本論文中研究三個主題，第一：針對具有多個時滯狀態的線性多時延系統進行穩定性分析，假設這些系統中時滯長度會在固定區間內變化，時延相關的指數穩定性分析條件是以線性矩陣不等式的型式呈現。為了降低條件的保守性，本論文使用新的李亞普諾夫函數，包含了更完整的狀態資訊，使得在定理推導過中可以將時變時滯視為不確定參數。第二：當考慮到時變時滯長度的變化下限大於或等於零時，建立一時延參數相關的李亞普諾夫函數，以分析具有一個非遞減時滯長度之時延系統。本論文所提出之理論藉由數值範例以及直流馬達模型之應用進行說明，以驗證其應用方式及效能。第三：針對連續時間系統提出一種數位準比例微分控制器，以達系統之指數穩定化。本論文所提出的控制方法則可視為傳統PDP控制之一種變形，不同點在於本論文提出的控制法則僅使用系統輸出的取樣訊號，因此較易於實現且實用。此部分所提出的控制器藉由應用於二階負阻尼系統及一雙倒單擺系統來驗證其效能及可行性。

This dissertation studies a general class of linear systems with multiple successive delay components. First of all, the delays are assumed to vary in intervals, and delay-dependent exponential stability conditions are derived in terms of linear matrix inequalities. To reduce conservativeness, a new Lyapunov-Krasovskii functional is designed to contain more complete state information, so that a derivation procedure with time-varying delays treated as uncertain parameters can be adopted. Secondly, when the lower bound of time-varying delay’s rate of change is greater than or equal to zero, a delay-parameter-dependent Lyapunov functional is built to analyze the stability of linear systems with a non-decreasing time-varying delay. Pure numerical examples as well as an example with a DC motor model are provided to demonstrate the effectiveness of the proposed stability criteria.
Finally, a digital quasi-PD controller is proposed to achieve exponential stabilization for linear continuous-time systems. As a variation of the traditional position and delayed position (PDP) control, the proposed controller uses samples of the output signals, which is easier to implement and more practical. A second-order negatively damped system and a double inverted pendulum system are tested to show the control method is effective.

誌謝 I
摘要 II
Abstract III
Content IV
List of Figures V
List of Tables V
Chapter 1 Introduction 1
1.1 Background 1
1.2 Organization 6
1.3 Notations 7
Chapter 2 Exponential stability analysis of linear systems with multiple successive delay components 8
2.1 Problem formulation 8
2.2 Exponential stability analysis 9
2.3 Two special cases and numerical examples 21
Chapter 3 Parameter-dependent Lyapunov functional for linear systems with a time-varying delay 29
3.1 Problem formulation 29
3.2 Stability analysis for both nominal and uncertain system 30
3.3 Numerical examples 38
Chapter 4 Stability analysis of sampled-data control systems 41
4.1 Problem formulation 41
4.2 A DC motor example 42
4.2.1. Considering non-uniform sampling 45
4.2.2. PID type controller with non-uniform sampling and delay rate 48
Chapter 5 Exponential stabilization of linear systems using discrete-time quasi-PD controller 53
5.1 Problem formulation 53
5.2 Synthesis conditions 57
5.3 Numerical examples 63
Chapter 6 Conclusions and future works 68
6.1 Conclusions 68
6.2 Future works 68
References 70

[1]K. Gu, V. L. Kharitonov, J. Chen, Stability of Time-Delay Systems, Birkha&;uuml;ser, Boston, MA, 2003.
[2]S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Springer, Berlin, CA, 2001.
[3]Y. S. Moon, P. Park, W. H. Kown, Y. S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, International Journal of Control 74 (2001) 1447-1455.
[4]P Park, Delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Transactions on Automatic Control 44 (1999) 876-877.
[5]S. Xu , J. Lam, Improved delay-dependent stability criteria for time-delay systems, IEEE Transactions on Automatic Control 50 (2005) 384-387.
[6]Y. He, Q.-G. Wang, L. Xie, C. Liu, Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE Transactions on Automatic Control 52 (2007) 293-299.
[7]X. Jiang, Q.-L. Han, On control for linear systems with interval time-varying delay, Automatica 41 (2005) 2099-2106.
[8]T. Zhang, Y. Li, G. Liu, Robust stabilisation of uncertain systems with interval time-varying state and input delays, International Journal of Systems and Science 40 (2009) 11-20.
[9]E. H. Tissir, Delay-dependent robust stability of linear systems with non-commensurate time varying delays, International Journal of Systems and Science 38 (2007) 749-757.
[10]C. Lin, Q.-G. Wang, T. H. Lee, A less conservative robust stability test for linear uncertain time-delay systems, IEEE Transactions on Automatic Control 51 (2006) 87-91.
[11]P. Park, J. W. Ko, Stability and robust stability for systems with a time-varying delay, Automatica 43 (2007) 1855-1858.
[12]M. Wu, Y. He, J.-H. She, G.-P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems, Automatica 40 (2004) 1435-1439.
[13]W. Zhang, M. S. Branicky, S. M. Phillips, Stability of networked control systems, IEEE Control Systems Magazine 21 (2001) 84-99.
[14]D. Yue, Q.-L. Han, C. Peng, State feedback controller design of networked control systems, IEEE Transactions on Circuits and Systems II: Express Briefs 51 (2004) 640-644.
[15]D. Yue, Q.-L. Han, J. Lam, Network-based robust control of systems with uncertainty, Automatica 41 (2005) 999-1007.
[16]H. Shao, New Delay-dependent stability criteria for systems with interval delay, Automatica 45 (2009) 744-749.
[17]Y. He, Q.-G. Wang, C. Lin, M. Wu, Delay-range-dependent stability for systems with time-varying delay, Automatica 43 (2007) 371-376.
[18]C. Peng, Y.-C. Tian, Improved delay-dependent robust stability criteria for uncertain systems with interval time-varying delay, IET Control Theory and Applications 2 (2008) 752-761.
[19]B. R. Barmish, New tools for robustness of linear systems, Macmillan, New York, New York, 1994.
[20]S. Xu, J. Lam, On equivalence and efficiency of certain stability criteria for time-delay systems, IEEE Transactions on Automatic Control 52 (2007) 95-101.
[21]J. Lam, H. Gao, C. Wang, Stability analysis for continuous systems with two additive time-varying delay components, Systems &; Control Letters 56 (2007) 16-24.
[22]B. Du, J. Lam, Z. Shu, Z. Wang, A delay-partitioning projection approach to stability analysis of continuous systems with multiple delay components, IET Control Theory and Applications 3 (2009) 383-390.
[23]G. Guo, Stability and performance of multiple-delay systems with successive delay components, International Journal of Adaptive Control and Signal Processing 24 (2010) 643-656.
[24]R. Dey, G. Ray, S. Ghosh, A. Rakshit, Stability analysis for continuous system with additive time-varying delays: a less conservative result, Applied Mathematics and Computation 215 (2010) 3740-3745.
[25]H. Gao, T. Chen, J. Lam, A new delay system approach to network- based control, Automatica 44 (2008) 39-52.
[26]H. Wu, X. Liao, W. Feng, S. Guo, W. Zhang, Robust stability analysis of uncertain systems with two additive time-varying delay components, Applied Mathematical Modelling 33 (2009) 4345-4353.
[27]E. Fridman, A. Seuret, J. P. Richard, Robust sampled-data stabilization of linear system: an input delay approach, Automatica 40 (2004) 1441-1446.
[28]B. Liu, H. Hu, Stabilization of linear undamped systems via position and delayed position feedbacks, Journal of Sound and Vibration 312 (2008) 509-525.
[29]F. M. Atay, Balancing the inverted pendulum using position feedback, Applied Mathematics Letters 12 (1999) 51-56.
[30]K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics Letters A 170 (1992) 421-428.
[31]A. Maccari, Vibration control for the primary resonance of a cantilever beam by a time delay state feedback, Journal of Sound and Vibration 259 (2003) 241-251.
[32]R. Sipahi, N. Olgac, Active vibration suppression with time delayed feedback, Journal of Vibration and Acoustics, Transactions of the ASME 125 (2003) 384-388.
[33]J. Das, A. K. Mallik, Control of friction driven oscillation by time-delayed state feedback, Journal of Sound and Vibration 297 (2006) 578-594.
[34]S. Chatterjee, P. Mahata, Time-delayed absorber for controlling friction-driven vibration, Journal of Sound and Vibration 322 (2009) 39-59.
[35]C. Raffel, J. Smith, Practical modeling of bucket-brigade device circuits, Proc. 13th International Conference on Digital Audio Effects, Graz, Australia, 1998.
[36]S.-I. Niculescu, memoryless control with an -stability constraint for time-delay systems: an LMI approach, IEEE Transactions on Automatic Control 43 (1998) 739-743.
[37]L. Xie, Output feedback control of systems with parameter uncertainty, International Journal of Control 63 (1996) 741-750.
[38]I-K. Fong, C.-P. Lin, J.-S. Huang, Linear matrix inequality stability conditions from a parameter-dependent Lyapunov functional for linear systems with a time-varying delay, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 224 (2010) 701-712.
[39]H. Du, N. Zong, F. Naghdy, Actuator saturation control of uncertain structures with input time delay, Journal of Sound and Vibration 330 (2011) 4399-4412.
[40]M. C. Berg, Multirate digital control system design, IEEE Transactions on Automatic Control 33 (1988) 1139-1150.
[41]A. Seuret, A novel stability analysis of linear systems under asynchronous samplings, Automatica 48 (2012) 177-182.
[42]P. Naghshtabrizi, J. P. Hespanha, A. R. Teel, Exponential stability of impulsive systems with application to uncertain sampled-data systems, Systems &; Control Letters 57 (2008) 378-385.
[43]E. Fridman, A refined input delay approach to sampled-data control, Automatica 46 (2010) 421-427.
[44]S. I. Niculescu, memoryless control with an -stability constraint for time-delay systems: an LMI approach, IEEE Transactions on Automatic Control 43 (1998) 739-743.
[45]C. A. R. Crusius, A. Trofino, Sufficient LMI conditions for output feedback control problems, IEEE Transactions on Automatic Control 44 (1999) 1053-1057.
[46]A. Trofino, A. S. Bazanella, A. Fischman, Designing robust controllers with operating point tracking, Proceedings of IFAC Conference on System Structure and Control, Nantes, France, 1998.
[47]M. C. de Oliveira, J. Bernussou, J. C. Geromel, A new discrete-time robust stability condition, Systems &; Control Letters 37 (1999) 261-265.