**3.4 Proposed method**

…………………………*:*

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

*<sup>N</sup>CL*ðÞ¼ *<sup>s</sup>* <sup>X</sup>*mcl*

numerator polynomial is given by

*DCL*ðÞ¼ *<sup>s</sup> <sup>s</sup>*

that the PID controller has the three terms.

obtained in a similar way and given by

*PCL*

**72**

*<sup>P</sup>CL*ðÞ¼ *<sup>s</sup> <sup>s</sup>*

obtaining the characteristic closed-loop polynomial.

*ncl* <sup>þ</sup> *n* X*cl*�1 *i*¼1

closed-loop plant and *ncl* is the order of referred polynomial.

*PCL*

*3.3.2 Characteristic polynomial*

*n*þ*nD*�*pid*

*i*¼0

when *nD*�*pid* <sup>¼</sup> 0, the proposal is to specify an additional *<sup>a</sup><sup>s</sup>*

þ *n*þ*n* X*<sup>D</sup>*�*pid*�<sup>1</sup>

system is given by

given by

þð Þ *bmKD* þ *bm*�<sup>1</sup>*KP* þ *bm*�<sup>2</sup>*KI s*

þð Þ *bm*þ<sup>1</sup>*KD* þ *bmKP* þ *bm*�<sup>1</sup>*KI s*

þð Þ *bm*þ<sup>2</sup>*KD* þ *bm*þ<sup>1</sup>*KP* þ *bmKI s*

*<sup>K</sup>pid*, *bk*�<sup>1</sup> � �*s*

where *mcl* <sup>¼</sup> *<sup>m</sup>* <sup>þ</sup> *mpid* and vector *bk* of the polynomial of zeros of the closed-loop

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

where *nD*�*pid* ¼ 0 or 1. When *nD*�*pid* ¼ 0, the PID controller structure have the

terms derivative and proportional. When *nD*�*pid* ¼ 1, the structure of the PID controller has an integrator term that increases the order of the system by 1, starting with the three terms: proportional, derivative and integrative [21]. In this work,

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

where *<sup>P</sup>CL*ð Þ*<sup>s</sup>* is the general form of the characteristic polynomial of the

the characteristic closed-loop polynomial for unit feedback (*H s*ðÞ¼ 1) is

*<sup>p</sup>* ð Þ*s* is the characteristic polynomial of the closed loop. The representation of the problem in the form of an internal product that relates the coefficients of the zero polynomial with the coefficients of the closed plant dynamics is the basis for

*ai* <sup>þ</sup> *<sup>K</sup>pid*, *bk*�<sup>1</sup> � � � � *s*

In terms of inner product, the general polynomial form of the closed-loop

*i*¼0

2

1

*:*

, (13)

*<sup>i</sup>* coefficient., to ensure

, (16)

0

*mcl*�*i*

*bk* ¼ ½ � *bk bk*�<sup>1</sup> *bk*�<sup>2</sup> *:* (14)

ð Þ *bkKD* <sup>þ</sup> *bk*�<sup>1</sup>*KP* <sup>þ</sup> *bk*�<sup>2</sup>*KI* <sup>þ</sup> *ai*þ<sup>1</sup> , (15)

*ncl*�*i*

*<sup>p</sup>* ðÞ¼ *s* 1 þ *CKpid* ð Þ*s G s*ð Þ*:* (17)

The problem is formulated based on the propagation matrix generated from the dot product between the terms of the earnings vector *Kpid* with the coefficients of the numerator *bk* associated with the coefficients of the denominator *ai* of the TF of plant. The propagation matrix product associated with the TF numerator coefficients of the plant, give rise to a new characteristic polynomial based on new specified operating points, which are imposed by new zeros and new poles.
