**4. Experiments**

The experimental results are evaluated in three plants with mathematical models in terms of TF obtained with real data, being: Plant I of second order, with a zero; Third order plan II, with two zeros and fourth order plant III, with three zeros.

### **4.1 Plant I**

Plant I, is a car dumper, which is used to unload solids in bulk, this equipment has the capacity to move up to 4,000 tons per hour (t/h). The general mathematical model of Plant I in TF is given by

$$G\_{p\_l}^G(\mathfrak{s}) = \frac{b\_0}{\mathfrak{s}^2 + a\_1 \mathfrak{s} + a\_2},\tag{21}$$

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

The product of the TF numerator of Plant I given in Eq. (21) associated with the TF numerator 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*

$$\left[\mathbf{C}\_{P\_I}^{\mathbf{K}^{\rm{mid}}}(\mathbf{s})\mathbf{G}\_{P\_I}^G(\mathbf{s})\right]\_N = \left(K\_D \mathbf{s}^2 + K\_P \mathbf{s} + K\_I\right) b\_0 = K\_D b\_0 \mathbf{s}^2 + K\_P b\_0 \mathbf{s} + K\_I b\_0. \tag{22}$$

The product of the TF denominator of Plant I given in Eq. (21) associated with the TF denominator of the controller given in Eq. (2) is given by

$$\left[\mathbf{C}\_{P\_l}^{R^{pid}}(\mathfrak{s})\mathbf{G}\_{P\_l}(\mathfrak{s})\right]\_D = \mathfrak{s}\left(\mathfrak{s}^2 + a\_1\mathfrak{s}^2 + a\_2\mathfrak{s}\right) = \mathfrak{s}^3 + a\_1\mathfrak{s}^2 + a\_2\mathfrak{s}.\tag{23}$$

The characteristic polynomial of Plant I is given by

$$P\_I(\mathbf{s}) = \mathbf{s}^3 + (a\_1 + K\_D b\_0)\mathbf{s}^2 + (a\_2 + K\_P b\_0)\mathbf{s} + K\_I b\_0. \tag{24}$$

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

$$\left< \mathcal{K}^{\mathrm{pid}}, \overline{b}\_{i} \right> = a\_{i}^{\varepsilon} \Rightarrow \begin{cases} a\_{1} + K\_{D}b\_{0} = a\_{1}^{\varepsilon}; \\ a\_{2} + K\_{D}b\_{0} = a\_{2}^{\varepsilon}; \\ a\_{3} + K\_{D}b\_{0} = a\_{3}^{\varepsilon}; \end{cases}$$

$$\Rightarrow \begin{cases} K\_{D}b\_{0} = a\_{1}^{\varepsilon} - a\_{1}; \\ K\_{P}b\_{0} = a\_{2}^{\varepsilon} - a\_{2}; \\ K\_{I}b\_{0} = a\_{3}^{\varepsilon} - a\_{3}; \end{cases} \tag{25}$$

$$\Rightarrow \begin{cases} 1) \, K\_{D}b\_{0} = a\_{1}^{\varepsilon}; \\ 2) \, K\_{P}b\_{0} = a\_{2}^{\varepsilon}; \\ 3) \, K\_{I}b\_{0} = a\_{3}^{\varepsilon}. \end{cases}$$

Placing the systems of equations given in (25) in matrix form, we have

$$
\langle \mathcal{K}^{pid}, \overline{\mathcal{B}} \rangle = a\_i^\epsilon \Rightarrow \begin{bmatrix} b\_0 & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & b\_0 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & b\_0 \end{bmatrix} \times \begin{bmatrix} K\_D \\ K\_P \\ K\_I \end{bmatrix} = \begin{bmatrix} a\_1^\epsilon \\ a\_2^\epsilon \\ a\_3^\epsilon \end{bmatrix}. \tag{26}
$$

The transfer function of Plant I related to Eq. (21) is given by

$$G\_{P\_l}(\mathbf{s}) = \frac{0.438}{\mathbf{1s}^2 + 0.086\mathbf{1s} + 0.042\mathbf{1s}}.\tag{27}$$

The *ai* coefficients of the transfer function of the Plant - I related to Eq. (27) are given by

$$a\_i \Rightarrow \begin{cases} a\_1 = 0.0861; \\ a\_2 = 0.0421; \\ a\_3 = 0. \end{cases} \tag{28}$$

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

$$a\_i^\epsilon \Rightarrow \begin{cases} a\_1^\epsilon = 0.8604; \\ a\_2^\epsilon = 0.421; \\ a\_3^\epsilon = 0.0641. \end{cases} \tag{29}$$

From Eq. (17), the equation system that has an unknown vector *Kpid* and design

Expanding the scalar representation of Eq. (19), the system of equations to be

*<sup>i</sup>* ) *<sup>K</sup>pid*, *bk*

where *bk* vector is assembled with the rows of the *B* matrix.

� � <sup>¼</sup> *as*

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

, *i* ¼ 1, 2, …, *n* þ 1 is assembled. In scalar form, this system of

*KDb*<sup>0</sup> <sup>þ</sup> *KP*<sup>0</sup> <sup>þ</sup> *KI*<sup>0</sup> <sup>¼</sup> *ae*

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

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

*KDbm* <sup>þ</sup> *KPbm*�<sup>1</sup> <sup>þ</sup> *KIbm*�<sup>2</sup> <sup>¼</sup> *ae*

*KDb*<sup>0</sup> <sup>þ</sup> *KPbm* <sup>þ</sup> *KIbm*�<sup>1</sup> <sup>¼</sup> *ae*

The formulation of the problem presented in Eq. (19) and expanded in Eq. (20) is the starting point for the development of forms of parametric variation problems

To determine the numerical values of the parameters *Kpid*, the following rules

characteristic polynomial are specified; rule-3) the dot product of the parameters of the gain vector *Kpid* is made with the rows of the matrix *B*, associated with the original *ai* parameters of the characteristic polynomial and with the specified *a<sup>s</sup>*

parameters; rule-4) the system of equations given in Eq. (20) and rule-5 the system of equations given in rule-4 (Eq. (20)) is solved to determine the numerical values

The experimental results are evaluated in three plants with mathematical models in terms of TF obtained with real data, being: Plant I of second order, with a zero; Third order plan II, with two zeros and fourth order plant III, with three zeros.

Plant I, is a car dumper, which is used to unload solids in bulk, this equipment has the capacity to move up to 4,000 tons per hour (t/h). The general mathematical

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

The product of the TF numerator of Plant I given in Eq. (21) associated with the

ð Þ*s* is the TF of Plant I, it is a second order plant with zero at infinity.

*G<sup>G</sup> PI* ðÞ¼ *s*

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

are presented: rule-1) the *B* matrix is assembled via Eq. (18), where *bk* is the

⋮ þ ⋮ þ ⋮ ¼ ⋮ *KD*<sup>0</sup> <sup>þ</sup> *KP*<sup>0</sup> <sup>þ</sup> *KIb*<sup>0</sup> <sup>¼</sup> *ae*

*<sup>i</sup>* � *ai* ) *<sup>K</sup>pid*, *bk*

� � <sup>¼</sup> *ae*

*<sup>i</sup> :* (19)

(20)

*i*

1

2

3

4

5

*n*

*<sup>i</sup>* parameters of the

, (21)

specifications *a<sup>s</sup>*

solved is given by

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

*i*

equations is represented by

*ai* <sup>þ</sup> *<sup>K</sup>pid*, *bk*

� � <sup>¼</sup> *as*

*<sup>i</sup>* )

of the parameters of the gains vector *Kpid*.

model of Plant I in TF is given by

**4. Experiments**

**4.1 Plant I**

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

**74**

*PI*

8

>>>>>>>>>>><

>>>>>>>>>>>:

of TFs, as well as, for the establishment of operational points.

coefficients of TF numerator polynomial; rule-2) the new *a<sup>s</sup>*

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

The error calculation *a<sup>e</sup> <sup>i</sup>* for Plant I associated with the coefficients *a<sup>s</sup> <sup>i</sup>* given in (29) and with the coefficients *ai* given in (28) is given by

$$a\_i^\epsilon \Rightarrow \begin{cases} a\_1^\epsilon - a\_1 = a\_1^\epsilon = 0.8604 - 0.0861 = 0.7743; \\ a\_2^\epsilon - a\_2 = a\_2^\epsilon = 0.421 - 0.0421 = 0.3789; \\ a\_3^\epsilon - a\_3 = a\_3^\epsilon = 0.0641 - 0 = 0.0641. \end{cases} \tag{30}$$

*Cpid*

**Figure 2.**

**77**

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

*PII* ð Þ*s GPII*ð Þ*s* h i

*PPII*ðÞ¼ *s s*

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

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

*<sup>D</sup>* <sup>¼</sup> *s s*<sup>3</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>s</sup>*

The characteristic polynomial of Plant II is given by

*<sup>i</sup>* )

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

8 >>>>><

>>>>>:

)

8 >>>>><

>>>>>:

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*<sup>s</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>3</sup> � � <sup>¼</sup> *<sup>s</sup>*

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

3

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

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

*<sup>a</sup>*<sup>4</sup> <sup>þ</sup> *KIb*<sup>0</sup> <sup>¼</sup> *as*

*KDb*<sup>1</sup> <sup>¼</sup> *as*

*KIb*<sup>0</sup> <sup>¼</sup> *as*

)

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

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

2

2;

3;

<sup>2</sup> � *a*2;

<sup>3</sup> � *a*3;

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

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

1;

4*:*

2;

3;

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

1;

4;

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

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

8 >>>>><

>>>>>:

<sup>1</sup> � *a*1;

<sup>4</sup> � *a*4;

<sup>1</sup><sup>Þ</sup> *KDb*<sup>1</sup> <sup>¼</sup> *ae*

<sup>4</sup><sup>Þ</sup> *KIb*<sup>0</sup> <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

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

*<sup>a</sup>*<sup>3</sup> <sup>þ</sup> *KPb*<sup>0</sup> <sup>þ</sup> *KIb*<sup>1</sup> <sup>¼</sup> *<sup>a</sup><sup>s</sup>*

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

þ *KIb*0*:*

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*3*s:* (35)

(36)

(37)

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 given by.

$$
\begin{bmatrix} 0.438 & 0 & 0 \\ 0 & 0.438 & 0 \\ 0 & 0 & 0.438 \end{bmatrix} \times \begin{bmatrix} K\_D \\ K\_P \\ K\_I \end{bmatrix} = \begin{bmatrix} 0.7743 \\ 0.3789 \\ 0.0641 \end{bmatrix} . \tag{31}
$$

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 (25), has three equations and three unknowns.

$$K^{pid} \Rightarrow \begin{cases} i) \, 0.438 K\_D = 0.7749 \Rightarrow K\_D = 0.7749/0.431 \Rightarrow K\_D = 1.78; \\ ii) \, 0.438 K\_P = 0.3789 \Rightarrow K\_P = 0.3789/0.431 \Rightarrow K\_P = 0.87; \\ iii) \, 0.438 K\_I = 0.0641 \Rightarrow K\_I = 0.0641/0.438 \Rightarrow K\_I = 0.15. \end{cases} \tag{32}$$

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 performance of the proposed method.

**Figure 2** 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.
