LMI approach to linear positive system analysis and synthesis
This paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities (LMIs). We first show that the celebrated Perron–Frobenius theorem can be proved concisely by a duality-based argument. Again by duality, we next clarify a necessary and sufficient...
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01.01.2014
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| Abstract | This paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities (LMIs). We first show that the celebrated Perron–Frobenius theorem can be proved concisely by a duality-based argument. Again by duality, we next clarify a necessary and sufficient condition under which a Hurwitz stable Metzler matrix admits a diagonal Lyapunov matrix with some identical diagonal entries as the solution of the Lyapunov inequality. This new result leads to an alternative proof of the recent result by Tanaka and Langbort on the existence of a diagonal Lyapunov matrix for the LMI characterizing the H∞ performance of continuous-time positive systems. In addition, we further derive a new LMI for the H∞ performance analysis where the variable corresponding to the Lyapunov matrix is allowed to be non-symmetric. We readily extend these results to discrete-time positive systems and derive new LMIs for the H∞ performance analysis and synthesis. We finally illustrate their effectiveness by numerical examples on robust state-feedback H∞ controller synthesis for discrete-time positive systems affected by parametric uncertainties. |
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| AbstractList | This paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities (LMIs). We first show that the celebrated Perron-Frobenius theorem can be proved concisely by a duality-based argument. Again by duality, we next clarify a necessary and sufficient condition under which a Hurwitz stable Metzler matrix admits a diagonal Lyapunov matrix with some identical diagonal entries as the solution of the Lyapunov inequality. This new result leads to an alternative proof of the recent result by Tanaka and Langbort on the existence of a diagonal Lyapunov matrix for the LMI characterizing the H infinity H infinity performance of continuous-time positive systems. In addition, we further derive a new LMI for the H infinity H infinity performance analysis where the variable corresponding to the Lyapunov matrix is allowed to be non-symmetric. We readily extend these results to discrete-time positive systems and derive new LMIs for the H infinity H infinity performance analysis and synthesis. We finally illustrate their effectiveness by numerical examples on robust state-feedback H infinity H infinity controller synthesis for discrete-time positive systems affected by parametric uncertainties. This paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities (LMIs). We first show that the celebrated Perron–Frobenius theorem can be proved concisely by a duality-based argument. Again by duality, we next clarify a necessary and sufficient condition under which a Hurwitz stable Metzler matrix admits a diagonal Lyapunov matrix with some identical diagonal entries as the solution of the Lyapunov inequality. This new result leads to an alternative proof of the recent result by Tanaka and Langbort on the existence of a diagonal Lyapunov matrix for the LMI characterizing the performance of continuous-time positive systems. In addition, we further derive a new LMI for the performance analysis where the variable corresponding to the Lyapunov matrix is allowed to be non-symmetric. We readily extend these results to discrete-time positive systems and derive new LMIs for the performance analysis and synthesis. We finally illustrate their effectiveness by numerical examples on robust state-feedback controller synthesis for discrete-time positive systems affected by parametric uncertainties. This paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities (LMIs). We first show that the celebrated Perron–Frobenius theorem can be proved concisely by a duality-based argument. Again by duality, we next clarify a necessary and sufficient condition under which a Hurwitz stable Metzler matrix admits a diagonal Lyapunov matrix with some identical diagonal entries as the solution of the Lyapunov inequality. This new result leads to an alternative proof of the recent result by Tanaka and Langbort on the existence of a diagonal Lyapunov matrix for the LMI characterizing the H∞ performance of continuous-time positive systems. In addition, we further derive a new LMI for the H∞ performance analysis where the variable corresponding to the Lyapunov matrix is allowed to be non-symmetric. We readily extend these results to discrete-time positive systems and derive new LMIs for the H∞ performance analysis and synthesis. We finally illustrate their effectiveness by numerical examples on robust state-feedback H∞ controller synthesis for discrete-time positive systems affected by parametric uncertainties. |
| Author | Arzelier, Denis Ebihara, Yoshio Peaucelle, Dimitri |
| Author_xml | – sequence: 1 givenname: Yoshio surname: Ebihara fullname: Ebihara, Yoshio email: ebihara@kuee.kyoto-u.ac.jp organization: Department of Electrical Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan – sequence: 2 givenname: Dimitri surname: Peaucelle fullname: Peaucelle, Dimitri organization: LAAS-CNRS, Université de Toulouse, 7 Av. du Colonel Roche, 31077, Toulouse Cedex 4, France – sequence: 3 givenname: Denis surname: Arzelier fullname: Arzelier, Denis organization: LAAS-CNRS, Université de Toulouse, 7 Av. du Colonel Roche, 31077, Toulouse Cedex 4, France |
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| Cites_doi | 10.1109/TAC.2007.900857 10.1109/TAC.2007.899057 10.1080/03081087808817203 10.1109/TAC.2002.806652 10.1016/0167-6911(95)00063-1 10.1109/ACC.2012.6314755 10.1080/00207179.2013.797106 10.1016/j.laa.2008.06.037 10.1109/TAC.2011.2157394 10.1007/978-3-540-30560-6_2 10.1109/CDC.2011.6161293 10.1109/MSP.2005.1406483 10.1109/CDC.2012.6426312 10.1080/00207170210140212 |
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| SubjectTerms | Automatic Control Engineering Computer Science Control systems Diagonal Lyapunov matrix Duality Inequalities Linear matrix inequalities LMI Mathematical models Positive system Synthesis Systems analysis Theorems Uncertainty |
| Title | LMI approach to linear positive system analysis and synthesis |
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