**1. Introduction**

The tuning of the parameters of PID controllers is challenging and requires expertise to achieve superior performance [1]. PID controllers are extensively used in the industries. However, the controllers are often implemented without a derivative action because of the highly sensitive tuning of parameters, which affects the efficiency of the controller [2]. This study presents a methodology for tuning three terms of the PID controller simultaneously to ensure overall efficiency of the controller [3].

The advantage of the PID controller tuning methodology, which is based on the internal product of the PID terms that generates the propagation matrix (PM), is that a vector of the specified parameters of a characteristic polynomial can be projected, and an error vector is obtained on comparison with the parameters of the characteristic plant polynomial [4, 5]. The method minimizes the error from the

specified parameters, thereby facilitating the design of a high-performance PID controller. The method also enables the allocation of the poles by direct replacement using specified parameters, thereby ensuring the desired operating point of the control system.

*Gg* ðÞ¼ *s*

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

**2.2 PID controller model**

real-world systems [18].

**69**

mizes efforts of computational cost and control.

parameters. TF in the form of a product is given by

*bos*

constant and adjustment are made only to the *ai* parameters.

PID actions are represented by the transfer function that is given by

*CKpid* ðÞ¼ *s*

started around 1920 and continues to the present days [11–13].

*<sup>m</sup>* <sup>þ</sup> *<sup>b</sup>*1*<sup>s</sup>*

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

*<sup>m</sup>*�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *bm*�1*<sup>s</sup>* <sup>þ</sup> *bm*

, (1)

*aosn* þ *a*1*sn*�<sup>1</sup> þ … þ *an*�<sup>1</sup> þ *an*

where *Gg* ð Þ*s* is the general TF of the control system blocks diagram, related to **Figure 1**, *n* is the order of the plant model and the number of poles that are entered into the system and *m* is the number of zeros, which is associated with the PID actions of the controller. As an imposition of the controller gains values, the coefficients *ai* and *bk* are the adjustable parameters to compensate for the parametric variations of the plant. In the proposed formulation, the *bk* coefficients are kept

The controller model associated with the TF given in Eq. (1) is customized to perform the actions of the controller's PID terms, where *n* ¼ 1, *m* ¼ 2, *a*<sup>0</sup> ¼ 1, the

*KDs*

where *CKpid* ð Þ*s* is the controller model associated with the TF given in Eq. (1). Adjustments of parameters that meet the project specifications, can be found in a large number of scientific and technical publications in controle specialized books, conferences and high quality journals [8]. The importance of developing methods for adjusting parameters of PID controllers and systematizing applications in industrial processes of real-world plants, has the objective of meeting the project specifications contained in technological advances, in order to guarantee the optimal adjustment of the parameters of the PID term of the controller [9, 10]. The challenge of tuning with optimal performance of the parameters of a PID controller,

The parameters of the PID controllers are adjusted to adapt to the tuning needs in a combination of proportionality associated with the proportional action, lead associated with the derivative action, and delay associated with the integral action of the error signal. However, there are still many problems that can be solved with computational intelligence-based algorithms. The purpose of this work is to contribute with a method of tuning PID controllers, which can support the development of electronic devices that contribute to technological advancement and the evolution of industry 4.0 with logical planning units, for optimal, robust decisionmaking and adaptability [14, 15]. Such units must be based on digital control technologies and embedded systems [16] in real time [17], to be reliably deployed in

To meet the demands of design specitifications, the proposed solution contributes to the evolution in approaches of optimal and adaptive control, providing the optimization of the figures of merit [19], ensuring a solution with satisfactory performance, meeting the requirements specified in projects, in a way that mini-

TF is specified in the factored form, that is, by the roots of the numerator and

Q*<sup>m</sup>*

Q*<sup>n</sup>*

*<sup>k</sup>*¼<sup>1</sup> *<sup>s</sup>* � *<sup>s</sup>* ð Þ *zk*

*<sup>i</sup>*¼<sup>1</sup> *<sup>s</sup>* � *spi* � � , (3)

denominator polynomials associated with Eq. (1). TF in the factored form is represented in terms of product, where the designer inserts the specified or desired

*Gg* ðÞ¼ *s K*

<sup>2</sup> <sup>þ</sup> *KPs* <sup>þ</sup> *KI*

*<sup>s</sup>* , (2)

This chapter presents a formulation proposal to resolve the PID controller tuning problem. The proposal is based on the dot product of the gain vector parameters of the controller and the rows of the propagation matrix. The dot product represents the changes in the behavior of the plant that are determined by the parametric variations in the coefficients of the TF polynomial characteristic.

The following topics and proposal development are presented in the remainder of this chapter. In Section 2, a preliminary on the transfer functions of the plant and PID controller are presented. In terms of internal product, the main properties of PID controllers and the development of proposed method are presented in Section 3. Taking into account three industrial plants of the mining sector, computational evaluation experiments of the PID tuning proposal are presented in Section 4. Finally, the conclusion of the work is presented in Section 5.
