**4.3 Plant III**

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 general mathematical model of Plant III in TF is given by

$$G\_{P\_{\rm III}}^{G}(\mathbf{s}) = \frac{b\_2 \mathbf{s}^2 + b\_1 \mathbf{s} + b\_0}{\mathbf{s}^4 + a\_1 \mathbf{s}^3 + a\_2 \mathbf{s}^2 + a\_3 \mathbf{s} + a\_4}.\tag{45}$$

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

The product of the TF numerator of Plant III given in Eq. (45) associated with the TF numerator of the controller given in Eq. (2) is given by

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

$$\begin{aligned} \left[\mathbf{C}\_{P\_{\rm lll}}^{\rm pid}(\mathbf{s})\mathbf{G}\_{P\rm ll}(\mathbf{s})\right]\_{N} &= \left(\mathbf{K}\_{D}\mathbf{s}^{2} + \mathbf{K}\_{P}\mathbf{s} + \mathbf{K}\_{I}\right) \left(\mathbf{b}\_{2}\mathbf{s}^{2} + \mathbf{b}\_{1}\mathbf{s} + \mathbf{b}\_{0}\right) \\ &= \mathbf{K}\_{D}\mathbf{b}\_{2}\mathbf{s}^{4} + (\mathbf{K}\_{D}\mathbf{b}\_{1} + \mathbf{K}\_{P}\mathbf{b}\_{2})\mathbf{s}^{3} \\ &+ (\mathbf{K}\_{D}\mathbf{b}\_{0} + \mathbf{K}\_{P}\mathbf{b}\_{1} + \mathbf{K}\_{I})\mathbf{s}^{2} \\ &+ (\mathbf{K}\_{P}\mathbf{b}\_{0} + \mathbf{K}\_{I}\mathbf{b}\_{1})\mathbf{s} \\ &+ \mathbf{K}\_{I}\mathbf{b}\_{0}. \end{aligned} \tag{4}$$

The transfer function of Plant III related to Eq. (45) is given by

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

0*:*959*s*

*ai* )

*as <sup>i</sup>* )

(53) and with the coefficients *ai* given in (52) is given by

*ae* <sup>1</sup> <sup>¼</sup> *as*

8 >>>>>><

>>>>>>:

*ae* <sup>2</sup> <sup>¼</sup> *<sup>a</sup><sup>s</sup>*

*ae* <sup>3</sup> <sup>¼</sup> *as*

*ae* <sup>4</sup> <sup>¼</sup> *as*

*ae* <sup>5</sup> <sup>¼</sup> *as*

of Eq. (51) in the system of equations given in (49) and (50).

0*:*1593 0 0 0*:*1698 0*:*1593 0 0*:*959 0*:*1698 0*:*1593 0 0*:*959 0*:*1698 0 00*:*959

The *ai* coefficients of the Plant TF - III related to Eq. (51) are given by

8 >>>>>><

>>>>>>:

<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*1698*<sup>s</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>1593</sup> *<sup>s</sup>*<sup>4</sup> <sup>þ</sup> <sup>0</sup>*:*1767*s*<sup>3</sup> <sup>þ</sup> <sup>0</sup>*:*3463*s*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*029*<sup>s</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>02331</sup> *:* (51)

(52)

(53)

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

(54)

*a*<sup>1</sup> ¼ 0*:*1767; *a*<sup>2</sup> ¼ 0*:*3463; *a*<sup>3</sup> ¼ 0*:*029; *a*<sup>4</sup> ¼ 0*:*2331; *a*<sup>5</sup> ¼ 0*:*

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

<sup>1</sup> ¼ 1*:*8358;

<sup>2</sup> ¼ 8*:*619;

<sup>3</sup> ¼ 8*:*9947;

<sup>4</sup> ¼ 1*:*4277;

<sup>5</sup> ¼ 1, 3254*:*

<sup>1</sup> � *a*<sup>1</sup> ¼ 1*:*8358 � 0*:*1767 ¼ 1*:*659;

<sup>2</sup> � *a*<sup>2</sup> ¼ 8*:*619 � 0*:*3463 ¼ 4*:*9066;

<sup>3</sup> � *a*<sup>3</sup> ¼ 8*:*9947 � 0*:*029 ¼ 8*:*9657;

<sup>5</sup> � *a*<sup>5</sup> ¼ 1*:*3254 � 0 ¼ 1*:*3254*:*

The calculation of the gain vector *Kpid* is done by replacing the numerical values

The Plant III given in Eq. (51) related to Eq. (45), has the coefficients *b*0, *b*<sup>1</sup> and *b*2, with that, the generated system of equations, for the system of equations related to the system of equations given in (49), has five equations and three unknowns.

<sup>1</sup> ) *KD* ¼ 1*:*6591*=*959 ) *KD* ¼ 1*:*73;

<sup>5</sup> ) *KI* ¼ 1*:*3254*=*0*:*1593 ) *KI* ¼ 8*:*32*:*

�

*KD KP KI*

<sup>2</sup> ) *KP* ¼ 4*:*6128*=*0*:*959 ) *KP* ¼ 4*:*81;

<sup>4</sup> ) *KI* ¼ 1*:*4127*=*0*:*1698 ) *KI* ¼ 8*:*32;

<sup>3</sup> ) *KI* ¼ 7*:*978*=*0*:*959 ) *KI* ¼ 8*:*32;

1*:*6591 4*:*9066 9*:*0712 2*:*7381 1*:*3254

*:* (55)

(56)

2 6 4

<sup>4</sup> � *a*<sup>4</sup> ¼ 1*:*4277 � 0*:*0233 ¼ 1*:*4044;

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

*as*

8 >>>>>><

>>>>>>:

*as*

*as*

*as*

*as*

*GPIII*ðÞ¼ *s*

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

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

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

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

*<sup>i</sup>*<sup>Þ</sup> *KDb*<sup>2</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

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

*ii*<sup>Þ</sup> *KDb*<sup>1</sup> <sup>þ</sup> *KPb*<sup>2</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

*iv*<sup>Þ</sup> *KPb*<sup>0</sup> <sup>þ</sup> *KIb*<sup>1</sup> <sup>¼</sup> *ae*

*iii*<sup>Þ</sup> *KDb*<sup>0</sup> <sup>þ</sup> *KPb*<sup>1</sup> <sup>þ</sup> *KIb*<sup>2</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

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

**81**

8 >>>>>><

>>>>>>:

The product of the TF denominator of Plant III given in Eq. (45) associated with the TF numerator of the controller given in Eq. (2) is given by

$$\begin{aligned} \left[ \mathbf{C}\_{P\_{\rm III}}^{\rm pid}(\mathbf{s}) \mathbf{G}\_{P\_{\rm III}}(\mathbf{s}) \right]\_{D} &= \mathbf{s} \left( \mathbf{s}^{4} + a\_{1} \mathbf{s}^{3} + a\_{2} \mathbf{s}^{2} + a\_{3} \mathbf{s} + a\_{4} \right) \\ &= \mathbf{s}^{5} + a\_{1} \mathbf{s}^{4} + a\_{2} \mathbf{s}^{3} + a\_{3} \mathbf{s}^{2} + a\_{4} \mathbf{s}. \end{aligned} \tag{47}$$

The characteristic polynomial of Plant III is given by

$$\begin{aligned} P\_{P\_{\rm ll}}(\mathbf{s}) &= \mathbf{s}^5 + \mathbf{a}\_1 + K\_D b\_2 \mathbf{s}^4 \\ &+ \mathbf{a}\_2 + (K\_D b\_1 + K\_P b\_2) \mathbf{s}^3 \\ &+ \mathbf{a}\_3 + (K\_D b\_0 + K\_P b\_1 + K\_I b\_2) \mathbf{s}^2 \\ &+ \mathbf{a}\_4 + (K\_P b\_0 + K\_I b\_1) \mathbf{s} \\ &+ K\_I b\_0. \end{aligned} \tag{48}$$

System equations da Planta III related to Eq. (19) in the form *Ax* ¼ *b* is given by

$$\begin{aligned} \langle K^{\text{int}}, \overline{b}\_{i} \rangle = a\_{i}^{\*} \Rightarrow \begin{cases} a\_{1} + K\_{D}b\_{2} = a\_{1}^{\*}; \\ a\_{2} + K\_{D}b\_{1} + K\_{P}b\_{2} = a\_{2}^{\*}; \\ a\_{3} + K\_{D}b\_{0} + K\_{P}b\_{1} + K\_{I}b\_{2} = a\_{3}^{\*}; \\ a\_{4} + K\_{P}b\_{0} + K\_{I}b\_{1} = a\_{4}^{\*}; \\ a\_{5} + K\_{I}b\_{0} = a\_{5}^{\*}. \end{cases} \\ \Rightarrow \begin{cases} K\_{D}b\_{2} = a\_{1}^{\*} - a\_{1}; \\ K\_{D}b\_{1} + K\_{P}b\_{2} = a\_{2}^{\*} - a\_{2}; \\ K\_{D}b\_{0} + K\_{P}b\_{1} + K\_{I}b\_{2} = a\_{3}^{\*} - a\_{3}; \\ K\_{P}b\_{0} + K\_{I}b\_{1} = a\_{4}^{\*} - a\_{4}; \\ K\_{I}b\_{0} = a\_{5}^{\*} - a\_{5}. \end{cases} \\ \Rightarrow \begin{cases} 1) K\_{D}b\_{2} = a\_{1}^{\*}; \\ 3) K\_{D}b\_{0} + K\_{P}b\_{1} + K\_{I}b\_{2} = a\_{3}^{\*}; \\ 4) K\_{P}b\_{0} + K\_{I}b\_{1} = a\_{4}^{\*}; \\ 5) K\_{P}b\_{0} + K\_{I}b\_{1} = a\_{5}^{\*}; \\ 5) K\_{I}b\_{0} = a\_{5}^{\*}. \end{cases} \end{aligned}$$

Placing the systems of equations given in (49) in the matrix form, we have

$$
\left< K^{pid}, \overline{B} \right> = a\_i^\epsilon \Rightarrow \begin{bmatrix} b\_0 & 0 & 0 \\ b\_1 & b\_0 & 0 \\ b\_2 & b\_1 & b\_0 \\ 0 & b\_2 & b\_1 \\ 0 & 0 & b\_2 \end{bmatrix} \times \begin{bmatrix} K\_D \\ K\_P \\ K\_I \end{bmatrix} = \begin{bmatrix} a\_1^\epsilon \\ a\_2^\epsilon \\ a\_3^\epsilon \\ a\_4^\epsilon \\ a\_5^\epsilon \end{bmatrix}. \tag{50}
$$

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

The transfer function of Plant III related to Eq. (45) is given by

$$G\_{\rm Pl}(s) = \frac{0.959s^2 + 0.1698s + 0.1593}{s^4 + 0.1767s^3 + 0.3463s^2 + 0.029s + 0.02331}.\tag{51}$$

The *ai* coefficients of the Plant TF - III related to Eq. (51) are given by

$$a\_i \Rightarrow \begin{cases} a\_1 = 0.1767; \\ a\_2 = 0.3463; \\ a\_3 = 0.029; \\ a\_4 = 0.2331; \\ a\_5 = 0. \end{cases} \tag{52}$$

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

$$a\_i^\circ \Rightarrow \begin{cases} a\_1^\circ = 1.8358; \\ a\_2^\circ = 8.619; \\ a\_3^\circ = 8.9947; \\ a\_4^\circ = 1.4277; \\ a\_5^\circ = 1,3254. \end{cases} \tag{53}$$

The error calculation *a<sup>e</sup> <sup>i</sup>* for Plant III associated with the coefficients *a<sup>s</sup> <sup>i</sup>* given in (53) and with the coefficients *ai* given in (52) is given by

$$a\_i^\epsilon \Rightarrow \begin{cases} a\_1^\epsilon = a\_1^\epsilon - a\_1 = 1.8358 - 0.1767 = 1.659; \\ a\_2^\epsilon = a\_2^\epsilon - a\_2 = 8.619 - 0.3463 = 4.9066; \\ a\_3^\epsilon = a\_3^\epsilon - a\_3 = 8.9947 - 0.029 = 8.9657; \\ a\_4^\epsilon = a\_4^\epsilon - a\_4 = 1.4277 - 0.0233 = 1.4044; \\ a\_5^\epsilon = a\_5^\epsilon - a\_5 = 1.3254 - 0 = 1.3254. \end{cases} \tag{54}$$

The calculation of the gain vector *Kpid* is done by replacing the numerical values of Eq. (51) in the system of equations given in (49) and (50).


The Plant III given in Eq. (51) related to Eq. (45), has the coefficients *b*0, *b*<sup>1</sup> and *b*2, with that, the generated system of equations, for the system of equations related to the system of equations given in (49), has five equations and three unknowns.

$$K^{pid} \Rightarrow \begin{cases} \text{i)} \, K\_D b\_2 = a\_1^e \Rightarrow K\_D = \text{1.6591/959} \Rightarrow K\_D = \text{1.73;}\\ \text{ii)} \, K\_D b\_1 + K\_P b\_2 = a\_2^e \Rightarrow K\_P = \text{4.6128/0.959} \Rightarrow K\_P = \text{4.81;}\\ \text{iii)} \, K\_D b\_0 + K\_P b\_1 + K\_I b\_2 = a\_3^e \Rightarrow K\_I = \text{7.978/0.959} \Rightarrow K\_I = \text{8.32;}\\ \text{iv)} \, K\_P b\_0 + K\_I b\_1 = a\_4^e \Rightarrow K\_I = \text{1.4127/0.1698} \Rightarrow K\_I = \text{8.32;}\\ \text{v)} \, K\_I b\_0 = a\_5^e \Rightarrow K\_I = \text{1.3254/0.1593} \Rightarrow K\_I = \text{8.32.} \end{cases} \text{(56)}$$

*Cpid*

*PIII*ð Þ*s GPIII*ð Þ*s* h i

*Cpid*

*PPIII*ðÞ¼ *s s*

*Kpid*, *bi* � � <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

*PIII*ð Þ*s GPIII*ð Þ*s* h i

<sup>5</sup> <sup>þ</sup> *<sup>a</sup>*<sup>1</sup> <sup>þ</sup> *KDb*2*<sup>s</sup>*

8 >>>>>><

>>>>>>:

)

*<sup>i</sup>* )

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

**80**

*<sup>i</sup>* )

*<sup>N</sup>* <sup>¼</sup> *KDs*

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

The characteristic polynomial of Plant III is given by

<sup>2</sup> <sup>þ</sup> *KPs* <sup>þ</sup> *KI* � � *b*2*s*

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

¼ *KDb*2*s*

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

<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*2*<sup>s</sup>*

<sup>4</sup> <sup>þ</sup> *<sup>a</sup>*2*<sup>s</sup>*

3 þ*a*<sup>3</sup> þ ð Þ *KDb*<sup>0</sup> þ *KPb*<sup>1</sup> þ *KIb*<sup>2</sup> *s*

2;

4;

<sup>2</sup> � *a*2;

<sup>4</sup> � *a*4;

1; <sup>2</sup>Þ*KDb*<sup>1</sup> <sup>þ</sup> *KPb*<sup>2</sup> <sup>¼</sup> *ae*

<sup>3</sup>Þ*KDb*<sup>0</sup> <sup>þ</sup> *KPb*<sup>1</sup> <sup>þ</sup> *KIb*<sup>2</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

*KD KP KI*

2 6 4

<sup>4</sup>Þ*KPb*<sup>0</sup> <sup>þ</sup> *KIb*<sup>1</sup> <sup>¼</sup> *ae*

5*:*

� �

<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*3*<sup>s</sup>*

þ*a*<sup>4</sup> þ ð Þ *KPb*<sup>0</sup> þ *KIb*<sup>1</sup> *s*

3;

<sup>3</sup> � *a*3;

2;

4;

*ae* 1 *ae* 2 *ae* 3 *ae* 4 *ae* 5

*:* (50)

3;

*<sup>D</sup>* <sup>¼</sup> *s s*<sup>4</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>s</sup>*

<sup>5</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>s</sup>*

System equations da Planta III related to Eq. (19) in the form *Ax* ¼ *b* is given by

1;

*<sup>a</sup>*<sup>3</sup> <sup>þ</sup> *KDb*<sup>0</sup> <sup>þ</sup> *KPb*<sup>1</sup> <sup>þ</sup> *KIb*<sup>2</sup> <sup>¼</sup> *as*

5*:*

*KDb*<sup>1</sup> <sup>þ</sup> *KPb*<sup>2</sup> <sup>¼</sup> *as*

*KPb*<sup>0</sup> <sup>þ</sup> *KIb*<sup>1</sup> <sup>¼</sup> *as*

<sup>1</sup> � *a*1;

*KDb*<sup>0</sup> <sup>þ</sup> *KPb*<sup>1</sup> <sup>þ</sup> *KIb*<sup>2</sup> <sup>¼</sup> *<sup>a</sup><sup>s</sup>*

<sup>5</sup> � *a*5*:*

<sup>1</sup>Þ*KDb*<sup>2</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

<sup>5</sup>Þ*KIb*<sup>0</sup> <sup>¼</sup> *ae*

�

*<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *KDb*<sup>1</sup> <sup>þ</sup> *KPb*<sup>2</sup> <sup>¼</sup> *as*

*<sup>a</sup>*<sup>4</sup> <sup>þ</sup> *KPb*<sup>0</sup> <sup>þ</sup> *KIb*<sup>1</sup> <sup>¼</sup> *<sup>a</sup><sup>s</sup>*

*KDb*<sup>2</sup> <sup>¼</sup> *as*

*KIb*<sup>0</sup> <sup>¼</sup> *<sup>a</sup><sup>s</sup>*

8 >>>>>><

>>>>>>:

Placing the systems of equations given in (49) in the matrix form, we have

*b*<sup>0</sup> 0 0 *b*<sup>1</sup> *b*<sup>0</sup> 0 *b*<sup>2</sup> *b*<sup>1</sup> *b*<sup>0</sup> 0 *b*<sup>2</sup> *b*<sup>1</sup> 0 0 *b*<sup>2</sup>

)

¼ *s*

þ*a*<sup>2</sup> þ ð Þ *KDb*<sup>1</sup> þ *KPb*<sup>2</sup> *s*

4

*<sup>a</sup>*<sup>1</sup> <sup>þ</sup> *KDb*<sup>2</sup> <sup>¼</sup> *<sup>a</sup><sup>s</sup>*

*<sup>a</sup>*<sup>5</sup> <sup>þ</sup> *KIb*<sup>0</sup> <sup>¼</sup> *as*

8 >>>>>><

>>>>>>:

<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>s</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>0</sup> � �

þð Þ *KDb*<sup>0</sup> þ *KPb*<sup>1</sup> þ *KI s*

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*3*<sup>s</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>4</sup>

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*4*s:*

2

þ*KIb*0*:*

3

þð Þ *KPb*<sup>0</sup> þ *KIb*<sup>1</sup> *s*

2

þ*KIb*0*:*

*:* (46)

(47)

(48)

(49)

<sup>4</sup> <sup>þ</sup> ð Þ *KDb*<sup>1</sup> <sup>þ</sup> *KPb*<sup>2</sup> *<sup>s</sup>*

and infrastructure. We acknowledge the Department of Computer Engineering of the UEMA for making this research possible. Finally, we also acknowledge CAPES and CNPq for promoting and supporting the advanced studies that contributed

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

to this work.

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

**Author details**

**83**

José Pinheiro de Moura1,2\*† and João Viana da Fonseca Neto2†

\*Address all correspondence to: josepinheiro@professor.uema.br

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

1 State University of Maranhão, Brazil

† These authors contributed equally.

2 Federal University of Maranhão, Brazil

provided the original work is properly cited.

**Figure 4.** *Plant III - PID-ZN x PID-specified.*

Solving the system of equations given in (56), 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 v) or replace the values of *KD* and *KP* in Equation iii) or you can substitute the value of *KP* in Equation iv). 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 4** 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.
