performs better than the already classical balanced truncation method even when its static matrix gain is optimized, procedure which involves greater computational efforts. Our method performs particularly well when approximations of large scale flexible structures are considered and we believe that it is a real alternative to determine reduced order models of this important class of linear time invariant systems. [3] Y. Ebihara and T. Hagiwara, "On H model reduction using LMIs," REFERENCESIEEE Trans. Autom. Control, vol. 49, no. 7, pp. 1187-1191 J. C. Geromel, "Convex analysis and global optimization of joint actuator location and control problems," IEEE Trans. Autom. Control, vol. 34, no. 7, pp. 711-720, Jul. 1989. [5] , "Optimal linear filtering under parameter uncertainty," IEEE Trans. Signal Process., vol. 47, no. 1, pp. 168-175, Jan. 1999. [6] J. C. Geromel, F. R. R. Kawaoka, and R. G. Egas, "Model reduction of discrete time systems through linear matrix inequalities," Int. J. Control, vol. 77, no. 10, pp. 978-984, Jul. 2004. [7] K. Glover, "All optimal Hankel-norm approximations of linear multivariable systems and their L -error bounds," Int. J. Control, vol. 39, no. 6, pp. 1115Control, vol. 39, no. 6, pp. -1193Control, vol. 39, no. 6, pp. , 1984 [8] K. M. Grigoriadis, "Optimal H model reduction via linear matrix inequalities: Continuous and discrete-time cases," Syst. Control Lett., vol. 26, no. 5, pp. 321-333, 1995. [9] W. M. Haddad and D. S. Bernstein, "Combined L =H model-reduction," Int. J. Control, vol. 49, no. 5, pp. 1523Control, vol. 49, no. 5, pp. -1535Control, vol. 49, no. 5, pp. , 1989 [10] L. Meirovitch, H. Baruh, and H. Öz, "A comparison of control techniques for large flexible systems," J. Guid., vol. 6, no. 4, 1983. [11] C. Scherer, "Mixed H =H Abstract-This note provides results on approximating the minimal robust positively invariant (mRPI) set (also known as the 0-reachable set) of an asymptotically stable discrete-time linear time-invariant system. It is assumed that the disturbance is bounded, persistent and acts additively on the state and that the constraints on the disturbance are polyhedral. Results are given that allow for the computation of a robust positively invariant, outer approximation of the mRPI set. Conditions are also given that allow one to a priori specify the accuracy of this approximation.

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Robust positively invariant (RPI) sets for linear difference inclusions are considered here under the assumption that the linear difference inclusion is absolutely asymptotically stable in the absence of additive state disturbances, which is the case for parametrically uncertain or switching linear discrete-time systems controlled by a stabilizing linear state feedback controller. The existence and uniqueness of the minimal RPI set and the minimal convex RPI set are studied. A new method for the computation of outer RPI approximations of the minimal RPI set for linear difference is presented; these approximations include a family of star-shaped RPI sets and two families of convex RPI sets. The use of a family of star-shaped RPI sets, and the characterization of the family, is reported for the first time.

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Abstract-This paper provides a solution to the problem of computing a robustly positively invariant outer approximation of the minimal robustly positively invariant set for a discretetime, linear, time-invariant system. It is assumed that the disturbance is additive and persistent, but bounded.