Abstract
The notion of stochastic optimal control as currently defined has its roots in statistical methods for dealing with certain tracking and signal estimation problems arising from the existence of uncertainties inherent either in the measurement or in the excitation that drives the evolution of systems, which involve prediction, filtering, and data smoothing. The pioneering work on these problems was done by the mathematician Wiener, who is accredited as the founder of control theory (Wiener 1949). A large number of research efforts were devoted to estimation problems of practical interest in electronics, communications and control engineering. An important attempt was the filtering and prediction theory by Kalman and Bucy in the early 1960s (Bucy and Kalman 1961). Almost in the same period, the introduction of the state-space method (Kalman 1960a, b), the developments of the stochastic maximum principle (Kushner 1962), and the stochastic dynamic programming (Florentin 1961) in the context of Itô calculus received great attention. The stochastic optimal control theorem was then developed into a rather integrated system in the early 1970s (Åström 1970). Thereafter, the duality methods, as a major branch of the stochastic optimal control theory, also known as the Martingale approach, have been paid extensive attention in recent years because they offered powerful tools for the study of some classes of stochastic optimal control problems (Josa-Fombellida and Rincón-Zapatero 2007).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Åström KJ (1970) Introduction to stochastic control theory. Academic Press, New York
Athans M, Falb P (1966) Optimal control: an introduction to the theory and its applications. McGraw Hill, New York
Bani-Hani KA, Alawneh MR (2007) Prestressed active post-tensioned tendons control for bridges under moving loads. Struct Control Health Monit 14:357–383
Bucy RS, Kalman RE (1961) New results in linear filtering and prediction theory. ASME Trans J Basic Eng 83:95–108
Chang CC, Yu LO (1998) A simple optimal pole location technique for structural control. Eng Struct 20(9):792–804
Chatfield C (1989) The analysis of time series-an introduction, 4th edn. Chapman and Hall, London
Chen JB, Li J (2008) Strategy for selecting representative points via tangent spheres in the probability density evolution method. Int J Numer Meth Eng 74(13):1988–2014
Chen JB, Liu WQ, Peng YB, Li J (2007) Stochastic seismic response and reliability analysis of base-isolated structures. J Earthq Eng 11(6):903–924.
Chen SP, Li XJ, Zhou XY (1998) Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J Control Optim 36:1685–1702
Chung LL, Reinhorn AM, Soong TT (1988) Experiments on active control of seismic structures. ASCE J Eng Mech 114(2):241–256
Crandall SH (1958) Random vibration. Technology Press of MIT, Wiley, New York
Florentin JJ (1961) Optimal control of continuous time, Markov, stochastic systems. J Electron Control 10:473–488
Ho CC, Ma CK (2007) Active vibration control of structural systems by a combination of the linear quadratic Gaussian and input estimation approaches. J Sound Vib 301:429–449
Josa-Fombellida R, Rincón-Zapatero JP (2007) New approach to stochastic optimal control. J Optimiz Theory and App 135(1):163–177
Kalman RE (1960a) On the general theory of control systems. In: Proceedings of 1st IFAC Moscow congress. Butterworth Scientific Publications
Kalman RE (1960b) A new approach to linear filtering and prediction problems. ASME Trans J Basic Eng 82:35–45
Kohiyama M, Yoshida M (2014) LQG design scheme for multiple vibration controllers in a data center facility. Earthq Struct 6(3):281–300
Kushner HJ (1962) Optimal stochastic control. IRE Trans Autom Control AC-7:120–122
Li J, Ai XQ (2006) Study on random model of earthquake ground motion based on physical process. Earthq Eng Eng Vib 26(5):21–26 (in Chinese)
Li J, Chen JB (2008) The principle of preservation of probability and the generalized density evolution equation. Struct Saf 30:65–77
Li J, Chen JB (2009) Stochastic dynamics of structures. Wiley, Singapore
Li J, Chen JB, Fan WL (2007) The equivalent extreme-value event and evaluation of the structural system reliability. Struct Saf 29(2):112–131
Li J, Liu ZJ (2006) Expansion method of stochastic processes based on normalized orthogonal bases. J Tongji Univ (Nat Sci) 34(10):1279–1283 (in Chinese)
Li J, Peng YB, Chen JB (2010) A physical approach to structural stochastic optimal controls. Probabilistic Eng Mech 25(1):127–141
Lin YK, Cai GQ (1995) Probabilistic structural dynamics: advanced theory and applications. McGraw-Hill, New York
Mathews JH Fink KD (2003) Numerical methods Using Matlab, 4th edn. Prentice-Hall
Øksendal B (2005) Stochastic differential equations: An introduction with applications, 6th edn, Springer-Verlag, Berlin
Roberts JB, Spanos PD (1990) Random vibration and statistical linearization. Wiley, West Sussex
Soong TT (1990) Active structural control: theory and practice. Longman Scientific & Technical, New York
Stengel RF (1986) Stochastic optimal control: theory and application. Wiley, New York
Stengel RF, Ray LR, Marrison CI (1992) Probabilistic evaluation of control system robustness. In: IMA workshop on control systems design for advanced engineering systems: complexity, uncertainty, information and organization, Minneapolis, MN
Sun JQ (2006) Stochastic dynamics and control. Elsevier, Amsterdam
Wiener N (1949) Extrapolation, interpolation and smoothing of stationary time series, with engineering applications. The MIT Press, Cambridge
Wiener N (1964) Time series. The MIT Press, Cambridge
Yang JN (1975) Application of optimal control theory to civil engineering structures. ASCE J Eng Mech Div 101(EM6):819–838
Yang JN, Akbarpour A, Ghaemmaghami P (1987) New optimal control algorithms for structural control. ASCE J Eng Mech 113(9):1369–1386
Yang JN, Li Z, Vongchavalitkul S (1994) Generalization of optimal control theory: linear and nonlinear control. ASCE J Eng Mech 120(2):266–283
Zhang WS, Xu YL (2001) Closed form solution for along-wind response of actively controlled tall buildings with LQG controllers. J Wind Eng Ind Aerodyn 89:785–807
Zhu WQ (2006) Nonlinear stochastic dynamics and control in Hamiltonian formulation. ASME Trans 59:230–248
Zhu WQ, Ying ZG, Soong TT (2001) An optimal nonlinear feedback control strategy for randomly excited structural systems. Nonlinear Dyn 24:31–51
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd. and Shanghai Scientific and Technical Publishers
About this chapter
Cite this chapter
Peng, Y., Li, J. (2019). Physically Based Stochastic Optimal Control. In: Stochastic Optimal Control of Structures. Springer, Singapore. https://doi.org/10.1007/978-981-13-6764-9_3
Download citation
DOI: https://doi.org/10.1007/978-981-13-6764-9_3
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-6763-2
Online ISBN: 978-981-13-6764-9
eBook Packages: EngineeringEngineering (R0)