PID Controller

12.19.1. Control Algorithm Description

Proportional integral-derivative controller is the most widely used type of automatic feedback controllers. The block interface is displayed in figure 28.

Figure 28. The PID Function Block Interface

Let us take a brief look at PID controller operation. Suppose a certain object has an input that allows to control this object and a sensor that estimates the object response (an output variable). Furthermore, various perturbing factors affect the object. This is how the output variable can be altered even by the constant reference input . The difference between the reference input and the output variable is the control error e ( t ) . The controller task is to change automatically the input action y ( t ) so that the perturbing effects will be minimized.

Controllers regulate any process which has a measurable output such as voltage, pressure, temperature, rate of travel, etc. A mathematical model of such an object involves the differential equation system. The optimal law of control can be derived from the object model. However, this is quite a difficult task. Practically, the all-purpose controller is applied for most industrial automation. The law of control is formed by tuning three constants, which are calculated from object model or fitted experimentally. The PID controller equation takes the form:

( )

( )

( )

( )

∫

where Y0 is zero error value (initial value), Kp is proportionality constant, Tn

is integral time, Tυ is derivative action time.

The proportional controller term produces an output value that is proportional to the current error value. The low proportional gain results in a small output response to an infinitely long-lasting error.

Even the small constant error integration results in a constant increasing of proportional gain. Thus, the controlling accuracy is attained on the change of object characteristics. The derivative term of a PID controller improves the dynamic characteristics, compensating delay of the control signal phase. The input variable descriptions of the PID controller are presented in tables 28 and 29.

Table 29

The Description of Block Inputs

Input Data Type Description

ACTUAL REAL response signal SET_POINT REAL desired value, task КР REAL transmission coefficient TN DWORD integral time (msec)

TV DWORD derivative action time (msec) Y_MANUAL REAL manual task

Y_OFFSET REAL offset for the manipulated variable Y Y_MIN REAL lower limit for the manipulated variable Y Y_MAX REAL upper limit for the manipulated variable Y MANUAL BOOL manual mode

RESET BOOL reset

Table 30

The Description of Block Outputs

Y REAL manipulated value

LIMITS_ACTIVE BOOL Indicator of achieving the prescribed _limits OVERFLOW BOOL indicator of overflow

12.19.2. PID Controller Setting

There are numbers of PID regulator setting, such as the Ziegler-Nichols method, the Kuhn method, the Schedel method, the Shubladze method.

The Ziegler-Nochols method was developed by American engineers John Ziegler and Nathaniel Nichols. Here are the main principles of this method:

• the integral and derivative gains are set to zero in the function system(ki = ∞, kd = 0), i.e., the system is transferred to P control;

• by gradually increasing kp and generating a jump signal the output

generates sustained oscillations with the Tcr period. The system is put

on the oscillation boundary. Thus, the critical gain Kcr and the

corresponding period in the system Tcr are determined. By critical

oscillations no variables can exceed the limiting level;

• the PID controller parameters are set according to the formula shown below:

0,6 , , .

2

8

p i

T

T

k

=

k k

=

k

=

This method have been widely used in process control systems when the plant dynamics are not known precisely. A majority of experts recognize the significant overshoot and putting the system on the stability limit as the main disadvantage of this method.

The general drawback of the existing methods is approximation of driving models, which contain neither zeroes nor lag units.

The considered engineering methods are easy to operate. However, let’s consider actual advantages and disadvantages.

The basic concept of the Ziegler-Nichols method is putting the control system on the stability limit. The sustained oscillations of the output signal are generated by increasing the proportionality coefficient. An obvious advantage is the simple realizing of this method because there is no need for opening the control system loop and further examining the object. However, not every object is allowed to exceed the stability limits according to the operating procedure. Oftentimes this condition is unrealizable. It should be noticed that the system with a PID controller that is set by this method can easier be overcontrolled.

According to the Shoubladze method of adjusting the PID controller the control object is approximated by a lag element of nth rank. The system gets the high level of stability, there is no possibility for overregulation, but the transient period is extended considerably. The Shoubladze method is

effective for inertial objects of low dynamics. A disadvantage of this method is the great number of calculations performed by determination of the controller coefficients.

The Kuhn method is based on the calculating the total time constant of the object. This method is effectively used by adjusting a controller for a low- order control object the first or the second-order object). Increasing the order demands the significant overshooting the synthesized system.

Using the Schedel method of a PID controller adjustment the system offers the shortest transient period as compared to other methods of PID controller adjustment. However, the increasing the overregulation of the control system also occurs by using the Schedel method.

The analysis of the existing methods of the PID controller adjustment revealed that decision for one or another method should be based on control object characteristics and requirements of the synthesized system. If the higher system performance is required, the Ziegler – Nichols method and the Schedel method are preferable. If the opportunity of overcontrol is of crucial importance, PID controllers, which are adjusted by the Shoubladze and the Kuhn methods, provide more efficient system operation.

The general drawback of the existing methods is approximation of driving models, which contain neither zeroes nor lag units. The considered methods assure the stability of the synthesized control system. However, to execute the process successfully a controlled parameter should be altered in the certain time period with the deviation within a certain limit. In other words, a control system should have the certain quality indicators: these are overcontrol and control time, which should be provided by PID controllers. So it is necessary to design a method of setting the PID controllers which ensure the certain quality indicators.

Developing this method a designer will face the issue which concerns the approach alternatives. This paper offers to use the root approach, because it allows to evaluate the dynamics of the synthesized control system.

The problem of developing the methods of PID controller adjustment continues to be relevant so far. Scientists develop different frequency, algebraic, root methods, link the controller settings to different quality indicators, which should provide the required quality of running the control loops.