**6. Comparison to some other methods**

In this sub-chapter the proposed method will be compared to some other tuning methods based on non-parametric description of the process. Besides the already introduced MOMI and DRMO methods, the DE-MOMI method will be compared to Åström and Hägglund's tuning method [1] (denoted as "AH") and to ADRC method [27].

The AH method [1] is based on the calculation of the maximum sensitivity index *MS*, which is the inverse of the smallest open-loop Nyquist curve distance to the critical point (1,0i). The method was developed for values *MS* = 1.4 and *MS* = 2. In this comparison we will use *MS* = 2, since it gives better disturbance-rejection performance. However, even though the process transfer function does not need to be derived, the method requires the identification of the process steady-state gain and the inflexion point along with maximum slope of the process output signal during the step-change of the process input signal. Note that those parameters usually require manual measurements and cannot be easily performed by using automatic calculation. The AH method is using the PID controller structure with adjustable reference-weighting factor *b*, and by fixing factor *c* =0(**Figure 2**).

*GMI*4*GFD*<sup>4</sup> <sup>¼</sup> <sup>0</sup>*:*36 1 <sup>þ</sup> <sup>3</sup>*:*06*<sup>s</sup>* <sup>þ</sup> <sup>2</sup>*:*69*<sup>s</sup>*

*The calculated controller parameters for the processes (20) for MOMI (6) and DRMO (9) method, taking into*

**Controller parameters** *KP KI KD* MOMI controller for *GP3* 2.35 1.88 0.48 DRMO controller for *GP3* 2.91 3.83 0.48 MOMI controller for *GP4* 0.84 0.26 0.77 DRMO controller for *GP4* 0.94 0.32 0.77

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

times higher than the measurement noise at high frequencies.

stability if the actual process and the process model differ.

**Figure 11.**

**58**

**Table 4.**

*account the calculated controller filters.*

The closed-loop responses for the MOMI, DRMO and the proposed DE-MOMI method, are given in **Figures 11** and **12**. Again, the disturbance rejection performance of the DE-MOMI method is the best (note that the unity-step process input disturbance signal was applied at the half of experiment time). The level of controller output (*u*) noise is close to the expected one taken into account that both, the PID controller (*uC*) and the disturbance estimator output (*dF*) noise should be 4-

The disturbance rejection performance of the DE-MOMI method can be additionally improved by increasing the high-frequency gain *KDEn*. However, increased gain is associated with higher controller output noise and decreased closed-loop

The computation of the controller and the DE parameters can be performed similarly as before on another OctaveOnline Bucket website [26]. The calculation of

*The closed-loop responses on the process* GP3*, when using the MOMI, DRMO and DE-MOMI method.*

<sup>2</sup> ð Þ

ð Þ <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*116*<sup>s</sup>* <sup>3</sup> (38)

The ADRC method [27–31] is based on a simple controller with three gains associated with extended state-observer (ESO), as shown in **Figure 13**.

The method does not require the process transfer function. However, few userdefined parameters, like the observer speed, the desired settling time and the main controller gain *KC*, should be defined by the user before calculating the rest of ADRC parameters. As shown in **Figure 13**, the ADRC method is using control structure which consists of an extended state observer (ESO) with three gains (*β*1, *β*<sup>2</sup> and *β*3) and three controller gains (*KC*, *KP* and *KD*) [27].

Since ADRC method depends on three user-defined parameters, which, in great extent, determine the closed-loop performance, we were limited to the set of processes tested in [27]. Someone would argue that, by limiting our choice to the mentioned processes, we are favouring the ADRC method. However, it should be noted that in [27], the ADRC method was tested on 8 different processes, so the choice of processes was actually not significantly limited. In this regard, the following two processes have been selected:

$$G\_{P\S} = \frac{1}{(1+s)(1+0.2s)(1+0.04s)(1+0.008s)}$$

$$G\_{P\6} = \frac{e^{-5t}}{\left(1+s\right)^3} \tag{39}$$

It can be seen that the proposed DE-MOMI method, when compared to some other methods, gives quite good responses. The AH method for process *GP5* gives somehow oscillatory response. For the same process, the ADRC method gives slightly oscillatory response during the reference change (see the process input signal). While DE\_MOMI and MOMI methods clearly give the best tracking responses on process *GP6*, all of the methods have similar disturbance-rejection performance. Only slightly oscillatory response can be observed for ADRC method. For more objective comparison between the methods, the integral of absolute error (IAE) measure is used. The IAE value has been measured on tracking response (unity step-change of the reference *r*) and on disturbance rejection response (unity step-change of the process input disturbance *d*). The results are given in **Table 8**. It can be seen that the best values (marked with greyed colour) were obtained with

**Process Tuning method** *KP KI KD TF b c GP5* MOMI 6.45 5.35 1.108 0.055 1 1

*Improving Disturbance-Rejection by Using Disturbance Estimator*

*DOI: http://dx.doi.org/10.5772/intechopen.95615*

*GP6* MOMI 0.53 0.126 0.66 0.165 1 1

*The calculated controller parameters for the processes (39) for MOMI, DRMO, DE-MOMI and AH method.*

**Process** *KPRM a1m a2m Tdelm TFD KFD GP5* 1 1.205 0.205 0.043 0.018 0.909 *GP6* 1 2.58 1.84 5.42 0.077 0.159

**Process** *KC KP KD β<sup>1</sup> β<sup>2</sup> β<sup>3</sup> GP5* 1/5 100 20 120 4800 19200 *GP6* 1/3 0.16 0.8 4.8 7.68 30.72

*The calculated disturbance estimator's parameters for the processes (39) for DE-MOMI method.*

*The calculated ADRC controller parameters for the processes (39).*

DRMO 9.69 23.71 1.108 0.055 0 0 DE-MOMI 6.45 5.35 1.108 0.055 1 1 AH 21.35 53.05 2.22 0.055 0.24 0

DRMO 0.57 0.140 0.66 0.165 0 0 DE-MOMI 0.53 0.126 0.66 0.165 1 1 AH 0.52 0.136 0.52 0.165 0.36 0

The DE-MOMI method, therefore, compares favourably with few other

The process closed-loop responses for all the process models tested in this chapter (*GP1* to *GP6*) revealed that the proposed method can significantly improve the disturbance-rejection performance of the lower-order processes with smaller delays, while the improvement of the higher-order processes and/or processes with higher delays is not so significant. Therefore, the application of the method for lower-order processes with smaller delays might be beneficial in practice.

methods, based on the non-parametric description of the process.

DE-MOMI method.

**61**

**Table 5.**

**Table 6.**

**Table 7.**

The PID controller parameters for the MOMI, DRMO, DE-MOMI and AH methods are given in **Tables 5** and **6**. The ADRC controller parameters are given in **Table 7**. The chosen high frequency gains for the PID controller and disturbance estimator are *KPIDn* = *KDEn* = 20 for *GP5* and *KPIDn* = *KDEn* = 4 for *GP6*. The higher gains were chosen for *GP5*, since the closed-loop tracking and control performance was substantially improved when using higher gains. Increasing the gains for *GP6* above 4 did not significantly improve the performance.

The sampling time for *GP5* is chosen as *TS* = 0.001 s and for *GP6* as *TS* = 0.01 s. The closed-loop process responses are given in **Figures 14** and **15**. In both experiments the unity-step process input disturbance signal was applied at the half of experiment time.

**Figure 13.** *The ADRC control structure with the controller gains (up) and the extended state observer (down).*

