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Sixty years of cybernetics: a comparison of approaches to solving the \(H_2\) control problem. (English) Zbl 1154.93415

Summary: The \(H_2\) control problem consists of stabilizing a control system while minimizing the \(H_2\) 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.

MSC:

93D15 Stabilization of systems by feedback
93C05 Linear systems in control theory
49N10 Linear-quadratic optimal control problems
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References:

[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 · Zbl 0786.93047 · doi:10.1080/00207179308923006
[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 · Zbl 0698.93031 · doi:10.1109/9.29425
[3] Kučera V.: Discrete Linear Control: The Polynomial Equation Approach. Wiley, Chichester 1979, pp. 115-118
[4] Kučera V.: The H\(_{2}\) control problem: a general transfer-function solution. Internat. J. Control 80 (2007), 800-815 · Zbl 1162.93395 · doi:10.1080/00207170701203590
[5] Kwakernaak H.: H\(_{2}\) optimization - Theory and applications to robust control design. Proc. 3rd IFAC Symposium on Robust Control Design, Prague 2000, pp. 437-448
[6] Meinsma G.: On the standard H\(_{2}\) problem. 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 · Zbl 0542.93014 · doi:10.1109/TAC.1984.1103674
[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 · Zbl 0682.93020 · doi:10.1109/9.24230
[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 · Zbl 1035.93503 · doi:10.1016/0005-1098(96)00052-0
[10] Stoorvogel A. A.: The singular H\(_{2}\) control problem. Automatica 28 (1992), 627-631 · Zbl 0766.93043 · doi:10.1016/0005-1098(92)90189-M
[11] Vidyasagar M.: Control System Synthesis: A Factorization Approach. MIT Press, Cambridge, Mass. 1985, pp. 108-116 · Zbl 0655.93001
[12] Zhou K.: Essentials of Robust Control\(. \) Prentice Hall, Upper Saddle River 1998, pp. 261-265
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