Finding the Minimum of H Infinity Norm in Novel State Derivative Space Form

Abstract:

H-infinity norm is the maximum system gain and is related to the system output response. Finding the minimum of H-infinity norm is essential in control design to avoid overshoot or saturation in implementation. The fundamental of finding the minimum of H-infinity norm is based on optimal control. There are three problems that people usually have to face when they carry out optimal control designs in state space system form or descriptor system form. They are no state derivative considered in cost functional, no well-developed state derivative feedback control algorithms and some systems cannot easily design optimal control in these two system forms. To overcome the problems, a novel state derivative space system (SDS) form is introduced. In this novel form, novel differential Lagrange multiplier should be used to adjoin SDS system constraint to the cost functional that is function of state derivative only to straightforwardly carry out optimal control design. Based on the optimal control design in SDS form, the process of finding the minimum H-infinity norm in SDS form is successfully developed and verified in this paper. That process in SDS form is actually as straightforward as its counterpart in state space form. Consequently, wider ranges of problems can be solved without much of mathematics overhead.

• Periodical: Advanced Materials Research (Volume 740)
• Pages: 45-50
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.740.45
• Online since: August 2013
• Authors: Yuan Wei Tseng
• Keywords: Descriptor System, Differential Lagrange Multiplier, H Infinity Norm, Optimal Control, Reciprocal State Space System, State Derivative Space System

References

[1] Dan Simon, Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches, Wiley, 2006.

[2] L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze, E. F. Mischenko, The Mathematical Theory of Optimal Processes. New York, NY: Wiley, ISBN 68981, 1962.

[3] Yuan-Wei Tseng, "Optimal Control in Novel State Derivative Space System", IEEE Conference on Industrial Electronics and Application (ICIEA 2012), Page(s): 371 – 376, Singapore, July, 2012.

DOI: 10.1109/iciea.2012.6360755

[4] R. W. Newcomb, "The Semistate description of nonlinear time variable circuits," IEEE Control Circuits System, CAS-28(1): pp.28-33, 1981.

DOI: 10.1109/tcs.1981.1084908

[5] D. G. Luenberger, "Dynamic equations in descriptor form," IEEE Transactions on Automatic Control, vol. AC-22(3), pp.312-321, 1977.

DOI: 10.1109/tac.1977.1101502

[6] F. B. Yeh and H. N. Huang, "H infinity state feedback control of smart beam-plates via the descriptor system approach". http://www2.thu.edu.tw/~sci/uploads/tadnews/000202.pdf

[7] D. Cobb, "State feedback impulse elimination for singular systems over a Hermite domain," SIAM J. Control and Optimization, vol. 44(6): pp.2189-2209, 2006.

DOI: 10.1137/040618515

[8] D. Liu, G. Zhang and Y. Xie , "Guaranteed cost control for a class of descriptor systems with uncertainties," International Journal of Information and Systems, vol. 5, no. 3-4, pp.430-435, 2009.

[9] D. Cobb, "Eigenvalue conditions for convergence of singularly perturbed matrix exponential functions," SIAM J. Control and Optimization, vol. 48(7), pp.4327-4351, 2010.

DOI: 10.1137/09075113x

[10] H. T. Yang, F. B. Yeh, Linear and Nonlinear control: Game Theoretic Approach, Chuan Hwa Book Co., Ltd, Taipei, Taiwan, 1997.

[11] J. C. Doyle, Lecture Notes in Advances in Multivariable Control ONR/Honeywell Workshop, Minneapolis, MN, 1984.