**3. PID adjustment problem**

The PID controller adjustment problem is formulated based on the parametric difference between the specified coefficients and the original coefficients of the TF denominator polynomial. This formulation is based on the references [20, 21]. Where the authors present the development of models for optimized online optimization that is based on computational intelligence approaches.

The formulation of the proposed PID controller adjustment problem is presented in this section and is illustrated by the block diagram of **Figure 1**. Thus, in the context of the proposal, the performance matrix of the control system provides the means to determine the values of the parameters of the gain vector of the PID controller *Kpid*.

The models are represented in the form of internal produc <,>, which is a notation widely used in this text as the product of two vectors (internal product) and the models are called internal product models of the plant. The internal product is the appropriate form for analysis, allowing the designer to observe the impact of the earnings vector parameters *Kpid* (*KD*, *KP* and *KI*), in the output of the dynamic system associated with the polynomial coefficients of the TF denominator.

The PID controller model, in terms of the ODE equations and the Laplace transform, obtains the TF of the PID controller in terms of the dot product, which is given by

$$C\_{K^{pid}}(\mathfrak{s}) = \frac{\left< K^{pid}, \mathfrak{s}^{pid} \right>}{\mathfrak{s}},\tag{4}$$

$$K^{pid} = \begin{bmatrix} K\_D \ K\_P \ K\_I \end{bmatrix}^T,\tag{5}$$

and *s pid* is vector-powered in *s* of the PID gains associated with transfer function numerator that is given by

$$\mathfrak{s}^{pid} = \begin{bmatrix} \mathfrak{s}^2 \ \mathfrak{s}^1 \ \mathfrak{s}^0 \end{bmatrix}^T. \tag{6}$$

### **3.1 PID model in the form of internal product**

Inserting the characteristics of the plant, through the values of the coefficients of the polynomials of the numerator and the denominator (poles and zeros), associated with the mathematical models in terms of TFs given in Eq. (1) in internal product form h i� and in Eq. (3) in generic form, which is given by

$$G\_p(s) = K \frac{\langle b\_k, s^m \rangle}{s^n + \langle a\_i, s^{n-1} \rangle},\tag{7}$$

where *Gp*ð Þ*s* is the TF in the form of internal product and the coefficients *ai* with *i* ¼ 1, …, *n* and *bk* with *k* ¼ 0, …, *m* are a combination of the *spi* poles and *szk* zeros.

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