integrator windup), causing overshoot when the actuator exits saturation. Include anti-windup analysis in theoretical design.

4. Using Ideal PID Equations Without Derivative Filtering

Implementing the pure derivative term (D) without a low-pass filter on the derivative path. Pure differentiation amplifies high-frequency noise — a 1 mV sensor noise spike becomes a large control output spike. Always include a first-order filter (time constant = D/5 to D/10) on the derivative term.

Label the Diagram

💻 Code Challenge

68.15 Summary

This chapter covered the theory of PID control:

• PID Equation: Combined P, I, and D terms with their respective gains
• Proportional (P): Responds to current error magnitude - fast but leaves steady-state error
• Integral (I): Accumulates past errors - eliminates steady-state error but can cause windup
• Derivative (D): Predicts future from rate of change - reduces overshoot but sensitive to noise
• Configuration Selection: P for simple systems, PI for most applications, full PID for precision control

68.16 What’s Next

The next chapter covers PID Tuning and Applications, including systematic tuning methods, real-world examples, and practical implementation considerations for IoT systems.

Previous	Current	Next
Open & Closed Loop Systems	PID Control Theory	PID Tuning and Applications

Related Chapters

• Feedback Fundamentals - Foundation concepts
• Open-Loop and Closed-Loop Systems - Control architectures
• PID Tuning and Applications - Practical implementation
• Sensor Fundamentals - Feedback sensors
• Actuators Introduction - Control outputs