How to tune PID controllers: open / closed loop control systems--part 2

Loop time constant (LTC) method

This method of tuning, as in the previous two examples, makes use of the reaction curve and is applicable when the system has a first order lag response as defined by a linear first order differential equation. This equation is expressed as:

Where c = output r = input K = gain t = time constant.

Inspection of a first order response curve will show that it’s always falling off, i.e. the rate of response is at maximum in the very beginning and is continuously decreasing from that time onward. If the system continued to change at its maximum response rate, the rate that occurs at the origin, it would reach its final value (100%) in one time constant (TINT time).

+=+=+=+ a first order curve derived from a step input. This curve gives numerical values to the change, and in the first period of time (in our case the integral time constant set by TINT) the change equals 63.2%. In the second time period 63.2% of the remaining 36.8% will take place, and so on in every time interval. Theoretically the response never reaches 100%, but it does approach it asymptotically.

+=+=+=+ Response of a first order lag to a step input. At T2 time the value equals 63.2% of the 36.8% remainder at T1 time

By measuring both the loop deadtime and the loop time constant, the time from a noticeable change in the PV value to the time (in minutes) that a value of 63.2% is reached, the following can be determined:

• PG = 1 / PG (open loop)

• IG = LTC

• D = 0.25 × I

+=+=+=+ First order lag response curve.

Exercise -- Open Loop Tuning Exercise Provides practice in the reaction curve method of tuning.

Hysteresis problems that may be encountered in open loop tuning

In the real operational world it’s good practice to perform open loop tuning with as big a step as possible, over the normal operating range, and in both directions, i.e. after making, say a 20% step up and recording the systems response, return the output back to its original starting value and again record the systems reaction response. In most systems the incremental and decremental responses will be different. If this difference is only a few percent (<5-6%), take the average values of the two recordings and apply the results to the tuning algorithms being used. If the differences are large, then tuning to either response can lead to instability or poor control when the process responds to the other response that was not used for tuning. Re-engineering of the process system itself, or introduction of corrective algorithms, will be required in order to reduce the hysteresis to an acceptable level. An example of one method to correct this problem 'Combined feedback and feedforward control', where correcting the time difference between heating and cooling a boiler is discussed. The PID controller itself cannot be set or tuned to alleviate this type of problem.

Continuous cycling method (Ziegler-Nichols)

This method of tuning requires that we determine the critical value of controller gain (KC) that will produce a continuous oscillation of a control loop. This will occur when the total loop gain (KLOOP) is equal to one. The controller gain value (KC) then becomes known as the ultimate gain (KU). Requirements needed for a system to be considered stable.

We have to remember here that the loop is made up of several component parts, all of which contribute to the total gain of the loop (KLOOP), and the only one that we can adjust is normally the controller's gain (KC). If we consider a basic liquid flow control loop consisting of:

1. A measuring device: A venturi flow meter with a 4-20 mA output signal, fed to a controller

2. A controller: A PID controller with 4-20 mA output signal, that is used to control an actuator

3. A control device: A valve actuator which controls the flow rate of the process fluid and

4. The process.

When the product of the gains of all four of these component parts equals one, the system will become unstable when a process disturbance occurs (a setpoint change). It will oscillate at its natural frequency which is determined by the process lag and response time, and caused by the loop gain becoming one.

E.g., if the system listed above, had the following gain characteristics:

Venturi gain = 0.75

Control valve gain = 1.12

Process gain = 0.98

... then the process gain (as 'seen' by the controller) is calculated as:

0.75 1.12 0.98 = 0.8232. ××

With KP equal to 0.8232, to make KLOOP equal to 1, the value of KC has to be LOOP 1/ 0.8232 1.215, giving = 0.75 1.12 0.98 1.215 = 1 K =×××

In order to observe the process dynamic characteristics only, we must not use any integral or derivative control during the process (as explained below) of determining the value of KC in order to obtain a total loop gain, KLOOP, equal to one (with no 'corrupting' phase shift introduced by the controller). We can then measure the frequency of oscillation (the period of one cycle of oscillation), this being the ultimate period PU. In addition, we know that the final value of KC is the critical gain of the controller (KU). This gain value when multiplied with the unknown process gain(s), will give a loop gain, KLOOP, of 1. From there we can stabilize the loop by reducing the value of KC.

The stages of obtaining closed loop tuning (continuous cycling method):

1. Put controller in P-control only: In order to avoid the controller influencing the assessment of the process dynamic, no integral or derivative control should be active. Make TINT = 999 and TDER = 0.

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

3. Put the controller into automatic mode: We need a closed loop situation to obtain continuous cycling at the critical gain setting.

4. Make a step change to the setpoint: To observe how the PV settles after a disturbance, change the SP value to simulate one. Before making this step change to the SP make sure the process is 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.

6. Conclusion of tuning procedure: Once you obtain continuous cycling of the process, measure the cycle time and the value of KC obtained for continuous cycling. This time is the ultimate period (PU), and the value of KC is the ultimate gain (KU). Reduce the value of KC by 50% to stop the oscillations and return the SP to its original value to stabilize the process.

Calculation of tuning constants (continuous cycling method):

We will obtain different tuning constants with P, PI and PID control modes. However, your attention is drawn to the fact that the same relationships as discovered in the reaction curve method of tuning re-appear here. Controller settings are determined below.

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===Controller Proportional Only Proportional Integral Proportional Integral Derivative

Gain KC; KC = 0.5 × KU KC = 0.45 × KU KC = 0.6 × KU

Integral time TINT

Derivative time TDER

Table: PID controller tuning parameter settings for closed loop using Zeigler Nichols continuous cycling method Note: There are no values given for PD-control with this method, but the ratios used for open loop tuning can be applied if required.

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Exercise -- Closed Loop Tuning Exercise -- For practice in the techniques of closed loop tuning.

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