# Model reduction with a finite-interval *H*_{∞} criterion

## Abstract

An important problem in flight control and flying qualities is the approximation of a complex high order system by a low order model. In this chapter, for a given reduced order model, we define the correlation measure between the plant and the model outputs to be the minimum of the ratio of weighted signal energy to weighted error energy. We give a criterion for the evaluation of the correlation measure in terms of minimization of a parameter occurring in a two-point boundary value problem. Once the correlation measure for a given reduced order model can be evaluated, a nonlinear programming algorithm can be used to select a model which maximizes the correlation between the plant and model outputs. The correlation index used can be regarded as an extension of the *H*_{∞} performance criterion to the finite-interval time-varying case. However, the usual *H*_{∞} problem seeks an optimal controller, whereas our problem is to select the reduced order model matrices which give the best correlation index. We also give an expression for the variation of the correlation owing to parameter variations and pose a robust model reduction problem. The utilization of the theory is demonstrated by means of some examples. In particular, a problem which involves the reduction of an unstable aircraft model with structural modes is worked out.

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## References

- [1]B. D. Anderson and L. Y. Liu, Controller reduction: Concepts and approaches,
*IEEE Trans. Automat. Contr.***34**, 1989, pp. 802–812.Google Scholar - [2]U.-L. Ly, “A Design Algorithm for Robust Low Order Controller,” Ph. D. dissertation, Department of Aeronautics and Astronautics, Stanford University, 1982.Google Scholar
- [3]D. S. Bernstein and D. C. Hyland, The optimal projection equations for fixed-order dynamic compensation,
*IEEE Trans. Automat. Contr.***29**, 1984, pp. 1034–1037.Google Scholar - [4]G. Tadmor,
*H*_{∞}in the time domain: The standard four block problem,*Mathematics of Control, Signals, and Systems*, to appear.Google Scholar - [5]M. B. Subrahmanyam, Synthesis of finite-interval
*H*_{∞}controllers by state space methods,”*AIAA Journal of Guidance, Control, Dynamics*, to appear.Google Scholar - [6]—M. B. Subrahmanyam, “Optimal disturbance rejection in time-varying linear systems,”
*Proceedings of the American Control Conference*, Vol.1, 1989, pp. 834–840.Google Scholar - [7]—, “Necessary conditions for the design of control systems with optimal disturbance rejection,”
*Proceedings of the 28th IEEE Conference on Decision and Control*, 1989.Google Scholar - [8]W. M. Haddad and D. S. Bernstein, Robust reduced-order modeling via the optimal projection equations with Petersen-Hollot bounds,”
*IEEE Trans. Automat. Contr.***33**, 1988, pp. 692–695.Google Scholar - [9]A. H. G. Rinnooy Kan and G. T. Timmer, “Global Optimization: A Survey,”
*New Methods in Optimization and their Industrial Uses*, J. Penot (ed.), Birkhäuser Verlag, Basel, Boston, 1989, pp. 133–155.Google Scholar - [10]J. L. Kuester and J. H. Mize, “Optimization Techniques with Fortran”, McGraw-Hill, New York, 1973.Google Scholar
- [11]R. D. Colgren, “Methods for model reduction,”
*Proceedings of the AIAA Guidance, Navigation and Control Conference*, Part 2, 1988, pp. 777–790.Google Scholar - [12]S. Kirkpatrick, C. D. Gelatt, Jr, and M. P. Vecchi, Optimization by simulated annealing,”
*Science***220**, No. 4598, 1983, pp. 671–680.Google Scholar