Cascade Controller

Master-slave PID cascade for improved disturbance rejection

Overview

The Cascade Controller implements a master-slave PID structure where the outer (master) controller's output becomes the setpoint for the inner (slave) controller. This arrangement provides superior disturbance rejection for nested control problems.

When to Use

• Nested process dynamics: When one variable affects another sequentially
• Fast inner loop: When inner variable responds faster than outer
• Better disturbance rejection: When upsets affect the inner loop first

Common Applications

Master Variable	Slave Variable	Application
MLSS	DO	Aeration control
Effluent TN	Methanol dose	Denitrification
SRT	WAS flow	Solids management
Temperature	Cooling water	Digester control

Control Theory

Concept

graph LR
    SP_M[Master SP] --> M_Sum((+/-))
    PV_M[Master PV] --> M_Sum
    M_Sum --> Master[Master PID]
    Master --> SP_S[Slave SP]
    SP_S --> S_Sum((+/-))
    PV_S[Slave PV] --> S_Sum
    S_Sum --> Slave[Slave PID]
    Slave --> MV[Output]
    MV --> Inner[Inner Process]
    Inner --> Outer[Outer Process]
    Inner --> PV_S
    Outer --> PV_M

Mathematical Model

Master Controller: $$u_m(t) = K_{pm} \cdot e_m + K_{im} \int e_m , dt + K_{dm} \frac{de_m}{dt}$$

Slave Controller: $$u_s(t) = K_{ps} \cdot e_s + K_{is} \int e_s , dt + K_{ds} \frac{de_s}{dt}$$

Where:

• $e_m = SP_{master} - PV_{master}$ (master error)
• $e_s = u_m - PV_{slave}$ (slave error = master output - slave PV)
• $u_s$ = Final output to manipulated variable

Parameters

Master Loop

Parameter	Symbol	Range	Default	Description
Setpoint	SP_m	Process dependent	-	Master setpoint
Proportional gain	K_pm	0.1 - 10	1.0	Master Kp
Integral gain	K_im	0.01 - 1	0.1	Master Ki
Derivative gain	K_dm	0 - 1	0	Master Kd
Output min	-	Process dependent	0	Min slave setpoint
Output max	-	Process dependent	100	Max slave setpoint

Slave Loop

Parameter	Symbol	Range	Default	Description
Proportional gain	K_ps	0.5 - 20	2.0	Slave Kp
Integral gain	K_is	0.1 - 5	0.5	Slave Ki
Derivative gain	K_ds	0 - 1	0.1	Slave Kd
Output min	-	Process dependent	0	Min MV output
Output max	-	Process dependent	100	Max MV output

Tuning Guidelines

Cascade Tuning Procedure

1. Place master in manual - Hold slave setpoint constant
2. Tune slave first - As standard PID loop
3. Verify slave response - Should be 3-5× faster than master
4. Close slave loop - Slave now tracks setpoint
5. Tune master - As if slave were the process

Rule of Thumb

The master loop should be 3-5× slower than the slave loop:

$$\tau_{master} \approx (3 \text{ to } 5) \times \tau_{slave}$$

Recommended Settings

Application	Master Kp	Master Ki	Slave Kp	Slave Ki
DO cascade	0.5	0.05	50	10
Temperature cascade	1.0	0.1	2.0	0.5
Pressure cascade	0.8	0.08	3.0	1.0

Configuration

Example: DO Cascade Control

{
  "id": "do-cascade-1",
  "type": "Cascade",
  "enabled": true,
  "master": {
    "setpoint": 3000,
    "kp": 0.5,
    "ki": 0.05,
    "kd": 0,
    "outputMin": 0.5,
    "outputMax": 4.0,
    "pvNodeId": "aerobic-reactor-1",
    "pvVariable": "MLSS"
  },
  "slave": {
    "kp": 50,
    "ki": 10,
    "kd": 5,
    "outputMin": 0,
    "outputMax": 1000,
    "pvNodeId": "aerobic-reactor-1",
    "pvVariable": "DO",
    "manipulatedVariable": "airflow",
    "targetNodeId": "aerobic-reactor-1"
  },
  "mode": "auto",
  "bumplessTransfer": true
}

Integration Example

MLSS-DO Cascade for Aeration

Scenario: Maintain MLSS at 3000 mg/L by adjusting DO setpoint, which in turn adjusts airflow.

1. Master loop monitors MLSS
2. If MLSS high, master reduces DO setpoint → less aeration → less oxygen for growth
3. Slave loop adjusts airflow to meet new DO setpoint
4. Fast DO control rejects disturbances before affecting MLSS

Benefits

• Disturbances to airflow are corrected quickly by slave
• MLSS remains stable despite blower variations
• More robust than single-loop MLSS control

Best Practices

1. Inner loop must be faster - Verify slave response time
2. Start with slave tuning - Get inner loop right first
3. Use bumpless transfer - Prevent bumps on mode changes
4. Monitor both loops - Track master and slave outputs
5. Decouple if needed - Consider if loops interact

Troubleshooting

Issue	Cause	Solution
Master oscillates	Master too fast	Reduce master Kp, Ki
Slave oscillates	Slave too aggressive	Retune slave alone
Slow response	Master too slow	Increase master gains
Bumps on startup	No bumpless transfer	Enable bumplessTransfer
Interaction	Loops similar speed	Slow down master 3-5×

Related Controllers

• PID-Controller - Base PID for slave tuning reference
• Feedforward-Controller - For additional disturbance rejection
• Ratio-Controller - For flow-based cascade alternative

References

1. Marlin, T.E. "Process Control" (2nd Edition), Chapter 14
2. Shinskey, F.G. "Process Control Systems" (4th Edition)