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Extra resources for Discrete-Time-Control System Analysis and Design
Automat. Control, vol. 34, No. 8, pp. 831-847, 1989. 34 TONGWEN CHEN AND BRUCE A . FRANCIS [ I l l 11. T. lity criterion," A ~ t o n z ~ t i cvol. a , 28, No. tems with an H 1, pp. 45-54, 1992. 1121 Y. Yamamoto, "A new approacll t o sampled-data control systems - a function space approach method," Proc. CDC, 1990.  B. Bamieh and J. B. Pearson, "A general framework for linear periodic systems with application t o H ' , sampled-data control," IEEE Trans. Autonzat. Control, vol. 37, pp. 418-435, 1992.
The constant weights were obtained by trial-and-error. 1 Note that (C1, A) is not detectable, reflecting our desire to have 01 and 02 settle to nonzero final values after a finite-duration torque is applied; this will necessitate some massaging below. First, the optimal analog state feedback controller is computed. The MATLAB command is (A, B2, CIIC1 -Jr-10-4I, F - -LQR ! D12D12), where the third matrix on the right-hand side has been perturbed to make it nonsingular. 1 and the result is shown in Figure 8 (01 solid, 02 dash.
Liapunov , D. L. M. Bianchini  and many others. He covers also minimum time optimization problems. Surprisingly enough, his excellent book is only published in italian. M. E. Evans  dealts with controllability of discrete time linear systems with control values constrained to a bounded convex set. The upshot of Evans' work is to connect the controllability properties of a constrained system to those of state eigen-subspaces. He states a decomposition of the given system in subsystems, in the sense that each of them accounts for part of the spectrum of the dynamical matrix, and the controllability properties of the system can be deduced from those of the subsystems.