Article
Full entry |
PDF
(3.9 MB)
Feedback
PDF
(3.9 MB)
Feedback
Keywords:
MIMO; distributed parameter system; sampled-data control; finite-dimensional controllers; finite-dimensional systems
MIMO; distributed parameter system; sampled-data control; finite-dimensional controllers; finite-dimensional systems
Summary:
This paper considers the problem of designing near-optimal finite-dimensional controllers for stable multiple-input multiple-output (MIMO) distributed parameter plants under sampled-data control. A weighted $ {\cal H}^\infty $-style mixed-sensitivity measure which penalizes the control is used to define the notion of optimality. Controllers are generated by solving a “natural” finite-dimensional sampled-data optimization. A priori computable conditions are given on the approximants such that the resulting finite- dimensional controllers stabilize the sampled-data controlled distributed parameter plant and are near-optimal. The proof relies on the fact that the control input is appropriately penalized in the optimization. This technique also assumes and exploits the fact that the plant can be approximated uniformly by finite-dimensional systems. Moreover, it is shown how the optimal performance may be estimated to any desired degree of accuracy by solving a single finite-dimensional problem using a suitable finite-dimensional approximant. The constructions given are simple. Finally, it should be noted that no infinite-dimensional spectral factorizations are required. In short, the paper provides a straight forward control design approach for a large class of MIMO distributed parameter systems under sampled-data control.
References:
[1] Bamieh B. A., Pearson J. B.: A general framework for linear periodic systems with applications to ${\mathcal H}^\infty $ sampled–data control. IEEE Trans. Automat. Control 37 (1992), 418–435 DOI 10.1109/9.126576 | MR 1153103
[2] Boyd, S .P., Balakrishnan V., Kabamba P.: A bisection method for computing the ${\mathcal H}^\infty $ norm of a transfer matrix and related problems. Math. Control Signals Systems 2 (1989), 207–219 DOI 10.1007/BF02551385 | MR 0997214
[3] Callier F. M., Desoer C. A.: An algebra of transfer functions for distributed linear time–invariant systems. IEEE Trans. Circuits and Systems 25 (1978), 9, 651–662 DOI 10.1109/TCS.1978.1084544 | MR 0510690 | Zbl 0468.93020
[4] Cantoni M. W., Glover K.: A design framework for continuous–time systems under sampled–data control. In: Proceedings of the 35th Conference on Decision and Control, Kobe 1996, pp. 458–463
[5] Chen T., Francis B.: ${\mathcal H}^2$-optimal Sampled–Data Control. Technical Report No. 9001, Dept. Elect. Eng., Univ. Toronto, 1990 MR 1097092
[6] Chen T., Francis B.: Optimal Sampled–Data Control Systems. Springer–Verlag, London – New York 1995 MR 1410060 | Zbl 0876.93002
[7] Conway J. B.: A Course in Functional Analysis. Springer–Verlag, Berlin 1990 MR 1070713 | Zbl 0706.46003
[8] Desoer C. A., Vidyasagar M.: Feedback Systems: Input–Output Properties. Academic Press, NY 1975 MR 0490289 | Zbl 1153.93015
[9] Dullerud G. E.: Control of Uncertain Sampled–Data Systems. Birkhäuser, Boston 1996 MR 1377267 | Zbl 0843.93006
[10] Flamm D. S., Mitter S. K.: Approximation of ideal compensators for delay systems. In: Linear Circuits, Systems and Signal Processing: Theory and Applications (C. I. Byrnes, C. F. Martin and R. E. Saeks, eds.), Elsevier Science Publishers B. V., 1988, pp. 517–524 MR 1031070 | Zbl 0675.93027
[11] Foias C., Francis B., Helton J. W., Kwakernaak H., Pearson J. B.: ${\mathcal H}^\infty $-Control Theory. (Lecture Notes in Mathematics 1496.) Springer-Verlag, Berlin 1991
[12] Foias C., Özbay H., Tannenbaum A.: Robust Control of Infinite Dimensional Systems. (Lecture Notes in Control and Information Sciences.) Springer–Verlag, Berlin 1996 MR 1369772 | Zbl 0839.93003
[13] Francis B. A.: A Course in Control Theory. Springer-Verlag, Berlin 1987 MR 0932459
[14] Gibson J. S., Adamian A.: Approximation theory for linear–quadratic–gaussian optimal control of flexible structures. SIAM J. Control Optim. 29 (1991), 1, 1–37 DOI 10.1137/0329001 | MR 1088217 | Zbl 0788.93027
[15] Gibson J. S., Rosen I. G.: Numerical approximation for the infinite–dimensional discrete–time optimal linear–quadratic regulator problem. SIAM J. Control Optim. 26 (1988), 2, 428–451 DOI 10.1137/0326025 | MR 0929811 | Zbl 0644.93013
[16] Glader C., Högnäs G., Mäkilä P. M., Toivonen H. T.: Approximation of delay systems – a case study. Internat. J. Control 53 (1991), 2, 369–390 DOI 10.1080/00207179108953623 | MR 1091150 | Zbl 0745.93016
[17] Glover K.: All optimal Hankel–norm approximations of linear multivariable systems and their ${\mathcal L}^\infty $-error bounds. Internat. J. Control 39 (1984), 6, 1115–1193 DOI 10.1080/00207178408933239 | MR 0748558
[18] Glover K., Curtain R. F., Partington J. R.: Realisation and approximation of linear infinite–dimensional systems with error bounds. SIAM J. Control Optim. 26 (1988), 4, 863–898 DOI 10.1137/0326049 | MR 0948650 | Zbl 0654.93011
[19] Glover K., Lam J., Partington J. R.: Rational approximation of a class of infinite–dimensional systems. I: singular values of Hankel operators. Math. Control Signals Systems 4 (1990), 325–344 DOI 10.1007/BF02551374 | MR 1066376 | Zbl 0727.41020
[20] Glover K., Lam J., Partington J. R.: Rational approximation of a class of infinite–dimensional systems. II: optimal convergence rates of ${\mathcal L}^\infty $ approximants. Math. Control Signals Systems 4 (1991), 233–246 DOI 10.1007/BF02551279 | MR 1107236 | Zbl 0733.41023
[21] Gu G., Khargonekar P. P., Lee E. B.: Approximation of infinite–dimensional systems. IEEE Trans. Automat. Control 34 (1989), 6, 610–618 DOI 10.1109/9.24229 | MR 0996150 | Zbl 0682.93035
[22] Ichikawa A.: The semigroup approach to ${\mathcal H}^2$ and ${\mathcal H}^\infty $-control for sampled–data systems with first–order hold. In: Proceedings of the 35th Conference on Decision and Control, Kobe 1996, pp. 452–457
[23] Ito K.: Finite–dimensional compensators for infinite-dimensional systems via Galerkin–type approximation. SIAM J. Control Optim. 28 (1990), 6, 1251–1269 DOI 10.1137/0328067 | MR 1075203 | Zbl 0733.93031
[24] Kabamba P. T., Hara S.: Worst–case analysis and design of sampled–data control systems. IEEE Trans. Automat. Control 38 (1993), 1337–1357 DOI 10.1109/9.237646 | MR 1240826 | Zbl 0787.93068
[25] Kamen E. W., Khargonekar P. P., Tannenbaum A.: Stabilization of time delay systems with finite–dimensional compensators. IEEE Trans. Automat. Control 30 (1985), 75–78 DOI 10.1109/TAC.1985.1103789 | MR 0777079
[26] Lenz K., Ozbay H., Tannenbaum A., Turi J., Morton B.: Robust control design for a flexible beam using a distributed parameter ${\mathcal H}^\infty $ method. In: CDC, Tampa 1989 MR 1039105
[27] Logemann H., Townley S.: Adaptive low–gain sampled–data control of DPS. In: Proceedings of the 34th Conference on Decision and Control, New Orleans 1995, pp, 2946–2947
[28] Mäkilä P. M.: Laguerre series approximation of infinite dimensional systems. Automatica 26 (1990), 6, 985–995 DOI 10.1016/0005-1098(90)90083-T | MR 1080985 | Zbl 0717.93028
[29] McFarlane D. C., Glover K.: Robust Controller Design Using Normalized Coprime Factor Plant Descriptions. Springer–Verlag, Berlin 1990 MR 1029524 | Zbl 0688.93044
[30] Rodriguez A. A., Dahleh M. A.: Weighted ${\mathcal H}^\infty $ optimization for stable infinite–dimensional systems using finite–dimensional techniques. In: Proceedings of the 29th IEEE CDC, Honolulu 1990, pp. 1814–1820
[31] Rodriguez A. A., Dahleh M. A.: On the computation of induced norms for non–compact Hankel operators arising from distributed control problems. Systems Control Lett. 19 (1992), 429–438 DOI 10.1016/0167-6911(92)90074-3 | MR 1195300 | Zbl 0787.47024
[32] Rosen I. G.: Optimal discrete–time LQR problems for parabolic systems with unbounded input – approximation and convergence. Control Theory Adv. Tech. 5 (1989), 227–300 MR 1020634
[33] Rosen I. G., Wang C.: On the continuous dependence with respect to sampling of the linear quadratic regulator problem for distributed parameter systems. SIAM J. Control Optim. 30 (1992), 4, 942–974 DOI 10.1137/0330052 | MR 1167820 | Zbl 0765.49021
[34] Rosen I. G., Wang C.: On stabilizability and sampling for infinite dimensional systems. IEEE Trans. Automat. Control 37 (1992), 10, 1653–1656 DOI 10.1109/9.256405 | MR 1188781 | Zbl 0770.93080
[35] Royden H. L.: Real Analysis. MacMillan Publishing Co, Inc, 1968 MR 0151555
[36] Sågfors M. F., Toivonen H. T.: The sampled–data ${\mathcal H}^\infty $ problem: The equivalence of discretization–based methods and a Riccati equation solution. In: Proceedings of the 35th Conference on Decision and Control, Kobe 1996, pp. 428–433
[37] Smith M.: Well–posedness of ${\mathcal H}^\infty $ optimal control problems. SIAM J. Control Optim. 28 (1990), 342–358 DOI 10.1137/0328018 | MR 1040463
[38] Sun W., Nagpal K. M., Khargonekar P. P.: ${\mathcal H}^\infty $ control and filtering for sampled–data systems. IEEE Trans. Automat. Control 38 (1993), 1162–1175 DOI 10.1109/9.233150 | MR 1235247
[39] Sz.-Nagy B., Foiaş C.: Harmonic Analysis of Operators on Hilbert Space. North–Holland, Amsterdam 1970 MR 0275190 | Zbl 1234.47001
[40] Tadmor G.: ${\mathcal H}^\infty $ optimal sampled–data control in continuous time systems. Internat. J. Control 56 (1992), 1, 99–141 DOI 10.1080/00207179208934306 | MR 1170889
[41] Toivonen H. T.: Sampled–data control of continuous–time systems with an ${\mathcal H}^\infty $ optimality criterion. Automatica 28 (1992), 45–54 DOI 10.1016/0005-1098(92)90006-2 | MR 1144109
[42] Vidyasagar M.: Control Systems Synthesis: A Factorization Approach. MIT Press, Cambrdige MA 1985
[43] Yamamoto Y.: A function space approach to sampled–data control systems and tracking problem. IEEE Trans. Automat. Control 39 (1994), 703–713 DOI 10.1109/9.286247 | MR 1276768
[44] Yue D., Liu Y., Xu S.: Finite–dimensional compensator for a class of uncertain distributed parameter systems. Internat. J. Systems Sci. 26 (1995), 12, 2383–2390 DOI 10.1080/00207729508929175 | Zbl 0853.93056
[45] Youla D. C., Jabr H. A., Bongiorno J. J.: Modern Wiener–Hopf design of optimal controllers. Part 2: The multivariable case. IEEE Trans. Automat. Control 21 (1976), 319–338 DOI 10.1109/TAC.1976.1101223 | MR 0446637 | Zbl 0339.93035
Search
Partner of
