Abstract
There is a dilemma concerning design of control systems. Due to increasing demands on quality and productivity of industrial systems and with deeper understanding of these systems, mathematical models derived to represent the system dynamics are more complete, usually of multi-input-multi-output form, and are of high orders. Consequently, the controllers designed are complex. The order of such controllers designed using, for instance, the \(\mathcal{H}_{\infty}\) optimization approach or the μ-method, is higher than, or at least similar to, that of the plant. On the other hand, in the implementation of controllers, high-order controllers will lead to high cost, difficult commissioning, poor reliability and potential problems in maintenance. Lower-order controllers are always welcomed by practicing control engineers. Hence, how to obtain a low-order controller for a high-order plant is an important and interesting task, and is the subject of the present chapter.
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References
Adamjan, V.M., Arov, D.Z., Krein, M.G.: Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur–Takagi problem. Math. USSR Sb. 15, 31–73 (1971)
Al-Saggaf, U.M., Franklin, G.F.: An error bound for a discrete reduced order model of a linear multivariable system. IEEE Trans. Autom. Control AC-32, 815–819 (1987)
Bernstein, D.S., Haddad, W.M.: LQG control with an \(\mathcal{H}^{\infty}\) performance bound: a Riccati equation approach. IEEE Trans. Autom. Control AC-34, 293–305 (1989)
Choi, B.W., Gu, D.W., Postlethwaite, I.: Low-order \(\mathcal{H}_{\infty}\) sub-optimal controllers. IEE Proc. Part D, Control Theory Appl. 141, 243–248 (1994)
Desai, U.B., Pal, D.: A transformation approach to stochastic model reduction. IEEE Trans. Autom. Control AC-29, 1097–1100 (1984)
Enns, D.: Model reduction for control systems design. PhD thesis, Department of Aeronautics and Astronautics, Stanford University, Stanford, CA (1984)
Enns, D.: Model reduction with balanced realizations: an error bound and a frequency weighted generalization. In: Proceedings of the 23rd IEEE Conference on Decision and Control, Las Vegas, NV, pp. 127–132 (1984)
Fernando, K.V., Nicholson, H.: Singular perturbational model reduction of balanced systems. IEEE Trans. Autom. Control AC-27, 466–468 (1982)
Fernando, K.V., Nicholson, H.: Singular perturbational approximation for discrete-time systems. IEEE Trans. Autom. Control AC-28, 240–242 (1983)
Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their l ∞-error bounds. Int. J. Control 39, 1115–1193 (1984)
Glover, K.: Multiplicative approximation of linear multivariable systems with \(\mathcal{L}_{\infty}\) error bounds. In: Proceedings of the 1986 American Control Conference, Minneapolis, MN, pp. 1705–1709 (1986)
Glover, K., Jonckheere, E.A.: A comparison of two Hankel norm methods for approximating spectra. In: Byrnes, C.I., Lindquist, A. (eds.) Modelling, Identification and Robust Control. North-Holland, Amsterdam (1986)
Green, M.: Balanced stochastic realizations. Linear Algebra Appl. 98, 211–247 (1988)
Green, M.: A relative error bound for balanced stochastic truncation. IEEE Trans. Autom. Control AC-33, 961–965 (1988)
Gu, D.W., Choi, B.W., Postlethwaite, I.: Low-order stabilizing controllers. IEEE Trans. Autom. Control AC-38, 1713–1717 (1993)
Hammarling, S.: Numerical solution of the stable non-negative definite Lyapunov equation. IMA J. Numer. Anal. 2, 303–323 (1982)
Hsu, C.S., Yu, X., Yeh, H.H., Banda, S.S.: \(\mathcal{H}_{\infty}\) compensator design with minimal order observer. In: Proceedings of the 1993 American Control Conference, San Francisco, CA, June 1993
Iwasaki, T., Skelton, R.E.: All low order \(\mathcal{H}_{\infty}\) controllers with covariance upper bound. In: Proceedings of the 1993 American Control Conference, San Francisco, CA, June 1993
Kim, S.W., Anderson, B.D.O., Madievski, A.G.: Error bound for transfer function order reduction using frequency weighted balanced truncation. Syst. Control Lett. 24, 183–192 (1995)
Kung, S.: A new low-order approximation algorithm via singular value decomposition. In: Proceedings of the 18th IEEE Conference on Decision and Control, Ft. Lauderdale, Florida, December 1979
Kung, S., Lin, D.W.: Optimal Hankel norm model reduction: multivariable systems. IEEE Trans. Autom. Control AC-26, 832–852 (1981)
Laub, A.J.: On computing balancing transformations. In: Proceedings of the Joint 1980 American Control Conference, San Francisco, CA, August 1980, p. 8 (1980)
Laub, A.J., Heath, M.T., Paige, C.C., Ward, R.C.: Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans. Autom. Control AC-32, 115–121 (1987)
Lin, C.-A., Chiu, T.-Y.: Model reduction via frequency weighted balanced realization. Control Theory Adv. Technol. 8, 341–351 (1992)
Lindquist, A., Picci, G.: On the stochastic realization problem. SIAM J. Control Optim. 17, 365–389 (1979)
Liu, Y., Anderson, B.D.O.: Singular perturbation of balanced systems. Int. J. Control 50(4), 1379–1405 (1989)
Meyer, D.G.: Fractional balanced reduction: Model reduction via fractional representation. IEEE Trans. Autom. Control AC-35(3), 1341–1345 (1990)
Moore, B.C.: Principle component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control AC-26, 17–31 (1981)
Murad, G.: Robust multivariable control of industrial production processes: a discrete-time multi-objective approach. PhD thesis, Department of Engineering, University of Leicester, Leicester, UK (1995)
Murad, G., Gu, D.-W., Postlethwaite, I.: A Direct Model Reduction Approach for Discrete-Time Non-Minimal State-Space Systems. Internal Report 94-23, University of Leicester, Leicester, UK, September 1994
Nehari, Z.: On bounded bilinear forms. Ann. Math. 65, 153–162 (1957)
Pernebo, L., Silverman, L.M.: Model reduction via balanced state space representations. IEEE Trans. Autom. Control AC-27, 382–387 (1982)
Safonov, M.G., Chiang, R.Y.: A Schur method for balanced-truncation model reduction. IEEE Trans. Autom. Control AC-34, 729–733 (1989)
Safonov, M.G., Chiang, R.Y., Limebeer, D.J.N.: Hankel model reduction without balancing—a descriptor approach. In: Proceedings of the 26th IEEE Conference on Decision and Control, Los Angeles, CA, December 1987
Samar, R., Postlethwaite, I., Gu, D.-W.: Model reduction with balanced realizations. Int. J. Control 62, 33–64 (1995)
Sreeram, V., Anderson, B.D.O., Madievski, A.G.: New results on frequency weighted balanced reduction technique. In: Proceedings of the 1995 American Control Conference, Seattle, WA, June 1995, pp. 4004–4009 (1995)
Tombs, M.S., Postlethwaite, I.: Truncated balanced realization of a stable non-minimal state space system. Int. J. Control 46(4), 1319–1330 (1987)
Varga, A.: Balancing free square-root algorithm for computing singular perturbation approximations. In: Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, UK, December 1991, pp. 1062–1065 (1991)
Varga, A.: On stochastic balancing related model reduction. In: Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December 2000, pp. 2385–2390 (2000)
Wang, G., Sreeram, V., Liu, W.Q.: A new frequency-weighted balanced truncation method and an error bound. IEEE Trans. Autom. Control 44, 1734–1737 (1999)
Zeiger, H.P., McEwen, A.J.: Approximate linear realization of given dimension via Ho’s algorithm. IEEE Trans. Autom. Control AC-19, 153 (1974)
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Gu, DW., Petkov, P.H., Konstantinov, M.M. (2013). Lower-Order Controllers. In: Robust Control Design with MATLAB®. Advanced Textbooks in Control and Signal Processing. Springer, London. https://doi.org/10.1007/978-1-4471-4682-7_7
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DOI: https://doi.org/10.1007/978-1-4471-4682-7_7
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