Continuous P Controller

Proportional-only with integrating output (velocity form)

Overview

The Continuous P Controller implements proportional-only control with an integrating (accumulating) output. Also known as "velocity form" or "incremental" control, the output changes proportionally to error rather than being directly proportional. This is ideal for integrating processes like valve positioning.

When to Use

• Integrating processes: Level, position, inventory
• Self-regulating outputs: Valve positions
• Noise immunity: When derivative causes issues
• Simple tuning: Single gain parameter

Common Applications

Application	Process Type	Typical Kp
Valve positioning	Integrating	0.5-2.0
Level control	Integrating	0.1-1.0
Tank inventory	Integrating	0.2-0.8
Dosing adjustment	Self-regulating	0.5-1.5

Control Theory

Concept

Unlike standard P control where output is directly proportional to error, Continuous P changes output proportionally to error:

Standard P: $$output = K_p \cdot error$$

Continuous P (Velocity Form): $$\Delta output = K_p \cdot error \cdot \Delta t$$ $$output_k = output_{k-1} + \Delta output$$

Block Diagram

graph LR
    SP[Setpoint] --> Sum((+/-))
    PV[Process Variable] --> Sum
    Sum -->|error| P[Kp × dt]
    P -->|Δoutput| Int[Integrator]
    Int --> Output[Output]
    Output --> Rate[Rate Limit]
    Rate --> MV[Manipulated Variable]

Mathematical Model

$$output_k = output_{k-1} + K_p \cdot e_k \cdot \Delta t$$

Subject to: $$rate_{min} \leq \frac{\Delta output}{\Delta t} \leq rate_{max}$$ $$output_{min} \leq output \leq output_{max}$$

Parameters

Parameter	Unit	Range	Default	Description
Setpoint	varies	-	-	Target value
Kp	-	0.1-10	1.0	Proportional gain
Rate Max	units/day	1-100	10	Maximum rate of change
Rate Min	units/day	-100 to -1	-10	Minimum rate of change
Output Min	%	0-100	0	Minimum output
Output Max	%	0-100	100	Maximum output
Bias	%	0-100	50	Initial/center output
Action	-	direct/reverse	direct	Control direction

Tuning Guidelines

Gain Selection

1. Start low: Begin with Kp = 0.5
2. Increase gradually: Until response is adequate
3. Watch for overshoot: Reduce if overshooting

Rate Limit Selection

1. Equipment capability: Actuator speed limit
2. Process dynamics: How fast process can respond
3. Stability: Prevent output racing

Recommended Settings

Application	Kp	Rate Limit	Bias
Valve position	1.0	10%/hr	50%
Level control	0.5	20%/hr	50%
Flow trim	1.5	30%/hr	0%

Configuration

Example: Valve Position Control

{
  "id": "valve-position-1",
  "type": "ContinuousP",
  "enabled": true,
  "targetNodeId": "control-valve-1",
  "pvNodeId": "flow-meter-1",
  "pvVariable": "flow",
  "manipulatedVariable": "position",
  "setpoint": 1000,
  "kp": 1.0,
  "rateLimit": {
    "maxRate": 10,
    "minRate": -10
  },
  "outputMin": 0,
  "outputMax": 100,
  "action": "direct",
  "bias": 50,
  "mode": "auto"
}

Integration Example

Flow Control via Valve Position

Scenario: Control flow by adjusting valve position

Error = Setpoint - Flow
        1000 - 900 = 100 m³/d (positive error)

Δposition = Kp × error × dt
          = 1.0 × 100 × (1/24) day
          = 4.2%

If previous position = 50%
New position = 50 + 4.2 = 54.2%

Rate check: 4.2% / (1/24 day) = 100%/day
If rate limit = 10%/day → limited to:
New position = 50 + 10 × (1/24) = 50.4%

Response Characteristics

Setpoint  1000 ──────────────────────────
                        ┌────────────────  Actual Flow
Flow       900 ─────────┘

Position   60% ─────────────────────────
                   ╭────────────────────  Valve Position
           50% ────╯                       (ramps up)

Time        0      1h      2h      3h

Comparison: Continuous P vs Standard PID

Aspect	Continuous P	Standard PID
Integrating action	Built-in	From Ki term
Complexity	Simple	More complex
Tuning	One parameter	Three parameters
Noise sensitivity	Low	Depends on Kd
Best for	Integrating processes	Self-regulating processes

Why Use Continuous P?

1. Natural for valves: Valve position is inherently integrating
2. Simple tuning: Only Kp to adjust
3. Smooth response: Rate limits prevent jumps
4. Bumpless restart: Starts from last position

Best Practices

1. Set appropriate bias

   • Center of operating range
   • Starting point after reset

2. Rate limit for stability

   • Match actuator capabilities
   • Prevent process upsets

3. Action direction

   • Direct: More output = more PV
   • Reverse: More output = less PV

4. Monitor for drift

   • Check output over time
   • Ensure reaching setpoint

Troubleshooting

Issue	Cause	Solution
Output drifts continuously	Kp too high	Reduce Kp
Slow response	Kp too low	Increase Kp
Output races to limit	Rate limit too high	Reduce rate limit
Stuck at bias	Kp = 0 or disabled	Check configuration
Wrong direction	Action incorrect	Toggle action

Advanced Features

Rate Limiting

Prevents rapid output changes:

const maxChange = maxRate * dt;
const minChange = minRate * dt;
outputChange = clamp(outputChange, minChange, maxChange);

Accumulated Output

Output accumulates over time:

accumulatedOutput += kp * error * dt;
output = clamp(accumulatedOutput + bias, outputMin, outputMax);

Manual Reset

Set output to specific value:

controller.setOutput(60); // Set valve to 60%

Mathematical Derivation

From PID to Velocity Form

Standard PID: $$u = K_p e + K_i \int e , dt + K_d \frac{de}{dt}$$

Velocity form (derivative of output): $$\frac{du}{dt} = K_p \frac{de}{dt} + K_i e + K_d \frac{d^2e}{dt^2}$$

For Continuous P (Ki = Kd = 0): $$\frac{du}{dt} = K_p \frac{de}{dt}$$

For constant setpoint: $$\frac{du}{dt} = -K_p \frac{dPV}{dt}$$

But we want response to error magnitude: $$\frac{du}{dt} = K_p \cdot e$$

This gives integrating behavior on position.

Related Controllers

• PID-Controller - Standard position-form PID
• Feedforward-Controller - Combine with FF for disturbances
• Ratio-Controller - For ratio-based control

References

1. Smith, C.L. "Digital Computer Process Control" (1972)
2. Shinskey, F.G. "Process Control Systems" (4th Edition)
3. Seborg, D.E. "Process Dynamics and Control" (4th Edition)