臺灣博碩士論文加值系統

本論文係研究正向系統之穩定性分析及穩定化控制設計，研究的範疇包含連續時間與離散時間兩大系統，系統中含有的區間跟凸形不確定量因子，是本文的主要探討之列。針對穩定性與正向性分析問題，而推導出新的充份與必要條件，基於此條件，搭配狀態迴授來設計控制器，便可應用於控制系統的穩定化設計上。利用前述之分析結果，配合線性歸劃方法來尋找控制器參數。最後以實際的正向系統為例，對此系統設計控制器，討論補償前與補償後系統性能的差異，經由模擬結果顯示，所設計的控制器是有效且適用的。

This thesis is concerned with stability and stabilization of interval and polytopic systems with the positivity constraints. Both continuous-time and discrete-time are discussed. For stability analysis and stabilization problems of those systems, we derive some new conditions. Then based on these conditions, a linear programming method is applied to design controllers. Several examples, including compartmental systems and Leslie systems are given to demonstrate the effectiveness and applicability of the proposed methods.

Contents I
List of Figures V
List of Tables VII
Chapter 1 Introductions 1
1.1 Motivations and Literature reviews 1
1.2 Organization 2
Chapter 2 Notations and Preliminaries 4
2.1 Notations 4
2.2 Positive Linear Systems 5
2.2.1 Continuous-time Positive Linear Systems 5
2.2.2 Discrete-time Positive Linear Systems 6
2.3 Metzler matrix 7
2.4 Asymptotically Stability 7
2.5 Lyapunov’s Direct Method 8
2.6 Summary 9
Chapter 3 Stability Analysis and Controller Design of Continuous-time Interval and Polytopic Systems under Positivity Constraint 10
3.1 Stability Analysis and Controller Design of Continuous-time Interval Systems under Positivity Constraint 10
3.1.1 Stability Analysis of Continuous-time Interval Linear Systems under Positivity Constraint 11
3.1.2 Controller Design of Continuous-time Interval Linear Systems under Positivity Constraint 14
3.2 Stability Analysis and Controller Design of Continuous-time Polytopic Systems under Positivity Constraint 17
3.2.1 Stability Analysis of Continuous-time Polytopic Linear Systems under Positivity Constraint 17
3.2.2 Controller Design of Continuous-time Polytopic Linear Systems under Positivity Constraint 20
3.3 Illustrative Examples 28
3.4 Summary 38
Chapter 4 Stability Analysis and Controller Design of Discrete-time Interval and Polytopic Systems under Positivity Constraint 39
4.1 Stability Analysis and Controller Design of Discrete-time Interval Systems under Positivity Constraint 39
4.1.1 Stability Analysis of Discrete-time Interval Linear Systems under Positivity Constraint 39
4.1.2 Controller Design of Discrete-time Interval Linear Systems under Positivity Constraint 43
4.2 Stability Analysis and Controller Design of Discrete-time Polytopic Systems under Positivity Constraint 46
4.2.1 Stability Analysis of Discrete-time Polytopic Linear Systems under Positivity Constraint 46
4.2.2 Controller Design of Discrete-time Polytopic Linear Systems under Positivity Constraint 50
4.3 Illustrative Examples 57
4.4 Summary 67
Chapter 5 Stabilization Problems on Compartmental Systems and Leslie Systems 68
5.1 Stabilization Problem on Compartmental Systems 68
5.1.1 Overview of Compartmental Systems 68
5.1.2 Stabilization of Compartmental Systems 69
5.2 Stabilization Problem on Leslie Systems 73
5.2.1 Overview of Leslie Systems 73
5.2.2 Stabilization of Leslie Systems 74
5.3 Summary 78
Chapter 6 Conclusions 79
References 80
Appendix 86
Publication List 103

[1] A. Berman, M. Neumann, and R. J. Stern, Nonnegative Matrices in Dynamic Systems, New York: Wiley, 1989.
[2] L. Farina and S. Rinaldi, Positive Linear System: Theory and Application, New York: Wiley, 2000.
[3] T. Kaczorek, 1-D and 2-D Systems, Berlin: Springer-Verlag, 2002.
[4] D. G. Luenberger, Introduction to Dynamic Systems, New York: Wiley, 1979.
[5] H. Minc, Nonnegative Matrices, New York: Wiley, 1987.
[6] P. H. Leslie, “On the use of matrices in certain population mathematics,” Biomerriko, vol. 33, pp. 183-212, 1945.
[7] P. D. Berk, J. R. Bloomer, and R. B. Howe, “Constitutional hepatic dysfunction (Gilbert’s syndrome),” The American Journal of Medicine, vol49, no. 3, pp.296-305, 1970.
[8] J. A. Jacquez, Compartmental Analysis in Biology and Medicine, Ann Arbor, MI: Univ. Michigan Press, 1985.
[9] H. Caswell, Matrix Population Model: Construction, Analysis and Interpretation, Sunderland, MA: Sinauer Assoc., 2001.
[10] L. Benvenuti and L. Farina, “Positive and compartmental systems,” IEEE Transactions on Automatic Control, vol.47, no.2, pp. 370-373, 2002.
[11] H. K. Khalil, Nonlinear Systems, third ed., Upper Saddle River, NJ: Prentice-Hall, Inc., 2002.
[12] M. Vidyasagar, Nonlinear Systems Analysis, New Jersey: Prentice-Hall, Inc., 2000.
[13] J. E. Slotine and W. Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice-Hall, Inc., 1991.
[14] T. Kaczorek, “Stabilization of positive linear systems,” Proceedings of the 37th IEEE Conference on Decision and Control, pp. 620-621, 1998.
[15] T. Kaczorek, “Stabilization of positive linear system by state-feedback,” Pomiary, Automatyka, Kontrola, vol. 3, pp. 2-5, 1999.
[16] P. D. Leenheer and D. Aeyels, “Stabilization of positive linear system,” Systems & Control Letters, vol. 44, no.4, pp. 259-271, 2001.
[17] H. Gao, J. Lam, C. Wang, and S. Xu, “Control for stability and positivity: Equivalent conditions and computation,” IEEE Transactions on Circuits and Systems Ⅱ, vol. 52, no. 9, pp. 540-544, Sep. 2005.
[18] M. A. Rami and F. Tadeo, “Controller synthesis for positive linear systems with bounded controls,” IEEE Transactions on Circuits and Systems Ⅱ, vol. 54, no. 2, pp. 151-155, 2007.
[19] B. Roszak and E. J. Davison, “Necessary and sufficient conditions for stabilizability of positive LTI systems,” Systems & Control Letters, vol. 58, pp. 474-481, 2009.
[20] Z. Shu, J. Lam, H. Gao, B. Du, and L. Wu, “Positive observers and dynamic output-feedback controllers for interval positive linear systems,” IEEE Transactions on Circuits and Systems Ⅰ, vol. 55, no. 10, pp. 3209-3222, Nov. 2008.
[21] P. Ling, J. Lam, and Z. Shu, “Positive observers for Positive interval linear discrete-time delay systems,” Proceedings of 48th IEEE Conference on Decision Control, pp. 6107-6112, Shanghai, P.R., China, 2009.
[22] X. Liu, “Stability analysis of switched positive systems: A switched linear copositive lyapunov function method,” IEEE Transactions on Circuits and Systems Ⅱ, vol. 56, no. 5, pp. 414-418, 2009.
[23] O. Mason and R. Shorten, “Some results on the stability of positive switched linear systems,” Proceedings of 43rd Conference on Decision and Control, 2004, pp. 4601-4606.
[24] O. Mason and R. Shorten, “The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem,” Electronic Journal of Linear Algebra, vol. 12, pp. 42-63, 2005.
[25] O. Mason and R. Shorten, “Quadratic and copositive Lyapunov functions and the stability of positive switched linear systems,” Proceedings of 2007 American Control Conference, pp. 657-662, 2007.
[26] O. Mason and R. Shorten, “On linear copositive Lyapunov functions and the stability of switched positive linear systems,” IEEE Transactions on Automatic Control, vol. 52, no. 7, pp. 1346-1349, 2007.
[27] F. Knorn, O. Mason, and R. Shorten, “On Linear Co-positive Lyapunov Functions for Sets of Linear Positive Systems,” Automatica, vol. 45, no. 8, pp. 1943-1947, 2009.
[28] M. A. Rami, F. Tadeo, and A. Benzaouia, “Control of constrained positive discrete systems,” Proceedings of 2007 American Control Conference, pp. 5851-5856, New York, USA, 2007.
[29] M. A. Rami, and F. Tadeo, “Positive observation problem for linear discrete positive systems,” Proceedings of 45th IEEE Conference on Decision and Control, pp. 4729-4733, Athens, Greece, 2007.
[30] X. Liu, L. Wang, W. Yu, and S. Zhong, “Constrained control of positive discrete-time systems with delays,” IEEE Transactions on Circuits and Systems Ⅱ, vol. 55, no. 2, pp. 193-197, 2008.
[31] X. Liu, “Constrained control of positive systems with delays,” IEEE Transactions on Automatic Control, vol. 54, no. 7, pp. 1596-1600, 2009.
[32] M. A. Rami, F. Tadeo, and A. Benzaouia, “Control of constrained positive discrete systems,” Proceeding of 2007 American Control Conference, pp. 5851-5856, New York, USA, 2007.
[33] M. A. Rami, and F. Tadeo, “Positive observation problem for linear discrete positive systems,” Proceeding of 45th IEEE Conference on Decision and Control, pp. 4729-4733, Athens, Greece, 2007.
[34] L. Caccetta, L. R. Foulds, and V. G. Rumchev, “A positive linear discrete-time model of capacity planning and its controllability properties,” Mathematical and Computer Modelling, vol. 40, no. 1-2, pp. 217-226, 2004.
[35] M. Busłowicz and T. Kaczorek, “Robust stability of positive discrete-time interval systems with time-delays,” Bulletin of The Polish Academy of Sciences: Technical Sciences, vol.52, no.2, 2004.
[36] N. Dautrebande and G. Bastin, “Positive linear observers for positive linear systems,” Proceeding of 1999 Europe Control Conference, 1999.
[37] L. Benvenuti and L. Farina, “Eigenvalue regions for positive systems,” Systems & Control Letters, vol. 51, pp. 325-330, 2004.
[38] J. Back and A. Astolfi, “Positive linear observers for positive linear systems: A Sylvester equation approach,” Proceeding of 2006 American Control Conference, pp. 4037-4042, Minneapolis, Minnesota, USA, 2006.
[39] J. Feng, J. Lam, P. Li, and Z. Shu, “Decay rate constrained stabilization of positive systems using static output feedback,” International Journal of Robust and Nonlinear Control, vol. 83, no. 3, pp. 575-584, 2010.
[40] T. Coleman, M. Branch, and A. Grace, Optimization Toolbox for Use with Matlab, Natick, MA: The Math Works Inc., 1999.