Previous |  Up |  Next


generalized sampled-data; pole placement problem; adaptive control
The use of generalized sampled-data hold functions, in order to synthesize adaptive pole placers for linear multiple-input, multiple-output systems with unknown parameters, is investigated in this paper, for the first time. Such a control scheme relies on a periodically varying controller, which suitably modulates the sampled outputs of the controlled plant. The proposed control strategy allows us to assign the poles of the sampled closed-loop system arbitrarily in desired locations, and does not make assumptions on the plant other than controllability and observability of the continuous and the sampled system, and the knowledge of a set of structural indices, namely the locally minimum controllability indices of the continuous-time plant. The indirect adaptive control scheme presented here, estimates the unknown plant parameters (and hence the parameters of the desired modulating matrix function) on line, from sequential data of the inputs and the outputs of the plant, which are recursively updated within the time limit imposed by a fundamental sampling period $T_0$. The controller determination is based on the transformation of the discrete analogue of the system under control to a phase-variable canonical form, prior to the application of the control design procedure. The solution of the problem can, then, be obtained by a quite simple utilization of the concept of state similarity transformation, whereas known indirect adaptive pole placement techniques require the solution of matrix polynomial Diophantine equations. Moreover, in many cases, the solution of the Diophantine equation for a desired set of closed-loop eigenvalues might yield an unstable controller, and the overall adaptive pole placement scheme is then unstable with unstable compensators because their outputs are unbounded. The proposed strategy avoids these problems, since here gain controllers are essentially needed to be designed. Moreover, persistency of excitation and, therefore, parameter convergence, of the continuous-time plant is provided without making assumptions either on the existence of specific convex sets in which the estimated parameters belong or on the coprimeness of the polynomials describing the ARMA model, or finally on the richness of the reference signals, as compared to known adaptive pole placement schemes.
[1] Abramovitch D. Y., Franklin G. F.: On the stability of adaptive pole placement controllers with a saturating actuator. IEEE Trans. Automat. Control AC-35 (1990), 303–306 DOI 10.1109/9.50341 | MR 1044025 | Zbl 0707.93030
[2] Al-Rahmani H. M., Franklin G. F.: Linear periodic systems: Eigenvalue assignment using discrete periodic feedback. IEEE Trans. Automat. Control 34 (1989), 99–103 DOI 10.1109/9.8657 | MR 0970938 | Zbl 0657.93024
[3] Anderson B. D. O., Johnstone R. M.: Global adaptive pole positioning. IEEE Trans. Automat. Control AC-30 (1985), 11–22 DOI 10.1109/TAC.1985.1103799 | MR 0777073 | Zbl 0553.93032
[4] Araki M., Hagiwara T.: Pole assignment by multirate sampled data output feedback. Internat. J. Control 44 (1986), 1661–1673 DOI 10.1080/00207178608933692 | Zbl 0613.93040
[5] Arvanitis K. G.: An indirect adaptive pole placer for MIMO systems, based on multirate sampling of the plant output. IMA J. Math. Control Inform. 12 (1995), 363–394 DOI 10.1093/imamci/12.4.363 | MR 1363319 | Zbl 0848.93033
[6] Arvanitis K. G.: An indirect model reference adaptive controller based on the multirate sampling of the plant output. Internat. J. Adaptive Control Signal Process. 10 (1996), 673–705 DOI 10.1002/(SICI)1099-1115(199611)10:6<673::AID-ACS405>3.0.CO;2-M | MR 1423712 | Zbl 0876.93056
[7] Arvanitis K. G.: An indirect model reference adaptive control algorithm based on multidetected–output controllers. Appl. Math. Comput. Sci. 6 (1996), 667–706 MR 1438134 | Zbl 0867.93048
[8] Arvanitis K. G.: Adaptive LQ regulation by multirate–output controllers. Found. Computing Dec. Sciences 21 (1996), 183–213 MR 1440295
[9] Arvanitis K. G.: An adaptive decoupling compensator for linear systems based on periodic multirate–input controllers. J. Math. Syst. Est. Control 8 (1998), 373–376 MR 1650078 | Zbl 1126.93358
[10] Arvanitis K. G., Kalogeropoulos G.: A new periodic multirate model reference adaptive controller for possibly nonstably invertible plants. Kybernetika 33 (1997), 203–220 MR 1454279
[11] Arvanitis K. G., Paraskevopoulos P. N.: Exact model matching of linear systems using multirate digital controllers: In: Proc. 2nd European Control Conference, Groningen 1993, vol. 3, pp. 1648–1652
[12] Arvanitis K. G., Paraskevopoulos P. N.: Discrete model reference adaptive control of continuous–time linear multi–input, multi–output systems using multirate sampled–data controllers. J. Optim. Theory Appl. 84 (1995), 471–493 DOI 10.1007/BF02191981 | MR 1326071
[13] Åstrom K. J., Wittenmark B.: Analysis of a self–tuning regulator for non–minimum phase systems. In: Proc. IFAC Stochast. Control Symposium, Budapest 1974, pp. 165–173
[14] Åstrom K. J., Wittenmark B.: Self–tuning controllers based on pole–zero placement. Proc. IEE–D 127 (1980), 120–130
[15] Chammas A. B., Leondes C. T.: On the design of linear time invariant systems by periodic output feedback. Parts I and II. Internat. J. Control 27 (1978), 885–903 DOI 10.1080/00207177808922419 | MR 0504221 | Zbl 0388.93022
[16] Das M., Cristi R.: Robustness of an adaptive pole placement algorithm in the presence of bounded disturbances and slow time variation of parameters. IEEE Trans. Automat. Control AC-35 (1990), 752–756 DOI 10.1109/9.53562 | Zbl 0800.93452
[17] Egardt B.: Stability analysis of discrete–time adaptive control scheme. IEEE Trans. Automomat. Control AC-25 (1980), 710–716 DOI 10.1109/TAC.1980.1102416 | MR 0583447
[18] Eising R., Hautus M. L. J.: Realizations Algorithms for Systems over a Principal Ideal Domain. Memorandum COSOR 78–25, Eindhoven University of Technology, Dept. of Mathematics, Eindhoven 1978
[19] Elliott H.: Direct adaptive pole placement with application to nonminimum phase systems. IEEE Trans. Automat. Control AC-27 (1982), 720–722 DOI 10.1109/TAC.1982.1102963 | Zbl 0493.93033
[20] Elliott H., Cristi R., Das M.: Global stability adaptive pole placement algorithms. IEEE Trans. Automat. Control AC-30 (1985), 348–356 DOI 10.1109/TAC.1985.1103954 | MR 0786712
[21] Elliott H., Wolovich W. A., Das M.: Arbitrary adaptive pole placement for linear multivariable systems. IEEE Trans. Automat. Control AC-29 (1984), 221–229 DOI 10.1109/TAC.1984.1103491 | Zbl 0534.93026
[22] Giri F., Dion J. M., Dugard L., M’Saad M.: Robust pole placement direct adaptive control. IEEE Trans. Automat. Control AC-34 (1989), 356–359 DOI 10.1109/9.16434 | MR 0980372 | Zbl 0666.93084
[23] Giri F., M’Saad M., Dugard L., Dion J. M.: Robust pole placement indirect adaptive controller. Internat. J. Adaptive Control Signal Process. 2 (1988), 33–47 DOI 10.1002/acs.4480020103 | MR 0941862
[24] Goodwin G. C., Sin K. S.: Adaptive Filtering, Prediction and Control. Prentice-Hall, Englewood Cliffs, N. J. 1984 Zbl 0653.93001
[25] Greshak J. P., Vergese G. C.: Periodically varying compensation of time–invariant systems. Systems Control Lett. 2 (1982), 88–93 DOI 10.1016/S0167-6911(82)80016-9 | MR 0671861
[26] Guidorzi R.: Canonical structures in the identification of multivariable systems. Automatica 11 (1975), 361–374 DOI 10.1016/0005-1098(75)90085-0 | MR 0444226 | Zbl 0309.93012
[27] Hagiwara T., Araki M.: Design of a stable state feedback controller based on the multirate sampling of the plant output. IEEE Trans. Automat. Control AC-33 (1988), 812–819 DOI 10.1109/9.1309 | MR 0953875 | Zbl 0648.93043
[28] Ho B. L., Kalman R. E.: Effective construction of linear state variable models from input/output functions. In: Proc. 3rd Allerton Conference, pp. 449–459; Regelungstechnik 14 (1966), 545–548 MR 0245360 | Zbl 0145.12701
[29] Kabamba P. T.: Control of linear systems using generalized sampled–data hold functions. IEEE Trans. Automat. Control AC-32 (1987), 772–783 DOI 10.1109/TAC.1987.1104711 | MR 0902487 | Zbl 0627.93049
[30] Kabamba P. T., Yang C.: Simultaneous controller design for linear time–invariant systems. IEEE Trans. Automat. Control 36 (1991), 106–111 DOI 10.1109/9.62275 | MR 1084253 | Zbl 0745.93046
[31] Kalman R. E., Ho Y. C., Narendra K. S.: Controllability of linear dynamical systems. Contrib. Diff. Equations 1 (1972), 189-213, 1962 MR 0155070
[32] Khargonekar P. P., Poola K., Tannenbaum A.: Robust control of linear time–invariant plants using periodic compensation. IEEE Trans. Automat. Control AC-30 (1985), 1088–1096 DOI 10.1109/TAC.1985.1103841 | MR 0810310
[33] Kinnaert M., Blondel V.: Discrete–time pole placement with stable controller. Automatica 28 (1992), 935–943 DOI 10.1016/0005-1098(92)90146-7 | MR 1179696 | Zbl 0766.93022
[34] Kim J.-H., Hong Y.-C., Choi K.-K.: Direct model reference adaptive pole placement control with exponential weighting properties. IEEE Trans. Automat. Control AC-36 (1991), 1073–1077 DOI 10.1109/9.83541 | MR 1122486 | Zbl 0754.93028
[35] Lozano-Leal R.: Robust adaptive regulation without persistent excitation. IEEE Trans. Automat. Control AC-34 (1989), 1260–1267 DOI 10.1109/9.40771 | MR 1029376 | Zbl 0689.93038
[36] Lozano-Leal R., Goodwin G. C.: A globally convergent adaptive pole placement algorithm without a persistency of excitation requirement. IEEE Trans. Automat. Control AC-30 (1985), 795-798 DOI 10.1109/TAC.1985.1104036 | MR 0794220
[37] McElveen J. K., Lee K. R., Bennett J. E.: Identification of multivariable linear systems from input/output measurements. IEEE Trans. Ind. Electr. 39 (1992), 189–193 DOI 10.1109/41.141619
[38] Mita T., Pang B. C., Liu K. Z.: Design of optimal strongly stable digital control systems and application to output feedback control of mechanical systems. Internat. J. Control 45 (1987), 2071–2082 DOI 10.1080/00207178708933868 | MR 0891798 | Zbl 0616.93051
[39] Mo L., Bayoumi M. M.: A novel approach to the explicit pole assignment self–tuning controller design. IEEE Trans. Automat. Control AC-34 (1989), 359–363 DOI 10.1109/9.16435 | MR 0980373 | Zbl 0666.93083
[40] Paraskevopoulos P. N., Arvanitis K. G.: Exact model matching of linear systems using generalized sampled–data hold functions. Automatica 30 (1994), 503–506 DOI 10.1016/0005-1098(94)90127-9 | MR 1268953 | Zbl 0800.93329
[41] Sastry S., Bodson M.: Adaptive Control: Stability, Convergence and Robustness. Prentice–Hall, Englewood Cliffs, N. J. 1989 Zbl 0721.93046
[42] Silverman L. M.: Realization of linear dynamical systems. IEEE Trans. Automat. Control AC-16 (1971), 554–567 DOI 10.1109/TAC.1971.1099821 | MR 0307749
[43] Wellstead P. E., Edmunds J. M., Prager D., Zanker P.: Self–tuning pole/zero assignment regulators. Internat. J. Control 30 (1979), 1–26 DOI 10.1080/00207177908922754 | Zbl 0422.93096
[44] Youla D. C., Bongiorno J. J., Jr., Lu C. N.: Single–loop feedback stabilization of linear multivariable dynamical systems. Automatica 10 (1974), 159–173 DOI 10.1016/0005-1098(74)90021-1 | MR 0490293
Partner of
EuDML logo