**3.2 Open-loop transfer function**

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

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

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>ζ</sup>*<sup>2</sup> *i*

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 opti-

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

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

*CKpid* ðÞ¼ *s*

*s pid* <sup>¼</sup> *<sup>s</sup>*

product form h i� and in Eq. (3) in generic form, which is given by

*Gp*ðÞ¼ *<sup>s</sup> <sup>K</sup> bk*, *<sup>s</sup>*

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

*Kpid*, *s pid* � �

*pid* is vector-powered in *s* of the PID gains associated with transfer function

2 *s* 1 *s* <sup>0</sup> � �*<sup>T</sup>*

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

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.

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

*<sup>s</sup>* , (4)

*:* (6)

*sn* <sup>þ</sup> *ai*, *sn*�<sup>1</sup> h i, (7)

*<sup>K</sup>pid* <sup>¼</sup> *KD KP KI* ½ �*<sup>T</sup>*, (5)

.

q

mization that is based on computational intelligence approaches.

frequency, *σ<sup>i</sup>* ¼ *ζiωni*, and *ωdi* ¼ *ωni*

**3. PID adjustment problem**

controller *Kpid*.

given by

and *s*

**70**

numerator that is given by

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

$$G\_p^{OL}(\mathbf{s}) = K \frac{\langle K^{pid}, \mathbf{s}^{pid} \rangle \langle b\_k, \mathbf{s}^m \rangle}{\mathbf{s}^{u+1} + \mathbf{s} \langle a\_i, \mathbf{s}^{u-1} \rangle},\tag{8}$$

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

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

$$m\_{cl} = m^{PID} + n,\tag{9}$$

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

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

$$G\_p^{CL}(s) = \frac{\mathcal{C}\_{K^{pd}}(s)G(s)}{1 + \mathcal{C}\_{K^{pd}}(s)G(s)H(s)}.\tag{10}$$
