A stepwise method for tuning PI controllers using ITAE criteria

Applications based on control theory have gone through major developments in the past decades. However 95% of the industrial controllers are of PID type even though most loops are actually PI control [3]. The main reasons for the choice are the simplicity of the control law and the fewer tuning parameters. Many theories and tools are available for this purpose.

But still, finding the optimum parameters for the PI controller is a daunting task and in practice control engineers often use trial and error method for the tuning process [4]. System nonlinearities, design method based on specific problem in hand, continuous parameter variations make tuning a “Difficult task”.

This article provides a brief over view of the ITAE [Integral of the time weighted absolute error],and then describes a stepwise method which can be followed to simplify the process of PI tuning for field oriented motor control.

Overview of PID control

Most processes are non-linear in some aspects. The process gain can vary with load or with time. Feedback control loops are designed so that the controller variable (process output) is maintained at a set point even if disturbances occur or the dynamics of the process changes.

In general the process nonlinearities are compensated by PI controller functions. Finding optimum values for the PI controller functions is a daunting task. There has been vast amount of literature, tools on the subject of PI controller tuning.

However many control system engineers still use the trial and error method for tuning the system. This paper describes the systematic approach to be followed to get the optimum tuning values based on ITAE criterion.

Minimum Optimization Criterion

Many methods are presented in literature that minimizes a certain error criterion [8]. One of the methods called, Minimum ITAE approximate model for controller tuning rules is described here. Integral of the time weighted absolute error [ITAE] is defined as,

The time weighting ‘t’ is used, because the initial error for a step response is always large, and for most set point cases it is reasonable to weigh this error less. Based on the below FOLPD model it is possible to mathematically determine the controller settings:

Where K is the gain, DT is the dead time, τ is the time constant.

A distinction is made between responding to a disturbance and responding to a set point. The controller settings that give a minimum ITAE is given in Table 1 below[1].

Table 1. Constants A and B for P, PI, PID control [8][9]

Terminology. Before tuning it is good to understand the terminologies used in this article. Figure 1 below shows the generic control loop.

on image to enlarge.

Figure 1. PI control loop

Process: It is the physical process that we want to control. The manipulation of the physical properties affects the output of the process.

Process variable (PV): It is the output of the process. PV has to be controlled so that it gets close enough to the set point.

Controller variable (CV): It is the output of the PI controller and this value is manipulated by the PI controller to obtain the desired set point.

Load: Disturbances which can move the PV from its setpoint. Control system has the job of counteracting the load.

Set point (SP): SP is the desired value for the control system.

Error: It subtracts the PV from the SP inside the controller. The difference between the SP and the PV is called the control error E. This is the input to the PID Algorithm.