Distributed stabilisation of spatially invariant systems: positive polynomial approach

  • Petr AugustaEmail author
  • Zdeněk Hurák


The paper gives a computationally feasible characterisation of spatially distributed controllers stabilising a linear spatially invariant system, that is, a system described by linear partial differential equations with coefficients independent on time and location. With one spatial and one temporal variable such a system can be modelled by a 2-D transfer function. Stabilising distributed feedback controllers are then parametrised as a solution to the Diophantine equation ax + by = c for a given stable bi-variate polynomial c. The paper is built on the relationship between stability of a 2-D polynomial and positiveness of a related polynomial matrix on the unit circle. Such matrices are usually bilinear in the coefficients of the original polynomials. For low-order discrete-time systems it is shown that a linearising factorisation of the polynomial Schur-Cohn matrix exists. For higher order plants and/or controllers such factorisation is not possible as the solution set is non-convex and one has to resort to some relaxation. For continuous-time systems, an analogue factorisation of the polynomial Hermite-Fujiwara matrix is not known. However, for low-order systems and/or controller, positivity conditions on the closed-loop polynomial coefficients can be invoked. Then the computational framework of linear matrix inequalities can be used to describe the stability regions in the parameter space using a convex constraint.


Multidimensional systems Algebraic approach Control design Positiveness Convex optimisation 


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  1. Augusta, P., Hurák, Z., & Rogers, E. (2007). An algebraic approach to the control of statially distributed systems—the 2-D systems case with a physical application. In: Preprints of the 3rd IFAC symposium on system, structure and control. IFAC.Google Scholar
  2. Bamieh B., Paganini F., Dahleh M. (2002) Distributed control of spatially invariant systems. Automatic Control, IEEE Transactions on 47(7): 1091–1107. doi: 10.1109/TAC.2002.800646 MathSciNetCrossRefGoogle Scholar
  3. Barnett S. (1983) Polynomials and linear control systems. Marcel Dekker Inc, New YorkzbMATHGoogle Scholar
  4. Bose, N. (Ed.). (1985). Multidimensional systems theory: Progress, directions and open problems in multidimensional systems. D. Riedel Publishing Company, iSBN 90-277-1764-8.Google Scholar
  5. Brockett R. W., Willems J. L. (1974) Discretized partial differential equations: Examples of control systems defined on modules. Automatica 10(5): 507–515. doi: 10.1016/0005-1098(74)90051-X MathSciNetzbMATHCrossRefGoogle Scholar
  6. Cichy B., Galkowski K., Rogers E., Kummert A. (2011) An approach to iterative learning control for spatio-temporal dynamics using nd discrete linear systems models. Multidimensional Systems and Signal Processing 22: 83–96MathSciNetzbMATHCrossRefGoogle Scholar
  7. Cichy, B., Gakowski, K., & Rogers, E. (to be published). Iterative learning control for spatio-temporal dynamics using crank-nicholson discretization. Multidimensional Systems and Signal Processing.Google Scholar
  8. D’Andrea, R., & Dullerud, G. E. (2003). Distributed control design for spatially interconnected systems. IEEE Transactions on Automatic Control 48, 9.Google Scholar
  9. Dudgeon D.E., Mersereau R.M. (1984) Multidimensional digital signal processing. Prentice-Hall, New Jersey ISBN 0-13-604959-1zbMATHGoogle Scholar
  10. Dumitrescu B. (2007) Positive trigonometric polynomials and signal processing applications (1st ed.). Springer, BerlinzbMATHGoogle Scholar
  11. Genin Y., Hachez Y., Nesterov Y., Stefan R., Van Dooren P., Xu S. (2002) Positivity and linear matrix inequalities. European Journal of Control 8: 275–298CrossRefGoogle Scholar
  12. Genin Y., Hachez Y., Nesterov Y., Van Dooren P. (2003) Optimization problems over positive pseudo-polynomial matrices. SIAM Journal on Matrix Analysis and Applications 25: 57–79MathSciNetzbMATHCrossRefGoogle Scholar
  13. Goodman D. (1977) Some stability properties of two-dimensional linear shift-invariant digital filters. Circuits and Systems, IEEE Transactions on 24(4): 201–208zbMATHCrossRefGoogle Scholar
  14. Gorinevsky D. (2002) Loop shaping for iterative control of batch processes. Control Systems Magazine, IEEE 22(6): 55–65. doi: 10.1109/MCS.2002.1077785 CrossRefGoogle Scholar
  15. Gorinevsky D., Stein G. (2003) Structured uncertainty analysis of robust stability for multidimensional array systems. Automatic Control, IEEE Transactions on 48(9): 1557–1568. doi: 10.1109/TAC.2003.816980 MathSciNetCrossRefGoogle Scholar
  16. Henrion D., Garulli A. (2005) Positive Polynomials in Control. Springer, BerlinzbMATHGoogle Scholar
  17. Henrion D., Tarbouriech S., Šebek M. (1999) Rank-one LMI approach to simultaneous stabilization of linear systems. Systems and Control Letters 38(2): 79–89MathSciNetzbMATHCrossRefGoogle Scholar
  18. Henrion D., Šebek M., Bachelier O. (2001) Rank-one LMI approach to stability of 2-D polynomial matrices. Multidimensional Systems and Signal Processing 12(1): 33–48MathSciNetzbMATHCrossRefGoogle Scholar
  19. Jovanovic M., Bamieh B. (2005) Lyapunov-based distributed control of systems on lattices. Automatic Control, IEEE Transactions on 50(4): 422–433. doi: 10.1109/TAC.2005.844720 MathSciNetCrossRefGoogle Scholar
  20. Jury E. I. (1978) Stability of multidimensional scalar and matrix polynomial. Proceedings of the IEEE 6(9): 1018–1047MathSciNetCrossRefGoogle Scholar
  21. Justice, J. H., & Shanks, J. L. (1973). Stability criterion for N-dimensional digital filters. IEEE Transaction on automatic control, pp 284–286.Google Scholar
  22. Kamen E. W. (1975) On an algebraic theory of systems defined by convolution operators. Theory of Computing Systems 9(1): 57–74. doi: 10.1007/BF01698126 MathSciNetzbMATHGoogle Scholar
  23. Kamen, E. W. (1978) Lectures on algebraic systems theory: Linear systems over rings. Contractor report 316, NASA.Google Scholar
  24. Kamen E. W., Khargonekar P. (1984) On the control of linear systems whose coefficients are functions of parameters. Automatic Control, IEEE Transactions on 29(1): 25–33MathSciNetzbMATHCrossRefGoogle Scholar
  25. Khargonekar P., Sontag E. (1982) On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings. Automatic Control, IEEE Transactions on 27(3): 627–638MathSciNetzbMATHCrossRefGoogle Scholar
  26. Krstic, M., & Smyshlyaev, A. (2008). Boundary control of PDEs: A course on backstepping designs. : SIAM.Google Scholar
  27. Kučera V. (1979) Discrete linear control. Wiley, New YorkzbMATHGoogle Scholar
  28. Löfberg, J. (2004). Yalmip: A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan,
  29. Mastorakis, N. E. (1997) A new stability test for 2-D systems. In: Proceedings of the 5th IEEE Mediterranean Conference on control and systems (MED ’97).Google Scholar
  30. Mastorakis N. E. (1999) A method for computing the 2-D stability margin based on a new stability test for 2-D systems. Multidimensinal systems and signal processing 10: 93–99MathSciNetzbMATHCrossRefGoogle Scholar
  31. Rami, M. A., & Henrion, D. (2010). Inner approximation of conically constrained sets with stability aplication. to be published.Google Scholar
  32. Rogers E., Gałkowski K., Owens D. H. (2007) Control systems theory and applications for linear repetitive processes, Lecture notes in control and information sciences, vol 349. Springer, BerlinGoogle Scholar
  33. Rouchaleau, Y. (1972) Linear, discrete time, finite dimensional, dynamical systems over some classes of commutative rings.Google Scholar
  34. Šebek, M. (1994). Multi-dimensional systems: Control via polynomial techniques. Prague, Czech Republic: Dr.Sc. thesis, Academy of Sciences of the Czech Republic.Google Scholar
  35. Serban, I., & Najim, M. (2007). A new multidimensional Schur-Cohn type stability criterion. In: Proceedings of the 2007 American Control Conference.Google Scholar
  36. Šiljak D. (1973) Algebraic criteria for positive realness relative to the unit circle. Journal of the Franklin Institute 296: 115–122MathSciNetzbMATHCrossRefGoogle Scholar
  37. Šiljak, D. (1975). Stability criteria for two-variable polynomials. IEEE Transaction on Circuits and Systems 22(3).Google Scholar
  38. Sontag E. (1976) Linear systems over commutative rings: A survey. Ricerche di Automatica 7: 1–34Google Scholar
  39. Stein G., Gorinevsky D. (2005) Design of surface shape control for large two-dimensional arrays. IEEE Transactions on Control Systems Technology 13(3): 422–433CrossRefGoogle Scholar
  40. Stewart G., Gorinevsky D., Dumont G. (2003) Feedback controller design for a spatially distributed system: the paper machine problem. Control Systems Technology, IEEE Transactions on 11(5): 612–628. doi: 10.1109/TCST.2003.816420 MathSciNetCrossRefGoogle Scholar
  41. Strikwerda J. C. (1989) Finite difference schemes and partial differential equations. Wadsworth and Brooks, BelmontzbMATHGoogle Scholar
  42. Strintzis M. G. (1977) Test of stability of multidimensional filters. IEEE Transaction on automatic control CAS- 24(8): 432–437MathSciNetzbMATHGoogle Scholar
  43. Sturm J. F. (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 11–12: 625–653MathSciNetCrossRefGoogle Scholar
  44. Trentelman H. L., Rapisarda P. (1999) New algorithms for polynomial J-spectral factorization. Math Control Signals Systems 12: 24–61MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPraha 8Czech Republic
  2. 2.Faculty of Electrical EngineeringCzech Technical University in PraguePraha 6Czech Republic

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