Kybernetika 52 no. 5, 824-835, 2016

On approximation of stability radius for an infinite-dimensional feedback control system

Hideki SanoDOI: 10.14736/kyb-2016-5-0824


In this paper, we discuss the problem of approximating stability radius appearing in the design procedure of finite-dimensional stabilizing controllers for an infinite-dimensional dynamical system. The calculation of stability radius needs the value of $H_\infty$-norm of a transfer function whose realization is described by infinite-dimensional operators in a Hilbert space. From the computational point of view, we need to prepare a family of approximate finite-dimensional operators and then to calculate the $H_\infty$-norm of their transfer functions. However, it is not assured that they converge to the value of $H_\infty$-norm of the original transfer function. The purpose of this study is to justify the convergence. In a numerical example, we treat parabolic distributed parameter systems with distributed control and distributed/boundary observation.


distributed parameter system, finite-dimensional controller, stability radius, transfer function, semigroup


93D15, 93C25


  1. M. J. Balas: Finite-dimensional controllers for linear distributed parameter systems: exponential stability using residual mode filters. J. Math. Anal. Appl. 133 (1988), 283-296.   DOI:10.1016/0022-247x(88)90401-5
  2. C. Chicone and Y. Latushkin: Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, 1999.   DOI:10.1090/surv/070
  3. R. F. Curtain: Finite dimensional compensators for parabolic distributed systems with unbounded control and observation. SIAM J. Control Optim. 22 (1984), 255-276.   DOI:10.1137/0322018
  4. R. Glowinski, J.-L. Lions and J. He: Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. In: Encyclopedia of Mathematics and Its Applications, vol. 117. Cambridge University Press, Cambridge 2008.   CrossRef
  5. A. El Jai and A. J. Pritchard: Sensors and Controls in the Analysis of Distributed Systems. English Language Edition. Ellis Horwood Limited, Chichester 1988.   CrossRef
  6. N. Guglielmi and M. Manetta: Approximating real stability radii. IMA J. Numerical Analysis 35 (2015), 1402-1425.   DOI:10.1093/imanum/dru038
  7. I. Lasiecka and R. Triggiani: Control Theory for Partial Differential Equations: Continuous and Approximation Theories I: Abstract Parabolic Systems. In: Encyclopedia of Mathematics and Its Applications, vol. 74. Cambridge University Press, Cambridge 2000.   DOI:10.1017/cbo9780511574801
  8. T. Nambu: On stabilization of partial differential equations of parabolic type: boundary observation and feedback. Funkcialaj Ekvacioj, Serio Internacia 28 (1985), 267-298.   CrossRef
  9. A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. In: Applied Mathematical Sciences, vol. 44. Springer-Verlag, New York 1983.   DOI:10.1007/978-1-4612-5561-1
  10. A. J. Pritchard and S. Townley: Robust compensator design via structured stability radii. Systems Control Lett. 11 (1988), 33-37.   DOI:10.1016/0167-6911(88)90108-9
  11. A. J. Pritchard and S. Townley: Robustness of linear systems. J. Differential Equations 77 (1989), 254-286.   DOI:10.1016/0022-0396(89)90144-7
  12. Y. Sakawa: Feedback stabilization of linear diffusion systems. SIAM J. Control Optim. 21 (1983), 667-676.   DOI:10.1137/0321040
  13. H. Sano and N. Kunimatsu: Feedback stabilization of infinite-dimensional systems with $A^{\gamma}$-bounded output operators. Appl. Math. Lett. 7 (1994), 5, 17-22.   DOI:10.1016/0893-9659(94)90065-5
  14. H. Sano: Finite-dimensional $H_{\infty}$ control of linear parabolic systems with unbounded output operators. Int. J. Control 72 (1999), 16, 1466-1479.   DOI:10.1080/002071799220119
  15. H. Sano: Stability-enhancing control of a coupled transport-diffusion system with Dirichlet actuation and Dirichlet measurement. J. Math. Anal. Appl. 388 (2012), 1194-1204.   DOI:10.1016/j.jmaa.2011.11.011
  16. J. M. Schumacher: A direct approach to compensator design for distributed parameter systems. SIAM J. Control Optim. 21 (1983), 823-836.   DOI:10.1137/0321050
  17. K. Zhou, J. Doyle and K. Glover: Robust and Optimal Control. (In Japanese, translated from the English by K. Z. Liu and Z. H. Luo.) Corona, Tokyo 1997.   CrossRef