Non-Linear Observer Design with Laguerre Polynomials
In this paper, a methodology for a non-linear system state estimation is demonstrated, exploiting the input and parameter observability. For this purpose, the initial system is transformed into the canonical observability form, and the function that aggregates the non-linear dynamics of the system,...
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| Format: | Article |
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MDPI AG
2022-06-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/7/913 |
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| author | Maria Trigka Elias Dritsas |
| author_facet | Maria Trigka Elias Dritsas |
| author_sort | Maria Trigka |
| collection | DOAJ |
| description | In this paper, a methodology for a non-linear system state estimation is demonstrated, exploiting the input and parameter observability. For this purpose, the initial system is transformed into the canonical observability form, and the function that aggregates the non-linear dynamics of the system, which may be unknown or difficult to be computed, is approximated by a linear combination of Laguerre polynomials. Hence, the system identification translates into the estimation of the parameters involved in the linear combination in order for the system to be observable. For the validation of the elaborated observer, we consider a biological model from the literature, investigating whether it is practically possible to infer its states, taking into account the new coordinates to design the appropriate observer of the system states. Through simulations, we investigate the parameter settings under which the new observer can identify the state of the system. More specifically, as the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> increases, the system converges more quickly to the steady-state, decreasing the respective distance from the system’s initial state. As for the first state, the estimation error is in the order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>=</mo><mn>15</mn></mrow></semantics></math></inline-formula>, and assuming <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>=</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mo>,</mo><msub><mi>c</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Under the same conditions, the estimation error of the system’s second state is in the order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula>, setting a performance difference of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> in relation to the first state. The outcomes show that the proposed observer’s performance can be further improved by selecting even higher values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>. Hence, the system is observable through the measurement output. |
| first_indexed | 2024-03-09T10:19:13Z |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T10:19:13Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
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| series | Entropy |
| spelling | doaj.art-5f8e83a208a04c7bae8d28e6176e6d812023-12-01T22:06:41ZengMDPI AGEntropy1099-43002022-06-0124791310.3390/e24070913Non-Linear Observer Design with Laguerre PolynomialsMaria Trigka0Elias Dritsas1Department of Computer Engineering and Informatics, University of Patras, 26504 Patras, GreeceDepartment of Computer Engineering and Informatics, University of Patras, 26504 Patras, GreeceIn this paper, a methodology for a non-linear system state estimation is demonstrated, exploiting the input and parameter observability. For this purpose, the initial system is transformed into the canonical observability form, and the function that aggregates the non-linear dynamics of the system, which may be unknown or difficult to be computed, is approximated by a linear combination of Laguerre polynomials. Hence, the system identification translates into the estimation of the parameters involved in the linear combination in order for the system to be observable. For the validation of the elaborated observer, we consider a biological model from the literature, investigating whether it is practically possible to infer its states, taking into account the new coordinates to design the appropriate observer of the system states. Through simulations, we investigate the parameter settings under which the new observer can identify the state of the system. More specifically, as the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> increases, the system converges more quickly to the steady-state, decreasing the respective distance from the system’s initial state. As for the first state, the estimation error is in the order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>=</mo><mn>15</mn></mrow></semantics></math></inline-formula>, and assuming <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>=</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mo>,</mo><msub><mi>c</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Under the same conditions, the estimation error of the system’s second state is in the order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula>, setting a performance difference of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> in relation to the first state. The outcomes show that the proposed observer’s performance can be further improved by selecting even higher values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula>. Hence, the system is observable through the measurement output.https://www.mdpi.com/1099-4300/24/7/913non-linear dynamicsidentifiabilityobservabilityLaguerre polynomial |
| spellingShingle | Maria Trigka Elias Dritsas Non-Linear Observer Design with Laguerre Polynomials Entropy non-linear dynamics identifiability observability Laguerre polynomial |
| title | Non-Linear Observer Design with Laguerre Polynomials |
| title_full | Non-Linear Observer Design with Laguerre Polynomials |
| title_fullStr | Non-Linear Observer Design with Laguerre Polynomials |
| title_full_unstemmed | Non-Linear Observer Design with Laguerre Polynomials |
| title_short | Non-Linear Observer Design with Laguerre Polynomials |
| title_sort | non linear observer design with laguerre polynomials |
| topic | non-linear dynamics identifiability observability Laguerre polynomial |
| url | https://www.mdpi.com/1099-4300/24/7/913 |
| work_keys_str_mv | AT mariatrigka nonlinearobserverdesignwithlaguerrepolynomials AT eliasdritsas nonlinearobserverdesignwithlaguerrepolynomials |