**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: *R s*ð Þ is reference input, *W s*ð Þ is disturbance signal and *V s*ð Þ is noise signal, *Y s*ð Þ is plant output and *Ym*ð Þ*s* is plant output measured by the sensor, U(s) and E(s) are the control effort and closed loop error, respectively, that are internal control system performance variables.

Applying the Laplace transforms to the control elements of **Figure 1**, the generalized TF with a polynomial structure in the *s* domain is obtained that is given by

**Figure 1.** *Canonical block diagram of the closed-loop control system.*

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

$$G\_{\mathbf{g}}(\mathbf{s}) = \frac{b\_o s^m + b\_1 s^{m-1} + \dots + b\_{m-1} \mathbf{s} + b\_m}{a\_o s^n + a\_1 s^{n-1} + \dots + a\_{n-1} + a\_n},\tag{1}$$

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

### **2.2 PID controller model**

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 Based on PID Framework - The Mutual Promotion of Control and Identification…*

variations in the coefficients of the TF polynomial characteristic.

Finally, the conclusion of the work is presented in Section 5.

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

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.

Adjusting the PID controller gain parameters is not a trivial task and requires indepth 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

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: *R s*ð Þ is reference input, *W s*ð Þ is disturbance signal and *V s*ð Þ is noise signal, *Y s*ð Þ is plant output and *Ym*ð Þ*s* is plant output measured by the sensor, U(s) and E(s) are the control effort and closed loop error, respectively, that are internal control

Applying the Laplace transforms to the control elements of **Figure 1**, the generalized TF with a polynomial structure in the *s* domain is obtained that is given by

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

control system.

**2. Preliminaries**

s domain [6, 7].

**Figure 1.**

**68**

system performance variables.

*Canonical block diagram of the closed-loop control system.*

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

$$\mathbf{C}\_{K^{p\bar{u}}}(\mathbf{s}) = \frac{K\_D \mathbf{s}^2 + K\_P \mathbf{s} + K\_I}{\mathbf{s}},\tag{2}$$

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

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 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 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

$$\mathbf{G}\_{\mathcal{g}}(\boldsymbol{s}) = K \frac{\prod\_{k=1}^{m} (\boldsymbol{s} - \boldsymbol{s}\_{xk})}{\prod\_{i=1}^{n} (\boldsymbol{s} - \boldsymbol{s}\_{pi})},\tag{3}$$

where *spi* ¼ *ωdi* are the poles and *szk* ¼ *σkωdk* are the zeros of the dynamic system. The poles and zeros of the system are represented by the pair (*ζ*, *ωn*), the first component is the damping factor and the second is the undamped natural ffiffiffiffiffiffiffiffiffiffiffiffi q

**3.2 Open-loop transfer function**

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

where *GOL*

of order *mPID* <sup>þ</sup> *<sup>m</sup>* .

*V s*ðÞ¼ 0 are given by

*3.3.1 Polynomial of zeros*

**71**

and *n*> *m*.

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

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

*Kpid*, *s pid bk*, *s*

The structure of TF is determined by the relationship *n* þ 1≥2 þ *m*. For *n* þ 1 ¼

where *mPID* can only assume 0 (zero) or 1 (one) values. The PID is observed to impose a proper TF, if the closed-loop system is of order *n* þ 1 and the numerator is

According to the block diagram of **Figure 1**, the TFs *Y s*ð Þ*=R s*ð Þ, *W s*ðÞ¼ 0, and

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.

When replacing Eqs. (1) and (2) in Eq. (10), the numerator polynomial of the

þð Þ *b*1*KD* þ *b*0*KP* þ *b*�<sup>1</sup>*KI s*

þð Þ *b*2*KD* þ *b*1*KP* þ *b*0*KI s*

þð Þ *b*3*KD* þ *b*2*KP* þ *b*1*KI s*

*<sup>N</sup>CL*ðÞ¼ *<sup>s</sup>* ð Þ *<sup>b</sup>*0*KD* <sup>þ</sup> *<sup>b</sup>*�<sup>1</sup>*KP* <sup>þ</sup> *<sup>b</sup>*�<sup>2</sup>*KI <sup>s</sup>*

*CKpid* ð Þ*s G s*ð Þ

2 þ *m*, the system is proper and for *n* þ 1<2 þ *m* the system is strictly proper, thereby establishing a general relationship between the order of the PID controller and the order of plant dynamics. This relationship ensures that the system structure is adequate, not allowing the system to present a nonpractical structure. In this way,

it establishes that the relationship of the closed loop system is given by

*<sup>m</sup>* h i

*<sup>p</sup>* ð Þ*s* is the TF of the plant in the open loop and *K* the gain of the plant

*sn*þ<sup>1</sup> <sup>þ</sup> *s ai*, *sn*�<sup>1</sup> h i , (8)

*ncl* <sup>¼</sup> *<sup>m</sup>PID* <sup>þ</sup> *<sup>n</sup>*, (9)

<sup>1</sup> <sup>þ</sup> *CKpid* ð Þ*<sup>s</sup> G s*ð Þ*H s*ð Þ*:* (10)

*<sup>N</sup>CL*ðÞ¼ *<sup>s</sup> CKpid* ð Þ*<sup>s</sup> G s*ð Þ*:* (11)

*<sup>m</sup>*þ*mpid*

*m*

*m*�1

*<sup>m</sup>*þ*mpid*�<sup>1</sup>

*:* (12)

*GOL <sup>p</sup>* ðÞ¼ *s K*

*GCL <sup>p</sup>* ðÞ¼ *s*

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

closed-loop TF is obtained, which is given by

Expanding and ordering Eq. (11), one obtains

frequency, *σ<sup>i</sup>* ¼ *ζiωni*, and *ωdi* ¼ *ωni* <sup>1</sup> � *<sup>ζ</sup>*<sup>2</sup> *i* .
