# Synthesis of Suboptimal *H*_{∞} Controllers over a Finite Horizon

- 142 Downloads

## Abstract

In this chapter a finite horizon *H* _{∞} optimal control problem is posed and solved. A criterion which is useful for the evaluation of the infimal *H* _{∞} norm in the finite horizon case is given. Also, a differential equation is derived for the measure of performance in terms of the final time. A general suboptimal control problem is then posed, and an expression for a suboptimal controller is derived solving the saddle point conditions. An expression for a feedback controller can be derived by solving a dynamic Riccati equation. Also, a criterion that yields the actual performance of the suboptimal controller is given. In the time-invariant case, the finite horizon controller converges to a static controller as the final time becomes large. Examples are given to illustrate the usefulness of the theory.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Doyle, J. C., Glover, K., Khargonekar, P. P., and Francis, B. A., “State-space solutions to standard H
_{2}and H_{∞}control problems,”*IEEE Transactions on Automatic Control*,**34**, 1989, pp. 831–847.MathSciNetzbMATHCrossRefGoogle Scholar - [2]Francis, B. A. and Doyle, J. C., “Linear Control Theory with an H∞ Optimality Criterion,”
*SIAM J. Control and Optimization*,**25**, 1987, pp. 815–844.MathSciNetzbMATHCrossRefGoogle Scholar - [3]Khargonekar, P. P., “State-space
*H∞*Control Theory,” in*Mathematical System Theory: The Influence of R. E. Kalman*, edited by A. C. Antoulas, Springer-Verlag, Berlin, 1991.Google Scholar - [4]Khargonekar, P. P., Nagpal, K. M., and Poolla, K. R., “
*H∞*control with transients, ”*SIAM Journal on Control and Optimization*,**29**, 1991, pp. 1373–1393.MathSciNetzbMATHCrossRefGoogle Scholar - [5]Perkins, W. R. and Medanic, J. V., “Systematic Low Order Controller Design for Disturbance Rejection with Plant Uncertainties,”
*Report #WRDC-TR-90-3036*, Wright-Patterson AFB, U.S.A., 1990.Google Scholar - [6]Pontryagin, L. S., Boltyanski, V. G., Gamkrelidze, R. V., and Mischenko, E. F.,
*The Mathematical Theory of Optimal Processes*, Interscience Publishers, New York, 1962.zbMATHGoogle Scholar - [7]Ravi, R., Nagpal, K. M., and Khargonekar, P. P., “
*H∞*control of linear time-varying systems: A state space approach,”*SIAM Journal on Control and Optimization*,**29**, 1991, pp. 1394–1413.MathSciNetzbMATHCrossRefGoogle Scholar - [8]Subrahmanyam, M. B., “Optimal disturbance rejection and performance robustness in linear systems,”
*Journal of Mathematical Analysis and Applications*,**164**, 1992, pp. 130–150.MathSciNetzbMATHCrossRefGoogle Scholar - [9]Subrahmanyam, M. B.,
*Optimal Control with a Worst-Case Performance Criterion and Applications*, Lecture Notes in Control and Information Sciences, No. 145, Springer-Verlag, Berlin, 1990.Google Scholar - [10]Subrahmanyam, M. B., “Synthesis of finite-interval H∞ controllers by state-space methods,
*” AIAA Journal of Guidance, Control, and Dynamics*,**13**, 1990, pp. 624 - 629.MathSciNetzbMATHCrossRefGoogle Scholar - [11]Subrahmanyam, M. B., “Worst-case performance measures for linear control problems,”
*Proc. IEEE Conference on Decision and Control*, Honolulu, U.S.A., 1990, pp. 2439–2443.Google Scholar - [12]Subrahmanyam, M. B. and Steinberg, M., “Model reduction with a finite-interval
*H*_{∞}criterion,”*Proc. AIAA Guidance, Navigation and Control Conference*, Portland, U.S.A., 1990, pp. 1419–1427.Google Scholar - [13]Tadmor, G., “Worst-case design in the time domain: the maximum principle and the standard H∞ problem,”
*Mathematics of Control, Signals, and Systems*,**3**, 1990, pp. 301–324.MathSciNetzbMATHCrossRefGoogle Scholar