## Abstract

The tuning of the gains of a controller with proportional-integral-derivative (PID) actions has been prevalent in the industry. The adjustment of these gains in PID controllers is often determined by classical methods, such as Ziegler-Nichols, and trial and error. However, these methods fail to deliver satisfactory performance and often do not meet specific project demands because of the inherent complexity of industrial processes, such as plant parameter variations. To solve the tuning problem in highly complex industrial processes, a controller adjustment method based on the internal product of PID terms is proposed, and a propagation matrix (PM) is generated by the numerator coefficients of the plant transfer function (TF). In the proposed method, each term of the PID controller is influenced by each of the numerator and the denominator coefficients. Mathematical models of practical plants, such as unloading and resumption of bulk solids by car dumpers and bucket wheel resumption, were employed to evaluate the proposed method. The obtained results demonstrated an assertive improvement in the adjustment gains from PID actions, thereby validating it as a promising alternative to conventional methods.

### Keywords

- parameter variations
- industrial processes
- internal product
- propagation matrix
- PID actions

## 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.

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.

## 2. Preliminaries

Adjusting the PID controller gain parameters is not a trivial task and requires in-depth knowledge from the experts. In this work, the problem of tuning PID controllers is based on original studies regarding polynomial compensators and problems strongly related to the specification of parameters to meet operational constraints in plant dynamics, presented as a particular form of compensators in the s domain [6, 7].

### 2.1 Mathematical model of the plant in terms of transfer function

The plant’s dynamic system is represented by ordinary differential equations (ODE), described by TFs in the s domain (Laplace transform). The ODE concept in terms of TFs established in this chapter is in accordance with the block diagram shown in Figure 1, where the closed loop system relates the input and output signals:

Applying the Laplace transforms to the control elements of Figure 1, the generalized TF with a polynomial structure in the

where

### 2.2 PID controller model

The controller model associated with the TF given in Eq. (1) is customized to perform the actions of the controller’s PID terms, where

where

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, started around 1920 and continues to the present days [11, 12, 13].

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 decision-making 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 real-world systems [18].

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 minimizes efforts of computational cost and control.

TF is specified in the factored form, that is, by the roots of the numerator and 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 parameters. TF in the form of a product is given by

where

## 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

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

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

and

### 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

where

### 3.2 Open-loop transfer function

The open-loop FT or direct branch of the control system is given by

where

The structure of TF is determined by the relationship

where

According to the block diagram of Figure 1, the TFs

### 3.3 Propagation of PID terms x b k coefficients

The development of the polynomials of the numerator (zeros) and the denominator (poles) consists of the propagation of the gain vector

#### 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

Expanding and ordering Eq. (11), one obtains

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

where

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

where

#### 3.3.2 Characteristic polynomial

The general form of the closed-loop denominator polynomial is given by

where

the characteristic closed-loop polynomial for unit feedback (

### 3.4 Proposed method

The problem is formulated based on the propagation matrix generated from the dot product between the terms of the earnings vector

#### 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

One case notice in [20] that the diagonals are not repeated, and they vary according to the order

The law of formation of the propagation Matrix (18) is ruled by

#### 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

where

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

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

## 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

where,

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

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

The characteristic polynomial of Plant I is given by

System equations of Plant I related to Eq. (19) in the form

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

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

The

The specified coefficients

The error calculation

The calculation of the

The Plant I given in Eq. (21) related to Eq. (21) has only the coefficient

Solving the system of equations given in (32), you can start with any of the equations to find the numerical values of

Figure 2 shows the performance of the PID-Specified controller, which has the transfer function parameters specified by the designer and the

### 4.2 Plant II

Plant II, is a solid bulk reclaimer, which is used to recover bulk for ship loading, this equipment has the capacity to move up to 8,000 tons per hour (t/h).

where

The product of the TF numerator of Plant II given in Eq. (33) associated with the TF numerator of the controller given in Eq. (2) is given by

The product of the TF denominator of Plant II given in Eq. (33) associated with the TF denominator of the controller given in Eq. (2) is given by

The characteristic polynomial of Plant II is given by

System equations of Plant II related to Eq. (19) in the form

Placing the systems of equations given in (37) in the matrix form, we have

The transfer function of Plant II related to Eq. (33) is given by

The

The specified coefficients

The error calculation

The calculation of the

The plant II given in Eq. (39) related to Eq. (33) has the

Solving the system of equations given in (44), first, solve Equation i) to find the numerical value of

Figure 3 shows the performance of the PID-Specified controller, which has the transfer function parameters specified by the designer and the

### 4.3 Plant III

Plant III, is a car dumper with two feeders, which is used to unload solids in bulk, this equipment has the capacity to move up to 8,000 tons per hour (t/h). The general mathematical model of Plant III in TF is given by

where,

The product of the TF numerator of Plant III given in Eq. (45) associated with the TF numerator of the controller given in Eq. (2) is given by

The product of the TF denominator of Plant III given in Eq. (45) associated with the TF numerator of the controller given in Eq. (2) is given by

The characteristic polynomial of Plant III is given by

System equations da Planta III related to Eq. (19) in the form

Placing the systems of equations given in (49) in the matrix form, we have

The transfer function of Plant III related to Eq. (45) is given by

The

The specified coefficients

The error calculation

The calculation of the gain vector

The Plant III given in Eq. (51) related to Eq. (45), has the coefficients

Solving the system of equations given in (56), first, solve Equation i) to find the numerical value of

Figure 4 shows the performance of the PID-Specified controller, which has the transfer function parameters specified by the designer and the

## 5. Conclusions

The study presented a methodology for adjusting the gains of a PID controller in terms of the internal product of the gains and the propagation matrix. In addition, the relevance of the matrix was shown, which enabled impact assessment of the PID actions associated with the plant parameters. The proposed methodology complied with the project specifications and ensured high controller efficiency without suppressing the PID terms caused by the adjustments. The three PID controller terms were adjusted. Therefore, this methodology can be considered as an alternative to conventional methods for the computation of

## Acknowledgments

The authors would like to thank PPGEE of the UFMA for the resources to develop this work. We are especially grateful to FAPEMA for research incentive and infrastructure. We acknowledge the Department of Computer Engineering of the UEMA for making this research possible. Finally, we also acknowledge CAPES and CNPq for promoting and supporting the advanced studies that contributed to this work.