×
Fleming, W. H.; James, M. R.

The risk-sensitive index and the \(H_ 2\) and \(H_ \infty\) norms for nonlinear systems. (English) Zbl 0854.93045

Math. Control Signals Syst. 8, No. 3, 199-221 (1995).
This paper deals with \(H_\infty\) and \(H_2\) norms which are important tools for determining the influence of disturbances on the performance. The authors introduce the \(H_\infty\) and \(H_2\) norms and the risk-sensitive index for nonlinear system, as a natural generalization of the linear case, and they investigate the relationship between them. In particular, they show that the \(H_\infty\) (resp. \(H_2)\) norm is the small noise (resp. risk) limit of the risk-sensitive index and they compute them using directional derivatives of the risk-sensitivity index.
Reviewer: M.Nisio (Osaka)

Cited in 15 Documents

MSC:

93B36 \(H^\infty\)-control
93C10 Nonlinear systems in control theory
93E20 Optimal stochastic control

Keywords:

asymptotic analysis; \(H_ 2\) norms; \(H_ \infty\) norms; risk-sensitive; nonlinear
Cite Review PDF
Full Text: DOI

References:

[1] M. D. Donsker and S. R. S. Varadhan, Asymptotic Evaluation of Certain Markov Process Expectations for Large Time, I, II, III,Comm. Pure Appl. Math.,28 (1975), 1-45, 279-301;29 (1976), 389-461. · Zbl 0323.60069 · doi:10.1002/cpa.3160280102
[2] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, State-Space Solutions to StandardH 2 and Ht8 Control Problems,IEEE Trans. Automat. Control,34(8) (1989), 831-847. · Zbl 0698.93031 · doi:10.1109/9.29425
[3] W. H. Fleming and W. M. McEneaney, Risk Sensitive Optimal Control and Differential Games,Proc. Conf. on Adaptive and Stochastic Control, University of Kansas, Springer-Verlag, New York, 1991. · Zbl 0788.90097
[4] W. H. Fleming and W. H. McEneaney, Risk Sensitive Control on an Infinite Time Horizon,SIAM J. Control Optim.,33 (1995), 1881-1915. · Zbl 0949.93079 · doi:10.1137/S0363012993258720
[5] W. H. Fleming and R. W. Rishel,Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975. · Zbl 0323.49001
[6] W. H. Fleming and H. M. Soner,Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993. · Zbl 0773.60070
[7] K. Glover and J. C. Doyle, State Space Formulae for all Stabilizing Controllers that Satisfy an H ? Norm Bound and Relations to Risk Sensitivity,Systems Control Lett.,11 (1988), 167-172. · Zbl 0671.93029 · doi:10.1016/0167-6911(88)90055-2
[8] M. Green and D. Limebeer,Linear Robust Control, Prentice-Hall, Englewood Cliffs, NJ, 1994. · Zbl 0951.93500
[9] D. J. Hill and P. J. Moylan, The Stability of Nonlinear Dissipative Systems,IEEE Trans. Automat. Control,21 (1976), 708-711. · Zbl 0339.93014 · doi:10.1109/TAC.1976.1101352
[10] D. H. Jacobson, Optimal Stochastic Linear Systems with Exponential Performance Criteria and Their Relation to Deterministic Differential Games,IEEE Trans. Automat. Control,18(2) (1973), 124-131. · Zbl 0274.93067 · doi:10.1109/TAC.1973.1100265
[11] M. R. James, Asymptotic Analysis of Nonlinear Stochastic Risk-Sensitive Control and Differential Games,Math. Control Signals Systems,5(4) (1992), 401-417. · Zbl 0761.93076 · doi:10.1007/BF02134013
[12] M. R. James, A Partial Differential Inequality for Dissipative Nonlinear Systems,Systems Control Lett.,21 (1993), 315-320. · Zbl 0793.93064 · doi:10.1016/0167-6911(93)90074-G
[13] M. R. James and S. Yuliar, Numerical Approximation of theH ?, Norm for Nonlinear Systems,Automatica,31(8) (1995), 1075-1086. · Zbl 0838.93028 · doi:10.1016/0005-1098(94)00161-B
[14] P. R. Kumar and P. Varaiya,Stochastic Systems: Estimation, Identification, and Adaptive Control, Prentice-Hall, Englewood Cliffs, NJ, 1986. · Zbl 0706.93057
[15] W. M. McEneaney, Uniqueness for Viscosity Solutions of Nonstationary HJB Equations under somea priori Conditions (with Applications),SIAM J. Control Optim.,33 (1995), 1560-1576. · Zbl 0833.49017 · doi:10.1137/S0363012994262695
[16] W. M. McEneaney, Robust Control and Differential Games on a Finite Time Horizon,Math. Control Signals Systems,8 (1995), 138-166. · Zbl 0849.90139 · doi:10.1007/BF01210205
[17] T. Runolfsson, The Equivalence Between Infinite Horizon Control of Stochastic Systems with Exponential-of-Integral Performance Index and Stochastic Differential Games, Technical Report JHU/ECE 91-07, Johns Hopkins University, 1991. · Zbl 0930.93084
[18] P. Whittle, Risk-Sensitive Linear/Quadratic/Gaussian Control,Adv. in Appl. Probab.,13 (1981), 764-777. · Zbl 0489.93067 · doi:10.2307/1426972
[19] P. Whittle, A Risk-Sensitive Maximum Principle,Systems Control Lett.,15 (1990), 183-192. · Zbl 0724.93084 · doi:10.1016/0167-6911(90)90110-G
[20] P. Whittle,Risk-Sensitive Optimal Control, Wiley, New York, 1990. · Zbl 0718.93068
[21] J. C. Willems, Dissipative Dynamical Systems, Part I: General Theory,Arch. Rational Mech. Anal,45 (1972), 321-351. · Zbl 0252.93002 · doi:10.1007/BF00276493
[22] W. M. Wonham, A Liapunov Method for Estimation of Statistical Averages,J. Differential Equations,2 (1966), 365-377. · Zbl 0168.15903 · doi:10.1016/0022-0396(66)90047-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.