Skip to main content
SpringerLink
Log in

Positive Real Systems

  • Chapter
  • First Online:
Dissipative Systems Analysis and Control

Part of the book series: Communications and Control Engineering ((CCE))

Abstract

Positive real systems were first discovered and studied in the Networks and Circuits scientific community, by the German scientist Wilhelm Cauer in his 1926 Ph.D. thesis [1,2,3,4]. However, the term positive real has been coined by Otto Brune in his 1931 Ph.D. thesis [5, 6], building upon the results of Ronald M. Foster [7] (himself inspired by the work in [8], and we stop the genealogy here). O. Brune was in fact the first to provide a precise definition and characterization of a positive real transfer function (see [6, Theorems II, III, IV, V]).

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 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
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

Change history

  • 21 May 2022

    The original version of the book was inadvertently published with incorrect text and equations. The incorrect text and equations were corrected. The erratum chapters and the book have been updated with the changes.

Notes

  1. 1.

    More details on \({\mathscr {L}}_{p}\) spaces can be found in Chap. 4.

  2. 2.

    Throughout the book, we shall deal only with causal (non-anticipative) systems.

  3. 3.

    Bounded Input-Bounded Output.

  4. 4.

    For a holomorphic function f(s), one defines an essential singularity as a point a where neither \(\lim _{s \rightarrow a} f(s)\) nor \(\lim _{s \rightarrow a} \frac{1}{f(s)}\) exist. The function \(e^{\frac{1}{s}}\) has an essential singularity at \(s=0.\) Rational functions do not have essential singularities; they have only poles.

  5. 5.

    Minors, or subdeterminants, are the determinants of the square submatrices of a matrix.

  6. 6.

    In the case of square matrices, the poles and their multiplicities can be determined from the fact that \(\mathrm{det}(H(s))=c\frac{n(s)}{d(s)}\) for some polynomials n(s) and d(s), after possible cancelation of the common factors. The roots of n(s) are the zeroes of H(s), the roots of d(s) are the poles of H(s) [31, Corollary 2.1].

  7. 7.

    Also called in this particular case the Cayley transformation.

  8. 8.

    As we shall see later, such a definition may not be entirely satisfactory, because some non-regular transfer matrices can be SPR according to it, while they should not, see Example 2.67, see also the paragraph after Definition 2.77.

  9. 9.

    As noted in [38], the third condition encompasses the other two, so that the first and the second conditions are presented only for the sake of clarity.

  10. 10.

    This is also proved in [26, Problem 5.2.4], which uses the fact that if (A, B, C, D) is a minimal realization of \(H(s) \in \mathbb {C}^{m \times m}\), then \((A-BD^{-1}C,BD^{-1},-C^{T}(D^{-1})^{T},D^{-1})\) is a minimal realization of \(H^{-1}(s).\) Then use the KYP Lemma (next chapter) to show that \(H^{-1}(s)\) is positive real.

  11. 11.

    Thus, it would certainly be more rigorous either to augment Definition 2.58 with regularity or to modify Definition 2.77. This was pointed out to us by Augusto Ferrante.

  12. 12.

    We use the notation \(\langle f_{t},g_{t}\rangle \) for \(\int _{0}^{t}f(s)g(s)ds.\)

  13. 13.

    Similarly as in the foregoing section, this is a consequence of the Kalman–Yakubovich–Popov Lemma for SPR systems.

  14. 14.

    Said otherwise, velocities and forces are reciprocal, or dual, variables.

References

  1. Cauer E, Mathis W, Pauli R (2000) Life and work of Wilhelm Cauer. In: Jaï AE, Fliess, MM (eds) Proceedings 14th international symposium of mathematical theory of networks and systems MTNS2000. Perpignan, France

    Google Scholar 

  2. Cauer W (1926) Die verwirklichung der wechselstromwiderstände vorgeschriebener frequenzabhängigkeit. Archiv für Elecktrotechnik 17:355–388

    Google Scholar 

  3. Cauer W (1929) Vierpole. Elektrische Nachrichtentechnik 6:272–282

    Google Scholar 

  4. Pauli R (2002) The scientic work of Wilhelm Cauer and its key position at the transition from electrical telegraph techniques to linear systems theory. In: Proceedings of 16th European meeting on cybernetics and systems research (EMCSR), vol 2. Vienna, Austria, pp. 934–939

    Google Scholar 

  5. Brune O (1931) The synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function frequency. J Math Phys 10:191–236

    MATH  Google Scholar 

  6. Brune O (1931) Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency. PhD thesis, Massachusetts Institute of Technology, Department of Electrical Engineering, USA (1931). http://hdl.handle.net/1721.1/10661

  7. Foster R (1924) A reactance theorem. Bell Syst Tech J 3(2):259–267

    Google Scholar 

  8. Campbell GA (1922) Physical theory of the electric wave filter. Bell Syst Tech J 1(2):1–32

    Google Scholar 

  9. Arimoto S (1996) Control theory of nonlinear mechanical systems: a passivity-based and circuit-theoretic approach. Oxford University Press, Oxford, UK

    MATH  Google Scholar 

  10. Moylan PJ, Hill DJ (1978) Stability criteria for large-scale systems. IEEE Trans Autom Control 23(2):143–149

    MathSciNet  MATH  Google Scholar 

  11. Willems JC (1971) The generation of Lyapunov functions for input-output stable systems. SIAM J Control 9:105–133

    MathSciNet  MATH  Google Scholar 

  12. Willems JC (1972) Dissipative dynamical systems, Part I: general theory. Arch Rat Mech Anal 45:321–351

    MATH  Google Scholar 

  13. Barb FD, Ionescu V, de Koning W (1994) A Popov theory based approach to digital \(h_{\infty }\) control with measurement feedback for Pritchard-Salamon systems. IMA J Math Control Inf 11:277–309

    Google Scholar 

  14. Weiss M (1997) Riccati equation theory for Pritchard-Salamon systems: a Popov function approach. IMA J Math Control Inf 14:45–83

    MathSciNet  MATH  Google Scholar 

  15. Sontag ED (1998) Mathematical control theory: deterministic finite dimensional systems, vol 6, 2nd edn. Texts in applied mathematics. Springer, New York, USA

    MATH  Google Scholar 

  16. Desoer CA, Vidyasagar M (1975) Feedback systems: input-output properties. Academic Press, New-York

    MATH  Google Scholar 

  17. Vidyasagar M (1981) Input-output analysis of large-scale interconnected systems. Decomposition, well-posedness and stability, vol 29. Lecture notes in control and information sciences. Springer, London

    MATH  Google Scholar 

  18. Vidyasagar M (1993) Nonlinear systems analysis, 2nd edn. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  19. Willems JC (1972) Dissipative dynamical systems, part II: linear systems with quadratic supply rates. Arch Rat Mech Anal 45:352–393

    MATH  Google Scholar 

  20. Hill DJ, Moylan PJ (1976) The stability of nonlinear dissipative systems. IEEE Trans Autom Control 21(5):708–711

    MathSciNet  MATH  Google Scholar 

  21. Hill DJ, Moylan, PJ (1975) Cyclo-dissipativeness, dissipativeness and losslessness for nonlinear dynamical systems, Technical Report EE7526, November, The university of Newcastle. Department of Electrical Engineering, New South Wales, Australia

    Google Scholar 

  22. Hill DJ, Moylan PJ (1980) Connections between finite-gain and asymptotic stability. IEEE Trans Autom Control 25(5):931–936

    MathSciNet  MATH  Google Scholar 

  23. Hill DJ, Moylan PJ (1980) Dissipative dynamical systems: Basic input-output and state properties. J Frankl Inst 30(5):327–357

    MathSciNet  MATH  Google Scholar 

  24. Bhowmick P, Patra S (2017) On LTI output strictly negative-imaginary systems. Syst Control Lett 100:32–42

    MathSciNet  MATH  Google Scholar 

  25. Hughes TH (2017) A theory of passive linear systems with no assumptions. Automatica 86:87–97

    MathSciNet  MATH  Google Scholar 

  26. Anderson BDO, Vongpanitlerd S (1973) Network analysis and synthesis: a modern systems theory approach. Prentice Hall, Englewood Cliffs, New Jersey, USA

    Google Scholar 

  27. Faurre P, Clerget M, Germain F (1979) Opérateurs Rationnels Positifs. Application à l’Hyperstabilité et aux Processus Aléatoires. Méthodes Mathématiques de l’Informatique. Dunod, Paris. In French

    Google Scholar 

  28. Willems JC (1976) Realization of systems with internal passivity and symmetry constraints. J Frankl Inst 301(6):605–621

    MathSciNet  MATH  Google Scholar 

  29. Desoer CA, Kuh ES (1969) Basic circuit theory. McGraw Hill International, New York

    Google Scholar 

  30. Guiver C, Logemann H, Opmeer MR (2017) Transfer functions of infinite-dimensional systems: positive realness and stabilization. Math Control Signals Syst 29(2)

    Google Scholar 

  31. Maciejowski J (1989) Multivariable feedback design. Electronic systems engineering. Addison Wesley, Boston

    MATH  Google Scholar 

  32. Reis T, Stykel T (2010) Positive real and bounded real balancing for model reduction of descriptor systems. Int J Control 83(1):74–88

    MathSciNet  MATH  Google Scholar 

  33. Ober R (1991) Balanced parametrization of classes of linear systems. SIAM J Control Optim 29(6):1251–1287

    MathSciNet  MATH  Google Scholar 

  34. Bernstein DS (2005) Matrix mathematics. Theory, facts, and formulas with application to linear systems theory. Princeton University Press, Princeton

    MATH  Google Scholar 

  35. Narendra KS, Taylor JH (1973) Frequency domain criteria for absolute stability. Academic Press, New York, USA

    MATH  Google Scholar 

  36. Ioannou P, Tao G (1987) Frequency domain conditions for SPR functions. IEEE Trans Autom Control 32:53–54

    MATH  Google Scholar 

  37. Corless M, Shorten R (2010) On the characterization of strict positive realness for general matrix transfer functions. IEEE Trans Autom Control 55(8):1899–1904

    MathSciNet  MATH  Google Scholar 

  38. Ferrante A, Lanzon A, Ntogramatzidis L (2016) Foundations of not necessarily rational negative imaginary systems theory: relations between classes of negative imaginary and positive real systems. IEEE Trans Autom Control 61(10):3052–3057

    MathSciNet  MATH  Google Scholar 

  39. Tao G, Ioannou PA (1990) Necessary and sufficient conditionsfor strictly positive real matrices. Proc Inst Elect Eng 137:360–366

    Google Scholar 

  40. Khalil HK (1992) Nonlinear systems. MacMillan, New York, USA. 2nd edn. published in 1996, 3rd edn. published in 2002

    Google Scholar 

  41. Wen JT (1988) Time domain and frequency domain conditions for strict positive realness. IEEE Trans Autom Control 33:988–992

    MathSciNet  MATH  Google Scholar 

  42. Ferrante A, Lanzon A, Brogliato B (2019) A direct proof of the equivalence of side conditions for strictly positive real matrix transfer functions. IEEE Trans Autom Control 65(10):450–452. https://hal.inria.fr/hal-01947938

  43. Tao G, Ioannou PA (1988) Strictly positive real matrices and the Lefschetz-Kalman-Yakubovich Lemma. IEEE Trans Autom Control 33(12):1183–1185

    MathSciNet  MATH  Google Scholar 

  44. Heemels WPMH, Camlibel MK, Schumacher JM, Brogliato B (2011) Observer-based control of linear complementarity systems. Int J Robust Nonlinear Control 21(10):1193–1218

    MathSciNet  MATH  Google Scholar 

  45. Wolovich WA (1974) Linear multivariable systems, vol 11. Applied mathematical sciences. Springer, Berlin

    Google Scholar 

  46. Ferrante A, Ntogramatzidis L (2013) Some new results in the theory of negative imaginary systems with symmetric transfer matrix function. Automatica 49(7):2138–2144

    MathSciNet  MATH  Google Scholar 

  47. Gregor J (1996) On the design of positive real functions. IEEE Trans Circuits Syst I Fundam Theory Appl 43(11):945–947

    MathSciNet  Google Scholar 

  48. Henrion D (2002) Linear matrix inequalities for robust strictly positive real design. IEEE Trans Circuits Syst I Fundam Theory Appl 49(7):1017–1020

    MathSciNet  MATH  Google Scholar 

  49. Dumitrescu B (2002) Parametrization of positive-real transfer functions with fixed poles. IEEE Trans Circuits Syst I Fundam Theory Appl 49(4):523–526

    MATH  Google Scholar 

  50. Yu W, Li X (2001) Some stability properties of dynamic neural networks. IEEE Trans Circuits Syst I Fundam Theory Appl 48(2):256–259

    MATH  Google Scholar 

  51. Patel VV, Datta KB (2001) Comments on “Hurwitz stable polynomials and strictly positive real transfer functions’’. IEEE Trans Circuits Syst I Fundam Theory Appl 48(1):128–129

    Google Scholar 

  52. Wang L, Yu W (2001) On Hurwitz stable polynomials and strictly positive real transfer functions. IEEE Trans Circuits Syst I Fundam Theory Appl 48(1):127–128

    MathSciNet  MATH  Google Scholar 

  53. Marquez HJ, Agathokis P (1998) On the existence of robust strictly positive real rational functions. IEEE Trans Circuits Syst I(45):962–967

    MathSciNet  MATH  Google Scholar 

  54. Zeheb E, Shorten R (2006) A note on spectral conditions for positive realness of single-input-single-output systems with strictly proper transfer functions. IEEE Trans Autom Control 51(5):897–900

    MathSciNet  MATH  Google Scholar 

  55. Shorten R, King C (2004) Spectral conditions for positive realness of single-input single-output systems. IEEE Trans Autom Control 49(10):1875–1879

    MathSciNet  MATH  Google Scholar 

  56. Fernandez-Anaya G, Martinez-Garcia JC, Kucera V (2006) Characterizing families of positive real matrices by matrix substitutions on scalar rational functions. Syst Control Lett 55(11):871–878

    MathSciNet  MATH  Google Scholar 

  57. Bai Z, Freund RW (2000) Eigenvalue based characterization and test for positive realness of scalar transfer functions. IEEE Trans Autom Control 45(12):2396–2402

    MathSciNet  MATH  Google Scholar 

  58. Shorten R, Wirth F, Mason O, Wulff K, King C (2007) Stability criteria for switched and hybrid systems. SIAM Rev 49(4):545–592

    MathSciNet  MATH  Google Scholar 

  59. Liu M, Lam J, Zhu B, Kwok KW (2019) On positive realness, negative imaginariness, and \({H}_{\infty }\) control of state-space symmetric systems. Automatica 101:190–196

    Google Scholar 

  60. Hakimi-Moghaddam M, Khaloozadeh H (2015) Characterization of strictly positive multivariable systems. IMA J Math Control Inf 32:277–289

    MathSciNet  MATH  Google Scholar 

  61. Skogestad S, Postlethwaite I (2005) Multivariable feedback control, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  62. Kailath T (1980) Linear systems. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  63. Kottenstette N, McCourt M, Xia M, Gupta V, Antsaklis P (2014) On relationships among passivity, positive realness, and dissipativity in linear systems. Automatica 50:1003–1016

    MathSciNet  MATH  Google Scholar 

  64. van der Schaft AJ (2017) \(L2\)-gain and passivity techniques in nonlinear control, 3rd edn. Communications and control engineering. Springer International Publishing AG, New York

    Google Scholar 

  65. Hodaka I, Sakamoto N, Suzuki M (2000) New results for strict positive realness and feedback stability. IEEE Trans Autom Control 45(4):813–819

    MathSciNet  MATH  Google Scholar 

  66. Sakamoto N, Suzuki M (1996) \(\gamma \)-passive system and its phase property and synthesis. IEEE Trans Autom Control 41(6):859–865

    MathSciNet  MATH  Google Scholar 

  67. Fernandez-Anaya G, Martinez-Garcia JC, Kucera V, Aguilar-George D (2004) MIMO systems properties preservation under SPR substitutions. IEEE Trans Circuits Syst-II Express Briefs 51(5):222–227

    Google Scholar 

  68. Joshi SM, Gupta S (1996) On a class of marginally stable positive-real systems. IEEE Trans Autom Control 41(1):152–155

    MathSciNet  MATH  Google Scholar 

  69. Parks PC (1966) Lyapunov redesigns of model reference adaptive control systems. IEEE Trans Autom Control 11:362–367

    Google Scholar 

  70. Johansson R, Robertsson A, Lozano R (1999) Stability analysis of adaptive output feedback control. In: Proceedings of the 38th IEEE conference on decision and control, vol 4. Phoenix, Arizona, USA, pp 3796–3801

    Google Scholar 

  71. Huang CH, Ioannou PA, Maroulas J, Safonov MG (1999) Design of strictly positive real systems using constant output feedback. IEEE Trans Autom Control 44(3):569–573

    MathSciNet  MATH  Google Scholar 

  72. Barkana I, Teixeira MCM, Hsu L (2006) Mitigation of symmetry condition in positive realness for adaptive control. Automatica 42(9):1611–1616

    MathSciNet  MATH  Google Scholar 

  73. Safonov MG, Jonckeere EA, Verma M, Limebeer DJN (1987) Synthesis of positive real multivariable feedback systems. Int J Control 45:817–842

    MathSciNet  MATH  Google Scholar 

  74. Haddad WM, Bernstein DS, Wang YW (1994) Dissipative \(H_{2}/H_{\infty }\) controller synthesis. IEEE Trans Autom Control 39:827–831

    Google Scholar 

  75. Sun W, Khargonekar PP, Shim D (1994) Solution to the positive real control problem for linear time-invariant systems. IEEE Trans Autom Control 39:2034–2046

    MathSciNet  MATH  Google Scholar 

  76. Wang Q, Weiss H, Speyer JL (1994) System characterization of positive real conditions. IEEE Trans Autom Control 39:540–544

    MathSciNet  MATH  Google Scholar 

  77. Weiss H, Wang Q, Speyer JL (1994) System characterization of positive real conditions. IEEE Trans Autom Control 39(3):540–544

    MathSciNet  MATH  Google Scholar 

  78. Alpay D, Lewkowicz I (2011) The positive real lemma and construction of all realizations of generalized positive rational functions. Syst Control Lett 60:985–993

    MathSciNet  MATH  Google Scholar 

  79. Fradkov AL (2003) Passification of non-square linear systems and feedback Yakubovich-Kalman-Popov Lemma. Eur J Control 6:573–582

    MATH  Google Scholar 

  80. Larsen M, Kokotovic PV (2001) On passivation with dynamic output feedback. IEEE Trans Autom Control 46(6):962–967

    MathSciNet  MATH  Google Scholar 

  81. Fradkov AL, Hill DJ (1998) Exponential feedback passivity and stabilizability of nonlinear systems. Automatica 34(6):697–703

    MathSciNet  MATH  Google Scholar 

  82. Barkana I (2004) Comments on “Design of strictly positive real systems using constant output feedback’’. IEEE Trans Autom Control 49(11):2091–2093

    MathSciNet  MATH  Google Scholar 

  83. Kaufman H, Barkana I, Sobel K (1998) Direct adaptive control algorithms. Theory and applications, 2nd edn. Springer, Berlin

    Google Scholar 

  84. Covacic MR, Teixeira MCM, Assunçao E, Gaino R (2010) LMI-based algorithm for strictly positive real systems with static output feedback. Syst Control Lett 61:521–527

    MathSciNet  MATH  Google Scholar 

  85. de la Sen M (1998) A method for general design of positive real functions. IEEE Trans Circuits Syst I Fundam Theory Appl 45(7):764–769

    MathSciNet  MATH  Google Scholar 

  86. Xie L, Soh YC (1995) Positive real control problem for uncertain linear time-invariant systems. Syst Control Lett 24:265–271

    MathSciNet  MATH  Google Scholar 

  87. Mahmoud MS, Soh YC, Xie L (1999) Observer-based positive real control of uncertain linear systems. Automatica 35:749–754

    MathSciNet  MATH  Google Scholar 

  88. Xu S, Lam J, Lin Z, Galkowski K (2002) Positive real control for uncertain two-dimensional systems. IEEE Trans Circuits Syst I Fundam Theory Appl 49(11):1659–1666

    MathSciNet  MATH  Google Scholar 

  89. Betser A, Zeheb E (1993) Design of robust strictly positive real transfer functions. IEEE Trans Circuits Syst I Fundam Theory Appl 40(9):573–580

    MathSciNet  MATH  Google Scholar 

  90. Bianchini G, Tesi A, Vicino A (2001) Synthesis of robust strictly positive real systems with \(l_{2}\) parametric uncertainty. IEEE Trans Circuits Syst I Fundam Theory Appl 48(4):438–450

    MATH  Google Scholar 

  91. Turan L, Safonov MG, Huang CH (1997) Synthesis of positive real feedback systems: a simple derivation via Parott’s Theorem. IEEE Trans Autom Control 42(8):1154–1157

    MATH  Google Scholar 

  92. Son YI, Shim H, Jo NH, Seo JH (2003) Further results on passification of non-square linear systems using an input-dimensional compensator. IEICE Trans Fundam E86–A(8):2139–2143

    Google Scholar 

  93. Bernussou J, Geromel JC, de Oliveira MC (1999) On strict positive real systems design: guaranteed cost and robustness issues. Syst Control Lett 36:135–141

    MathSciNet  MATH  Google Scholar 

  94. Antoulas AC (2005) A new result on passivity preserving model reduction. Syst Control Lett 54(4):361–374

    MathSciNet  MATH  Google Scholar 

  95. Anderson BDO, Landau ID (1994) Least squares identification and the robust strict positive real property. IEEE Trans Cicuits Syst I 41(9):601–607

    MathSciNet  Google Scholar 

  96. Haddad WM, Bernstein DS (1991) Robust stabilization with positive real uncertainty: Beyond the small gain theorem. Syst Control Lett 17:191–208

    MathSciNet  MATH  Google Scholar 

  97. Lanzon A, Petersen IR (2007) A modified positive-real type stability condition. In: Proceedings of European control conference. Kos, Greece, pp 3912–3918

    Google Scholar 

  98. Lanzon A, Petersen IR (2008) Stability robustness of a feedback interconnection of systems with negative imaginary frequency response. IEEE Trans Autom Control 53(4):1042–1046

    MathSciNet  MATH  Google Scholar 

  99. Petersen IR, Lanzon A (2010) Feedback control of negative-imaginary systems. IEEE Control Syst Mag 30(5):54–72

    MathSciNet  Google Scholar 

  100. Petersen IR (2016) Negative imaginary systems theory and applications. Annu Rev Control 42:309–318

    Google Scholar 

  101. Lanzon A, Chen HJ (2017) Feedback stability of negative imaginary systems. IEEE Trans Autom Control 62(11):5620–5633

    MathSciNet  MATH  Google Scholar 

  102. Xiong J, Petersen IR, Lanzon A (2012) On lossless negative imaginary systems. Automatica 48:1213–1217

    MathSciNet  MATH  Google Scholar 

  103. Song Z, Lanzon A, Patra S, Petersen IR (2012) A negative-imaginary lemma without minimality assumptions and robust state-feedback synthesis for uncertain negative-imaginary systems. Syst Control Lett 61:1269–1276

    MathSciNet  MATH  Google Scholar 

  104. Xiong J, Petersen IR, Lanzon A (2010) A negative imaginary lemma and the stability of interconnections of linear negative imaginary systems. IEEE Trans Autom Control 55(10):2342–2347

    MathSciNet  MATH  Google Scholar 

  105. Liu M, Xiong J (2017) Properties and stability analysis of discrete-time negative imaginary systems. Automatica 83:58–64

    MathSciNet  MATH  Google Scholar 

  106. Ferrante A, Lanzon A, Ntogramatzidis L (2017) Discrete-time negative imaginary systems. Automatica 79:1–10

    MathSciNet  MATH  Google Scholar 

  107. Liu M, Xiong J (2018) Bilinear transformation for discrete-time positive real and negative imaginary systems. IEEE Trans Autom Control. https://doi.org/10.1109/TAC.2018.2797180

    Article  MathSciNet  MATH  Google Scholar 

  108. Mabrok MA, Kallapur AG, Petersen IR, Lanzon A (2014) Locking a three-mirror optical cavity using negative imaginary systems approach. Quantum Inf Rev 1:1–8

    Google Scholar 

  109. Mabrok MA, Kallapur AG, Petersen IR, Lanzon A (2014) Spectral conditions for negative imaginary systems with applications to nanopositioning. IEEE/ASME Trans Mechatron 19(3):895–903

    Google Scholar 

  110. Cai C, Hagen G (2010) Stability analysis for a string of coupled stable subsystems with negative imaginary frequency response. IEEE Trans Autom Control 55:1958–1963

    MathSciNet  MATH  Google Scholar 

  111. Ahmed B, Pota H (2011) Dynamic compensation for control of a rotary wing UAV using positrive position feedback. J Intell Robot Syst 61:43–56

    Google Scholar 

  112. Diaz M, Pereira E, Reynolds P (2012) Integral resonant control scheme for cancelling human-induced vibations in light-weight pedestrian structures. Struct Control Health Monit 19:55–69

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA Grenoble Rhône-Alpes, Université Grenoble Alpes, Grenoble, France

    Bernard Brogliato

  2. Centre de Recherche de Royalieu Heuristique et Diagnostic des Systèmes, Université de Technologie de Compiègne UMR-CNRS 6599, Compiègne, France

    Rogelio Lozano

  3. LAGEP, Université Lyon 1, Villeurbanne, France

    Bernhard Maschke

  4. Department of Production and Quality Engineering, Norwegian University of Science and Technology, Trondheim, Norway

    Olav Egeland

Authors
  1. Bernard Brogliato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rogelio Lozano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bernhard Maschke
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Olav Egeland
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernard Brogliato .

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Brogliato, B., Lozano, R., Maschke, B., Egeland, O. (2020). Positive Real Systems. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-19420-8_2

Download citation

Publish with us

Policies and ethics

Search

Navigation