**8. Conclusions**

72 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

case, the PM tuning method is used with a specification *PMdes*=0.15 rad. The obtained controller is presented in Table 6. Figures 9a and 9b, show the set-point step response from 6.5mm to 7.5mm and the regulatory control around the operating point *x0*=7mm using the new controller. Clearly, the obtained responses are significantly faster, as it was expected from the design of the PID controller (smaller *τIm*, larger *KCm*). Moreover, in the case of regulatory control the maximum error produced in the present case is significantly smaller (at least three times smaller) than the maximum error produced when the robust

Time in sec.

0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12

Time in sec.

Fig. 9. Position response of MagLev system using a fast controller designed with the PM-

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>4</sup>

Fig. 8. Position response of MagLev system using the robust controller designed with the GM-method: (a) Set-point response and (b) Load step response (current load disturbance

0 2 4 6 8 10 12 14 16 18 20

controller is used.

a Position in mm

6 8 10

6.5

6.5

method. Other legend as in Fig. 8.

7.5

Position in mm

7

b

7

7.5

a

amplitude 20% or 0.2 A).

b

New methods for tuning PID-like controllers for USOPDT systems have been developed in this work. These methods are based on various criteria, such as the appropriate assignment of the dominant poles of the closed-loop system, the attainment of various time-domain closed-loop characteristics, as well as the satisfaction of gain and phase margins specifications of the closed-loop system. In the general case, where the derivative action of the controller is selected arbitrarily, the tuning methods require the use of iterative algorithms for the solution of nonlinear systems of equations. In the special case where the controller derivative time constant is selected equal to the stable time constant of the system, the solutions of the nonlinear system of equations involved in the tuning methods are given in the form of quite accurate analytic approximations and, thus, the iterative algorithms can be avoided. In this case the tuning methods can readily be used for on-line applications. The proposed tuning methods have successfully been applied to the control of an experimental magnetic levitation system that is modelled as an USOPDT process. The obtained experimental results verify the efficiency of the proposed tuning methods that provide a very satisfactory performance of the closed-loop system.

#### **9. References**


4

Magnitude Optimum Techniques

Today, most tuning rules for PID controllers are based either on the process step response or else on relay-excitation experiments. Tuning methods based on the process step response are usually based on the estimated process gain and process lag and rise times (Åström & Hägglund, 1995). The relay-excitation method is keeping the process in the closed-loop configuration during experiment by using the on/off (relay) controller. The measured data

The experiments mentioned are popular in practice due to their simplicity. Namely, it is easy to perform them and get the required data either from manual or from automatic experiments on the process. However, the reduction of process time-response measurement

Therefore, more sophisticated tuning approaches have been suggested. They are usually based on more demanding process identification methods (Åström et al., 1998; Gorez, 1997; Huba, 2006). One such method is a magnitude optimum method (MO) (Whiteley, 1946). The MO method results in a very good closed-loop response for a large class of process models frequently encountered in the process and chemical industries (Vrančić, 1995; Vrančić et al., 1999). However, the method is very demanding since it requires a reliable estimation of quite a large number of process parameters, even for relatively simple controller structures (like a PID controller). This is one of the main reasons why the method is not frequently

Recently, the applicability of the MO method has been improved by using the concept of 'moments', which originated in identification theory (Ba Hli, 1954; Strejc, 1960; Rake, 1987). In particular, the process can be parameterised by subsequent (multiple) integrals of its input and output time-responses. Instead of using an explicit process model, the new tuning method employs the mentioned multiple integrals for the calculation of the PID controller parameters and is, therefore, called the "Magnitude Optimum Multiple Integration" (MOMI) tuning method (Vrančić, 1995; Vrančić et al., 1999). The proposed approach therefore uses information from a relatively simple experiment in a time-domain while

The deficiency of the MO (and consequently of the MOMI) tuning method is that it is designed for optimising tracking performance. This can lead to the poor attenuation of load disturbances (Åström & Hägglund, 1995). Disturbance rejection performance is particularly

into two or three parameters may lead to improperly tuned controller parameters.

is the amplitude of input and output signals and the oscillation period.

1. Introduction

used in practice.

retaining all the advantages of the MO method.

for PID Controllers

Damir Vrančić Jožef Stefan Institute

Slovenia

