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博碩士論文 etd-0710103-152954 詳細資訊
Title page for etd-0710103-152954
論文名稱
Title
以線性矩陣不等式求解描述系統之正實性分析與設計問題
LMI Approach to Positive Real Analysis and Design for Descriptor Systems
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
140
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2003-07-09
繳交日期
Date of Submission
2003-07-10
關鍵字
Keywords
正實性、描述系統、線性矩陣不等式
Descriptor systems, Positive real, LMI
統計
Statistics
本論文已被瀏覽 5728 次,被下載 2241
The thesis/dissertation has been browsed 5728 times, has been downloaded 2241 times.
中文摘要
本論文將利用線性矩陣不等式(LMI)的方法分別針對連續廣義系統ESPR性質的設計問題和數位廣義系統SPR性質的分析和設計問題做研究。在連續系統ESPR性質的設計問題上,分別考慮靜態狀態迴授控制器、估測狀態迴授控制器、跟動態輸出迴授控制器三種架構,針對這三種不同的控制器架構,根據ESPR引理提出LMI型式的設計準則。而且,在動態輸出迴授控制器的架構上,將更進一步的發展出標準系統形式的控制器。在數位系統SPR性質分析方面,經由SPR和SBR的定義建立轉移矩陣SPR性質和SBR性質的轉換關係,並據此利用SBR引理推得LMI型式的SPR引理。此外,我們推導出系統經由SVD座標轉換後其SPR引理的LMI解為一區塊對角矩陣。經由此結果,我們考慮靜態狀態迴授控制器的設計問題,提出一個LMI的設計準則。

再者,不論是連續或數位廣義系統,本論文考慮系統內部包含了有界之非結構不確定項時,系統的ESPR性質或SPR性質的分析與設計。在分析上,根據ESPR引理或SPR引理,定義強韌ESPR性質或強韌SPR性質,並據此推導LMI 型式的分析準則。在連續系統的控制器設計上,分別考慮靜態狀態迴授控制器、跟動態輸出迴授控制器兩種架構,提出LMI型式的設計準則。在數位系統的控制器設計上,只考慮靜態狀態迴授控制器的架構,提出LMI型式的設計準則。

最後,根據ESPR引理(或SPR引理),本論文針對不確定連續(或數位)廣義系統提出一個新的LMI型式分析準則,保證當系統內部包含了多邊形結構之不確定項時,系統仍將保有原有的正則性、無脈衝行為、及穩定性等特性。在控制器的設計方面,我們根據此分析準則發展一個靜態狀態迴授控制器。此外,本論文將SPR性質應用在絕對穩定性分析上,亦即,考慮一個非線性閉迴路系統--其中一個子系統為線性數位廣義系統,而另一個子系統代表一任意的輸入與輸出之間的非線性時變對應關係,惟其輸入-輸出對的內積限制在一、三象限--的穩定性分析問題。由於本論文所推得的分析方法和控制器的設計準則皆為LMI型式,因此皆可藉著現有解LMI的電腦軟體計算而求得結果。這可以從本論文所提供的數值模擬例子得到驗證。



Abstract
For linear time-invariant descriptor models, this dissertation studies the extended strictly positive real (ESPR) design of continuous-time systems and the strictly positive real (SPR) analysis and design of discrete-time systems, respectively, all in the LMI framework. For a continuous-time system, by the LMI-based ESPR Lemma, a controller is designed such that the closed-loop system has its transfer matrix being ESPR while admissibility of the compensated descriptor system is guaranteed. Three forms of synthesis are considered, i.e. the static state feedback synthesis, estimated state feedback synthesis, and the dynamic output feedback synthesis. Moreover, design criterion of a dynamic output feedback controller in the state-space model is also addressed. For a discrete-time system, an LMI-based SPR characterization is developed. After giving the definition of SPR, the Cayley transformation is used to establish formulas bridging the admissible realizations for SPR and strictly bounded real (SBR) transfer matrices. Based on them, an LMI-based necessary and sufficient condition for a descriptor system to be, simultaneously, admissible and SPR is derived. When the descriptor variables are transformed into the SVD coordinate, it is shown that such a condition will have solution in the block diagonal form. Based on this result, the problem of static state feedback design to make transfer matrix of the closed-loop systems SPR is tackled.

The problems of robust ESPR and SPR analysis and design when the considered systems have norm-bounded unstructured uncertainty are also addressed. Similarly, LMI-based conditions to guarantee robust admissibility with transfer matrices being ESPR for continuous systems or being SPR for discrete systems are proposed. Based on them, for continuous systems, a static state feedback controller and a dynamic output feedback controller are designed to make the entire family of uncertain closed-loop systems robustly admissible with transfer matrices being ESPR. While for discrete systems, only static state feedback controller is designed to achieve the robust admissibility and robust SPR property.

Finally, based on ESPR lemma (or SPR lemma), we propose a new LMI-based robust admissibility analysis for a class of LTI continuous-time (or discrete-time) descriptor systems with convex polytopic uncertainties appearing on all the system matrices. Moreover, the development of state feedback controllers stemmed from these analysis results is also investigated. It is shown that the provided method has the capability to tackle the problem of computing a required feedback gain matrix for systems with either constant or polytopically dependent derivative (or advanced) state matrix in a unified way. Besides, the application of SPR property to absolute stability problem involving an LTI discrete-time descriptor system and a memoryless time-varying nonlinearity is also addressed. Since all conditions are expressed in LMIs, the obtained results are numerically tractable. It is illustrated by several numerical examples.



目次 Table of Contents
ch. 1 Introduction
ch. 2 Preliminaries
ch. 3 ESPR Design for Continuous Descriptor Systems
Ch. 4 Robust ESPR Analysis and Design for Continuous Descriptor Systems
ch. 5 SPR Control Problem for Discrete Descriptor Systems
ch. 6 Robust SPR Analysis and Design for Discrete Descriptor Systems
ch. 7 Miscellaneous Related Topics
Ch. 8 Conclusions
參考文獻 References
ibitem{AizGan_64} M. A. Aizerman and F. R. Gantmacher,
{it Absolute Stability of Regulator Systems},
Holden-Day, San Francisco, 1964.

ibitem{And_67} B. D. O. Anderson,
``A system theory criterion for positive real matrices,"
{it J. SIAM Control}, vol. 5, pp. 171-182, 1967.

ibitem{AmdMoo_90} B. D. O. Anderson and J. B. Moore,
{it Optimal Control: Linear Quadratic Methods},
Prentice Hall, New Jersey, 1990.

ibitem{Apl_91} J. D. Aplevich,
{it Implicit Linear Systems}, Springer-Verlag, Berlin, 1991.

ibitem{BalKas_99} V. Balakrishnan and R. L. Kashyap,
``Robust stability and performance analysis of uncertain systems using linear matrix
inequalities,"
{it JOTA}, vol. 100, pp. 457-478, 1999.

ibitem{Baretal_96} N. E. Barabanov, A. K. Gelig, G. A. Leonov,
A. L. Likhtarnikov, A. S. Matveev, V. B. Smirnova, and A. L. Fradkov,
``The frequency theorem (Kalman-Yakubovich Lemma) in control theory,"
{it Automat. Remote Control}, vol. 57, pp. 1377-1407, 1996.

ibitem{BenLau_87} D. J. Bender and A. J. Laub,
``The linear-quadratic optimal regulator for descriptor systems,"
{it IEEE Trans. Automat. Contr.}, vol. 32, pp. 672-688, 1987.

ibitem{Boyetal_94} S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
{it Linear Matrix Inequalities in Systems and Control Theory}, SIAM,
Philadelphia, 1994.

ibitem{BowChu_96} J. W. Brown and R. V. Churchill,
{it Complex Variables and Applications},
Sixth Edition, McGraw-Hill, Singapore, 1996.

ibitem{Chen_84} C. T. Chen,
{it Linear System Theory and Design},
Holt Rinehart and Winston, New York, 1984.

ibitem{CheLee_99} J. L. Chen and L. Lee,
``$H_infty$ control for discrete-time descriptor systems,"
{it Proc. of the 38th CDC}, pp. 4100-4105, 1999.

ibitem{ChoLia_99} J. H. Chou and W. H. Liao,
``Stability robustness of continuous-time perturbed descriptor systems,"
{it IEEE Trans. Circuits Syst. I}, vol. 46, pp. 1153-1155, 1999.

ibitem{Cob_84} J. D. Cobb,
``Controllability, observability, and duality in singular systems,"
{it IEEE Trans. Automat. Contr.}, vol. 29, pp. 1076-1082, 1984.

ibitem{Dai_89} L. Dai,
{it Singular Control Systems-Lecture notes in control and information
sciences},
Springer-Verlag, Berlin, 1989.

ibitem{Fanetal_94} C. H. Fang, L. Lee, and F. R. Chang,
``Robust control analysis and design for discrete-time singular systems,"
{it Automatica}, vol. 30, pp. 1741-1750, 1994.

ibitem{FanLee_97} C. H. Fang and L. Lee,
``Robustness of generalized state-space systems with unidirectional
perturbations,"
{it J. of Contr. Syst. and Technology}, vol. 5, pp. 221-234, 1997.

ibitem{Fanetal_02} C. H. Fang, L. Hong, S. W. Kau, and L. Lee,
``An LMI approach for robust stability of continuous-time descriptor
system families,"
{it Proceedings of 2002 R.O.C. Automatic Control Conference},
pp. 203-207, 2002.

ibitem{FanLee_02} C. H. Fang and L. Lee,
``Stability robustness analysis of uncertain discrete-time descriptor systems,"
{it Proceedings of the 15th IFAC}, pp. 21-26, 2002.

ibitem{Gahetal_95} P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali,
{it The LMI Control Toolbox}, The MathWorks, Inc. 1995.

ibitem{Geretal_98} J. C. Geromel, M. C. De Oliveira, and L. Hsu,
``LMI characterization of structural and robust stability,"
{it Linear Algebra and Its Applications}, vol. 285, pp. 69-80, 1998.

ibitem{Ghaetal_95} L. El Ghaoui, R. Nikoukhah, and F. Delebecque,
``{ t LMITOOL}: a package for LMI optimization,"
{it Proc. of the 34th IEEE CDC}, pp. 3096-3101, 1995.

ibitem{GhaNic_00} L. El Ghaoui and S. L. Niculescu,
{it Advances in Linear Matrix Inequality Methods in Control}, SIAM,
Philadelphia, 2000.

ibitem{HadBer_91} W. M. Haddad and D. S. Bernstein,
``Robust stabilization with positive real uncertainty:
beyond the small gain theorem,"
{it Syst. Contr. Letters}, vol. 17, pp. 191-208, 1991.

ibitem{HadBer_93} W. M. Haddad and D. S. Bernstein,
``Explicit construction of quadratic Lyapunov functions for the
small gain, positivity, circle, and Popov theorems and their
application to robust stability. Part I: Continuous-time theory,"
{it Int. J. Robust and Nonlinear Control}, vol. 3, pp. 313-339, 1993.

ibitem{HadBer_94} W. M. Haddad and D. S. Bernstein,
``Explicit construction of quadratic Lyapunov functions for the
small gain, positivity, circle, and Popov theorems and their
application to robust stability. Part II: Discrete-time theory,"
{it Int. J. Robust and Nonlinear Control}, vol. 4, pp. 249-265, 1994.

ibitem{HitAnd_69} L. Hitz and B. D. O. Anderson,
``Discrete positive-real functions and their application to
system stability,"
{it Proc. IEE}, vol. 116, pp. 153-155, 1969.

ibitem{HsiLee_97} K. L. Hsiung and L. Lee,
``Pole-clustering characterization via LMI for descriptor systems,"
{it Proc. of the 36th IEEE CDC}, pp. 1313-1314, 1997.

ibitem{HsiLee_99} K. L. Hsiung and L. Lee,
``Lyapunov inequality and bounded real lemma for discrete-time
descriptor systems,"
{it IEE Proc. Control Theory and Applications},
vol. 146, pp. 327-331, 1999.

ibitem{HuDav_96} Y. Z. Hu and E. J. Davison,
``A study of the stability radius for descriptor systems,"
{it Proc. of the 35th IEEE CDC}, pp. 4256-4261, 1996.

ibitem{Hauetal_00} J. C. Huang, H. S. Wang, and F. R. Chang,
``Robust $H_infty$ control for uncertain linear time-invariant descriptor systems,"
{it IEE Proc. Control Theory and Applications}, vol. 147, pp. 648-654, 2000.

ibitem{Kha_96} H. K. Khalil,
{it Nonlinear Systems}, 2nd ed.,
Prentice-Hall, New Jersey, 1996.

ibitem{Lan_79} Y. D. Landau,
{it Adaptive Control},
Marcel Dekker, Inc., New York, 1979.

ibitem{LeeChe_03} L. Lee and J. L. Chen,
``Strictly positive real lemma and absolute stability for discrete-time
descriptor systems,"
{it IEEE Trans. Circuits Syst. I}, 2003 (to appear in the June issue).

ibitem{Lew_86} F. L. Lewis, ``A survey of linear singular systems,'
{it J. Circuits, Systems, and Signal Processing},
vol. 5, pp. 3-36, 1986.

ibitem{LewMer_89} F. L. Lewis and V. G. Mertzios,
``Recent advances in singular systems,"
{it Circuits, Systems, and Signal Processing}, vol. 8, 1989.

ibitem{Linetal_97} C. Lin, J. Wang, D. Wang, and C. B. Soh,
``Robustness of uncertain descriptor systems,"
{it Syst. Contr. Letters}, vol. 31, pp. 129-138, 1997.

ibitem{Linetal_01} C. Lin, J. Lam, J. Wang, and G. H. Yang,
``Analysis on robust stability for interval descriptor systems,"
{it Syst. Contr. Letters}, vol. 42, pp. 267-278, 2001.

ibitem{LozJos_90} R. Lozano-Leal and S. M. Joshi,
``Strictly positive real transfer functions revisited",
{it IEEE Trans. Automat. Contr.}, vol. 35, pp. 1243-1245, 1990.

ibitem{LueAre_77} D. G. Luenberger and A. Arbel,
``Singular dynamic Leontief systems,"
{it Econometrica}, vol. 45, pp. 991-995, 1977.

ibitem{Lue_78} D. G. Luenberger,
``Time-invariant descriptor systems,"
{it Automatica}, vol. 14, pp. 473-480, 1978.

ibitem{Lue_79}D. G. Luenberger,
``Nonlinear descriptor systems,'
{it J. Economic Dynamics and Control}, vol. 1, pp. 219-242, 1979.

ibitem{MahXie_00} M. S. Mahmoud and L. H. Xie,
``Positive real analysis and synthesis of uncertain discrete
time systems,"
{it IEEE Trans. Circuits Syst. I}, vol. 47, pp. 403-406, 2000.

ibitem{Masetal_97} I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda,
``The $H_infty$ control for descriptor systems: a matrix inequalities
approach,"
{it Automatica}, vol. 33, pp. 669-673, 1997.

ibitem{Mer_84} B. G. Mertzios,
``On the sensitivity analysis of linear time-invariant singular systems,"
{it IEEE Trans. Circuits and Systems}, vol. 31, pp. 978-982, 1984.

ibitem{MilGol_89} J. K. Mills and A. A. Goldenberg,
``Force and position control of manipulators during constrained motion tasks,"
{it IEEE Trans. Robot. Automat.}, vol. 5, pp.30-46, 1989.

ibitem{NarAnn_89} K. S. Narendra and A. M. Annaswamy,
{it Stable Adaptive Systems},
Prentice Hall, New Jersey, 1989.

ibitem{New_66} R. W. Newcomb,
{it Linear Multiport Synthesis},
McGraw-Hill, New York, 1966.

ibitem{Olietal_99} M. C. De OliveiRa, J. Bernussou, and J. C. Geromel,
``A new discrete-time robust stability condition,"
{it Systems $&$ Control Letters}, vol. 37, pp. 261-265, 1999.

ibitem{Olietal_99b} M. C. De Oliveira, J. C. Geromel, and L. Hsu,
``LMI characterization of structural and robust stability: the discrete-time case,"
{it Linear Algebra and Its Applications}, vol. 296, pp. 27-38, 1999.

ibitem{Peaetal_00} D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou,
``A new ${cal D}$-stability condition for real convex polytopic uncertainty,"
{it Syst. Contr. Letters}, vol. 40, pp. 21-30, 2000.

ibitem{Pop_61} V. M. Popov,
``Absolute stability of nonlinear systems of automatic control,"
{it Automat. Remote Control}, vol. 22, pp. 857-875, 1961.

ibitem{Pop_73} V. M. Popov,
{it Hyperstability of Control Systems},
Springer, New York, 1973.

ibitem{QiuDav_92} L. Qiu and E. J. Davison,
``The stability robustness of generalized eigenvalues,"
{it IEEE Trans. Automat. Contr.}, vol. 37, pp. 886-891, 1992.

ibitem{RehAll_00} A. Rehm and F. Allg"{o}wer,
``Self-scheduled $H_{infty}$ output feedback control of
descriptor systems,"
{it Computers and Chemical Engineering}, vol. 24, pp. 279-284, 2000.

ibitem{RehAll_02} A. Rehm and F. Allg"{o}wer,
``An LMI approach towards stabilization of discrete-time
descriptor systems,"
{it Proceedings of the 15th World Congress of IFAC}, Barcelona, Spain, 2002.

ibitem{RehAll_02b} A. Rehm and F. Allg"{o}wer,
``An LMI approach towards $H_{infty}$ control of discrete-time
descriptor systems,"
{it Proc. American Control Conference}, pp. 614-619, 2002.

ibitem{Ros_74} H. H. Rosenbrock,
``Structure properties of linear dynamical systems,'
{it Int. J. Contr.}, vol. 20, pp. 191-202, 1974.

ibitem{Schetal_97} C. Scherer, P. Gahient and M. Chilali,
``Multiobjective output-feedback control via LMI optimization,"
{it IEEE Trans. Automat. Contr.}, vol. 42, pp. 896-910, 1997.

ibitem{SinLiu_73} S. P. Singh and R. W. Liu,
``Existence of state equation representation of linear large-scale dynamic systems,"
{it IEEE Trans. Circuit Theory}, vol. 20, pp. 239-246, 1973.

ibitem{Sto_92} A. A. Stoorvogel,
{it The $H_{infty}$ Control Problem},
Prentice Hall, Now York, 1992.

ibitem{Sunetal_94} W. Sun, P. P. Khargonekar and D. Shim,
``Solution to the positive real control problem for linear time-invariant systems,"
{it IEEE Trans. Automat. Contr.}, vol. 39, pp. 2034-2046, 1994.

ibitem{TakKat_98} K. Takaba and T. Katayama,
``$H^{2}$ output feedback control for descriptor systems,"
{it Automatica}, vol. 34, pp. 841-850, 1998.

ibitem{Tak_99} K. Takaba,
``Linear quadratic optimal control for linear implicit system,"
{it Proc. of the 38th CDC}, pp. 4074-4079, 1999.

ibitem{TaoIoa_90} G. Tao and P. A. Ioannou,
``Necessary and sufficient conditions for strictly positive real
matrices,"
{it IEE Proc. Part G}, vol. 137, pp. 360-366, 1990.

ibitem{Wanetal_97} H. S. Wang and F. R. Chang,
``The generalized state-space description of positive realness and bounded realness,"
{it Proc. IEEE 39th Midwest Symposium on Circuits and System},
pp. 893-896, 1997.

ibitem{Wanetal_98} H. S. Wang, C. F. Yung, and F. R. Chang,
``Bounded real lemma and $H_infty$ control for descriptor systems,"
{it IEE Proc. Control Theory and Applications}, vol. 145, pp. 316-322, 1998.

ibitem{Wanetal_01} H. S. Wang, C. F. Yung, and F. R. Chang,
``The positive real control problem and the generalized algebraic Riccati
equation for descriptor systems,"
{it Journal of the Chinese Institute of Engineerings}, vol. 24, pp. 203-220, 2001.

ibitem{Wei_91} A. Weinmann,
{it Uncertain Models and Robust Control},
Springer-Verlag, New York, 1991.

ibitem{XiaHil_98} C. Xiao and D. J. Hill,
``Concepts of strict positive realness and the absolute stability
problem of continuous-time systems,"
{it Automatica}, vol. 34, pp. 1071-1082, 1998.

ibitem{XiaHil_99} C. Xiao and D. J. Hill,
``Generalizations and new proof of the discrete-time positive
real lemma and bounded real lemma,"
{it IEEE Trans. Circuits Syst. I}, vol. 46, pp. 740-743, 1999.

ibitem{Xie_96} L. Xie,
``Output feedback $H_infty$ control of systems with parameter uncertainty,"
{it Int. J. Contr.}, vol. 63, pp. 741-750, 1996.

ibitem{XuYan_99} S. Xu and C. Yang,
``Stabilization of discrete-time singular systems: a matrix inequalities
approach,"
{it Automatica}, vol. 35, pp. 1613-1617, 1999.

ibitem{XuYan_99b} S. Xu and C. Yang,
``Robust stabilization for generalized state-space system with uncertainty,"
{it Int. J. Control}, vol. 72, pp. 1659-1664, 1999.

ibitem{XuYan_00} S. Xu and C. Yang,
``$H_infty$ state feedback control for discrete singular systems,"
{it IEEE Trans. Automat. Contr.}, vol. 45, pp. 1405-1409, 2000.

ibitem{Yunetal_99} C. F. Yung, H. S. Wang, and F. R. Chang,
``Passivity technique in control and network synthesis of descriptor systems,"
{it Proc. European Control Conference}, BA-9, 1999.

ibitem{Zhaetal_02} L. Zhang, J. Lam, and S. Xu,
`On positive realness of descriptor systems',
{it IEEE Trans. Circuits Syst. I}, vol. 49, pp. 401-407, 2002.

ibitem{Zhoetal_96} K. Zhou, J. C. Doyle, and K. Glover,
{it Robust and Optimal Control},
Prentice-Hall, New Jersey, 1996.
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