How to tune PID controllers: open / closed loop control systems--part 3

Damped cycling tuning method

This method is a variation of the continuous cycling method. It’s used whenever continuous cycling imposes danger to the process, but a damped oscillation of some extent is acceptable. The steps of closed loop tuning (damped cycling method) are as follows:

Tuning method:

1. Put the controller into P-control only: In order to avoid the controller influencing the assessment of the process dynamics, no I-control or D-control must be active.

2. P-control on ERR = (SP - PV): Make sure that the P-control is working with PV changes as well as with SP changes. This enables us to make changes to the ERR term by changing the SP value.

3. Put controller in automatic mode: We need a closed loop situation to obtain damped cycling.

4. Step change to the setpoint: A step change to the SP causes a disturbance and we observe how the PV settles. Before making a step change to the SP, the process must be steady with only minor dynamic fluctuations visible.

5. Actions based on the observation: If any oscillations that occur settle down quickly (or indeed there are no oscillation at all), then increase the value of KC. The amount of increase to KC depends on the rate and magnitude of change of the PV as a result of the last SP change. Then repeat 4 above, returning the setpoint back to its original value. When oscillations appear, and if they seem to be increasing in amplitude, terminate the exercise immediately and reduce the value of KC to enable the process to stabilize. The total loop gain was >1, hence it amplified the SP change value. Repeat the exercise again being more cautious with high values of KC. When a damped oscillation is obtained note the value of KC, this now being denoted as KD. Then terminate the test by reducing the value of KC. KD is used to determine the gain later in this exercise.

Rise time = t1 Time to first peak= t2 Settling time = t3 Overshoot = Decay ratio=

+=+=+=+ Damped oscillation decay ratio

Calculations: By measuring and dividing the amplitude of the first overshoot by the amplitude of the second overshoot the delay ratio P is found. The time (in minutes) between these two measured points gives a value for Pd (period of damping).

P = Decay ratio = 1st overshoot 2nd overshoot

Then calculate the damping ratio d from:

…and then:

In most cases the damping factor, d , having a value of around 0.5 for a damped oscillation is acceptable. We then need to evaluate Rd from U / PP where PU represents the ultimate period and P represents the actual period:

This leads to the following formula for TINT and TDER: PI-control:

PID-control:

Next, we have to turn our attention to calculating the gain setting for the controller (KC). Some manuals inform us that KC is determined by good operator judgment; however this would be very much a hit- and -miss approach. As we have a value of controller gain (KD), used to obtain the damped cycle response used to evaluate the integral and derivative time constants. We can use this to obtain a value for KU.

First we need to calculate the overshoot ratio; this is the result of Overshoot Steady-state change

We then calculate KU from:

Overshoot ratio; Achieving a value for KU will let us use the Ziegler-Nichols closed loop formulas.

These being P-control:

PI-control: CU =0.45 KK × and PID-control:

+=+=+=+ gives a graphical representation to obtain the damping ratio directly from the % overshoot that occurred in the PV as a result of a step change made to the controller output.

% Overshoot vs damping ratio d System with dominant 2nd order character (damped response to step input)

% Overshoot vs damping ratio system with dominant 2nd order character (damped response to step input)

Step responses:

+=+=+=+ is used to determine the ultimate period (PU) from the damped cycle period (P). Damped step response of system with dominant 2nd order character; To determine ultimate period (PU) from damped cycling period (P) for use of Ziegler-Nichols continuous cycling PID tuning method with damped cycling data: 1% 10% 100%; % Overshoot; Ratio of ultimate period/ actual period and damping ratio d Rd =(PU/P) Rd = PU/P = -ln(OS) (thin line)

ln2(OS) 1 p 2+ln2(OS) p 2+ln2(OS) d PU/P d

+=+=+=+ Damped step response of system with a dominant 2nd order characteristic 8.10 Tuning for no overshoot on start-up (Pessen)

This method is a variation of the continuous cycling method and it’s used whenever no overshoot is permitted, even in the extreme case of start-up of the process. With start-up, we mean the transition from manual to automatic control. An extreme start-up situation exists, if the setpoint and PV are very different when changing from manual to automatic control. In contrast to a change of setpoint, the change from manual to automatic control does not cause a step change in ERR. Therefore, the change does not directly affect P or D-control. Examples for applying this tuning procedure, according to Pessen, is a closed tank that could burst or an open tank that could overflow.

The steps of closed loop tuning for no overshoot are the same as the ones for continuous cycling method. The formulas developed for this case by Pessen are as follows: PID-control:

Tuning for some overshoot on start-up (Pessen)

This method is a variation of the continuous cycling method. It’s used whenever no overshoot during normal modulating control is desired, but some overshoot at start-up is acceptable.

The steps of closed loop tuning for some overshoot are the same as the ones for continuous cycling method. The formulae developed for this case by Pessen are as follows:

The tuning constants PID-control:

Summary of important closed loop tuning algorithms

Tuning for Continuous Oscillation Pessen Some Overshoot Pessen No Overshoot

Table: Summary of closed loop PID controller tuning parameter settings for different controller responses.

PID equations: dependent and independent gains

The general PID equation as applicable to digital (PLC) systems is the sum of four terms: OP = Proportional + integral + derivative + bias (MANUAL) value This equation can be represented in two ways, ISA (Instrument Society of America) (dependant gains) and independent gains.

In the independent gains equation, as the name suggests, all three PID terms operate independently. In the ISA equation a change in the proportional term also effects the integral and derivative terms.

ISA Independent gains Output Controller gain KP Reset term TI Rate term TD Output Proportional term KP; Integral term KI Derivative term KD PV SP Process variable Setpoint Process Bias SS

+=+=+=+ Closed loop control showing terms and comparison between ISA and independent gains equations.

ISA equation:

The ISA equation is interactive, that is, it contains dependent terms that mean if the controller gain KC is changed, the integral and derivative terms also change.

Where:

CV = Output

= Controller gain constant (unitless)

= Integral time constant (minutes per repeat)

= Derivative time constant (minutes) d = Time between samples (minutes) Bias = Feedforward or output bias = Error = to PV SP or SP PV; PV = Process variable

PV( 1) = PV value from last sample (1) = Error value from last sample.

Independent gains equation:

This equation is non-interactive. As such P, I and D terms are adjusted independently.

Where:

PID CV = Output

= Proportional gain constant (unitless)

= Integral gain constant (1/sec)

= Derivative gain constant (seconds) d = Time between samples (seconds) Bias = Feedforward or output bias.

E Error = to PV SP or SP PV; PV = Process variable PV( 1) = PV value from last sample ( 1) = Error value from last sample.

The ISA and independent gains constants can be compared as follows: ISA Constants Independent Gains Constants

Controller gain KC (dimensionless) Proportional gain KP (dimensionless)

Reset term TINT (minutes per repeat)

Integral gain KI (inverse seconds)

Rate term TDER (minutes)

Derivative term KD (seconds)

To convert from ISA terms to independent gain terms:

1: ISA Dependant Gains 0: AB Independent Gains Setpoint (Scaled) Setpoint Proportional gain (KC) (0.01) Proportional gain (KP) (0.01)

Reset time (T1) (0.01 min/repeat) Integral gain (KI) (0.001/s)

Derivative rate (T2) (0.01 min) Derivative gain (KD) (0.01 s)

Loop update time (0.01 s) Loop update time (0.01 s)

Derivative Error 0:PV 1: Error: Example

Proportional band applications PB%:

NEXT: Controller output modes, operating equations, cascade control
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