Mixed H2/H Control

  • Carsten Scherer


In this article we provide a solution to the mixed H 2 /H problem with reduced order controllers for time-varying systems in terms of the solvability of differential linear matrix inequalities and rank conditions, including a detailed discussion of how to construct a controller. Immediate specializations lead to a solution of the full order problem and the mixed H 2 /H problem for linear systems whose description depends on unknown but in real-time measurable time-varying parameters. As done in the literature for the H problem, we resolve the quadratic mixed H 2 /H problem by reducing it to the solution of a finite number of algebraic linear matrix inequalities. Moreover, we point out directions how to overcome the conservatism caused by assuming a particular parameter dependence or by using constant solutions of the differential matrix inequalities. For linear time-invariant systems, we reveal how to incorporate robust asymptotic tracking or disturbance rejection as an objective in the mixed H 2 /H problem. Finally, we address the specializations to the fully general pure H or generalized H 2 problem, and provide quadratically convergent algorithms to compute optimal values. Our techniques do not only lead to insights into the structure of the solution sets of the corresponding linear matrix inequalities, but they also allow to explicitly describe the influence of various system zeros on the optimal values.


Linear Matrix Inequality Full Column Rank Convergent Algorithm Positive Definite Solution Robust Regulation 
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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  1. 1.
    Abedor, J., Nagpal, K., Khargonekar, P.P., Poolla, K.: Robust regulation in the presence of norm-bounded uncertainty. IEEE Trans. Automat. Control 40 (1995) 147–153MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Abedor, J., Nagpal, K., Poolla, K.: Robust regulation with H 2 performance. Systems Control Lett. 23 (1994) 431–443MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Apkarian, P., Gahinet, P.: A convex characterization of gain-scheduled H controllers. IEEE Trans. Automat. Control (to appear)Google Scholar
  4. 4.
    Apkarian, P., Gahinet, P.: H control of linear parameter-varying systems: A design example. Preprint (1993)Google Scholar
  5. 5.
    Apkarian, P., Gahinet, P., Becker, G.: Self-scheduled H control of linear parameter-varying systems. Preprint (1993)Google Scholar
  6. 6.
    Basar, T., Bernhard, P.: H -Optimal Control and Related Minimax Design Problems, A Dynamic Game Approach. Birkhäuser, Basel (1991)Google Scholar
  7. 7.
    Becker, G., Packard, A., Philbrick, D., Balas, G.: Control of parametrically-dependent linear systems: A single quadratic Lyapunov approach, Proc. Amer. Contr. Conf., San Francisco, CA (1993) 2795–2799Google Scholar
  8. 8.
    Becker, G., Packard A.: Robust performance of linear parametrically varying systems using parametrically dependent linear feedback. Systems Control Lett. 23 (1994) 205–215MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bernstein, D.S., Haddad, W.M.: LQG control with an H performance bound: A Riccati equation approach. IEEE Trans. Automat. Control 34 (1989) 293–305MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Boyd, S., Balakrishnan, V.: A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L -norm. Proc. 28th IEEE Conf. Decision Contr. (1989) 954–955Google Scholar
  11. 11.
    Boyd, S.P., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in Systems and Control Theory. SIAM Studies in Applied Mathematics 15, SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  12. 12.
    Davison, E. J.: The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. Automat. Control 21 (1976) 25–34MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Desoer, C.A., Vidyasagar, M.: Feedback Synthesis: Input-Output Properties. Academic Press, New York (1975)MATHGoogle Scholar
  14. 14.
    Doyle, J., Glover, K.: State-space formulae for all stabilizing controllers that satisfy an H norm bound and relations to risk sensitivity. Systems Control Lett. 11 (1988) 167–172MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Doyle, J., Glover, K., Khargonekar, P., Francis, B.: State-space solutions to standard H and H 2 control problems. IEEE Trans. Automat. Control 34 (1989) 831–847MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Doyle, J.C., Packard, A., Zhou, K.: Review of LFTs, LMIs, and μ. Proc. 30th IEEE Conf. Decision Contr. (1991) 1227–1232Google Scholar
  17. 17.
    Doyle J., Zhou, K., Glover, K., Bodenheimer, B.: Mixed H 2 and H performance objectives II: Optimal control. IEEE Trans. Automat. Control 39 (1994) 1575–1586MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Francis, B.A.: A Course in H Control Theory. Lect. N. Contr. Inform. Sci. No. 88, Springer-Verlag, Berlin (1987)MATHCrossRefGoogle Scholar
  19. 19.
    Freiling, G., Jank, G., Abou-Kandil, H.: On global existence of solutions to coupled matrix Riccati equations in closed loop Nash games. Preprint (1994)Google Scholar
  20. 20.
    Gahinet, P.: On the game Riccati equation arising in H control problmes. SIAM J. Control Optim. 32 (1994) 635–647MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Gahinet, P.: Explicit controller formulas for LMI-based H control. Proc. Amer. Contr. Conf., Baltimore (1994) 2396–2400Google Scholar
  22. 22.
    Gahinet, P., Apkarian, P.: A linear matrix inequality approach to H control. Int. J. of Robust and Nonlinear Control 4 (1994) 421–448MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Gahinet, P., Laub, A.J.: Numerically reliable computation of γopt in singular H control. Preprint (1994)Google Scholar
  24. 24.
    Gahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M.: The LMI control toolbox. Proc. 33rd IEEE Conf. Decision Contr. (1994) 2038–2041Google Scholar
  25. 25.
    Haddad, W.M., Bernstein, D.S., Mustafa, D.: Mixed-norm H 2 /H regulation and estimation: The discrete time case. Systems Control Lett. 16 (1991) 235–247MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Iwasaki, T., Skelton, R.E.: All controllers for the general H control problem: LMI existence conditions and state space formulas. Automatica 30 (1994) 1307–1317MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Iwasaki, T., Skelton, R.E.: A unified approach to fixed order controller design via linear matrix inequalities. Proc. Amer. Contr. Conf., Baltimore (1994) 35–39Google Scholar
  28. 28.
    Khargonekar, P.P., Rotea, M.A.: Mixed H 2 /H control: a convex optimization approach. IEEE Trans. Automat. Control 36 (1991) 824–837MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Khargonekar, P.P., Rotea, M.A., Sivashankar, N.: Exact and approximate solutions to a class of multiobjective controller synthesis problems. Proc. Amer. Contr. Conf., San Francisco, CA (1993) 1602–1606Google Scholar
  30. 30.
    Knobloch, H.W., Kwakernaak, H.: Lineare Kontrolltheorie. Springer-Verlag, Berlin (1985)MATHCrossRefGoogle Scholar
  31. 31.
    Limebeer, D.J.N., Anderson, B.D.O., Hendel, B.: A Nash game approach to mixed H 2 /H control. IEEE Trans. Automat. Control 39 (1994) 69–82MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Lu, W.M., Doyle, J.C., Robustness analysis and synthesis for uncertain nonlinear systems. Proc. 33rd IEEE Conf. Decision Contr. (1994) 787–792Google Scholar
  33. 33.
    Mustafa, D., Glover, K.: Minimum Entropy H Control. Lect. N. Contr. Inform. Sci. No. 146, Springer-Verlag, Berlin (1990)MATHCrossRefGoogle Scholar
  34. 34.
    Nagpal, K.M., Khargonekar, P.P.: Filtering and smoothing in an H setting. IEEE Trans. Automat. Control 36 (1991) 152–166MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Nestereov, Y., Nemirovsky, A.: Interior point polynomial methods in convex programming: Theory and applications. SIAM Studies in Applied Mathematics 13, SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  36. 36.
    Packard, A.: Gain-scheduling via linear fractional transformations. Systems Control Lett. 22 (1994) 79–92MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Packard, A., Doyle, J.: The complex structured singular value. Automatica 29 (1993) 71–109MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Packard, A., Pandey, P., Leonhardson, J., Balas, G.: Optimal, constant I/O similarity scaling for full-information and state-feedback control problems. Systems Control Lett. 19 (1992) 271–280MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Ravi, R., Nagpal, K.M., Khargonekar, P.P.: H control of linear time-varying systems: A state-space approach. SIAM J. Control Optim. 29 (1991) 1394–1413MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Rotea, M.A.: The generalized H 2 control. Automatica 29 (1993) 373–385MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Rotea, M.A., Iwasaki, T.: An alternative to the D - K iteration? Proc. Amer. Contr. Conf., Baltimore (1994) 53–57Google Scholar
  42. 42.
    Rotea, M.A., Khargonekar, P.P.: H 2-optimal control with an H -constraint: The state feedback case. Automatica 27 (1991) 307–316MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Rotea, M.A., Khargonekar, P.P.: Generalized H 2 /H control via convex optimization. Proc. 30th IEEE Conf. Decision Contr. (1991) 2719–2720Google Scholar
  44. 44.
    Rotea, M.A., Prasanth, R.K.: The ρ performance measure: A new tool for controller design with multiple frequency domain specifications. Proc. Amer. Contr. Conf., Baltimore (1994) 430–435Google Scholar
  45. 45.
    Sampei, M., Mita, T., Nakamichi, M.: An algebraic approach to H output feedback control problems. Systems Control Lett. 14 (1990) 13–24MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Scherer, C.W.: The Riccati Inequality and State-Space H -Optimal Control. Ph.D. thesis, University of Würzburg (1990)Google Scholar
  47. 47.
    Scherer, C.W.: H -eontrol by state-feedback and fast algorithms for the computation of optimal H -norms. IEEE Trans. Automat. Control 35 (1990) 1090–1099Google Scholar
  48. 48.
    Scherer, C.W.: H -control by state-feedback for plants with zeros on the imaginary axis. SIAM J. Control Optim. 30 (1992) 123–142Google Scholar
  49. 49.
    Scherer, C.W.: H -optimization without assumptions on finite or infinite zeros. SIAM J. Control Optim. 30 (1992) 143–166MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Shamma, J.F., Athans, M.: Guaranteed properties of gain-scheduled control for linear parameter-varying plants. Automatica 27 (1991) 559–564MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Steinbuch, M., Bosgra, O.H.: Necessary conditions for static and fixed order dynamic mixed H 2 /H optimal control. Proc. Amer. Contr. Conf. (1991) 1137–1143Google Scholar
  52. 52.
    Stoorvogel, A.A.: The H control problem: a state space approach. Prentice Hall, Hemel Hempstead, UK (1992)MATHGoogle Scholar
  53. 53.
    Stoorvogel, A.A.: The singular H 2 control problem. Automatica 28 (1992) 627–631MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Szanier, M.: An exact solution to general SISO mixed H 2 /H problems via convex optimization. IEEE Trans. Automat. Control 39 (1995) 2511–2517CrossRefGoogle Scholar
  55. 55.
    Tadmor, G.: Worst-case design in the time domain: the maximum principle and the standard H problem. Math. Control Signals Systems 3 (1990) 301–324MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Trentelman, H.L.: Almost invariant subspaces and high gain feedback. CWI Tract No. 29, Amsterdam (1986)MATHGoogle Scholar
  57. 57.
    Willems, J.C.: Least-squares stationary optimal control and the algebraic Riccati equation. IEEE Trans. Automat. Control 21 (1971) 319–338Google Scholar
  58. 58.
    Willems, J.C.: Almost invariant subspaces: An approach to high gain feedback design-Part II: Almost conditioned invariant subspaces. IEEE Trans. Autom. Control 27 (1982) 1071–1085MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Yeh, H., Banda, S., Chang, B.: Necessary and sufficient conditions for mixed H 2 and H control. Proc. 29th IEEE Conf. Decision Contr. (1990) 1013–1017Google Scholar
  60. 60.
    Zhou, K., Glover, K., Bodenheimer, B., Doyle, J.: Mixed H 2 and H performance objectives I: Robust performance analysis. IEEE Trans. Automat. Control 39 (1994) 1564–1574MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 1995

Authors and Affiliations

  • Carsten Scherer
    • 1
  1. 1.Mechanical Engineering Systems and Control GroupDelft University of TechnologyDelftThe Netherlands

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