New Tools for Tuning PID Controllers Using Orthogonal Polynomials
This paper presents an algorithm to determine PID controller tuning regions that guarantee the stability of closed-loop control systems using the pole placement method, assigning a known robustly stable polynomial that depends on a vector of uncertain parameters as the desired closed-loop characteri...
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| Format: | Article |
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IEEE
2025-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/10887206/ |
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| author | Alejandro Arceo Luis E. Garza Gerardo Romero Jose Luis Valdez |
| author_facet | Alejandro Arceo Luis E. Garza Gerardo Romero Jose Luis Valdez |
| author_sort | Alejandro Arceo |
| collection | DOAJ |
| description | This paper presents an algorithm to determine PID controller tuning regions that guarantee the stability of closed-loop control systems using the pole placement method, assigning a known robustly stable polynomial that depends on a vector of uncertain parameters as the desired closed-loop characteristic polynomial. The algorithm generates PID gains contingent on uncertain parameters. This robustly stable polynomial is constructed using modified classical weights and relies on well-known properties of the theory of orthogonal polynomials, including the recurrence relation. In addition, it considers linear combinations of two orthogonal polynomials with consecutive degrees. An advantage of the proposed approach is the considerable flexibility in selecting both the closed-loop characteristic polynomial and number of uncertain parameters. As this problem arises from the challenge of assigning n closed-loop poles with only three free gains, the methodology uses the Moore-Penrose generalized inverse of a certain matrix to obtain expressions of the PID gains. Therefore, the proposed methodology does not work in certain cases of linear time-invariant systems. Three examples were provided to illustrate the design algorithm. |
| format | Article |
| id | doaj-art-b9d0e58be3ec4f899f52954877db9f53 |
| institution | DOAJ |
| issn | 2169-3536 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj-art-b9d0e58be3ec4f899f52954877db9f532025-08-20T03:00:25ZengIEEEIEEE Access2169-35362025-01-0113319923200310.1109/ACCESS.2025.354235810887206New Tools for Tuning PID Controllers Using Orthogonal PolynomialsAlejandro Arceo0https://orcid.org/0000-0002-2867-9903Luis E. Garza1https://orcid.org/0000-0002-2569-186XGerardo Romero2https://orcid.org/0000-0003-3007-9365Jose Luis Valdez3https://orcid.org/0000-0002-5601-7333Escuela de Ingeniería y Ciencias, Tecnológico de Monterrey, Monterrey, MexicoFacultad de Ciencias, Universidad de Colima, Colima, MexicoUnidad Académica Multidisciplinaria Reynosa Rodhe, Universidad Autónoma de Tamaulipas, Reynosa, MexicoFacultad de Ingeniería Mecánica y Eléctrica, Universidad de Colima, Colima, MexicoThis paper presents an algorithm to determine PID controller tuning regions that guarantee the stability of closed-loop control systems using the pole placement method, assigning a known robustly stable polynomial that depends on a vector of uncertain parameters as the desired closed-loop characteristic polynomial. The algorithm generates PID gains contingent on uncertain parameters. This robustly stable polynomial is constructed using modified classical weights and relies on well-known properties of the theory of orthogonal polynomials, including the recurrence relation. In addition, it considers linear combinations of two orthogonal polynomials with consecutive degrees. An advantage of the proposed approach is the considerable flexibility in selecting both the closed-loop characteristic polynomial and number of uncertain parameters. As this problem arises from the challenge of assigning n closed-loop poles with only three free gains, the methodology uses the Moore-Penrose generalized inverse of a certain matrix to obtain expressions of the PID gains. Therefore, the proposed methodology does not work in certain cases of linear time-invariant systems. Three examples were provided to illustrate the design algorithm.https://ieeexplore.ieee.org/document/10887206/Hurwitz polynomialsorthogonal polynomialsPID controllersrobust stability |
| spellingShingle | Alejandro Arceo Luis E. Garza Gerardo Romero Jose Luis Valdez New Tools for Tuning PID Controllers Using Orthogonal Polynomials IEEE Access Hurwitz polynomials orthogonal polynomials PID controllers robust stability |
| title | New Tools for Tuning PID Controllers Using Orthogonal Polynomials |
| title_full | New Tools for Tuning PID Controllers Using Orthogonal Polynomials |
| title_fullStr | New Tools for Tuning PID Controllers Using Orthogonal Polynomials |
| title_full_unstemmed | New Tools for Tuning PID Controllers Using Orthogonal Polynomials |
| title_short | New Tools for Tuning PID Controllers Using Orthogonal Polynomials |
| title_sort | new tools for tuning pid controllers using orthogonal polynomials |
| topic | Hurwitz polynomials orthogonal polynomials PID controllers robust stability |
| url | https://ieeexplore.ieee.org/document/10887206/ |
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