Abstract
In this paper it is shown that if costs are associated to sampling operations which are added to a performance criterion, the minimization of this new performance criterion results in a controller operated at an optimal sampling rate. This, under the assumptions that the system is periodically sampled, the applied control is kept fixed between every two sampling instances and some technical conditions are met. In case the considered planning horizon in the performance criterion is finite an algorithm is devised which calculates in a finite number of steps the optimal sampling period. It is shown that the technical conditions mentioned above are satisfied by the finite planning horizon time-varying LQG tracking problem. Since stability is a major requirement in controller design we also consider the case of an infinite planning horizon. This analysis is focused on the time-invariant digital LQ tracking problem. Given some mild regularity conditions a numerical algorithm is presented that approximates the optimal solution within any prespecified error norm. It is shown that also in this case an optimal sampling-rate exists. The algorithm for determining the optimal sampling period if the planning horizon is finite is illustrated in an economic example.
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References
Anderson, B.D.O. and Moore, J.B., 1989, Optimal Control: Linear Quadratic Methods, Prentice-Hall, London.
Aoki, M., 1973, Notes and comments: On sufficient conditions for optimal stabilization policies,Review of Economic Studies 47, 131–138.
Aström, K.J. and Wittenmark, B., 1990, Computer Controlled Systems, Prentice-Hall, Englewood Cliffs, N.J.
Chow, G.C., 1975, Analyses and Control of Dynamic Economic Systems, John Wiley & Sons, New York.
De Koning, W.L., 1980, Equivalent discrete optimal control problem for randomly sampled digital control systems,Int. J. Sys. Sci.,11, 841–855.
Dorato, P. and Levis, A.H., 1971, Optimal Linear Regulators: The Discrete-Time Case,IEEE Trans. Aut. Contr.,AC-16,6, 613–620.
Engwerda, J.C., 1990a, The solution of the infinite horizon tracking problem for discrete-time systems possessing an exogenous component,Journal of Economic Dynamics and Control 14, 741–762.
Engwerda, J.C. and Van Willigenburg, L.G., 1992, LQ-control of sampled continuous-time systems, In V. Strejc (Ed.), Proceedings of the 2nd IFAC workshop on system structure and control, IFAC, Prague, Czechoslovakia, pp. 128–131.
Franklin, G.F., and Powell D., 1980, Digital Control of Dynamic Systems, Addison and Wesley, London.
Kabamba, P.J., 1987, Control of linear systems using generalized sampled-data hold functions,I.E.E.E. Trans. Aut. Control,AC32 772–783.
Kalman, R.E., Ho, Y.C. and Narendra, K.S., 1963, Contributions to Differential Equations, vol. I, Interscience, New York.
Kwakernaak, H. and Sivan, R., 1972, Linear Optimal Control System, John Wiley and Sons, New York.
Levis, A.H., Schlueter, R.A. and Athans, M., 1971, On the behavior of optimal linear sampled data regulators,Int. J. Contr.,13, 343–361.
Lewis, F.L., 1986, Optimal Control, John Wiley & Sons, New York.
Maybeck, P.S., 1982, Stochastic Models Estimation and Control Vol. 141-3 in the series of Mathematics in Science and Engineering, Academic Press, London.
Pindyck, R.S., 1973, Optimal planning for Economic Stabilization, North Holland, Amsterdam.
Powell, S.R., 1967, B.S. Thesis, Department of Electrical Engineering, M.I.T.
Preston, A.J., and Pagan, A.R., 1982, The Theory of Economic Policy, Cambridge University Press, New York.
Pitchford, J.D., and Turnovsky, S.J., 1977, Applications of Control Theory to Economic Analysis, North-Holland, Amsterdam.
Rosen, I.G. and Wang, C., 1992, Sampling of the linear quadratic regulator problem for distributed parameter systems,Siam J. Control and Optimization,30,4 942–974.
Shats, S. and Shaked, U., 1991, Discrete-time filtering of noise correlated continuous-time processes: modeling and derivation of the sampling period sensitivities,IEEE Trans. Auto. Contr. AC36, 115–119.
Tinbergen, J., 1952, On the Theory of Economic Policy, North Holland, Amsterdam.
Turnovsky, S.J., 1972, Optimal stabilization policies for deterministic and stochastic linear economic systems,Review of Economic Studies, pp. 79–95.
Urikura, S. and Nagata, A., 1987, Ripple-free deadbeat control for sampled-data systems,IEEE Trans. Aut. Contr.,AC32,6, 474–482.
Van Willigenburg, L.G. and De Koning W.L., 1992, The digital optimal regulator and tracker for stochastic time-varying systems,International Journal of Systems Science,23,12, 2309–2322.
Van Willigenburg, L.G., 1992, Computation of the digital LQG regulator and tracker for time-varying systems,Optimal Control Applications and Methods,13, 289–299.
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Engwerda, J., van Willigenburg, G. Optimal sampling-rates and tracking properties of digital LQ and LQG tracking controllers for systems with an exogenous component and costs associated to sampling. Comput Econ 8, 107–125 (1995). https://doi.org/10.1007/BF01299713
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DOI: https://doi.org/10.1007/BF01299713