**4.2 Plant II**

Plant II, is a solid bulk reclaimer, which is used to recover bulk for ship loading, this equipment has the capacity to move up to 8,000 tons per hour (t/h).

$$G\_{P\_{\rm II}}(\mathbf{s}) = \frac{b\_1 \mathbf{s} + b\_0}{\mathbf{s}^3 + a\_1 \mathbf{s}^2 + a\_2 \mathbf{s} + a\_3},\tag{33}$$

where *G<sup>G</sup> PII*ð Þ*s* is the TF of Plant II, it is a third order plant with zero at infinity.

The product of the TF numerator of Plant II given in Eq. (33) associated with the TF numerator of the controller given in Eq. (2) is given by

$$\begin{aligned} \left[ \mathbf{C}\_{P\_{ll}}^{pid}(\mathbf{s}) \mathbf{G}\_{P\_{ll}}(\mathbf{s}) \right]\_{N} &= \left( \mathbf{K}\_{D} \mathbf{s}^{2} + \mathbf{K}\_{P} \mathbf{s} + \mathbf{K}\_{I} \right) (b\_{1} + b\_{0}) \\ &= \mathbf{K}\_{D} b\_{1} \mathbf{s}^{3} + (\mathbf{K}\_{D} b\_{0} + \mathbf{K}\_{I} b\_{1}) \mathbf{s}^{2} + \mathbf{K}\_{I} b\_{0}. \end{aligned} \tag{34}$$

The product of the TF denominator of Plant II given in Eq. (33) associated with the TF denominator of the controller given in Eq. (2) is given by

*Adjustment of the PID Gains Vector Due to Parametric Variations in the Plant Model… DOI: http://dx.doi.org/10.5772/intechopen.95051*

$$\left[\mathbf{C}\_{P\_{\rm II}}^{pid}(\mathbf{s})\mathbf{G}\_{P\rm II}(\mathbf{s})\right]\_{\rm D} = \mathfrak{s}\left(\mathbf{s}^{3} + a\_{1}\mathfrak{s}^{2} + a\_{2}\mathfrak{s} + a\_{3}\right) = \mathfrak{s}^{4} + a\_{1}\mathfrak{s}^{3} + a\_{2}\mathfrak{s}^{2} + a\_{3}\mathfrak{s}.\tag{35}$$

The characteristic polynomial of Plant II is given by

The error calculation *a<sup>e</sup>*

*ae i* )

2 6 4

given by.

*<sup>K</sup>pid* )

method of ZN.

where *G<sup>G</sup>*

**76**

**4.2 Plant II**

8 ><

>:

mance of the proposed method.

(29) and with the coefficients *ai* given in (28) is given by

<sup>1</sup> � *<sup>a</sup>*<sup>1</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

<sup>2</sup> � *<sup>a</sup>*<sup>2</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

<sup>3</sup> � *<sup>a</sup>*<sup>3</sup> <sup>¼</sup> *ae*

0*:*438 0 0 0 0*:*438 0 0 00*:*438

*as*

8 ><

>:

(25), has three equations and three unknowns.

*as*

*as*

*<sup>i</sup>* for Plant I associated with the coefficients *a<sup>s</sup>*

<sup>1</sup> ¼ 0*:*8604 � 0*:*0861 ¼ 0*:*7743;

<sup>2</sup> ¼ 0*:*421 � 0*:*0421 ¼ 0*:*3789;

<sup>3</sup> ¼ 0*:*0641 � 0 ¼ 0*:*0641*:*

*KD KP KI*

2 6 4

0*:*7743 0*:*3789 0*:*0641 3 7

<sup>5</sup> *:* (31)

2 6 4

The calculation of the *Kpid* gains vector of Plant I, is done by replacing the numerical values of Eq. (27) in the system of equations given in (25) and (26), is

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

The Plant I given in Eq. (21) related to Eq. (21) has only the coefficient *b*0, with that, the system of equations generated, related to the system of equations given in

> *i*Þ 0*:*438*KD* ¼ 0*:*7749 ) *KD* ¼ 0*:*7749*=*0*:*431 ) *KD* ¼ 1*:*78; *ii*Þ 0*:*438*KP* ¼ 0*:*3789 ) *KP* ¼ 0*:*3789*=*0*:*431 ) *KP* ¼ 0*:*87; *iii*Þ 0*:*438*KI* ¼ 0*:*0641 ) *KI* ¼ 0*:*0641*=*0*:*438 ) *KI* ¼ 0*:*15*:*

Solving the system of equations given in (32), you can start with any of the equations to find the numerical values of *KD*, *KP* and *KI*, since they are independent. With the numerical values of the earnings *KD*, *KP* and *KI*, it replaces in the simulator developed in the MATLAB/SIMULINK software to monitor the perfor-

**Figure 2** shows the performance of the PID-Specified controller, which has the

Plant II, is a solid bulk reclaimer, which is used to recover bulk for ship loading,

*b*1*s* þ *b*<sup>0</sup> *s*<sup>3</sup> þ *a*1*s*<sup>2</sup> þ *a*2*s* þ *a*<sup>3</sup>

*PII*ð Þ*s* is the TF of Plant II, it is a third order plant with zero at infinity. The product of the TF numerator of Plant II given in Eq. (33) associated with the

> <sup>2</sup> <sup>þ</sup> *KPs* <sup>þ</sup> *KI* � �ð Þ *<sup>b</sup>*<sup>1</sup> <sup>þ</sup> *<sup>b</sup>*<sup>0</sup>

The product of the TF denominator of Plant II given in Eq. (33) associated with

<sup>3</sup> <sup>þ</sup> ð Þ *KDb*<sup>0</sup> <sup>þ</sup> *KIb*<sup>1</sup> *<sup>s</sup>*

determined by the internal product of the vector of gains with the propagation matrix in purchase with the controller with the gains determined by the second

this equipment has the capacity to move up to 8,000 tons per hour (t/h).

*GPII*ðÞ¼ *s*

TF numerator of the controller given in Eq. (2) is given by

*<sup>N</sup>* <sup>¼</sup> *KDs*

the TF denominator of the controller given in Eq. (2) is given by

¼ *KDb*1*s*

*Cpid*

*PII* ð Þ*s GPII*ð Þ*s* h i

transfer function parameters specified by the designer and the *Kpid*

*<sup>i</sup>* given in

(30)

(32)

*Specified* gain vector

, (33)

(34)

<sup>2</sup> <sup>þ</sup> *KIb*0*:*

$$\begin{split} P\_{Pl}(\mathbf{s}) = \mathbf{s}^4 + (a\_1 + K\_D b\_1)\mathbf{s}^3 \\ &+ (a\_2 + K\_D b\_0 + K\_I b\_1)\mathbf{s}^2 \\ &+ (a\_3 + (K\_D b\_0 + K\_I b\_1)\mathbf{s} \\ &+ K\_I b\_0. \end{split} \tag{36}$$

System equations of Plant II related to Eq. (19) in the form *Ax* ¼ *b* is given by

$$\begin{aligned} \left< K^{\mathrm{pid}}, \overline{b}\_{i} \right> = a\_{i}^{\epsilon} \Rightarrow \begin{cases} a\_{1} + K\_{D}b\_{1} = a\_{1}^{\epsilon}; \\ a\_{2} + K\_{D}b\_{0} + K\_{P}b\_{1} = a\_{2}^{\epsilon}; \\ a\_{3} + K\_{P}b\_{0} + K\_{I}b\_{1} = a\_{3}^{\epsilon}; \\ a\_{4} + K\_{I}b\_{0} = a\_{4}^{\epsilon}; \end{cases} \\ \Rightarrow \begin{cases} K\_{D}b\_{1} = a\_{1}^{\epsilon} - a\_{1}; \\ K\_{D}b\_{0} + K\_{P}b\_{1} = a\_{2}^{\epsilon} - a\_{2}; \\ K\_{P}b\_{0} + K\_{I}b\_{1} = a\_{3}^{\epsilon} - a\_{3}; \\ K\_{I}b\_{0} = a\_{4}^{\epsilon} - a\_{4}; \end{cases} \\ \Rightarrow \begin{cases} 1) K\_{D}b\_{1} = a\_{1}^{\epsilon}; \\ 2) K\_{D}b\_{0} + K\_{P}b\_{1} = a\_{2}^{\epsilon}; \\ 3) K\_{P}b\_{0} + K\_{I}b\_{1} = a\_{3}^{\epsilon}; \\ 4) K\_{I}b\_{0} = a\_{4}^{\epsilon}. \end{cases} \end{aligned} \tag{37}$$

**Figure 2.** *Plant I - PID-ZN x PID-specified.*

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

Placing the systems of equations given in (37) in the matrix form, we have

$$
\langle K^{pid}, \overline{B} \rangle = a\_i^{\epsilon} \Rightarrow \begin{bmatrix} b\_0 & 0 & 0 \\ b\_1 & b\_0 & 0 \\ 0 & b\_1 & b\_0 \\ 0 & 0 & b\_1 \end{bmatrix} \times \begin{bmatrix} K\_D \\ K\_P \\ K\_I \end{bmatrix} = \begin{bmatrix} a\_1^{\epsilon} \\ a\_2^{\epsilon} \\ a\_3^{\epsilon} \\ a\_4^{\epsilon} \end{bmatrix}. \tag{38}
$$

The transfer function of Plant II related to Eq. (33) is given by

$$G\_{Pl}(s) = \frac{0.1812s + 0.087}{s^3 + 0.3853s^2 + 0.117s + 0.01567}.\tag{39}$$

The *ai* coefficients of the Plant TF - II related to Eq. (39) are given by

$$a\_i \Rightarrow \begin{cases} a\_1 = 0.3853; \\ a\_2 = 0.117; \\ a\_3 = 0.01567; \\ a\_4 = 0. \end{cases} \tag{40}$$

The specified coefficients *a<sup>s</sup> <sup>i</sup>* of Plant II are given by

$$a\_{4i}^{\epsilon s} \Rightarrow \begin{cases} a\_1^{\epsilon} = 1.7133; \\ a\_2^{\epsilon} = 0.8542; \\ a\_3^{\epsilon} = 0.2670; \\ a\_4^{\epsilon} = 0.0478. \end{cases} \tag{41}$$

Solving the system of equations given in (44), first, solve Equation i) to find the numerical value of *KD*. Then, replace the numerical value of *KD* in Equation ii) and find the numerical value of *KP*. To find the numerical value of *KI*, solve equation iv) or replace the values of *KP* in equation iii). With the

*Adjustment of the PID Gains Vector Due to Parametric Variations in the Plant Model…*

*DOI: http://dx.doi.org/10.5772/intechopen.95051*

numerical values of the gains *KD*, *KP* and *KI*, it replaces in the simulator developed in the MATLAB/SIMULINK software to monitor the performance of the proposed

**Figure 3** shows the performance of the PID-Specified controller, which has the

determined by the internal product of the vector of gains with the propagation matrix in purchase with the controller with the gains determined by the second

Plant III, is a car dumper with two feeders, which is used to unload solids in bulk, this equipment has the capacity to move up to 8,000 tons per hour (t/h). The

*b*2*s*

The product of the TF numerator of Plant III given in Eq. (45) associated with

<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>s</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>0</sup> *s*<sup>4</sup> þ *a*1*s*<sup>3</sup> þ *a*2*s*<sup>2</sup> þ *a*3*s* þ *a*<sup>4</sup>

*PIII*ð Þ*s* is the TF of Plant III, it is a fourth order plant with zero at infinity.

*Specified* gain vector

*:* (45)

transfer function parameters specified by the designer and the *Kpid*

general mathematical model of Plant III in TF is given by

the TF numerator of the controller given in Eq. (2) is given by

*G<sup>G</sup> PIII*ðÞ¼ *s*

method.

**Figure 3.**

*Plant II - PID-ZN x PID-specified.*

method of ZN.

**4.3 Plant III**

where, *G<sup>G</sup>*

**79**

The error calculation *a<sup>e</sup> <sup>i</sup>* for Plant II associated with the coefficients *as <sup>i</sup>* given in (41) and with the coefficients *ai* given in (40) is given by

$$a\_{4i}^{\epsilon \epsilon} \Rightarrow \begin{cases} a\_1^{\epsilon} = a\_1^{\epsilon} - a\_1 = 1.7133 - 0.3853 = 1.328; \\ \mathbf{a}\_2^{\epsilon} = a\_2^{\epsilon} - a\_2 = 0.7372 - 0.117 = 0.6202; \\ a\_3^{\epsilon} = a\_3^{\epsilon} - a\_3 = 0.2514 - 0.01567 = 0.2358; \\ a\_4^{\epsilon} = a\_4^{\epsilon} - a\_4 = 0.0478 - 0 = 0.0478. \end{cases} \tag{42}$$

The calculation of the *Kpid* gains vector is performed by replacing the numerical values of Eq. (39) in the system of equations given in (37) and (38).

$$
\begin{bmatrix} 0.087 & 0 & 0 \\ 0.1812 & 0.087 & 0 \\ 0 & 0.1812 & 0.087 \\ 0 & 0 & 0.1812 \end{bmatrix} \times \begin{bmatrix} K\_D \\ K\_P \\ K\_I \end{bmatrix} = \begin{bmatrix} 1.3280 \\ 0.9538 \\ 0.0996 \\ 0.0478 \end{bmatrix}. \tag{43}
$$

The plant II given in Eq. (39) related to Eq. (33) has the *b*<sup>0</sup> coefficients and *b*1, with this, the system of equations generated, for the system of equations related to the system of equations given in (37), has four equations and three unknowns.

$$K^{\text{pid}} \Rightarrow \begin{cases} \text{i)} \ K\_{\text{D}}b\_{1} = a\_{1}^{\epsilon} \Rightarrow K\_{\text{D}} = 1.328/0.1812 \Rightarrow K\_{\text{D}} = 7.329; \\ \text{ii)} \ K\_{\text{D}}b\_{0} + K\_{\text{P}}b\_{1} = a\_{2}^{\epsilon} \Rightarrow K\_{\text{P}} = (0.9538 - 0.6376)/0.1812 \Rightarrow K\_{\text{P}} = 1.745; \\ \text{iii)} \ K\_{\text{P}}b\_{0} + K\_{\text{I}}b\_{1} = a\_{3}^{\epsilon} \Rightarrow K\_{\text{I}} = (2514 - 0.1518)/0.1812 \Rightarrow K\_{\text{I}} = 0.549; \\ \text{iv)} \ K\_{\text{I}}b\_{0} = a\_{4}^{\epsilon} \Rightarrow K\_{\text{I}} = 0.0478/0.087 \Rightarrow K\_{\text{I}} = 0.549. \end{cases} \text{(44)}$$

*Adjustment of the PID Gains Vector Due to Parametric Variations in the Plant Model… DOI: http://dx.doi.org/10.5772/intechopen.95051*

**Figure 3.** *Plant II - PID-ZN x PID-specified.*

Placing the systems of equations given in (37) in the matrix form, we have

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

*b*<sup>0</sup> 0 0 *b*<sup>1</sup> *b*<sup>0</sup> 0 0 *b*<sup>1</sup> *b*<sup>0</sup> 0 0 *b*<sup>1</sup>

0*:*1812*s* þ 0*:*087

*a*<sup>1</sup> ¼ 0*:*3853; *a*<sup>2</sup> ¼ 0*:*117; *a*<sup>3</sup> ¼ 0*:*01567;

*a*<sup>4</sup> ¼ 0*:*

*<sup>i</sup>* of Plant II are given by

<sup>1</sup> ¼ 1*:*7133;

<sup>2</sup> ¼ 0*:*8542;

<sup>3</sup> ¼ 0*:*2670;

<sup>4</sup> ¼ 0*:*0478*:*

<sup>1</sup> � *a*<sup>1</sup> ¼ 1*:*7133 � 0*:*3853 ¼ 1*:*328;

<sup>2</sup> � *a*<sup>2</sup> ¼ 0*:*7372 � 0*:*117 ¼ 0*:*6202;

<sup>4</sup> � *a*<sup>4</sup> ¼ 0*:*0478 � 0 ¼ 0*:*0478*:*

The calculation of the *Kpid* gains vector is performed by replacing the numerical

The plant II given in Eq. (39) related to Eq. (33) has the *b*<sup>0</sup> coefficients and *b*1, with this, the system of equations generated, for the system of equations related to the system of equations given in (37), has four equations and three unknowns.

<sup>1</sup> ) *KD* ¼ 1*:*328*=*0*:*1812 ) *KD* ¼ 7*:*329;

<sup>4</sup> ) *KI* ¼ 0*:*0478*=*0*:*087 ) *KI* ¼ 0*:*549*:*

values of Eq. (39) in the system of equations given in (37) and (38).

<sup>3</sup> � *a*<sup>3</sup> ¼ 0*:*2514 � 0*:*01567 ¼ 0*:*2358;

*KD KP KI*

<sup>2</sup> ) *KP* ¼ ð Þ 0*:*9538 � 0*:*6376 *=*0*:*1812 ) *KP* ¼ 1*:*745;

<sup>3</sup> ) *KI* ¼ ð Þ 2514 � 0*:*1518 *=*0*:*1812 ) *KI* ¼ 0*:*549;

1*:*3280 0*:*9538 0*:*0996 0*:*0478

*:* (43)

(44)

2 6 4

*<sup>i</sup>* for Plant II associated with the coefficients *as*

*as*

8 >>><

>>>:

*as*

*as*

*as*

*KD KP KI*

*<sup>s</sup>*<sup>3</sup> <sup>þ</sup> <sup>0</sup>*:*3853*s*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*117*<sup>s</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>01567</sup> *:* (39)

*ae* 1 *ae* 2 *ae* 3 *ae* 4

*:* (38)

(40)

(41)

(42)

*<sup>i</sup>* given in

2 6 4

*<sup>K</sup>pid*, *<sup>B</sup>* � � <sup>¼</sup> *ae*

*GPII*ðÞ¼ *s*

The specified coefficients *a<sup>s</sup>*

The error calculation *a<sup>e</sup>*

*ae* 4 *e <sup>i</sup>* )

*<sup>i</sup>*<sup>Þ</sup> *KDb*<sup>1</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

*iv*<sup>Þ</sup> *KIb*<sup>0</sup> <sup>¼</sup> *ae*

*ii*<sup>Þ</sup> *KDb*<sup>0</sup> <sup>þ</sup> *KPb*<sup>1</sup> <sup>¼</sup> *ae*

*iii*<sup>Þ</sup> <sup>K</sup>*Pb*<sup>0</sup> <sup>þ</sup> *KIb*<sup>1</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

*<sup>K</sup>pid* )

**78**

8 >>><

>>>:

*i* )

The transfer function of Plant II related to Eq. (33) is given by

*ai* )

*ae* 4 *s <sup>i</sup>* )

(41) and with the coefficients *ai* given in (40) is given by

*ae* <sup>1</sup> <sup>¼</sup> *<sup>a</sup><sup>s</sup>*

8 >>><

>>>:

a*e* <sup>2</sup> <sup>¼</sup> *as*

*ae* <sup>3</sup> <sup>¼</sup> *<sup>a</sup><sup>s</sup>*

*ae* <sup>4</sup> <sup>¼</sup> *<sup>a</sup><sup>s</sup>*

0*:*087 0 0 0*:*1812 0*:*087 0 0 0*:*1812 0*:*087 0 00*:*1812

The *ai* coefficients of the Plant TF - II related to Eq. (39) are given by

8 >>><

>>>:

Solving the system of equations given in (44), first, solve Equation i) to find the numerical value of *KD*. Then, replace the numerical value of *KD* in Equation ii) and find the numerical value of *KP*. To find the numerical value of *KI*, solve equation iv) or replace the values of *KP* in equation iii). With the numerical values of the gains *KD*, *KP* and *KI*, it replaces in the simulator developed in the MATLAB/SIMULINK software to monitor the performance of the proposed method.

**Figure 3** shows the performance of the PID-Specified controller, which has the transfer function parameters specified by the designer and the *Kpid Specified* gain vector determined by the internal product of the vector of gains with the propagation matrix in purchase with the controller with the gains determined by the second method of ZN.
