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Structural Methods for Linear Systems: An Introduction

  • Nicos Karcanias
  • Efstathios Milonidis
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 367)

Abstract

This paper assumes familiarity with the basic Control and Dynamics, as covered in undergraduate courses. It introduces the different alternative system representations for linear systems and provides a quick review of the fundamental mathematical tools, which are essential for the treatment of the more advanced notions in Linear Systems. The paper focuses on some fundamental concepts underpinning the study of linear systems and dynamics and which play a crucial role in the analysis and design of control systems; thus, the paper deals with notions such as those of Controllability, Observability, Stability, poles and zeros and re lated dynamics and their properties under different compensation schemes. There is a concrete flavour in the current approach which runs through this presentation and this is that of the underlying algebraic structure. The term “structure” refers to aspects of the state space/transfer function description, which remain invariant under a variety of transformations. The set of transformations considered here are of the compensation type and include state feedback and output injection, dynamic compensation, as well as of the representation type transformations that include state, input, and output co-ordinate transformations. This structure stems from the system description and defines the nature of the dynamics and the related geometric proper ties and these in turn define what it is possible to achieve under feedback; such an approach is known as a structural approach. Central to our analysis are the notions of poles and zeros. The poles of a system are crucial characteristics of the internal system dynamics, characterise system free response, stability and general aspects of the performance of a system. The poles of a system are affected by the different compensation schemes and their assignment is the subject of many design methodologies aiming at shaping the internal system dynamics under different compensation schemes. The notion of zeros is more complex, since they express the interaction between internal dynamics and the effort to control and observe the system and they are thus products of overall system design, that apart from process synthesis involves selection of actuation and measurement schemes for the system. The significance of zeros is mainly due to that they remain invariant under a large set of compensation schemes, as well as that they define limits of what can be achieved under compensation. This makes zeros crucial for design, since they are part of those factors characterising the potential of a given system to achieve certain design objectives under compensation. The invariance of zeros implies that their design is an issue that has to be addressed outside the traditional control design; this requires understanding of the zero formation process and involves early design stages mechanisms such as process instrumentation. Poles and zeros are conceptually inverse concepts (resonances, antiresonances) and such mechanisms are highlighted throughout the paper. The role of system structure in characterising different system properties is central to this paper and it is defined by a set of discrete and continuous invariants; these invariants characterise a variety of key system properties and their type/values define the structure of canonical forms and determine somehow the potential of a given system for compensation. Invariants and canonical forms under the general transformation group are linked to compensation theory, whereas those associated with representation transformations play a key role in system identification. The emphasis in this article is to provide an overview of the fundamentals concepts, the back ground mathematical tools, explain their dynamic significance and link them to problems of control and systems design.

Keywords

State Space Model Singular System Root Locus Elementary Divisor Matrix Pencil 
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.

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References

  1. 1.
    Antsaklis, P.J., Michael, A.N.: Linear Systems. McGraw-Hill, New York (1997)Google Scholar
  2. 2.
    Aplevich, J.D.: Implicit Linear Systems. Lecture Notes in Control and Information Sciences, vol. 152. Springer, Berlin (1991)zbMATHGoogle Scholar
  3. 3.
    Basile, G., Marro, G.: Controlled and Conditioned Invariants in Linear Systems. Prentice-Hall, Englewood Cliffs (1992)Google Scholar
  4. 4.
    Brockett, R.W., Byrnes, C.I.: Multivariable Nyquist Criterion, Root Loci and Pole Placement: A geometric viewpoint. IEEE Trans. Aut. Control 26, 271–283 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brunovsky, P.: A classification of linear controllable systems. Kybernetica 3, 173–187 (1970)MathSciNetGoogle Scholar
  6. 6.
    Callier, F.M., Desoer, C.A.: Multivariable Feedback Systems. Springer, New York (1982)Google Scholar
  7. 7.
    Chen, C.T.: Linear Systems Theory and Design. Holt-Rinehart and Winston, New York (1984)Google Scholar
  8. 8.
    Denham, M.J.: Canonical forms for the identification of multivariable linear systems. IEEE Trans. Aut. Control 19, 645–656 (1974)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Desoer, C.A., Schulman, J.D.: Zeros and Poles of Matrix Transfer Functions and their Dynamical Interpretation. IEEE Trans. Circ. and Systems 21, 3–8 (1974)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Dickinson, B.W., Kailath, T., Morf, M.: Canonical matrix fraction and state-space descriptions for deterministic and stochastic linear systems. IEEE Trans. Automatic Control 19, 656–667 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Doyle, J.C., Stein, G.: Multivariable Feedback Design: Concepts for a Classical /Modern Synthesis. IEEE Trans. Aut. Control 26, 4–16 (1981)zbMATHCrossRefGoogle Scholar
  12. 12.
    Forney, D.G.: Minimal bases of rational vector spaces with applications to multivariable linear systems. SIAM J. Control 13, 493–520 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gantmacher, G.: Theory of Matrices, vol. 2. Chelsea, New York (1975)Google Scholar
  14. 14.
    Giannacopoulos, C., Karcanias, N.: Pole assignment of strictly proper systems by constant output feedback. Int. J. Control 42, 543 (1985)CrossRefGoogle Scholar
  15. 15.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, Chichester (1978)zbMATHGoogle Scholar
  16. 16.
    Hodge, W.V.D., Pedoe, P.D.: Methods of Algebraic Geometry, vol. 2. Cambridge Univ. Press, Cambridge (1952)zbMATHGoogle Scholar
  17. 17.
    Jaffe, S., Karcanias, N.: Matrix Pencil Characterisation of Almost (A,B) – invariant subspaces: A classification of geometric concepts. Int. J. Control 33, 51–53 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kailath, T.: Linear Systems. Prentice-Hall, Englewood Cliffs (1980)zbMATHGoogle Scholar
  19. 19.
    Kalman, R.E.: Canonical Structure of linear dynamical systems. Proc. Nat. Ac. Sci. 48, 596–600 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kalman, R.E.: Mathematical description of linear dynamical systems. SIAM J. Control 1, 152–192 (1963)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Kalman, R.E.: Kronecker invariants and Feedback. In: Weiss, L. (ed.) Ordinary Differential Equat, pp. 459–471. Academic Press, New York (1972)Google Scholar
  22. 22.
    Karcanias,: Matrix pencil approach to geometric system theory. Proc. IEEE 126, 585–590 (1979)MathSciNetGoogle Scholar
  23. 23.
    Karcanias, N.: Minimal Bases of Matrix Pencils: Algebraic, Toeplitz Structure and Geometric Properties. Linear Algebra and its Applications. Special Issue on Linear Systems 205–206, 831–865 (1994)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Karcanias, N.: Global Process Instrumentation: Issues and Problems of a System and Control Theory Framework. Measurement 14, 103–113 (1994)CrossRefGoogle Scholar
  25. 25.
    Karcanias, N.: Control Problems in Global Process Instrumentation: A Structural Approach. Proc. of ESCAPE-6. Comp. Chem. Eng. 20, 1101–1106 (1996)CrossRefGoogle Scholar
  26. 26.
    Karcanias, N.: Multivariable Poles and Zeros. In: Control Systems, Robotics and Automation, from Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Oxford (2002), http://www.eolss.net [Retrieved October 26, 2005]Google Scholar
  27. 27.
    Karcanias, N., Giannakopoulos, C.: On Grassman Invariants and Almost Zeros of Linear Systems and the Determinantal Zero, Pole Assignment Problem. Int. J. Control 40(4), 673–698 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Karcanias, N., Giannakopoulos, C.: Necessary and Sufficient Conditions for Zero Assignment by Constant Squaring Down. Linear Algebra and its Applications 122/123, 415–446 (1989)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Karcanias, N., Hayton, G.E.: Generalised Autonomous Dynamical Systems, Algebraic Duality and Geometric Theory. In: Proc. of 8th IFAC World Congress, Kyoto, Japan, pp. 289–294. Pergamon Press, Oxford (1981)Google Scholar
  30. 30.
    Karcanias, N., Hayton, G.E.: State Space and Transfer Function Infinite Zeros; A Unified Approach. In: Proc. Joint Aut. Cont. Conf. Univ. of Virginia (1981)Google Scholar
  31. 31.
    Karcanias, N., Kalogeropoulos, G.: Geometric Theory and Feedback Invariants of Generalised Linear Systems: A Matrix Pencil Approach. Circuits, Systems and Signal Process 8(3), 375–395 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Karcanias, N., Kouvaritakis, B.: The Output Zeroing Problem and its Relationship to the Invariant Zero Structure. Int. J. Control 30, 395–415 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Karcanias, N., Leventides, J.: Grassman Invariants, Matrix pencils and Linear System Properties. Linear Algebra and its Applications 241/243, 705–731 (1995)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Karcanias, A., MacBean, P.: Structural invariants and canonical forms of linear multivariable systems. In: Proc. 3rd IMA Conf. on Control Theory, pp. 257–282. Academic Press, London (1980)Google Scholar
  35. 35.
    Karcanias, N., Mitrouli, M.: Minimal Bases of Matrix Pencils and coprime Matrix Fraction Descriptions. IMA Journal of Math. Control Theory and Inform. 19, 245–278 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Karcanias, N., Vafiadis, D.: Canonical Forms for State Space Descriptions. In: Control Systems, Robotics and Automation, from Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Oxford (2002), http://www.eolss.net [Retrieved October 26, 2005]Google Scholar
  37. 37.
    Kouvaritakis, B., MacFarlane, A.G.J.: Geometric approach to analysis and synthesis of system zeros: Part II: non-square systems. Int. J. Control 23, 167–181 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Kouvaritakis, B., Shaked, U.: Asymptotic Behavior of Root-Loci of Multivariable Systems. Int. J. Control 23, 297–340 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Kucera, V.: Discrete Linear Control: The Polynomial Equation Approach. John Wiley & Sons, Chichester (1979)zbMATHGoogle Scholar
  40. 40.
    Lewis, L.: A survey of Linear Singular Systems. Circuits, Systems and Signal Processes 8, 375–397 (1989)CrossRefGoogle Scholar
  41. 41.
    Leventides, J., Karcanias, N.: The Pole PlacementMap, its Properties and Relationships to System Invariants. IEEE Trans. on Aut. Control 38, 1266–1270 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Leventides, J., Karcanias, N.: Global asymptotic linearisation of the pole placement map: Aclosed form solution for the output feedback problem. Automatica 31, 1303–1309 (1993)CrossRefMathSciNetGoogle Scholar
  43. 43.
    Leventides, J., Karcanias, N.: Dynamic Pole Assignment using Global, Blow up Linearisation: LowComplexity Solutions. Journal of Optimisation Theory and Applications 96, 57–86 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Loiseau, J.J., Ozcaldiran, K., Malabre, M., Karcanias, N.: Feedback canonical Forms of Singular Systems. Kybernetica 27, 289–305 (1991)zbMATHMathSciNetGoogle Scholar
  45. 45.
    Luenberger, D.G.: Canonical forms for linear multivariable systems. IEEE Trans. Aut. Control 12, 290–293 (1967)CrossRefMathSciNetGoogle Scholar
  46. 46.
    MacFarlane, A.G.J., Karcanias, N.: Poles and Zeros of Linear Multivariable Systems: A survey of the Algebraic, Geometric and Complex Variable Theory. Int. J. Control 24, 33–74 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    MacFarlane, A.G.J., Karcanias, N.: Relations Between State Space and Frequency Response Concepts. In: Proc.7th IFAC Cong. Part 43B, Helsinki, Finland (1978)Google Scholar
  48. 48.
    MacFarlane, A.G.J., Postlewaite, I.: The generalised Nyquist stability Criterion and Multivariable Root Loci. Int. J. Control 25, 581–622 (1977)Google Scholar
  49. 49.
    Marcus, M.: Finite dimensional multilinear algebra (in two parts). Marcel Deker, New York (1973)Google Scholar
  50. 50.
    Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Bacon (1964)zbMATHGoogle Scholar
  51. 51.
    Moore, B.C.: Principal Component Analysis in Linear Systems: Controllability, Observability and Model Reduction. IEEE Trans. Autom. Control 26, 17–32 (1981)zbMATHCrossRefGoogle Scholar
  52. 52.
    Morse, S.: Structural invariants of linear multivariable systems. SIAM J. Control 11, 446–465 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Owens, D.H.: Feedback and Multivariable Systems. Peter Peregrinus Ltd. IEE Publication, Stevenage (1978)zbMATHGoogle Scholar
  54. 54.
    Popov, V.M.: Some properties of the control systems with irreducible matrix transfer functions. In: Lecture Notes in Mathematics, vol. 144, pp. 250–261. Springer, Heidelberg (1969)Google Scholar
  55. 55.
    Popov, V.M.: Invariant descriptions of linear time invariant controllable systems. SIAM J. Control 10, 252–264 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Postlewaite, I., Edmunds, J.M., MacFarlane, A.G.J.: Principle Gains and Principle Phases in the analysis of Linear Multivariable Feedback Systems. IEEE Trans. Aut. Control 26, 32–46 (1981)CrossRefGoogle Scholar
  57. 57.
    Pugh, C., Krishnaswamy, V.: Algebraic and dynamic characterisations of poles and zeros at infinity. Int. J. Control 42, 1145–1153 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Rosenbrock, H.H.: State Space and Multivariable Theory. Nelson, London (1970)zbMATHGoogle Scholar
  59. 59.
    Rosenbrock, H.H.: Structural properties of linear dynamical systems. Int. J. Control 20, 191–202 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    Rosenbrock, H.H.: Order, degree and complexity. Int. J. Control 19, 323–331 (1979)CrossRefMathSciNetGoogle Scholar
  61. 61.
    Saberi, A., Sannuti, P.: Squaring Down of non-strictly proper systems. Int. J. Control 51, 621–629 (1990)zbMATHCrossRefGoogle Scholar
  62. 62.
    Saeks, R., DeCarlo, R.A.: Interconnected Dynamical Systems. Marcel Dekker, New York (1981)Google Scholar
  63. 63.
    Safonov, G., Laub, A.J., Hartmann, G.L.: Feedback Properties of Multivariable Systems: The Role and Use of the Return Difference Matrix. IEEE Trans. Auto. Control 26, 47–65 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Shaked, U., Karcanias, N.: The use of zeros and zero directions in model reduction. Int. Journal Control 23, 113–135 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    Skelton, R.E.: Dynamic Systems Control. John Wiley, Chichester (1988)Google Scholar
  66. 66.
    Skogestad, S., Postlethwaite, I.: Multivariable Feedback Control. John Wiley, Chichester (1996)Google Scholar
  67. 67.
    Smith, M.C.: Multivariable root-locus behaviour and the relationship to transfer function pole-zero structure. Int. J. Control 43, 497–515 (1986)zbMATHCrossRefGoogle Scholar
  68. 68.
    Thorp, J.P.: The singular pencil of linear dynamical systems. Int. J. Control 18, 577–596 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    Wang, S.H., Davison, E.J.: Canonical forms of linear multivariable systems. SIAM J. Control & Optimisation 14, 236–250 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    Warren, M.E., Eckberg, A.E.: On the dimensions of controllability subspaces: A characterisation via polynomial matrices and Kronecker invariants. SIAM J. Control 131, 434–445 (1975)CrossRefMathSciNetGoogle Scholar
  71. 71.
    Willems, J.C.: Almost Invariant Subspaces: An Approach to High Gain Feedback Design, Part I: Almost Controlled Subspaces. IEEE Trans. Aut. Control 26, 235–252 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  72. 72.
    Wolovich, W.A.: Linear multivariable systems. Applied Maths. Sciences, vol. 11. Springer, Heidelberg (1974)zbMATHGoogle Scholar
  73. 73.
    Wolowich, W.A., Falb, P.L.: On the structure of multivariable systems. SIAM J. Control 7, 437–451 (1969)CrossRefMathSciNetGoogle Scholar
  74. 74.
    Wonham, W.M.: Linear Multivariable Control: A Geometric Approach, 2nd edn. Springer, New York (1979)zbMATHGoogle Scholar
  75. 75.
    Vafiadis, D., Karcanias, N.: Canonical Forms for Descriptor Systems Under Restricted Systems Equivalence. Automatica 33, 955–958 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    Vardulakis, A.I. G., Limebeeer, D.J.N., Karcanias, N.: Structure and Smith–McMillan form of a Rational Matrix at Infinity. Int. J. Control 35, 701–725 (1982)zbMATHCrossRefGoogle Scholar
  77. 77.
    Vardulakis, A.I.G., Karcanias, N.: Structure, Smith–McMillan form and coprimeMFDs of a rational matrix inside a region  ∪ { ∞ }. Int. J. Control. 38, 927–957 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  78. 78.
    Vardulakis, A.I.G., Karcanias, N.: Relations between strict Equivalence Invariants and Structure at Infinity of Matrix Pencil. IEEE Trans. Aut. Control 28, 514–516 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    Verghese, G., Kailath, T.: Rational matrix structure. IEEE Trans. Aut. Control 26, 434–439 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    Vidyasagar, M.: Control System Synthesis: A Factorization Approach. MIT Press, Cambridge (1985)zbMATHGoogle Scholar

Copyright information

© Springer London 2007

Authors and Affiliations

  • Nicos Karcanias
    • 1
  • Efstathios Milonidis
    • 1
  1. 1.City University, Northampton Square, London, EC1V 0HB 

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