How to Tune the PID Controller

Before we discuss how to tune the PID controller, let’s look at some basic facts about PID Controller.

Basic Facts about PID Controller

Many control systems use a combinational of three types of control i.e. Proportional + Integral + Derivative (PID) control. The foundation of the system is the Proportional Control, adding Integral Control provides a means to eliminate steady-state error but may increase overshoot. Derivative Control is good for getting slow systems moving faster, and reduces the tendency to overshoot. The response of the PID systems can be described by the following equation:

Where,

 Output_PID = Output of the PID Controller

K_P = Proportional Control gain

K_I = Integral Control gain often seen as 1/T_I

K_D = Derivative Control gain (often seen as T_D)

E = Error (deviation from the setpoint)

∑(EΔt) = Sum of all past errors (Area under the error x time curve)

ΔE/Δt = Rate of change of error (slope of the error curve)

Tuning the PID Controller

The method of arriving at numerical values for the constants K_P, K_I and K_D depends on the application. A practical step-by-step procedure can be used to arrive at the PID constants. First the constants K_P, K_I, and K_D are set to initial values, and the controller is connected to the system. The system could consist of the actual hardware or a computer simulation of the same. Then the system is operated, and the response is noted. Based on the response, adjustments are made to K_P, K_I and K_D and the system is operated again. This iterative process of adjusting each constant in an orderly manner until the desired system response is achieved is called tuning. To make the system stable under all conditions, certain modification may be necessary to the basic PID equation. The two most common methods of tuning PID Controllers are the ones that were developed by Ziegler and Nichols, which are:

1. Continuous-cycle method
2. Reaction-curve method

Continuous-cycle method

This method is used for closed loop systems. It is used when there is no harm done if the system goes into oscillation. This method will produce a system with a quick response, which implies a step-function will cause a shift overshoot that settles out very quickly.

The tuning procedure is as follows:

• Set K_P = 1, K_I = 0, and K_D = 0 and connect the controller to the system.
• Using manual control, adjust the system until it is operating in the middle of its range. Then increase the proportional gain K´_P while forcing small disturbances to the set point (or the process) until the system oscillates with constant amplitude as illustrated below in figure 1(a). Record the values of K´_P and T_C for this condition.

    

• Based on the values of K´_P  and T_C from step 2, calculate the initial settings of K_P, K_I, and K_D as follows:

• Using the settings calculated above of K_P, K_I and K_D, operate the system, and observe the response, and make the adjustments as called for.  Increasing K_P will produce a stiffer and quicker response, increasing K_I will reduce the time it takes to settle out the zero error, and increasing K_D will decrease the overshoot. It is obvious that, K_P, K_I and K_D do not act independently therefore changing one constant will have an effect across the board on system response. Tuning the system is an iterative process of making smaller and smaller adjustments until the desired response is achieved as shown below:

Reaction-curve method

Reaction-curve method (open loop method) does not require driving the system to oscillation. Instead, the feedback loop is opened, and the controller is manually directed to output a small step function to the actuator as illustrated in figure 1(c).

The system response as reported by the sensor  is used to calculate K_P, K_I and K_D. You can notice that the actuator, the process itself and the sensor are operational in this test, so their individual characteristics are accounted for. Since the loop is open, this procedure will only work for systems that are inherently stable.

In the test setup in figure 1(c), the loop was opened by placing the controller into manual mode, and then a small step function was manually introduced. This signal caused the controlled variable to move a little, and the resulting position response was recorded. A typical response curve is illustrated in figure (d). Note that the vertical axis corresponds to the range of the process variable (in %)

The system constants are calculated based on the response curve, as outlined below:

• Draw a line tangent to the rising part of the curve as shown in figure 1(d). This line defines the lag time (L) and rise time (T) values. Lag time is the time delay between the controller output and the controlled variable response.
• Calculate the slope of the curve:

      Where,

       N = slope of the system-response curve

      ΔPV = change in the process variable as reported by the  sensor (in %).

      T = rise time, from response curve

• Calculate PID constants as follows:

K_D = 0.5L

Where, ΔCV = Percentage step change in the control variable (output of the controller)

N = slope as determined in step 2

L = lag time

Autotuning Controllers

Autotuning is the ability of the some digital controllers to monitor their own output and make minor changes in the gain constants (K_P, K_I, and K_D) e.g. a temperature control system may decrease its proportional gain slightly if the overshoot is beyond a certain threshold and increase the gain if the response is too sluggish. In the same way as manual tuning, autotuning is an iterative process, however, since it is ongoing, the system can adapt to the changes in the process. Hence, controllers that use autotuning are known as adaptive controllers.