Two-Degree-of-Freedom Heater PI: Separate Setpoint Tracking from Disturbance Rejection

System boundary

A linearized heater PI loop with proportional setpoint weighting, valid measurement and unsaturated actuator. Derivative behavior, feedforward heat calculation, gain scheduling and protection are separate design functions.

Integration interfaces

Integration risks

Ask whether the unwanted response follows the target or the load

A setpoint step asks the assembly to move to a new temperature. A load disturbance changes the heat balance while the desired temperature remains fixed. Examples include a changed process contact or inlet condition within the qualified operating envelope. A controller can respond differently to those two events, so record which event produces the unacceptable behavior before changing gains.

The comparison should use the same assembled heater, sensor, output limits and operating point. If one test is performed unloaded and the other with a flowing process, their difference is not evidence about setpoint weighting alone. Establish a local operating condition and use changes small enough that the linear comparison is meaningful and the actuator retains control authority.

Weight the proportional target, not the entire error

Write the proportional contribution as Kp times the quantity br minus y, where b is the setpoint weight. The integral contribution still accumulates r minus y. In contrast, multiplying the complete error by b changes both the target and measurement paths and is simply a different feedback gain allocation. It does not preserve the original disturbance response.

Use deviation variables around a documented operating point for the analysis below. This avoids hiding the steady output bias inside a weighted absolute-temperature expression. A practical implementation using absolute temperatures must initialize its bias and integral state consistently. Switching controller structures while retaining an incompatible state can create a command jump unrelated to the benefit being evaluated.

The two inputs have different controller numerators

For PI control without derivative action, the command path is Cr equal to bKp plus Ki divided by s, while the magnitude of the negative measurement-feedback path is Cy equal to Kp plus Ki divided by s. The actuator command is Cr times r minus Cy times y. Only Cr contains b. This separation makes the implementation review much clearer than calling every adjustment a softer PID setting.

Let the plant transfer be G, and introduce disturbance d additively at the plant input in the same units as the actuator command. Solving the closed loop gives the relationships below. A disturbance entering somewhere else, such as sensor measurement noise, has a different numerator, so always name the injection point before comparing response curves.

Changing the weight does not change the linear characteristic equation

For a first-order plant G equal to K divided by one plus tau s, the common closed-loop denominator after clearing the controller integrator is tau s squared plus one plus K Kp times s plus K Ki. The setpoint numerator is K times bKp s plus Ki. The plant-input disturbance numerator is K s. The denominator and disturbance numerator contain no b.

This does not mean every measured behavior is unchanged. The setpoint response has a different zero and initial command action, and different commands can reach actuator limits at different times. The statement is narrower: with fixed gains and the stated linear assumptions, changing proportional setpoint weight does not alter the internal closed-loop characteristic equation or the plant-input disturbance transfer.

Compare an immediate target step with the same measurement movement

Take a hypothetical Kp of two output percentage points per kelvin. At the instant of a five-kelvin setpoint step, before the measured temperature or integral output changes, b equal to one gives a ten-point proportional command increase. A weight of 0.3 gives a three-point increase. A weight of zero gives no immediate proportional increase from the target alone.

Now hold the setpoint constant and consider a one-kelvin rise in measured temperature. The proportional command decreases by two percentage points in all three cases. The measurement term has not been weighted. This is the practical difference from reducing Kp to 0.6, which would also reduce the immediate measurement response to only 0.6 percentage point per kelvin.

Zero setpoint weight is not an integral-only feedback loop

With b equal to zero, the proportional contribution is minus Kp y in deviation coordinates. It still reacts to measurement changes. The setpoint initially acts through the integral path, which continues to integrate the full r minus y difference. Calling this integral-only control would incorrectly suggest that the proportional measurement feedback has been removed.

For another illustrative check, use plant gain K equal to one kelvin per percentage point, tau equal to ten seconds, Kp equal to two and Ki equal to 0.2 in consistent units. The characteristic polynomial is 10s squared plus 3s plus 0.2 for every b. Its roots are minus 0.1 and minus 0.2 per second. These are model calculations, not tuned settings or measured heater performance.

Keep target weighting separate from filters, ramps and derivative choices

A target ramp limits the rate of the requested temperature trajectory. A setpoint prefilter introduces its own dynamics. Derivative on measurement changes a different controller term. Each can be useful, but combining changes in one experiment prevents attribution. Verify the proportional weighting alone first, then evaluate any additional target-management behavior as a separately identified function.

Record requested target, any processed target, measurement, proportional output, integral output and applied actuator command. Compare the immediate command increment with the simple calculation, then evaluate the full response. Confirm that neither a hidden output clamp nor a different anti-windup path is responsible for the apparent improvement. A smooth target trace is not sufficient evidence that the two controller input paths are implemented correctly.

Accept the structure with both command and disturbance tests

Retain the selected b together with Kp, Ki, sample period, controller form and operating-point definition. Test target movement and a separately controlled load disturbance using the same configuration. Specify acceptable overshoot, recovery and actuator behavior for the actual application; there is no universal setpoint weight that is best for every ceramic heater assembly.

The final engineering result is a controller structure whose target response can be adjusted without inadvertently weakening its intended measurement feedback. It does not create extra heating or cooling authority, repair sensor lag or validate an unsafe load. Those physical limits remain with the thermal assembly and its protective system, while the weighting choice has a precise and testable role.