**3. MOMI and DRMO tuning methods**

The MOMI and DRMO methods, as mentioned earlier, are based on the Magnitude Optimum (MO) method, which goes back to Whitley in 1946 [10]. The MO method shapes the closed-loop amplitude frequency response equal to one in a wide frequency range [6, 7, 10, 12–14, 21]. Such a closed-loop frequency response is usually "mirrored" into a fast and stable closed-loop time response.

The calculation of controller parameters has been simplified when using the MO method by determining the process characteristic areas or moments, which can be measured directly from the time responses during the change of the process steadystate [12, 15, 21, 22]. The mentioned areas or moments (*A1* to *Ak*) can also be calculated from the process model:

$$A\_0 = K\_{PR}$$

$$A\_1 = K\_{PR}(a\_1 - b\_1 + T\_{dd})$$

$$A\_2 = K\_{PR} \left[ b\_2 - a\_2 - b\_1 T\_{dd} + \frac{T\_{dd}^2}{2!} \right] + A\_1 a\_1$$

$$\vdots$$

$$A\_k = K\_{PR} \left[ (-1)^{k+1} (a\_k - b\_k) + \sum\_{i=1}^k (-1)^{k+i} \frac{T\_{dd}^i b\_{k-i}}{i!} \right] + \sum\_{i=1}^{k-1} (-1)^{k+i-1} A\_i a\_{k-i} \quad \text{(5)}$$

The controller parameters, for a given filter time constant *TF*, are then calculated as follows:

$$
\begin{bmatrix} K\_I \\ K\_P \\ K\_D \end{bmatrix} = \begin{bmatrix} -A\_1^\* & A\_0^\* & \mathbf{0} \\ -A\_3^\* & A\_2^\* & -A\_1^\* \\ -A\_5^\* & A\_4^\* & -A\_3^\* \end{bmatrix}^{-1} \begin{bmatrix} -\mathbf{0}.5 \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix},\tag{6}
$$

where the modified areas A0\* to A5\* are:

$$\begin{aligned} A\_0^\* &= A\_0\\ A\_1^\* &= A\_1 + A\_0 T\_F \end{aligned}$$

$$\begin{aligned} A\_2^\* &= A\_2 + A\_1 T\_F + A\_0 T\_F^2\\ &\vdots \end{aligned} \tag{7}$$

The reference-weighting factors are *b* = *c* = 1. Note that the areas (moments) in expression (6) apply areas of the process including the controller filter *GF* with time constant *TF* (4):

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

$$G\_F = \frac{1}{(1 + sT\_F)}\tag{8}$$

by using expression (7) [9]. The aforementioned modification of the method, referred to as the MOMI method, allowed the controller parameters to be computed directly from the process time response [12, 21] or from the process transfer function.

Since the MOMI method aims at optimising the tracking performance, the disturbance rejection performance may be degraded for some types of processes.

To improve the disturbance-rejection performance, the optimisation criteria of the MOMI method were modified accordingly. The new method, referred to as the DRMO (Disturbance-Rejection-Magnitude-Optimum) method, achieved significantly improved disturbance rejection performance [9, 16, 17].

Similar to the MOMI method, the controller parameters in the DRMO method are also based on characteristic areas or moments. Therefore, the controller parameters can be calculated either from the process time-response or from the process transfer function.

The PID controller parameters are calculated according to the following expressions when using the DRMO method [9, 16, 17]:

$$K\_P = \frac{\beta - \sqrt{\beta^2 - \alpha \gamma}}{a}$$

$$K\_I = \frac{\left(1 + K\_P A\_0^\*\right)^2}{2\left(K\_D A\_0^{\*,2} + A\_1^\*\right)}\tag{9}$$

where

*u* ¼ *GCR*ð Þ*s r* � *GCY*ð Þ*s y*

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

2

, (4)

2

*GCR* <sup>¼</sup> *KI* <sup>þ</sup> *bKPs* <sup>þ</sup> *cKDs*

*GCY* <sup>¼</sup> *KI* <sup>þ</sup> *KPs* <sup>þ</sup> *KDs*

as shown in **Figure 2**, where parameters *b* and *c* are reference-weighting

parameters for the proportional and derivative terms, respectively.

usually "mirrored" into a fast and stable closed-loop time response.

*A*<sup>2</sup> ¼ *KPR b*<sup>2</sup> � *a*<sup>2</sup> � *b*1*Tdel* þ

X *k*

" #

*i*¼1

�*A*<sup>∗</sup> <sup>1</sup> *A*<sup>∗</sup>

2 6 4

�*A*<sup>∗</sup> <sup>3</sup> *A*<sup>∗</sup>

�*A*<sup>∗</sup> <sup>5</sup> *A*<sup>∗</sup>

*A*<sup>∗</sup>

*A*<sup>∗</sup>

*A*<sup>∗</sup> <sup>0</sup> ¼ *A*<sup>0</sup>

<sup>1</sup> ¼ *A*<sup>1</sup> þ *A*0*TF*

<sup>2</sup> <sup>¼</sup> *<sup>A</sup>*<sup>2</sup> <sup>þ</sup> *<sup>A</sup>*1*TF* <sup>þ</sup> *<sup>A</sup>*0*T*<sup>2</sup>

The reference-weighting factors are *b* = *c* = 1. Note that the areas (moments) in expression (6) apply areas of the process including the controller filter *GF* with time

ð Þþ *ak* � *bk*

*KI KP KD*

where the modified areas A0\* to A5\* are:

2 6 4

**3. MOMI and DRMO tuning methods**

calculated from the process model:

*Ak* <sup>¼</sup> *KPR* ð Þ �<sup>1</sup> *<sup>k</sup>*þ<sup>1</sup>

as follows:

constant *TF* (4):

**46**

*s*ð Þ 1 þ *sTF*

*s*ð Þ 1 þ *sTF*

The MOMI and DRMO methods, as mentioned earlier, are based on the Magnitude Optimum (MO) method, which goes back to Whitley in 1946 [10]. The MO method shapes the closed-loop amplitude frequency response equal to one in a wide frequency range [6, 7, 10, 12–14, 21]. Such a closed-loop frequency response is

The calculation of controller parameters has been simplified when using the MO method by determining the process characteristic areas or moments, which can be measured directly from the time responses during the change of the process steadystate [12, 15, 21, 22]. The mentioned areas or moments (*A1* to *Ak*) can also be

> *A*<sup>0</sup> ¼ *KPR A*<sup>1</sup> ¼ *KPR*ð Þ *a*<sup>1</sup> � *b*<sup>1</sup> þ *Tdel*

� �

⋮

ð Þ �<sup>1</sup> *<sup>k</sup>*þ*<sup>i</sup> <sup>T</sup><sup>i</sup>*

The controller parameters, for a given filter time constant *TF*, are then calculated

<sup>0</sup> 0

<sup>2</sup> �*A*<sup>∗</sup> 1

<sup>4</sup> �*A*<sup>∗</sup> 3

*T*2 *del* 2!

*delbk*�*<sup>i</sup> i*!

> 3 7 5

þ *A*1*a*<sup>1</sup>

þ<sup>X</sup> *k*�1

�<sup>1</sup> �0*:*<sup>5</sup> 0 0

3 7

⋮ (7)

2 6 4

*F*

*i*¼1

ð Þ �<sup>1</sup> *<sup>k</sup>*þ*i*�<sup>1</sup>

*Aiak*�*<sup>i</sup>* (5)

<sup>5</sup>, (6)

$$a = A\_1^{\*\ 3} + A\_0^{\*\ 2} A\_3^\* - 2A\_0^\* A\_1^\* A\_2^\*$$

$$\beta = A\_1^{\*\ } A\_2^\* - A\_0^\* A\_3^\* + 2K\_D(A\_0^\* A\_1^{\*\ 2} - A\_0^{\*\ 2} A\_2^\*)$$

$$\gamma = K\_D^3 A\_0^{\*\ 4} + 3K\_D^2 A\_0^{\*\ 2} A\_1^\* + K\_D \left(2A\_0^\* A\_2^\* + A\_1^{\*\ 2}\right) + A\_3^\* \tag{10}$$

and the derivative gain *KD* is calculated directly from expression (6). The reference-weighting factors are *b* = *c* = 0.

The DRMO tuning method significantly improved the disturbance rejection performance, especially for the lower-order processes. However, the reference tracking becomes slower due to the reference-weighting factors *b* = *c* = 0 in the 2-DOF control structure (4). The problem can be circumvented by including a simple disturbance estimator in the control scheme. Such a solution is denoted as DE-MOMI method.
