1932

Abstract

Internal models are nowadays customarily used in different domains of science and engineering to describe how living organisms or artificial computational units embed their acquired knowledge about recurring events taking place in the surrounding environment. This article reviews the internal model principle in control theory, bioengineering, and neuroscience, illustrating the fundamental concepts and theoretical developments of the few last decades of research.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-control-042920-102205
2022-05-03
2024-05-13
Loading full text...

Full text loading...

/deliver/fulltext/control/5/1/annurev-control-042920-102205.html?itemId=/content/journals/10.1146/annurev-control-042920-102205&mimeType=html&fmt=ahah

Literature Cited

  1. 1. 
    Huang J, Isidori A, Marconi L, Mischiati M, Sontag E, Wonham WM 2018. Internal models in control, biology and neuroscience. 2018 IEEE Conference on Decision and Control (CDC)5370–90 Piscataway, NJ: IEEE
  2. 2. 
    Twain M. 2009 (1883). Life on the Mississippi New York: Signet
  3. 3. 
    Craik KJW. 1967 (1943). The Nature of Explanation Cambridge, UK: Cambridge Univ. Press
  4. 4. 
    Webb B. 2004. Neural mechanisms for prediction: Do insects have forward models?. Trends Neurosci 27:278–82
    [Google Scholar]
  5. 5. 
    Mischiati M, Lin HT, Herold P, Imler E, Olberg R, Leonardo A 2015. Internal models direct dragonfly interception steering. Nature 517:333–38
    [Google Scholar]
  6. 6. 
    Sontag E. 2003. Adaptation and regulation with signal detection implies internal model. Syst. Control Lett. 50:119–26
    [Google Scholar]
  7. 7. 
    Andrews BW, Sontag E, Iglesias PA 2008. An approximate internal model principle: applications to nonlinear models of biological systems. IFAC Proc. Vol. 41:215873–78
    [Google Scholar]
  8. 8. 
    Alon U, Surette MG, Barkai N, Leibler S. 1999. Robustness in bacterial chemotaxis. Nature 397:168–71
    [Google Scholar]
  9. 9. 
    Smith OJM. 1958. Feedback Control Theory New York: McGraw-Hill
  10. 10. 
    Francis BA, Wonham WM. 1975. The internal model principle for linear multivariable regulators. Appl. Math. Optim. 2:170–94
    [Google Scholar]
  11. 11. 
    Davison E. 1976. The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. Autom. Control 21:25–34
    [Google Scholar]
  12. 12. 
    Hepburn JSA, Wonham WM. 1984. Structurally stable nonlinear regulation with step inputs. Math. Syst. Theory 17:319–33
    [Google Scholar]
  13. 13. 
    Hepburn JSA, Wonham WN. 1984. Error feedback and internal models on differentiable manifolds. IEEE Trans. Autom. Control 29:397–403
    [Google Scholar]
  14. 14. 
    Wonham WM. 1985. Linear Multivariable Control: A Geometric Approach New York: Springer, 3rd ed..
  15. 15. 
    Isidori A, Byrnes C. 1990. Output regulation of nonlinear systems. IEEE Trans. Autom. Control 35:131–40
    [Google Scholar]
  16. 16. 
    Huang J, Rugh WJ. 1990. On a nonlinear multivariable servomechanism problem. Automatica 26:963–72
    [Google Scholar]
  17. 17. 
    Sepulchre R. 2021. To know or to predict?. IEEE Control Syst. Mag. 41:24–5
    [Google Scholar]
  18. 18. 
    Francis BA. 1977. The linear multivariable regulator problem. SIAM J. Control Optim. 15:486–505
    [Google Scholar]
  19. 19. 
    Francis BA, Wonham WM. 1976. The internal model principle of control theory. Automatica 12:457–65
    [Google Scholar]
  20. 20. 
    Desoer C, Lin CA. 1985. Tracking and disturbance rejection of MIMO nonlinear systems with PI controller. IEEE Trans. Autom. Control 30:861–67
    [Google Scholar]
  21. 21. 
    Huang J, Lin CF. 1991. On a robust nonlinear servomechanism problem. Proceedings of the 30th IEEE Conference on Decision and Control (CDC)2529–30 Piscataway, NJ: IEEE
  22. 22. 
    Byrnes C, Isidori A. 1989. Nonlinear output regulation: remarks on robustness. 27th Allerton Conference on Communications, Control, and Computing150–58 Urbana: Univ. Ill. Urbana-Champaign
  23. 23. 
    Huang J. 1995. Asymptotic tracking and disturbance rejection in uncertain nonlinear systems. IEEE Trans. Autom. Control 40:1118–22
    [Google Scholar]
  24. 24. 
    Huang J, Lin CF. 1993. Internal model principle and robust control of nonlinear systems. Proceedings of the 32nd IEEE Conference on Decision and Control (CDC) 21501–6 Piscataway, NJ: IEEE
  25. 25. 
    Byrnes CI, Priscoli FD, Isidori A, Kang W. 1997. Structurally stable output regulation of nonlinear systems. Automatica 33:369–85
    [Google Scholar]
  26. 26. 
    Khalil HK. 1994. Robust servomechanism output feedback controllers for feedback linearizable systems. Automatica 30:1587–99
    [Google Scholar]
  27. 27. 
    Byrnes C, Isidori A. 2003. Limit sets, zero dynamics, and internal models in the problem of nonlinear output regulation. IEEE Trans. Autom. Control 48:1712–23
    [Google Scholar]
  28. 28. 
    Huang J, Chen Z. 2004. A general framework for tackling the output regulation problem. IEEE Trans. Autom. Control 49:2203–18
    [Google Scholar]
  29. 29. 
    Marconi L, Praly L, Isidori A. 2007. Output stabilization via nonlinear Luenberger observers. SIAM J. Control Optim. 45:2277–98
    [Google Scholar]
  30. 30. 
    Isidori A, Byrnes C. 2008. Steady-state behaviors in nonlinear systems, with an application to robust disturbance rejection. Annu. Rev. Control 32:1–16
    [Google Scholar]
  31. 31. 
    Sontag E. 1989. Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34:435–43
    [Google Scholar]
  32. 32. 
    Nikiforov VO. 1998. Adaptive non-linear tracking with complete compensation of unknown disturbances. Eur. J. Control 4:132–39
    [Google Scholar]
  33. 33. 
    Serrani A, Isidori A. 2000. Global robust output regulation for a class of nonlinear systems. Syst. Control Lett. 39:133–39
    [Google Scholar]
  34. 34. 
    Serrani A, Isidori A, Marconi L. 2001. Semi-global nonlinear output regulation with adaptive internal model. IEEE Trans. Autom. Control 46:1178–94
    [Google Scholar]
  35. 35. 
    Liu L, Chen Z, Huang J 2009. Parameter convergence and minimal internal model with an adaptive output regulation problem. Automatica 45:1306–11
    [Google Scholar]
  36. 36. 
    Ye X, Huang J 2001. Decentralized adaptive output regulation for large-scale nonlinear systems. IFAC Proc. Vol. 34:6627–32
    [Google Scholar]
  37. 37. 
    Bin M, Marconi L, Teel AR. 2019. Adaptive output regulation for linear systems via discrete-time identifiers. Automatica 105:422–32
    [Google Scholar]
  38. 38. 
    Forte F, Marconi L, Teel AR. 2017. Robust nonlinear regulation: continuous-time internal models and hybrid identifiers. IEEE Trans. Autom. Control 62:3136–51
    [Google Scholar]
  39. 39. 
    Bin M, Marconi L. 2020.. “ Class-type” identification-based internal models in multivariable nonlinear output regulation. IEEE Trans. Autom. Control 65:4369–76
    [Google Scholar]
  40. 40. 
    Marconi L, Praly L. 2008. Uniform practical nonlinear output regulation. IEEE Trans. Autom. Control 53:1184–202
    [Google Scholar]
  41. 41. 
    Pyrkin A, Isidori A. 2017. Output regulation for robustly minimum-phase multivariable nonlinear systems. 2017 IEEE 56th Conference on Decision and Control (CDC)873–78 Piscataway, NJ: IEEE
  42. 42. 
    Bernard P, Bin M, Marconi L. 2020. Adaptive output regulation via nonlinear Luenberger observer-based internal models and continuous-time identifiers. Automatica 122:109261
    [Google Scholar]
  43. 43. 
    Bin M, Bernard P, Marconi L. 2021. Approximate nonlinear regulation via identification-based adaptive internal models. IEEE Trans. Autom. Control 66:3534–49
    [Google Scholar]
  44. 44. 
    Isidori A, Marconi L. 2012. Shifting the internal model from control input to controlled output in nonlinear output regulation. 2012 IEEE 51st IEEE Conference on Decision and Control (CDC)4900–5 Piscataway, NJ: IEEE
  45. 45. 
    Bin M, Marconi L. 2020. Output regulation by postprocessing internal models for a class of multivariable nonlinear systems. Int. J. Robust Nonlinear Control 30:1115–40
    [Google Scholar]
  46. 46. 
    Astolfi D, Praly L, Marconi L 2015. Approximate regulation for nonlinear systems in presence of periodic disturbances. 54th IEEE Conference on Decision and Control (CDC)7665–70 Piscataway, NJ: IEEE
  47. 47. 
    Astolfi D, Praly L, Marconi L 2019. Francis-Wonham nonlinear viewpoint in output regulation of minimum phase systems. IFAC-PapersOnLine 52:16532–37
    [Google Scholar]
  48. 48. 
    Wang L, Marconi L, Wen C, Su H 2020. Pre-processing nonlinear output regulation with non-vanishing measurements. Automatica 111:108616
    [Google Scholar]
  49. 49. 
    Bin M, Marconi L. 2018. The chicken-egg dilemma and the robustness issue in nonlinear output regulation with a look towards adaptation and universal approximators. 2018 IEEE Conference on Decision and Control (CDC)5391–96 Piscataway, NJ: IEEE
  50. 50. 
    Bin M, Astolfi D, Marconi L, Praly L. 2018. About robustness of internal model-based control for linear and nonlinear systems. 2018 IEEE Conference on Decision and Control (CDC)5397–402 Piscataway, NJ: IEEE
  51. 51. 
    Bin M, Astolfi D, Marconi L 2022. About robustness of control systems embedding an internal model. IEEE Trans. Autom. Control In press https://doi.org/10.1109/TAC.2022.3151574
    [Crossref] [Google Scholar]
  52. 52. 
    Hara S, Yamamoto Y, Omata T, Nakano M. 1988. Repetitive control system: a new type servo system for periodic exogenous signals. IEEE Trans. Autom. Control 33:659–68
    [Google Scholar]
  53. 53. 
    Ghosh J, Paden B. 2000. Nonlinear repetitive control. IEEE Trans. Autom. Control 45:949–54
    [Google Scholar]
  54. 54. 
    Califano F, Bin M, Macchelli A, Melchiorri C 2018. Stability analysis of nonlinear repetitive control schemes. IEEE Control Syst. Lett. 2:773–78
    [Google Scholar]
  55. 55. 
    Del Vecchio D, Qian Y, Murray R, Sontag E 2018. Future systems and control research in synthetic biology. Annu. Rev. Control 45:5–17
    [Google Scholar]
  56. 56. 
    Yi TM, Huang Y, Simon MI, Doyle J 2000. Robust perfect adaptation in bacterial chemotaxis through integral feedback control. PNAS 97:4649–53
    [Google Scholar]
  57. 57. 
    Shoval O, Alon U, Sontag E 2011. Symmetry invariance for adapting biological systems. SIAM J. Appl. Dyn. Syst. 10:857–86
    [Google Scholar]
  58. 58. 
    Sontag E. 2013. Mathematical Control Theory: Deterministic Finite Dimensional Systems New York: Springer, 2nd ed..
  59. 59. 
    Kim D, Kwon YK, Cho KH. 2008. The biphasic behavior of incoherent feed-forward loops in biomolecular regulatory networks. BioEssays 30:1204–11
    [Google Scholar]
  60. 60. 
    Sasagawa S, Ozak Yi, Fujita iK, Kuroda S 2005. Prediction and validation of the distinct dynamics of transient and sustained ERK activation. Nat. Cell Biol. 7:365–73
    [Google Scholar]
  61. 61. 
    Nagashima T, Shimodaira H, Ide K, Nakakuki T, Tani Y et al. 2007. Quantitative transcriptional control of ErbB receptor signaling undergoes graded to biphasic response for cell differentiation. J. Biol. Chem. 282:4045–56
    [Google Scholar]
  62. 62. 
    Menè P, Pugliese G, Pricci F, Mario UD, Cinotti GA, Pugliese F. 1997. High glucose level inhibits capacitative Ca2+ influx in cultured rat mesangial cells by a protein kinase C-dependent mechanism. Diabetologia 40:521–27
    [Google Scholar]
  63. 63. 
    Nesher R, Cerasi E. 2002. Modeling phasic insulin release: immediate and time-dependent effects of glucose. Diabetes 51:Suppl. 1S53–59
    [Google Scholar]
  64. 64. 
    Mahaut-Smith MP, Ennion SJ, Rolf MG, Evans RJ 2000. ADP is not an agonist at P2X1 receptors: evidence for separate receptors stimulated by ATP and ADP on human platelets. Br. J. Pharmacol. 131:108–14
    [Google Scholar]
  65. 65. 
    Marsigliante S, Elia MG, Di Jeso B, Greco S, Muscella A, Storelli C 2002. Increase of [Ca2+]i via activation of ATP receptors in PC-Cl3 rat thyroid cell line. Cell. Signal. 14:61–67
    [Google Scholar]
  66. 66. 
    Ridnour LA, Windhausen AN, Isenberg JS, Yeung N, Thomas DD et al. 2007. Nitric oxide regulates matrix metalloproteinase-9 activity by guanylyl-cyclase-dependent and -independent pathways. PNAS 104:16898–903
    [Google Scholar]
  67. 67. 
    Tsang J, Zhu J, van Oudenaarden A. 2007. MicroRNA-mediated feedback and feedforward loops are recurrent network motifs in mammals. Mol. Cell 26:753–67
    [Google Scholar]
  68. 68. 
    Tyson JJ, Chen KC, Novak B 2003. Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr. Opin. Cell Biol. 15:221–31
    [Google Scholar]
  69. 69. 
    Sontag E. 2009. Remarks on feedforward circuits, adaptation, and pulse memory. IET Syst. Biol. 4:39–51
    [Google Scholar]
  70. 70. 
    Yang L, Iglesias PA 2006. Positive feedback may cause the biphasic response observed in the chemoattractant-induced response of Dictyostelium cells. Syst. Control Lett. 55:329–37
    [Google Scholar]
  71. 71. 
    Levchenko A, Iglesias PA. 2002. Models of eukaryotic gradient sensing: application to chemotaxis of amoebae and neutrophils. Biophys. J. 82:50–63
    [Google Scholar]
  72. 72. 
    Feng-Dan X, Zeng-Rong L, Zhi-Yong Z, Jian-Wei S. 2009. Robust and adaptive microRNA-mediated incoherent feedforward motifs. Chin. Phys. Lett. 26:028701
    [Google Scholar]
  73. 73. 
    Kremling A, Bettenbrock K, Gilles ED. 2008. A feed-forward loop guarantees robust behavior in Escherichia coli carbohydrate uptake. Bioinformatics 24:704–10
    [Google Scholar]
  74. 74. 
    Voit E, Neves AR, Santos H. 2006. The intricate side of systems biology. PNAS 103:9452–57
    [Google Scholar]
  75. 75. 
    Bleris L, Xie Z, Glass D, Adadey A, Sontag E, Benenson Y 2011. Synthetic incoherent feedforward circuits show adaptation to the amount of their genetic template. Mol. Syst. Biol. 7:519
    [Google Scholar]
  76. 76. 
    Shoval O, Goentoro L, Hart Y, Mayo A, Sontag E, Alon U 2010. Fold-change detection and scalar symmetry of sensory input fields. PNAS 107:15995–6000
    [Google Scholar]
  77. 77. 
    Briat C, Gupta A, Khammash M. 2016. Antithetic integral feedback ensures robust perfect adaptation in noisy biomolecular networks. Cell Syst 2:15–26
    [Google Scholar]
  78. 78. 
    Aoki SK, Lillacci G, Gupta A, Baumschlager A, Schweingruber D, Khammash M. 2019. A universal biomolecular integral feedback controller for robust perfect adaptation. Nature 570:533–37
    [Google Scholar]
  79. 79. 
    Agrawal D, Marshall R, Noireaux V, Sontag E 2019. In vitro implementation of robust gene regulation in a synthetic biomolecular integral controller. Nat. Commun. 10:5760
    [Google Scholar]
  80. 80. 
    Kim J, Khetarpal I, Sen S, Murray RM 2014. Synthetic circuit for exact adaptation and fold-change detection. Nucleic Acids Res 42:6078–89
    [Google Scholar]
  81. 81. 
    Huang HH, Qian Y, Del Vecchio D. 2018. A quasi-integral controller for adaptation of genetic modules to variable ribosome demand. Nat. Commun. 9:5415
    [Google Scholar]
  82. 82. 
    von Holst E, Mittelstaedt H. 1950. The principle of reafference. Naturwissenschaften 37:464–76
    [Google Scholar]
  83. 83. 
    Wolpert DM, Ghahramani Z, Jordan MI 1995. An internal model for sensorimotor integration. Science 269:1880–82
    [Google Scholar]
  84. 84. 
    Mehta B, Schaal S. 2002. Forward models in visuomotor control. J. Neurophysiol. 88:942–53
    [Google Scholar]
  85. 85. 
    Scarchilli K, Vercher JL. 1999. The oculomanual coordination control center takes into account the mechanical properties of the arm. Exp. Brain Res. 124:42–52
    [Google Scholar]
  86. 86. 
    Miall RC, Christensen LOD, Cain O, Stanley J 2007. Disruption of state estimation in the human lateral cerebellum. PLOS Biol 5:e316
    [Google Scholar]
  87. 87. 
    Therrien AS, Bastian AJ. 2015. Cerebellar damage impairs internal predictions for sensory and motor function. Curr. Opin. Neurobiol. 33:127–33
    [Google Scholar]
  88. 88. 
    Brooks JX, Carriot J, Cullen KE 2015. Learning to expect the unexpected: rapid updating in primate cerebellum during voluntary self-motion. Nat. Neurosci. 18:1310–17
    [Google Scholar]
  89. 89. 
    Shadmehr R, Krakauer JW. 2008. A computational neuroanatomy for motor control. Exp. Brain Res. 185:359–81
    [Google Scholar]
  90. 90. 
    Mulliken GH, Musallam S, Andersen RA 2008. Forward estimation of movement state in posterior parietal cortex. PNAS 105:8170–77
    [Google Scholar]
  91. 91. 
    Mulliken G, Andersen R 2009. Forward models and state estimation in posterior parietal cortex. The Cognitive Neurosciences MS Gazzaniga 599–611 Cambridge, MA: MIT Press, 4th ed..
    [Google Scholar]
  92. 92. 
    Wolpert DM, Miall RC, Kawato M. 1998. Internal models in the cerebellum. Trends Cogn. Sci. 2:338–47
    [Google Scholar]
  93. 93. 
    Ebner TJ, Hewitt AL, Popa LS. 2011. What features of limb movements are encoded in the discharge of cerebellar neurons?. Cerebellum 10:683–93
    [Google Scholar]
  94. 94. 
    Medina JF, Lisberger SG. 2008. Links from complex spikes to local plasticity and motor learning in the cerebellum of awake-behaving monkeys. Nat. Neurosci. 11:1185–92
    [Google Scholar]
  95. 95. 
    Sawtell NB. 2017. Neural mechanisms for predicting the sensory consequences of behavior: insights from electrosensory systems. Annu. Rev. Physiol. 79:381–99
    [Google Scholar]
  96. 96. 
    Sauerbrei BA, Guo JZ, Cohen JD, Mischiati M, Guo W et al. 2020. Cortical pattern generation during dexterous movement is input-driven. Nature 577:386–91
    [Google Scholar]
  97. 97. 
    Poulet JFA, Hedwig B. 2006. The cellular basis of a corollary discharge. Science 311:518–22
    [Google Scholar]
  98. 98. 
    Wiederman SD, Fabian JM, Dunbier JR, O'Carroll DC. 2017. A predictive focus of gain modulation encodes target trajectories in insect vision. eLife 6:e26478
    [Google Scholar]
  99. 99. 
    Kim AJ, Fitzgerald JK, Maimon G 2015. Cellular evidence for efference copy in Drosophila visuomotor processing. Nat. Neurosci. 18:1247–55
    [Google Scholar]
  100. 100. 
    Lacquaniti F, Maioli C. 1989. The role of preparation in tuning anticipatory and reflex responses during catching. J. Neurosci. 9:134–48
    [Google Scholar]
  101. 101. 
    McIntyre J, Zago M, Berthoz A, Lacquaniti F. 2001. Does the brain model Newton's laws?. Nat. Neurosci. 4:693–94
    [Google Scholar]
  102. 102. 
    Zago M, Bosco G, Maffei V, Iosa M, Ivanenko YP, Lacquaniti F. 2004. Internal models of target motion: expected dynamics overrides measured kinematics in timing manual interceptions. J. Neurophysiol. 91:1620–34
    [Google Scholar]
  103. 103. 
    Gawthrop P, Loram I, Lakie M, Gollee H 2011. Intermittent control: a computational theory of human control. Biol. Cybernet. 104:31–51
    [Google Scholar]
  104. 104. 
    Kettner RE, Mahamud S, Leung HC, Sitkoff N, Houk JC et al. 1997. Prediction of complex two-dimensional trajectories by a cerebellar model of smooth pursuit eye movement. J. Neurophysiol. 77:2115–30
    [Google Scholar]
  105. 105. 
    Cerminara NL, Apps R, Marple-Horvat DE. 2009. An internal model of a moving visual target in the lateral cerebellum. J. Physiol. 587:429–42
    [Google Scholar]
  106. 106. 
    Cutlip S, Freudenberg J, Cowan N, Gillespie RB. 2019. Haptic feedback and the internal model principle. 2019 IEEE World Haptics Conference (WHC)568–73 Piscataway, NJ: IEEE
/content/journals/10.1146/annurev-control-042920-102205
Loading
/content/journals/10.1146/annurev-control-042920-102205
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error