Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Analysis and Synthesis of Positive Systems Under ℓ1 and L1 Performance

Part of the book series: Springer Theses ((Springer Theses))

Abstract

A control system is an interconnection of components forming a system configuration that will provide a desired system response. It is a arrangement of physical components and is composed of inputs, outputs and state variables. An input is a channel where changes can be injected into a system so as to activate or manipulate a process. An output is a channel where the response can be measured or observed. A state is a set of mathematical functions or physical variables which can be used to describe completely the future behaviour of a dynamic system if all the inputs future inputs are known. In many practical systems, there is such a kind of systems whose state variables and outputs are always positive (or at least nonnegative) for nonnegative inputs and initial conditions. These systems are commonly referred to as positive systems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Haddad WM, Chellaboina VS (2005) Stability and dissipativity theory for nonnegative dynamical systems: a unified analysis framework for biological and physiological systems. Nonlinear Anal: Real World Appl 6(1):35–65

    Article  MathSciNet  MATH  Google Scholar 

  2. Muratori S, Rinaldi S (1990) Equilibria, stability, and reachability of Leslie systems with nonnegative inputs. IEEE Trans Autom Control 35(9):1065–1068

    Article  MATH  Google Scholar 

  3. Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences. SIAM, Philadephia

    Book  MATH  Google Scholar 

  4. Luenberger D (1979) Introduction to dynamic systems: theory, models, and applications. Wiley, New York

    MATH  Google Scholar 

  5. Rami MA, Schönlein M, Jordan J (2013) Estimation of linear positive systems with unknown time-varying delays. Eur J Control 19(3):179–187

    Article  MathSciNet  MATH  Google Scholar 

  6. Feng J, Lam J, Li P et al (2011) Decay rate constrained stabilization of positive systems using static output feedback. Int J Robust Nonlinear Control 21(1):44–54

    Article  MathSciNet  MATH  Google Scholar 

  7. Fornasini E, Valcher ME (2010) Linear copositive Lyapunov functions for continuous-time positive switched systems. IEEE Trans Autom Control 55:1933–1937

    Article  MathSciNet  Google Scholar 

  8. Kaczorek T (2002) Positive 1D and 2D Systems. Springer, Berlin

    Book  MATH  Google Scholar 

  9. Li P, Lam J, Shu Z et al (2010) \({H}_{\infty }\) positive filtering for positive linear discrete-time systems: an augmentation approach. IEEE Trans Autom Control 55(10):2337–2342

    Article  MathSciNet  Google Scholar 

  10. Shu Z, Lam J, Gao H et al (2008) Positive observers and dynamic output-feedback controllers for interval positive linear systems. IEEE Trans Circuits Syst (I) 55(10):3209–3222

    Article  MathSciNet  Google Scholar 

  11. Valcher ME (2009) Reachability properties of continuous-time positive systems. IEEE Trans Autom Control 54:1586–1590

    Article  MathSciNet  Google Scholar 

  12. Farina L (1996) On the existence of a positive realization. Syst Control Lett 28:219–226

    Article  MathSciNet  MATH  Google Scholar 

  13. Benvenuti L, Farina L (2004) A tutorial on the positive realization problem. IEEE Trans Autom Control 49(5):651–664

    Article  MathSciNet  Google Scholar 

  14. Haddad WM, Chellaboina VS (2004) Stability theory for nonnegative and compartmental dynamical systems with time delay. Syst Control Lett 51(5):355–361

    Article  MathSciNet  MATH  Google Scholar 

  15. Haddad WM, Chellaboina VS, Rajpurohit T (2004) Dissipativity theory for nonnegative and compartmental dynamical systems with time delay. IEEE Trans Autom Control 49(5):747–751

    Article  MathSciNet  Google Scholar 

  16. Liu X, Lam J (2013) Relationships between asymptotic stability and exponential stability of positive delay systems. Int J Gen Syst 42(2):224–238

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu X, Yu W, Wang L et al (2009) Stability analysis of positive systems with bounded time-varying delays. IEEE Trans Circuits Syst (II) 56(7):600–604

    Article  MathSciNet  Google Scholar 

  18. Shen J, Lam J (2014) \({L}_{\infty }\)-gain analysis for positive systems with distributed delays. Automatica 50(1):175–179

    Article  MathSciNet  MATH  Google Scholar 

  19. Wu L, Lam J, Shu Z et al (2009) On stability and stabilizability of positive delay systems. Asian J Control 11(2):226–234

    Article  MathSciNet  Google Scholar 

  20. Kurek JE (2002) Stability of positive 2-D system described by the Roesser model. IEEE Trans Circuits Syst (I) 49:531–533

    Article  MathSciNet  Google Scholar 

  21. Valcher ME, Fornasini E (1995) State models and asymptotic behaviour of 2D positive systems. IMA J Math Control Inf 12(1):17–36

    Article  MathSciNet  MATH  Google Scholar 

  22. Farina L, Rinaldi S (2011) Positive linear systems: theory and applications. Wiley, New York

    MATH  Google Scholar 

  23. Son NK, Hinrichsen D (1996) Robust stability of positive continuous-time systems. Numer Funct Anal Opt 17:649–659

    Article  MathSciNet  MATH  Google Scholar 

  24. Feng J, Lam J, Shu Z et al (2010) Internal positivity preserved model reduction. Int J Control 83(3):575–584

    Article  MathSciNet  MATH  Google Scholar 

  25. Li P, Lam J, Wang Z et al (2011) Positivity-preserving \({H}_{\infty }\) model reduction for positive systems. Automatica 47(7):1504–1511

    Article  MathSciNet  MATH  Google Scholar 

  26. Kaczorek T (1999) Stabilization of positive linear systems by state feedback. Pomiary, Automatyka, Kontrola 3:2–5

    Google Scholar 

  27. De Leenheer P, Aeyels D (2001) Stabilization of positive linear systems. Syst Control Lett 44(4):259–271

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu X, Yu W, Wang L et al (2010) Stability analysis for continuous-time positive systems with time-varying delays. IEEE Trans Autom Control 55(4):1024–1028

    Article  MathSciNet  Google Scholar 

  29. Gurvits L, Shorten R, Mason O (2007) On the stability of switched positive linear systems. IEEE Trans Autom Control 52(6):1099–1103

    Article  MathSciNet  Google Scholar 

  30. Valcher ME (1997) On the internal stability and asymptotic behavior of 2-D positive systems. IEEE Trans Circuits Syst (I) 44(7):602–613

    Article  MathSciNet  MATH  Google Scholar 

  31. Kaczorek T (2009) Asymptotic stability of positive 2D linear systems with delays. Bull Polish Acad Sci: Tech Sci 57(2):133–138

    MATH  Google Scholar 

  32. Kaczorek T (2009) LMI approach to stability of 2D positive systems. Multidimens Syst Signal Process 20(1):39–54

    Article  MathSciNet  MATH  Google Scholar 

  33. He Z, Wu L, Su X (2009) \({H}_{\infty }\) control of linear positive systems: continuous-and discrete-time cases. Int J Innov Comput Inf Control 5(6):1747–1756

    Google Scholar 

  34. Tong Y, Zhang L, Basin M et al (2014) Weighted \({H}_{\infty }\) control with D-stability constraint for switched positive linear systems. Int J Robust Nonlinear Control 24(4):758–774

    Article  MathSciNet  MATH  Google Scholar 

  35. Lu WW, Balas GJ (1988) A comparison between Hankel norms and induced system norms. IEEE Trans Autom Control 43(11):1658–1662

    Article  MathSciNet  MATH  Google Scholar 

  36. Rami MA, Tadeo F (2007) Controller synthesis for positive linear systems with bounded controls. IEEE Trans Circuits Syst (II) 54(2):151–155

    Article  Google Scholar 

  37. Haddad WM, Chellaboina VS, Hui Q (2010) Nonnegative and compartmental dynamical systems. Princeton University Press, Princeton

    Book  MATH  Google Scholar 

  38. Ebihara Y, Peaucelle D, Arzelier D (2012) Optimal \({L}_{1}\)-controller synthesis for positive systems and its robustness properties. In: Proceedings of the 2012 American control conference

    Google Scholar 

  39. Chen X, Lam J, Li P et al (2013) \(\ell _{1}\)-induced norm and controller synthesis of positive systems-induced norm and controller synthesis of positive systems. Automatica 49:1377–1385

    Article  MathSciNet  MATH  Google Scholar 

  40. Chen X, Lam J, Li P et al (2014) Output-feedback control for continuous-time interval positive systems under \(L_{1}\) performance. Asian J Control 16(6):1592–1601

    Article  MathSciNet  MATH  Google Scholar 

  41. Briat C (2013) Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: \({L}_{1}\)-gain and \({L}_{\infty }\)-gain characterization. Int J Robust Nonlinear Control 23(17):1932–1954

    Article  MathSciNet  MATH  Google Scholar 

  42. David G (1971) An introduction to observers. IEEE Trans Autom Control 16(6):596–602

    Article  Google Scholar 

  43. Gauthier JP, Hammouri H, Othman S (1992) A simple observer for nonlinear systems applications to bioreactors. IEEE Trans Autom Control 37(6):875–880

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang Z, Huang B, Unbehauen H et al (1999) Robust \({H}_{\infty }\) observer design of linear state delayed systems with parametric uncertainty: the discrete-time case. Automatica 35(6):1161–1167

    Article  MathSciNet  MATH  Google Scholar 

  45. Rami MA, Cheng CH, De Prada C (2008) Tight robust interval observers: an LP approach. In: Proceedings of the 47th IEEE conference on decision and control

    Google Scholar 

  46. Rami MA, Helmke U, Tadeo F (2007) Positive observation problem for linear time-delay positive systems. In: Mediterranean conference on control and automation

    Google Scholar 

  47. Rami MA, Helmke U, Tadeo F (2007) Positive observation problem for linear time-lag positive systems. In: Proceedings of the third IFAC symposium on system structure and control

    Google Scholar 

  48. Van Den Hof JM (1998) Positive linear observers for linear compartmental systems. SIAM J Control Optim 36(2):590–608

    Article  MathSciNet  MATH  Google Scholar 

  49. Back J, Astolfi A (2008) Design of positive linear observers for positive linear systems via coordinate transformations and positive realizations. SIAM J Control Optim 47(1):345–373

    Article  MathSciNet  MATH  Google Scholar 

  50. Mazenc F, Bernard O (2011) Interval observers for linear time-invariant systems with disturbances. Automatica 47(1):140–147

    Article  MathSciNet  MATH  Google Scholar 

  51. Lemesle V, Gouzé JL (2005) Hybrid bounded error observers for uncertain bioreactor models. Bioprocess Biosyst Eng 27(5):311–318

    Article  Google Scholar 

  52. Moisan M, Bernard O, Gouzé JL et al (2009) Near optimal interval observers bundle for uncertain bioreactors. Automatica 45(1):291–295

    Article  MathSciNet  MATH  Google Scholar 

  53. Li P, Lam J (2012) Positive state-bounding observer for positive interval continuous-time systems with time delay. Int J Robust Nonlinear Control 22:1244–1257

    Article  MathSciNet  MATH  Google Scholar 

  54. Chen X, Lam J, Li P et al (2013) \(L_{1}\)-induced performance analysis and sparse controller synthesis for interval positive systems. In: Proceedings of the international conference of applied and engineering mathematics

    Google Scholar 

  55. Chen X, Lam J (2014) Positive state-bounding observer for interval positive systems under \(L_{1}\) performance. In: Proceedings of the 33rd chinese control conference

    Google Scholar 

  56. Anderson BDO, Moore JB (2012) Optimal filtering. Courier Dover Publications, Mineola

    MATH  Google Scholar 

  57. Chui CK, Chen G (1999) Kalman filtering. Springer, Berlin

    Book  MATH  Google Scholar 

  58. Grewal MS, Andrews AP (2011) Kalman filtering: theory and practice using MATLAB. Wiley, Canada

    Book  MATH  Google Scholar 

  59. Haykin SS (2001) Kalman filtering and neural networks. Wiley Online Library, New York

    Book  Google Scholar 

  60. Sinopoli B, Schenato L, Franceschetti M et al (2004) Kalman filtering with intermittent observations. IEEE Trans Autom Control 49(9):1453–1464

    Article  MathSciNet  Google Scholar 

  61. Gao H, Wang C (2004) A delay-dependent approach to robust \({H}_{\infty }\) filtering for uncertain discrete-time state-delayed systems. IEEE Trans Signal Process 52(6):1631–1640

    Article  MathSciNet  Google Scholar 

  62. Gershon E, Limebeer DJN, Shaked U et al (2001) Robust \({H}_{\infty }\) filtering of stationary continuous-time linear systems with stochastic uncertainties. IEEE Trans Autom Control 46(11):1788–1793

    Article  MathSciNet  MATH  Google Scholar 

  63. Chen X, Lam J, Li P (2014) Positive filtering for continuous-time positive systems under \({L}_{1}\) performance. Int J Control 87(9):1906–1913

    Article  MathSciNet  MATH  Google Scholar 

  64. Babuska R (1998) Fuzzy modeling for control. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  65. Bellman RE, Zadeh LA (1970) Decision-making in a fuzzy environment. Manag Sci 17(4):B141

    Article  MathSciNet  MATH  Google Scholar 

  66. Dutta S (1993) Fuzzy logic applications: technological and strategic issues. IEEE Trans Eng Manag 40(3):237–254

    Article  Google Scholar 

  67. Keller J, Krisnapuram R, Pal NR et al (2005) Fuzzy models and algorithms for pattern recognition and image processing. Springer, Berlin

    MATH  Google Scholar 

  68. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MathSciNet  MATH  Google Scholar 

  69. Zadeh LA (1968) Fuzzy algorithms. Inf Control 12(2):94–102

    Article  MathSciNet  MATH  Google Scholar 

  70. Zadeh LA (1971) Similarity relations and fuzzy orderings. Inf Sci 3(2):177–200

    Article  MathSciNet  MATH  Google Scholar 

  71. Zadeh LA (1973) Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans Syst Man Cybern 1:28–44

    Article  MathSciNet  MATH  Google Scholar 

  72. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8(3):199–249

    Article  MathSciNet  MATH  Google Scholar 

  73. Lee CC (1990) Fuzzy logic in control systems: fuzzy logic controller. II. IEEE Trans Syst Man Cybern 20(2):419–435

    Article  MathSciNet  MATH  Google Scholar 

  74. Su X, Shi P, Wu L et al (2013) A novel control design on discrete-time Takagi-Sugeno fuzzy systems with time-varying delays. IEEE Trans Fuzzy Syst 21(4):655–671

    Article  Google Scholar 

  75. Wang L (1999) A course in fuzzy systems. Prentice-Hall press, USA

    Google Scholar 

  76. Guerra TM, Vermeiren L (2004) LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form. Automatica 40(5):823–829

    Article  MathSciNet  MATH  Google Scholar 

  77. Tanaka K, Ikeda T, Wang HO et al (1998) Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs. IEEE Trans Fuzzy Syst 6(2):250–265

    Article  Google Scholar 

  78. Tuan HD, Apkarian P, Narikiyo T et al (2001) Parameterized linear matrix inequality techniques in fuzzy control system design. IEEE Trans Fuzzy Syst 9(2):324–332

    Article  Google Scholar 

  79. Teixeira M, Assunção E, Avellar RG et al (2003) On relaxed LMI-based designs for fuzzy regulators and fuzzy observers. IEEE Trans Fuzzy Syst 11(5):613–623

    Article  Google Scholar 

  80. Wang Z, Ho DWC, Liu X et al (2004) A note on the robust stability of uncertain stochastic fuzzy systems with time-delays. IEEE Trans Syst Man Cybern - Part A 34(4):570–576

    Article  Google Scholar 

  81. Tanaka K, Sugeno M (1992) Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst 45(2):135–156

    Article  MathSciNet  MATH  Google Scholar 

  82. Feng G (2004) Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions. IEEE Trans Fuzzy Syst 12(1):22–28

    Article  Google Scholar 

  83. Johansson M, Rantzer A, Arzen K (1999) Piecewise quadratic stability of fuzzy systems. IEEE Trans Fuzzy Syst 7(6):713–722

    Article  Google Scholar 

  84. Cao SG, Rees NW, Feng G (1996) Stability analysis and design for a class of continuous-time fuzzy control systems. Int J Control 64(6):1069–1087

    Article  MathSciNet  MATH  Google Scholar 

  85. Cao SG, Rees NW, Feng G (1997) Analysis and design for a class of complex control systems part II: fuzzy controller design. Automatica 33(6):1029–1039

    Article  MathSciNet  MATH  Google Scholar 

  86. Choi DJ, Park PG (2003) \({H}_{\infty }\) state-feedback controller design for discrete-time fuzzy systems using fuzzy weighting-dependent Lyapunov functions. IEEE Trans Fuzzy Syst 11(2):271–278

    Article  MathSciNet  Google Scholar 

  87. Tanaka K, Hori T, Wang HO et al (2003) A multiple Lyapunov function approach to stabilization of fuzzy control systems. IEEE Trans Fuzzy Syst 11(4):582–589

    Article  Google Scholar 

  88. Chen X, Lam J, Gao H et al (2013) Stability analysis and control design for 2-D fuzzy systems via basis-dependent Lyapunov functions. Multidimens Syst Signal Process 24(3):395–415

    Article  MathSciNet  MATH  Google Scholar 

  89. Jadbabaie A (1999) A reduction in conservatism in stability and \({L}_{2}\) gain analysis of Takagi-Sugeno fuzzy systems via linear matrix inequalities. In: Proceedings of 14th IFAC Triennial world congress. Beijing, China, pp 285–289

    Google Scholar 

  90. Tanaka K, Wang HO (2001) Fuzzy control systems design and analysis: a linear matrix inequality approach. Wiley, New York

    Book  Google Scholar 

  91. Tanaka K, Hori T, Wang HO (2001) A fuzzy Lyapunov approach to fuzzy control system design. In: Proceedings of the American control conference. Arlington, VA, pp 4790–4795

    Google Scholar 

  92. Zhou S, Li T (2005) Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent Lyapunov-Krasovskii function. Fuzzy Sets Syst 151(1):139–153

    Article  MathSciNet  MATH  Google Scholar 

  93. Benzaouia A, Mehdi D, El Hajjaji A et al (2007) Piecewise quadratic Lyapunov function for nonlinear systems with fuzzy static output feedback control. In: Proceedings of the European control conference, Kos, Greece

    Google Scholar 

  94. Benzaouia A, Hmamed A, Hajjaji EL, Hajjaji EL A (2010) Stabilization of controlled positive discrete-time T-S fuzzy systems by state feedback control. Int J Adap Control Signal Process 24:1091–1106

    Article  MathSciNet  MATH  Google Scholar 

  95. Mao Y, Zhang H, Dang C et al (2012) Stability analysis and constrained control of a class of fuzzy positive systems with delays using linear copositive Lyapunov functional. Circuits Syst Signal Process 31(5):1863–1875

    Article  MathSciNet  MATH  Google Scholar 

  96. Wu Y, Luo H, Zhang H et al (2013) Stability analysis of discrete-time fuzzy positive systems with time delays. J Intell Fuzzy Syst 25(4):893–905

    MathSciNet  MATH  Google Scholar 

  97. Chen X, Lam J, Lam HK (2015) Positive filtering for positive Takagi-Sugeno fuzzy systems under \(\ell _{1}\) performance. Inf Sci 299:32–41

    Article  MathSciNet  Google Scholar 

  98. Bapat RB, Raghavan TES (1997) Nonnegative matrices and applications. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  99. Fazel M (2002) Matrix rank minimization with applications. PhD thesis, Department of Electrical Engineering, Stanford University

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China

    Xiaoming Chen

Authors
  1. Xiaoming Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoming Chen .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Science+Business Media Singapore

About this chapter

Cite this chapter

Chen, X. (2017). Introduction. In: Analysis and Synthesis of Positive Systems Under ℓ1 and L1 Performance. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-10-2227-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-2227-2_1

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-2226-5

  • Online ISBN: 978-981-10-2227-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation