Automation and Remote Control

, Volume 77, Issue 2, pp 191–225 | Cite as

Design of interval observers for uncertain dynamical systems

  • D. Efimov
  • T. Raïssi
Topical Issue


Interval state observers provide an estimate on the set of admissible values of the state vector at each instant of time. Ideally, the size of the evaluated set is proportional to the model uncertainty, thus interval observers generate the state estimates with estimation error bounds, similarly to Kalman filters, but in the deterministic framework. Main tools and techniques for design of interval observers are reviewed in this tutorial for continuous-time, discrete-time and time-delayed systems.


Remote Control Interval Estimation Observer Gain Chemical Master Equation Interval Observer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, in Lecture Notes in Control and Information Sciences, vol. 322, Meurer, T., Graichen, K., and Gilles, E.-D., Eds., New York: Springer, 2005.Google Scholar
  2. 2.
    Fossen, T. and Nijmeijer, H., New Directions in Nonlinear Observer Design, New York: Springer, 1999.zbMATHGoogle Scholar
  3. 3.
    Nonlinear Observers and Applications, in Lecture Notes in Control and Information Sciences, vol. 363, Besançon, G., Ed., New York: Springer, 2007.Google Scholar
  4. 4.
    Esfandiari, F. and Khalil, H., Output Feedback Stabilization of Fully Linearizable Systems, Int. J. Control, 1992, vol. 56, pp. 1007–1037.CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Levant, A., Higher-Order Sliding Modes: Differentiation and Output Feedback Control, Int. J. Control, 2003, vol. 76, nos. 9–10, pp. 924–941.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Golubev, A., Krishchenko, A., and Tkachev, S., Stabilization of Nonlinear Dynamic Systems Using the System State Estimates Made by the Asymptotic Observer, Autom. Remote Control, 2005, vol. 66, no. 7, pp. 1021–1058.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Shamma, J., Control of Linear Parameter Varying Systems with Applications, Overview of LPV Systems, New York: Springer, 2012, pp. 1–22.Google Scholar
  8. 8.
    Marcos, A. and Balas, J., Development of Linear-Parameter-Varying Models for Aircraft, J. Guidance, Control, Dynamics, 2004, vol. 27, no. 2, pp. 218–228.CrossRefGoogle Scholar
  9. 9.
    Shamma, J. and Cloutier, J., Gain-Scheduled Missile Autopilot Design Using Linear Parameter-Varying Transformations, J. Guidance, Control, Dynamics, 1993, vol. 16, no. 2, pp. 256–261.CrossRefGoogle Scholar
  10. 10.
    Tan, W., Applications of Linear Parameter-Varying Control Theory, PhD Dissertation, Dept. of Mechanical Engineering, University of California at Berkeley, 1997.Google Scholar
  11. 11.
    Shtessel, Y., Edwards, C., Fridman, L., et al., Sliding Mode Control and Observation. Control Engineering, Basel: Birkhäuser, 2013.Google Scholar
  12. 12.
    Guanrong, C., Jianrong, W., and Leang, S., Interval Kalman Filtering, IEEE Trans. Aerospace Electron. Syst., 1997, vol. 33, no. 1, pp. 250–259.CrossRefGoogle Scholar
  13. 13.
    Jaulin, L., Nonlinear Bounded-Error State Estimation of Continuous-Time Systems, Automatica, 2002, vol. 38, no. 2, pp. 1079–1082.CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Kieffer, M. and Walter, E., Guaranteed Nonlinear State Estimator for Cooperative Systems, Numer. Algorithms, 2004, vol. 37, pp. 187–198.CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Olivier, B. and Gouzé, J., Closed Loop Observers Bundle for Uncertain Biotechnological Models, J. Process Control, 2004, vol. 14, no. 7, pp. 765–774.CrossRefGoogle Scholar
  16. 16.
    Milanese, M. and Novara, C., Unified Set Membership Theory for Identification, Prediction and Filtering of Nonlinear Systems, Automatica, 2011, vol. 47, no. 10, pp. 2141–2151.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Moisan, M., Bernard, O., and Gouzé, J., Near Optimal Interval Observers Bundle for Uncertain Bioreactors, Automatica, 2009, vol. 45, no. 1, pp. 291–295.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Raïssi, T., Videau, G., and Zolghadri, A., Interval Observers Design for Consistency Checks of Nonlinear Continuous-Time Systems, Automatica, 2010, vol. 46, no. 3, pp. 518–527.CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Raïssi, T., Efimov, D., and Zolghadri, A., Interval State Estimation for a Class of Nonlinear Systems, IEEE Trans. Automat. Control, 2012, vol. 57, no. 1, pp. 260–265.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Efimov, D., Fridman, L., Raïssi, T., et al., Interval Estimation for LPV Systems Applying High Order Sliding Mode Techniques, Automatica, 2012, vol. 48, pp. 2365–2371.CrossRefzbMATHGoogle Scholar
  21. 21.
    Gouzé, J., Rapaport, A., and Hadj-Sadok, M., Interval Observers for Uncertain Biological Systems, Ecolog. Modell., 2000, vol. 133, pp. 46–56.CrossRefGoogle Scholar
  22. 22.
    Mazenc, F. and Bernard, O., Interval Observers for Linear Time-Invariant Systems with Disturbances, Automatica, 2011, vol. 47, no. 1, pp. 140–147.CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Combastel, C., Stable Interval Observers in C for Linear Systems with Time-Varying Input Bounds, IEEE Trans. Automat. Control, 2013, vol. 58, no. 2, pp. 481–487.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Farina, L. and Rinaldi, S., Positive Linear Systems. Theory and Applications, New York: Wiley, 2000.CrossRefzbMATHGoogle Scholar
  25. 25.
    Smith, H., Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, in Surveys and Monographs, vol. 41, Providence: AMS, 1995.Google Scholar
  26. 26.
    Briat, C., Robust Stability Analysis of Uncertain Linear Positive Systems via Integral Linear Constraints: l 1- and l -gain Characterizations, Proc. 50th IEEE CDC and ECC, Orlando, 2011, pp. 6337–6342.Google Scholar
  27. 27.
    Ebihara, Y., Peaucelle, D., and Arzelier, D., L1 Gain Analysis of Linear Positive Systems and Its Application, Proc. 50th IEEE CDC and ECC, Orlando, 2011, pp. 4029–4035.Google Scholar
  28. 28.
    Khalil, H.K., Nonlinear Systems, Upper Saddle River: Prentice Hall, 2002, 3rd ed.zbMATHGoogle Scholar
  29. 29.
    Hirsch, M.W. and Smith, H.L., Monotone Maps. A Review, J. Differ. Equat. Appl., 2005, vol. 11, nos. 4–5, pp. 379–398.CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Haddad, W. and Chellaboina, V., Stability Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay, Syst. Control Lett., 2004, vol. 51, pp. 355–361.CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Dambrine, M. and Richard, J.-P., Stability Analysis of Time-Delay Systems, Dynamic Syst. Appl., 1993, vol. 2, pp. 405–414.MathSciNetzbMATHGoogle Scholar
  32. 32.
    Efimov, D., Polyakov, A., Fridman, E.M., et al., Delay-Dependent Positivity. Application to Interval Observers, Proc. ECC 2015, Linz, 2015.Google Scholar
  33. 33.
    McCarthy, P., Nielsen, C., and Smith, S., Cardinality Constrained Robust Optimization Applied to a Class of Interval Observers, Am. Control Conf. (ACC), 2014, pp. 5337–5342.Google Scholar
  34. 34.
    Chebotarev, S., Efimov, D., Raïssi, T., et al., Interval Observers for Continuous-Time LPV Systems with l 1/l 2 Performance, Automatica, 2015, vol. 58, pp. 82–89.CrossRefGoogle Scholar
  35. 35.
    Cacace, F., Germani, A., and Manes, C., A New Approach to Design Interval Observers for Linear Systems, IEEE Trans. Automat. Control, 2015, vol. 99, no. 99, p. 1.MathSciNetGoogle Scholar
  36. 36.
    Mazenc, F. and Bernard, O., Asymptotically Stable Interval Observers for Planar Systems with Complex Poles, IEEE Trans. Automat. Control, 2010, vol. 55, no. 2, pp. 523–527.CrossRefMathSciNetGoogle Scholar
  37. 37.
    Efimov, D., Perruquetti, W., Raïssi, T., et al., On Interval Observer Design for Time-Invariant Discrete- Time Systems, Proc. Eur. Control Conf. (ECC), Zurich, 2013.Google Scholar
  38. 38.
    Mazenc, F., Dinh, T.N., and Niculescu, S.I., Interval Observers for Discrete-Time Systems, 51th IEEE Conf. Decision and Control, Hawaii, 2012, 6755–6760.Google Scholar
  39. 39.
    Mazenc, F., Dinh, T.N., and Niculescu, S.I., Interval Observers for Discrete-Time Systems, Int. J. Robust Nonlin. Control, 2014, vol. 24, pp. 2867–2890.CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    Farina, L., and Rinaldi, S., Positive Linear Systems—Theory and Applications, New York: Wiley, 2000.CrossRefzbMATHGoogle Scholar
  41. 41.
    Efimov, D., Perruquetti, W., Raïssi, T., et al., Interval Observers for Time-Varying Discrete-Time Systems, IEEE Trans. Automat. Control 2013, vol. 58, no. 12, pp. 3218–3224.CrossRefMathSciNetGoogle Scholar
  42. 42.
    Efimov, D., Perruquetti, W., and Richard, J.-P., Interval Estimation for Uncertain Systems with Time- Varying Delays, Int. J. Control, 2013, vol. 86, no. 10, pp. 1777–1787.CrossRefMathSciNetGoogle Scholar
  43. 43.
    Efimov, D., Perruquetti, W., and Richard, J.-P., On Reduced-Order Interval Observers for Time-Delay Systems, Proc. Eur. Control Conf. (ECC), Zurich, 2013.Google Scholar
  44. 44.
    Kolmanovskii, V. and Myshkis, A., Introduction to the Theory and Applications of Functional Differential Equations, Dordrecht: Kluwer, 1999.CrossRefzbMATHGoogle Scholar
  45. 45.
    Pepe, P. and Jiang, Z.-P., A Lyapunov–Krasovskii Methodology for ISS and iISS of Time-Delay Systems, Syst. Control Lett., 2006, vol. 55, pp. 1006–1014.CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    Rami, M., Helmke, U., and Tadeo, F., Positive Observation Problem for Linear Time-Delay Positive Systems, Proc. Mediterranean Conf. Control & Automation (MED’07), 2007, 1–6.CrossRefGoogle Scholar
  47. 47.
    Mazenc, F., Niculescu, S.I., and Bernard, O., Exponentially Stable Interval Observers for Linear Systems with Delay, SIAM J. Control Optim., 2012, vol. 50, pp. 286–305.CrossRefMathSciNetzbMATHGoogle Scholar
  48. 48.
    Efimov, D., Polyakov, A., and Richard, J.-P., Interval Observer Design for Estimation and Control of Time-Delay Descriptor Systems, Eur. J. Control, 2015, vol. 23, pp. 26–35.CrossRefMathSciNetGoogle Scholar
  49. 49.
    Churilov, A., Medvedev, A., and Shepeljavyi, A., Mathematical Model of Non-basal Testosterone Regulation in the Male by Pulse Modulated Feedback, Automatica, 2009, vol. 45, pp. 78–85.CrossRefMathSciNetzbMATHGoogle Scholar
  50. 50.
    Greenhalgh, D. and Khan, Q., A Delay Differential Equation Mathematical Model for the Control of the Hormonal System of the Hypothalamus, the Pituitary and the Testis in Man, Nonlin. Anal., 2009, vol. 71, pp. e925–e935.CrossRefzbMATHGoogle Scholar
  51. 51.
    Bolajraf, M., Ait Rami, M., and Helmke, U.R., Robust Positive Interval Observers for Uncertain Positive Systems, Proc. 18th IFAC World Congr., 2011, 14330–14334.Google Scholar
  52. 52.
    Chen, K., Goh, L., He, G., et al., Identification of Nucleation Rates in Droplet-Based Microfluidic Systems, Chem. Eng. Sci., 2012, vol. 77, pp. 235–241.CrossRefGoogle Scholar
  53. 53.
    Goh, L., Chen, K., Bhamidi, V., et al., A Stochastic Model for Nucleation Kinetics Determination in Droplet-Based Microfluidic Systems, Crystal Growth Design, 2010, vol. 10, no. 6, pp. 2515–2521.CrossRefGoogle Scholar
  54. 54.
    Efimov, D., Raïssi, T., Chebotarev, S., et al., Interval State Observer for Nonlinear Time-Varying Systems, Automatica, 2013, vol. 49, no. 1, pp. 200–205.CrossRefMathSciNetzbMATHGoogle Scholar
  55. 55.
    Back, J. and Astolfi, A., Design of Positive Linear Observers for Positive Linear Systems via Coordinate Transformations and Positive Realizations, SIAM J. Control Optim., 2008, vol. 47, no. 1, pp. 345–373.CrossRefMathSciNetzbMATHGoogle Scholar
  56. 56.
    Ait Rami, M., Cheng, C., and de Prada, C., Tight Robust Interval Observers: An LP Approach, Proc. 47th IEEE Conf. Decision and Control, Cancun, Mexico, 2008, 2967–2972.Google Scholar
  57. 57.
    Löfberg, J., Automatic Robust Convex Programming, Optim., Methods Software, 2012, vol. 27, no. 1, pp. 115–129.CrossRefzbMATHGoogle Scholar
  58. 58.
    Efimov, D., Raïssi, T., Perruquetti, W., et al., Estimation and Control of Discrete-Time LPV Systems Using Interval Observers, Proc. 52nd IEEE Conf. Decision and Control, Florence, 2013.Google Scholar
  59. 59.
    Polyak, B.T., Nazin, S.A., Durieu, C., et al., Ellipsoidal Parameter or State Estimation under Model Uncertainty, Automatica, 2004, vol. 40, no. 7, pp. 1171–1179.CrossRefMathSciNetzbMATHGoogle Scholar
  60. 60.
    Alamo, T., Bravo, J., and Camacho, E., Guaranteed State Estimation by Zonotopes, Automatica, 2005, vol. 41, no. 6, pp. 1035–1043.CrossRefMathSciNetzbMATHGoogle Scholar
  61. 61.
    Raïssi, T., Ramdani, N., and Candau, Y., Set Membership State and Parameter Estimation for Systems Described by Nonlinear Differential Equations, Automatica, 2004, vol. 40, pp. 1771–1777.CrossRefzbMATHGoogle Scholar
  62. 62.
    Jaulin, L., Kieffer, M., Didrit, O., et al., Applied Interval Analysis, London: Springer, 2001.CrossRefzbMATHGoogle Scholar
  63. 63.
    Thabet, R.E.H., Raïssi, T., Combastel, C., et al., An Effective Method to Interval Observer Design for Time-Varying Systems, Automatica, 2014, vol. 50, no. 10, pp. 2677–2684.CrossRefMathSciNetzbMATHGoogle Scholar
  64. 64.
    Amato, F., Pironti, A., and Scala, S., Necessary and Sufficient Conditions for Quadratic Stability and Stabilizability of Uncertain Linear Time-Varying Systems, IEEE Trans. Automat. Control, 1996, vol. 41, pp. 125–128.CrossRefMathSciNetzbMATHGoogle Scholar
  65. 65.
    Zhu, J. and Johnson, C.D., Unified Canonical Forms for Linear Time-Varying Dynamical Systems under D-similarity Transformations, part I, Southeastern Sympos. Syst. Theory, 1989, 74–81.Google Scholar
  66. 66.
    Zhu, J. and Johnson, C.D., Unified Canonical Forms for Linear Time-Varying Dynamical Systems under D-similarity Transformations, part II, Southeastern Sympos. Syst. Theory, 1989, 57–63.Google Scholar
  67. 67.
    Zhu, J. and Johnson, C.D., Unified Canonical Forms for Matrices Over a Differential Ring, Linear Algebra Appl., 1991, vol. 147, no. 0, pp. 201–248.CrossRefMathSciNetzbMATHGoogle Scholar
  68. 68.
    Efimov, D., Raïssi, T., and Zolghadri, A., Stabilization of Nonlinear Uncertain Systems Based on Interval Observers, Proc. 50th IEEE CDC-ECC, Orlando, 2011, 8157–8162.Google Scholar
  69. 69.
    Efimov, D., Raïssi, T., and Zolghadri, A., Control of Nonlinear and LPV Systems: Interval Observer-Based Framework, IEEE Trans. Automat. Control, 2013, vol. 58, no. 3, pp. 773–782.CrossRefGoogle Scholar
  70. 70.
    Cai, X., Lv, G., and Zhang, W., Stabilisation for a Class of Nonlinear Uncertain Systems Based on Interval Observers, Control Theory Appl. IET, 2012, vol. 6, no. 13, pp. 2057–2062.CrossRefMathSciNetGoogle Scholar
  71. 71.
    Mazenc, F., Dinh, T.N., and Niculescu, S.I., Robust Interval Observers and Stabilization Design for Discrete-time Systems with Input and Output, Automatica, 2013, vol. 49, pp. 3490–3497.CrossRefMathSciNetGoogle Scholar
  72. 72.
    Mazenc, F. and Bernard, O., ISS Interval Observers for Nonlinear Systems Transformed into Triangular Systems, Int. J. Robust Nonlin. Control, 2014, vol. 24, no. 7, pp. 1241–1261.CrossRefMathSciNetzbMATHGoogle Scholar
  73. 73.
    Mazenc, F. and Malisoff, M., New Technique for Stability Analysis for Time-Varying Systems with Delay, Proc. 53th IEEE Conf. Decision and Control, Los Angeles, 2014, 1215–1220.Google Scholar
  74. 74.
    Puig, V., Stancu, A., Escobet, T., et al., Passive Robust Fault Detection Using Interval Observers. Application to the DAMADICS Benchmark Problem, Control Eng. Practice, 2006, vol. 14, pp. 621–633.CrossRefGoogle Scholar
  75. 75.
    Blesa, J., Puig, V., and Bolea, Y., Fault Detection Using Interval LPV Models in an Open-Flow Canal, Control Eng. Practice, 2010, vol. 18, no. 5, pp. 460–470.CrossRefGoogle Scholar
  76. 76.
    Blesa, J., Rotondo, D., Puig, V., et al., FDI and FTC of Wind Turbines Using the Interval Observer Approach and Virtual Actuators/Sensors, Control Eng. Practice, 2014, vol. 24, pp. 138–155.CrossRefGoogle Scholar
  77. 77.
    Efimov, D., Li, S., Hu, Y., et al., Application of Interval Observers to Estimation and Control of Air-Fuel Ratio in a Direct Injection Engine, Proc. ACC, Chicago, 2015.Google Scholar
  78. 78.
    Goffaux, G., Remy, M., andWouwer, A.V., Continuous-Discrete Confidence Interval Observer—Application to Vehicle Positioning, Inform. Fusion, 2013, vol. 14, no. 4, pp. 541–550.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.l’Institut National de Recherche en Informatique et en Automatique (INRIA)LilleFrance
  2. 2.ITMO UniversitySt. PetersburgRussia
  3. 3.Centre for Research in Computer Science and Telecommunications (CNAM)ParisFrance

Personalised recommendations