*Improving Disturbance-Rejection by Using Disturbance Estimator DOI: http://dx.doi.org/10.5772/intechopen.95615*

**Figure 6.** *The closed-loop responses on the process* GP1*, when using the MOMI, DRMO and DE-MOMI method.*

**Figure 7.** *The closed-loop responses on the process* GP2*, when using the MOMI, DRMO and DE-MOMI method.*

*GM*<sup>1</sup> <sup>¼</sup> *<sup>e</sup>*�0*:*5*<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 *GP1* 1.81 0.89 0.93 DRMO controller for *GP1* 2.25 1.49 0.93 MOMI controller for *GP2* 1.61 0.64 1.08 DRMO controller for *GP2* 1.93 0.98 1.08

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

*GMI*<sup>1</sup> ¼ 1 þ 2*s* þ *s*

*GMI*<sup>2</sup> ¼ 1 þ 2*:*58*s* þ 1*:*84*s*

Finally, the disturbance filter gain *KFD*, when taking into account the chosen

*KFD*<sup>1</sup> ¼ 0*:*57

Therefore, the complete inverse of the models with accompanying disturbance

<sup>2</sup> ð Þ ð Þ <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*1*<sup>s</sup>* <sup>3</sup>

<sup>2</sup> ð Þ

*GMI*1*GFD*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*57 1 <sup>þ</sup> <sup>2</sup>*<sup>s</sup>* <sup>þ</sup> *<sup>s</sup>*

*GMI*2*GFD*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*59 1 <sup>þ</sup> <sup>2</sup>*:*58*<sup>s</sup>* <sup>þ</sup> <sup>1</sup>*:*84*<sup>s</sup>*

The closed-loop responses, obtained with the calculated controller, model and filter parameters, for the MOMI, DRMO and the proposed DE-MOMI method, are given in **Figures 6** and **7**. At *t* = 0 s, the reference value (*r*) was changed from 0 to 1 and at half of experiment time the process input disturbance (*d*) was changed from 0 to 1. It is obvious that the disturbance rejection performance of the DE-MOMI method is the best. Note that when applying the DE-MOMI method, due to the difference between the actual process and the process model in the second example (*GP2*), the process input signal, during the reference change, is not smooth. This is expected, since the inverse process model with filter is amplifying the difference between the actual process and the process model. In this case, the response can be made smoother by increasing the disturbance filter time constant (*TFD*). Note that a possible limitation of the control signal can also help to smooth out the oscillations

The disturbance rejection performance of the DE-MOMI method can be increased by decreasing the disturbance filter time constant *TFD*. However, as already mentioned above, the process input signal can become oscillatory when the actual process and the process model differ. In this case, too small *TFD* can even render the closed-loop system unstable. Besides that, the process noise (signal *n* in

*GM*<sup>2</sup> <sup>¼</sup> *<sup>e</sup>*�0*:*616*<sup>s</sup>*

*TFD* = 0.1, is then calculated from (18):

*account the chosen controller filter TF = 0.1.*

**Table 2.**

filters (see **Figure 3**) are the following:

after the reference step [24].

**52**

1 þ 2*s* þ *s*<sup>2</sup>

2

<sup>1</sup> <sup>þ</sup> <sup>2</sup>*:*58*<sup>s</sup>* <sup>þ</sup> <sup>1</sup>*:*84*s*<sup>2</sup> (22)

*KFD*<sup>2</sup> ¼ 0*:*59 (23)

ð Þ <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*1*<sup>s</sup>* <sup>3</sup> (24)

2

measurement noise *n* and the frequency properties of noise, PID controller and disturbance estimator. The relation between the filters (*TF* and *TFD*) time constants and the controller output noise is rather complex, but can be calculated according to Parseval's theorem if the measurement noise frequency characteristics are known. However, this relation is higher-order and non-linear. Therefore, the search for adequate filter time constants *TF* and *TFD* would require optimisation procedure,

In practice, on the other side, it is enough to keep the noise sufficiently low at some sufficiently high frequency. The definition of "high frequency" is arguable. In

> *<sup>f</sup> <sup>S</sup>* <sup>¼</sup> <sup>1</sup> *TS*

where *TS* is the controller's sampling time. The highest signal, which may be sent to discrete function is, due to Shannon's theorem, *fS*/2. Therefore, any frequency close to *fS*/2 can be considered as high frequency. In this research we have arbitrarily decided that the "high frequency" *fHF* is the quarter of controller's sampling

*f HF* ¼ 0*:*25 *f <sup>S</sup>*

*<sup>ω</sup>HF* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>f</sup> HF* <sup>¼</sup> <sup>0</sup>*:*5*<sup>π</sup>*

*uPIDn*ð Þ¼ *ωHF KPIDnn*ð Þ *ωHF*

where *KPIDn* and *KDEn* are the high-frequency gains (around frequency *ωHF*) of

In practical applications of the DE-MOMI method, the noise specifications (limitations) should be given in as simple form as possible for the user (operator). We decided that the actual parameters, given by the user should be the high-frequency gains of the controller (*KPIDn*) and the disturbance estimator (*KDEn*). Therefore, in practice, by selecting the mentioned two gains, the user would limit the amount of

The actual gain of the PID controller around the chosen high frequency *ωHF* can

*KI* � *KDω*<sup>2</sup> *HF* � �<sup>2</sup> <sup>þ</sup> *<sup>K</sup>*<sup>2</sup>

*ωHF*

*HF K*<sup>2</sup>

*ω*2 *HFKPIDn*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> *<sup>T</sup>*<sup>2</sup> *Fω*<sup>2</sup>

*<sup>P</sup>* � <sup>2</sup>*KIKD* � *<sup>K</sup>*<sup>2</sup> *PIDn* � � <sup>þ</sup> *<sup>K</sup>*<sup>2</sup>

*HF* <sup>q</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

the PID controller and the disturbance estimator, respectively.

be calculated from the controller transfer function (2):

*KPIDn* ¼

*ω*4 *HFK*<sup>2</sup>

q

The controller filter time constant can then be calculated as:

*<sup>D</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

noise *n* (**Figure 3**). In DE-MOMI controller, the overall high-frequency control noise consists of the PID controller (*uPIDn*) and the disturbance estimator (*uDEn*)

As already mentioned above, the source of controller noise is the process output

*TS*

*uDEn*ð Þ¼ *ωHF KDEnn*ð Þ *ωHF* , (27)

*Pω*<sup>2</sup>

*HF* <sup>q</sup> (28)

*I*

(29)

(25)

(26)

which would significantly complicate the otherwise simple method.

discrete-realisation of the controller, the sampling frequency is

*Improving Disturbance-Rejection by Using Disturbance Estimator*

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

frequency *fS*:

high-frequency noise:

controller noise at high frequencies.

*TF* ¼

**55**


**Figure 8.**

*The website layout for the calculation of the controller and the DE parameters.*

**Figure 3**) is also amplified via block *GMIGFD*, so small *TFD* can cause excessive noise of signal *dF*. The selection of *TFD* is, therefore, important in practical realisation of the DE-MOMI method.

Calculating the controller and DE parameters is a relatively simple process. However, to simplify it even further, all Matlab/Octave scripts are available on the OctaveOnline Bucket website [25]. The layout of the website is shown in **Figure 8**. To calculate the controller and DE parameters, the user must 1) change the process and filter parameters, 2) press the "Save" button, and 3) press the "Run" button. The script will be executed and on the right side of the web screen all calculated parameters will be displayed. Note that users can change the content of the script only temporarily.
