A New Algorithm to Design Minimal Multi-Functional Observers for Linear Systems
Designing minimum possible order (minimal) observers for multi‐input multi‐output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi‐functional observers for linear time invariant (LTI) systems is proposed. The approach is applicab...
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| Veröffentlicht in: | Asian journal of control Jg. 18; H. 3; S. 842 - 857 |
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| Abstract | Designing minimum possible order (minimal) observers for multi‐input multi‐output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi‐functional observers for linear time invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings. |
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| AbstractList | Designing minimum possible order (minimal) observers for multi-input multi-output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi-functional observers for linear time invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings. |
| Author | Abdi, Hamid Nahavandi, Saeid Mohajerpoor, Reza |
| Author_xml | – sequence: 1 givenname: Reza surname: Mohajerpoor fullname: Mohajerpoor, Reza – sequence: 2 givenname: Hamid surname: Abdi fullname: Abdi, Hamid – sequence: 3 givenname: Saeid surname: Nahavandi fullname: Nahavandi, Saeid |
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| Cites_doi | 10.1049/ip-cta:19990585 10.1080/00207179.2014.891293 10.1109/TAC.2011.2160019 10.1111/j.1934-6093.2004.tb00372.x 10.1080/00207179.2012.715802 10.1109/CDC.2011.6160360 10.1109/TAC.1971.1099826 10.1002/asjc.889 10.1016/j.automatica.2010.10.027 10.1080/00207170600905337 10.1080/00207721.2010.521866 10.1109/TAC.2010.2042761 10.1109/TAC.1980.1102514 10.1111/j.1934-6093.2007.tb00430.x 10.1007/s00034-013-9618-z 10.1109/TAC.1966.1098323 10.1016/j.aml.2012.09.002 10.1109/9.855556 |
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| References_xml | – reference: Lungu, M. and R. Lungu, "Design of full-order observers for systems with unknown inputs by using the eigenstructure assignment," Asian J. Control, Vol. 16, No. 5, pp. 1470-1481, (2014). – reference: Teh, P. S. and H. Trinh, "Partial state and unknown input estimation for time-delay systems," Int. J. Syst. Sci., Vol. 43, No. 4, pp. 748-763 (2012). – reference: Rotella, F. and I. Zambettakis, "Minimal single linear functional observers for linear systems," Automatica, Vol. 47, No. 1, pp. 164-169 (2011). – reference: Darouach, M., "Unknown inputs observers design for delay systems," Asian J. Control, Vol. 9, No. 4, pp. 426-434 (2007). – reference: Trinh, H., T. Fernando, and S. Nahavandi, "Design of reduced-order functional observers for linear systems with unknown inputs," Asian J. Control, Vol. 6, No. 4, pp. 514-520 (2004). – reference: Fernando, T. and H. Trinh, "A system decomposition approach to the design of functional observers," Int. J. Control, Vol. 87, No. 9, pp. 1846-1860 (2014). – reference: Eskandari, Neda, Z. Jane Wang, and G. A. Dumont, "On the existence and design of functional observers for lti systems, with application to user modeling," Asian J. Control (2014). DOI: 10.1002/asjc.1002. – reference: Trinh, H., S. Nahavandi, and T. D. Tran, "Algorithms for designing reduced-order functional observers of linear systems," Int. J. Innov. Comput. I., Vol. 4, No. 2, pp. 321-334 (2008). – reference: Trinh, Hieu, TD Tran, and TD Nahavandi, "Design of scalar functional observers of order less than (1)," Int. J. Control, Vol. 79, No. 12, pp. 1654-1659 (2006). – reference: Darouach, M., "Existence and design of functional observers for linear systems," IEEE Trans. Autom. Control, Vol. 45, No. 5, pp. 940-943 (2000). – reference: Lovass-Nagy, V., R. Miller, and R. Mukundan, "On the application of matrix generalized inverses to the design of observers for time-varying and time-invariant linear systems," IEEE Trans. Autom. Control, Vol. 25, No. 6, pp. 1213-1218 (1980). – reference: Luenberger, D., "An introduction to observers," IEEE Trans. Autom. Control, Vol. 16, No. 6, pp. 596-602 (1971). – reference: Rao, C. R. and S. K. Mitra, Generalized inverse of matrices and its applications, Vol. 7, Wiley, New York (1971). – reference: Trinh, H. and T. Fernando, Functional Observers for Dynamical Systems, Vol. 420, Springer-Verlag, Berlin Heidelberg (2011). – reference: Fernando, T. L., T. Hieu Minh, and L. Jennings, "Functional observability and the design of minimum order linear functional observers," IEEE Trans. Autom. Control, Vol. 55, No. 5, pp. 1268-1273 (2010). – reference: Aldeen, M and H Trinh, "Reduced-order linear functional observer for linear systems," IEE P-Contr. Theor. Ap., Vol. 146, No. 5, pp. 399-405 (1999). – reference: Fernando, Tyrone and Hieu Trinh, "A procedure for designing linear functional observers," Appl. Math. Lett., Vol. 26, No. 2, pp. 240-243 (2013). – reference: Luenberger, D, "Observers for multivariable systems," IEEE Trans. Autom. Control, Vol. 11, No. 2, pp. 190-197 (1966). – reference: O'Reilly, J., Observers for linear systems, Vol. 170, Academic Press, Society for Industrial and Applied Mathematics, New York (1983). – reference: Fernando, T., S. MacDougall, V. Sreeram, and H. Trinh, "Existence conditions for unknown input functional observers," Int. J. Control, Vol. 86, No. 1, pp. 22-28 (2012). – reference: Jennings, L. S., T. L. Fernando, and H. M. Trinh, "Existence conditions for functional observability from an eigenspace perspective," IEEE Trans. Autom. Control, Vol. 56, No. 12, pp. 2957-2961 (2011). – reference: Lungu, Mihai and Romulus Lungu, "Reduced order observer for linear time-invariant multivariable systems with unknown inputs," Circuits Syst. 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Control – volume: 55 start-page: 1268 issue: 5 year: 2010 end-page: 1273 article-title: Functional observability and the design of minimum order linear functional observers publication-title: IEEE Trans. Autom. Control – volume: 56 start-page: 2957 issue: 12 year: 2011 end-page: 2961 article-title: Existence conditions for functional observability from an eigenspace perspective publication-title: IEEE Trans. Autom. Control – volume: 4 start-page: 321 issue: 2 year: 2008 end-page: 334 article-title: Algorithms for designing reduced‐order functional observers of linear systems publication-title: Int. J. Innov. Comput. I. – volume: 9 start-page: 426 issue: 4 year: 2007 end-page: 434 article-title: Unknown inputs observers design for delay systems publication-title: Asian J. Control – volume: 43 start-page: 748 issue: 4 year: 2012 end-page: 763 article-title: Partial state and unknown input estimation for time‐delay systems publication-title: Int. J. Syst. 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Control – volume: 26 start-page: 240 issue: 2 year: 2013 end-page: 243 article-title: A procedure for designing linear functional observers publication-title: Appl. Math. Lett. – year: 1971 – volume: 47 start-page: 164 issue: 1 year: 2011 end-page: 169 article-title: Minimal single linear functional observers for linear systems publication-title: Automatica – volume: 16 start-page: 596 issue: 6 year: 1971 end-page: 602 article-title: An introduction to observers publication-title: IEEE Trans. Autom. Control – volume: 16 start-page: 1470 issue: 5 year: 2014 end-page: 1481 article-title: Design of full‐order observers for systems with unknown inputs by using the eigenstructure assignment publication-title: Asian J. Control – volume: 6 start-page: 514 issue: 4 year: 2004 end-page: 520 article-title: Design of reduced‐order functional observers for linear systems with unknown inputs publication-title: Asian J. Control – ident: e_1_2_7_12_1 doi: 10.1049/ip-cta:19990585 – ident: e_1_2_7_6_1 doi: 10.1080/00207179.2014.891293 – ident: e_1_2_7_7_1 doi: 10.1109/TAC.2011.2160019 – ident: e_1_2_7_13_1 doi: 10.1111/j.1934-6093.2004.tb00372.x – ident: e_1_2_7_15_1 doi: 10.1080/00207179.2012.715802 – ident: e_1_2_7_20_1 doi: 10.1109/CDC.2011.6160360 – volume-title: Observers for linear systems year: 1983 ident: e_1_2_7_23_1 – ident: e_1_2_7_19_1 doi: 10.1109/TAC.1971.1099826 – ident: e_1_2_7_5_1 doi: 10.1002/asjc.889 – ident: e_1_2_7_8_1 doi: 10.1016/j.automatica.2010.10.027 – ident: e_1_2_7_11_1 doi: 10.1080/00207170600905337 – volume-title: Generalized inverse of matrices and its applications year: 1971 ident: e_1_2_7_21_1 – volume-title: Functional Observers for Dynamical Systems year: 2011 ident: e_1_2_7_3_1 – ident: e_1_2_7_16_1 article-title: On the existence and design of functional observers for lti systems, with application to user modeling publication-title: Asian J. Control – ident: e_1_2_7_22_1 doi: 10.1080/00207721.2010.521866 – volume: 4 start-page: 321 issue: 2 year: 2008 ident: e_1_2_7_9_1 article-title: Algorithms for designing reduced‐order functional observers of linear systems publication-title: Int. J. Innov. Comput. I. – ident: e_1_2_7_4_1 doi: 10.1109/TAC.2010.2042761 – ident: e_1_2_7_24_1 doi: 10.1109/TAC.1980.1102514 – ident: e_1_2_7_14_1 doi: 10.1111/j.1934-6093.2007.tb00430.x – ident: e_1_2_7_17_1 doi: 10.1007/s00034-013-9618-z – ident: e_1_2_7_18_1 doi: 10.1109/TAC.1966.1098323 – ident: e_1_2_7_10_1 doi: 10.1016/j.aml.2012.09.002 – ident: e_1_2_7_2_1 doi: 10.1109/9.855556 |
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| SubjectTerms | Algorithms Computer simulation constrained Sylvester equations Control systems functional observability and detectability Functional observer generalized inverse Input output Linear programming Linear systems Mathematical analysis MIMO (control systems) Observability (systems) Observers Recursive algorithms Studies |
| Title | A New Algorithm to Design Minimal Multi-Functional Observers for Linear Systems |
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