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linear systems; feedback control; stability; norm minimization
The H2 control problem consists of stabilizing a control system while minimizing the H2 norm of its transfer function. Several solutions to this problem are available. For systems in state space form, an optimal regulator can be obtained by solving two algebraic Riccati equations. For systems described by transfer functions, either Wiener-Hopf optimization or projection results can be applied. The optimal regulator is then obtained using operations with proper stable rational matrices: inner-outer factorizations and stable projections. The aim of this paper is to compare the two approaches. It is well understood that the inner-outer factorization is equivalent to solving an algebraic Riccati equation. However, why are the stable projections not needed in the state-space approach? The difference between the two approaches derives from a different construction of doubly coprime, proper stable matrix fractions used to represent the plant. The transfer-function approach takes any fixed doubly coprime fractions, while the state-space approach parameterizes all such representations and those selected then obviate the need for stable projections.
[1] Chen B. M., Saberi A.: Necessary and sufficient conditions under which an H$_{2}$ optimal control problem has a unique solution. Internat. J. Control 58 (1993), 337–348 MR 1229854
[2] Doyle J. C., Glover K., Khargonekar P. P., Francis B. A.: State space solutions to standard H$_{2}$ and H$_{\infty }$ control problems. IEEE Automat. Control 34 (1989), 831–847 MR 1004301
[3] Kučera V.: Discrete Linear Control: The Polynomial Equation Approach. Wiley, Chichester 1979, pp. 115–118 MR 0573447
[4] Kučera V.: The H$_{2}$ control problem: a general transfer-function solution. Internat. J. Control 80 (2007), 800–815 MR 2316383 | Zbl 1162.93395
[5] Kwakernaak H.: H$_{2}$ optimization – Theory and applications to robust control design. In: Proc. 3rd IFAC Symposium on Robust Control Design, Prague 2000, pp. 437–448
[6] Meinsma G.: On the standard H$_{2}$ problem. In: Proc. 3rd IFAC Symposium on Robust Control Design, Prague 2000, pp. 681–686
[7] Nett C. N., Jacobson C. A., Balas N. J.: A connection between state-space and doubly coprime fractional representations. IEEE Automat. Control 29 (1984), 831–832 MR 0756933 | Zbl 0542.93014
[8] Park K., Bongiorno J. J.: A general theory for the Wiener–Hopf design of multivariable control systems. IEEE Automat. Control 34 (1989), 619–626 MR 0996151 | Zbl 0682.93020
[9] Saberi A., Sannuti, P., Stoorvogel A. A.: H$_{2}$ optimal controllers with measurement feedback for continuous-time systems – Flexibility in closed-loop pole placement. Automatica 32 (1996), 1201–1209 MR 1409674 | Zbl 1035.93503
[10] Stoorvogel A. A.: The singular H$_{2}$ control problem. Automatica 28 (1992), 627–631 MR 1166033
[11] Vidyasagar M.: Control System Synthesis: A Factorization Approach. MIT Press, Cambridge, Mass. 1985, pp. 108–116 MR 0787045 | Zbl 0655.93001
[12] Zhou K.: Essentials of Robust Control$. $ Prentice Hall, Upper Saddle River 1998, pp. 261–265
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