*3.4.1 Propagation matrix of PID design*

The design is based on the propagation matrix, allowing the designer to specify new points of operation that improve the performance of the controller acting on the plant dynamics, where changes in the order and coefficients of the characteristic polynomial can be observed through the internal product of the zero and gain coefficients of PID controllers.

The problem is formulated based on the propagation matrix *B*, which is a consequence of the interaction between the parameters of the gain vector *Kpid* (*KD*, *KP*, *KI*) with the coefficients of the plant TF numerator, this matrix is represented by

$$
\overline{B} = \begin{bmatrix} b\_0 & 0 & 0 \\ b\_1 & b\_0 & 0 \\ b\_2 & b\_1 & b\_0 \\ b\_3 & b\_2 & b\_1 \\ \dots & \dots & \dots \\ b\_m & b\_{m-1} & b\_{m-2} \\ 0 & b\_m & b\_{m-1} \\ 0 & 0 & b\_m \end{bmatrix} . \tag{18}
$$

One case notice in [20] that the diagonals are not repeated, and they vary according to the order *n* of the system's characteristic polynomial. The propagation of the gains is weighted by the coefficients of the numerator polynomial. The closed-loop TF of the plant is given in terms of the product of the gains *KD*, *KP*, and *KI* with the coefficients *bk*, *bk*�1, and *bk*�<sup>2</sup> of the closed-loop TF.

The law of formation of the propagation Matrix (18) is ruled by *m* þ 2 rows and 3 columns. The rows represent the order of the system, starting with the propagation in the dynamics of order *s ncl* and ending in the dynamics of order *s* <sup>0</sup> zero. The columns represent the gains of the controller in the poles and zeros of the plant dynamics.

## *3.4.2 Proposed characteristic polynomial*

The proposed characteristic polynomial based on propagation matrix of PID controller gains idea is presented. From the system of equations that represents the actions of the PID controller in the plant dynamics, the formulation of the adjustment problem is established from the perspective of the inner product of the gains and the coefficients of the polynomials of the zeros of the closed-loop TF. In the case of the characteristic polynomial, the inner product is added to its coefficients. This way, the mechanism of gain adjustment is represented for allocations of zeros or poles.

From Eq. (17), the equation system that has an unknown vector *Kpid* and design specifications *a<sup>s</sup> i* , *i* ¼ 1, 2, …, *n* þ 1 is assembled. In scalar form, this system of equations is represented by

$$\left\langle a\_i + \left\langle \mathcal{K}^{\mathrm{pid}}, \overline{b}\_k \right\rangle = a\_i^\epsilon \Rightarrow \left\langle \mathcal{K}^{\mathrm{pid}}, \overline{b}\_k \right\rangle = a\_i^\epsilon - a\_i \Rightarrow \left\langle \mathcal{K}^{\mathrm{pid}}, \overline{b}\_k \right\rangle = a\_i^\epsilon. \tag{19}$$

*CKpid PI* ð Þ*<sup>s</sup> <sup>G</sup><sup>G</sup> PI* ð Þ*s*

h i

*CKpid*

*<sup>N</sup>* <sup>¼</sup> *KDs*

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

*PI* ð Þ*s GPI*ð Þ*s* h i

*PI*ðÞ¼ *s s*

*Kpid*, *bi* � � <sup>¼</sup> *ae*

*<sup>K</sup>pid*, *<sup>B</sup>* � � <sup>¼</sup> *<sup>a</sup><sup>e</sup>*

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

given by

**75**

*<sup>i</sup>* )

*GPI*ðÞ¼ *s*

2 6 4

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

*ai* )

*as <sup>i</sup>* ) 8 ><

>:

8 ><

>:

<sup>2</sup> <sup>þ</sup> *KPs* <sup>þ</sup> *KI* � �*b*<sup>0</sup> <sup>¼</sup> *KDb*0*<sup>s</sup>*

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

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

8 >><

>>:

)

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

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

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

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

*<sup>i</sup>* )

The characteristic polynomial of Plant I is given by

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

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

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

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

*<sup>a</sup>*<sup>3</sup> <sup>þ</sup> *KDb*<sup>0</sup> <sup>¼</sup> *as*

8 >><

>>:

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

*KPb*<sup>0</sup> <sup>¼</sup> *as*

*KIb*<sup>0</sup> <sup>¼</sup> *as*

8 >><

>>:

)

0*:*438 1*s*<sup>2</sup> þ 0*:*0861*s* þ 0*:*0421*s*

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

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

*as*

*as*

*as*

*a*<sup>1</sup> ¼ 0*:*0861; *a*<sup>2</sup> ¼ 0*:*0421; *a*<sup>3</sup> ¼ 0*:*

<sup>1</sup> ¼ 0*:*8604;

<sup>2</sup> ¼ 0*:*421;

<sup>3</sup> ¼ 0*:*0641*:*

1;

2;

3;

<sup>1</sup> � *a*1;

<sup>2</sup> � *a*2;

<sup>3</sup> � *a*3;

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

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

<sup>3</sup><sup>Þ</sup> *KIb*<sup>0</sup> <sup>¼</sup> *ae*

*KD KP KI*

2 6 4 1;

2;

3*:*

*ae* 1 *ae* 2 *ae* 3 3 7

<sup>5</sup>*:* (26)

(28)

(29)

*:* (27)

2 6 4

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

<sup>2</sup> <sup>þ</sup> *KPb*0*<sup>s</sup>* <sup>þ</sup> *KIb*0*:* (22)

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2*s:* (23)

(25)

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

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *KPb*<sup>0</sup> *<sup>s</sup>* <sup>þ</sup> *KIb*0*:* (24)

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

Expanding the scalar representation of Eq. (19), the system of equations to be solved is given by

$$
\left< K^{\mathrm{pid}}, \overline{b}\_{k} \right> = a\_i^{\epsilon} \Rightarrow \begin{cases}
K\_D b\_0 & + & K\_P \mathbf{0} & + & K\_I \mathbf{0} & = & a\_1^{\epsilon} \\
K\_D b\_1 & + & K\_P b\_0 & + & K\_I \mathbf{0} & = & a\_2^{\epsilon} \\
K\_D b\_2 & + & K\_P b\_1 & + & K\_I b\_0 & = & a\_3^{\epsilon} \\
K\_D b\_m & + & K\_P b\_{m-1} & + & K\_I b\_{m-2} & = & a\_4^{\epsilon} \\
K\_D b\_0 & + & K\_P b\_m & + & K\_I b\_{m-1} & = & a\_5^{\epsilon} \\
\vdots & + & \vdots & + & \vdots & = & \vdots \\
K\_D \mathbf{0} & + & K\_P \mathbf{0} & + & K\_I b\_0 & = & a\_n^{\epsilon}
\end{cases} \tag{20}
$$

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

To determine the numerical values of the parameters *Kpid*, the following rules are presented: rule-1) the *B* matrix is assembled via Eq. (18), where *bk* is the coefficients of TF numerator polynomial; rule-2) the new *a<sup>s</sup> <sup>i</sup>* parameters of the 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> i* 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 of the parameters of the gains vector *Kpid*.
