**3.3 Propagation of PID terms x** *bk* **coefficients**

The development of the polynomials of the numerator (zeros) and the denominator (poles) consists of the propagation of the gain vector *Kpid* of the controller by the numerator coefficients (*bk*) associated with the coefficients of the denominated (*ai*) TF of the plant. The equationing of the problem is given in the form of an internal product that weights the coefficients of the polynomial of zeros in the closed-loop and additive to the dynamics of the closed-loop transfer function.

## *3.3.1 Polynomial of zeros*

When replacing Eqs. (1) and (2) in Eq. (10), the numerator polynomial of the closed-loop TF is obtained, which is given by

$$N^{\rm CL}(\mathfrak{s}) = \mathcal{C}\_{\mathcal{K}^{\rm mid}}(\mathfrak{s}) \mathcal{G}(\mathfrak{s}).\tag{11}$$

Expanding and ordering Eq. (11), one obtains

$$\begin{aligned} N^{CL}(s) &= (b\_0 K\_D + b\_{-1} K\_P + b\_{-2} K\_I) s^{m + m^{\overline{r}d}} \\ &+ (b\_1 K\_D + b\_0 K\_P + b\_{-1} K\_I) s^{m + m^{\overline{r}d} - 1} \\ &+ (b\_2 K\_D + b\_1 K\_P + b\_0 K\_I) s^{m} \\ &+ (b\_3 K\_D + b\_2 K\_P + b\_1 K\_I) s^{m - 1} \end{aligned} \tag{12}$$

$$\begin{aligned} &\mathbf{a} + (\mathbf{b}\_m K\_D + \mathbf{b}\_{m-1} K\_P + \mathbf{b}\_{m-2} K\_I) \mathbf{s}^2 \\ &+ (\mathbf{b}\_{m+1} K\_D + \mathbf{b}\_m K\_P + \mathbf{b}\_{m-1} K\_I) \mathbf{s}^1 \\ &+ (\mathbf{b}\_{m+2} K\_D + \mathbf{b}\_{m+1} K\_P + \mathbf{b}\_m K\_I) \mathbf{s}^0 \end{aligned}$$

In terms of inner product, the general polynomial form of the closed-loop numerator polynomial is given by

$$\mathcal{N}^{CL}(s) = \sum\_{i=0}^{m\_{cl}} \langle \mathcal{K}^{pid}, \overline{b}\_{k-1} \rangle s^{m\_{cl}-i},\tag{13}$$

**3.4 Proposed method**

*3.4.1 Propagation matrix of PID design*

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

coefficients of PID controllers.

tion in the dynamics of order *s*

*3.4.2 Proposed characteristic polynomial*

dynamics.

or poles.

**73**

represented by

The problem is formulated based on the propagation matrix generated from the dot product between the terms of the earnings vector *Kpid* with the coefficients of the numerator *bk* associated with the coefficients of the denominator *ai* of the TF of plant. The propagation matrix product associated with the TF numerator coefficients of the plant, give rise to a new characteristic polynomial based on new specified operating points, which are imposed by new zeros and new poles.

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

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

> *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> *b*<sup>3</sup> *b*<sup>2</sup> *b*<sup>1</sup> ⋯⋯ ⋯ *bm bm*�<sup>1</sup> *bm*�<sup>2</sup> 0 *bm bm*�<sup>1</sup> 0 0 *bm*

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

columns represent the gains of the controller in the poles and zeros of the plant

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

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 propaga-

*ncl* and ending in the dynamics of order *s*

*:* (18)

<sup>0</sup> zero. The

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

*B* ¼

*KI* with the coefficients *bk*, *bk*�1, and *bk*�<sup>2</sup> of the closed-loop TF.

where *mcl* <sup>¼</sup> *<sup>m</sup>* <sup>þ</sup> *mpid* and vector *bk* of the polynomial of zeros of the closed-loop system is given by

$$
\overline{b}\_k = [b\_k \quad b\_{k-1} \quad b\_{k-2}].\tag{14}
$$

In similar way Eq. (13), one obtains the closed-loop denominator polynomial is given by

$$\begin{split} D^{CL}(\mathfrak{s}) &= \mathfrak{s}^{n+n\_{D-\mathrm{pid}}} \\ &+ \sum\_{i=0}^{n+n\_{D-\mathrm{pid}}-1} (b\_k K\_D + b\_{k-1} K\_P + b\_{k-2} K\_I + a\_{i+1}), \end{split} \tag{15}$$

where *nD*�*pid* ¼ 0 or 1. When *nD*�*pid* ¼ 0, the PID controller structure have the terms derivative and proportional. When *nD*�*pid* ¼ 1, the structure of the PID controller has an integrator term that increases the order of the system by 1, starting with the three terms: proportional, derivative and integrative [21]. In this work, when *nD*�*pid* <sup>¼</sup> 0, the proposal is to specify an additional *<sup>a</sup><sup>s</sup> <sup>i</sup>* coefficient., to ensure that the PID controller has the three terms.
