This paper investigates a delay-dependent robust control problem of discrete-time uncertain stochastic systems with delays. The uncertainty considered in this paper is time-varying but norm-bounded, and the delays are considered as interval time-varying case for both state and input. According to the considerations of uncertainty, stochastic behavior, and time delays, the problem considered in this paper is more general than the existing works for uncertain stochastic systems. Via the proposed Lyapunov–Krasovskii function, some sufficient conditions are derived into the extended linear matrix inequality form. Moreover, Jensen inequality and free matrix equation are employed to reduce conservatism of those conditions. Through using the proposed design method, a gain-scheduled controller is designed to guarantee asymptotical stability of uncertain stochastic systems in the sense of mean square. Finally, two numerical examples are provided to demonstrate applicability and effectiveness of the proposed design method.
Issue Section:
Research Papers
References
1.
Cerone
, V.
, and Regruto
, D.
, 2008
, “Set-Membership Identification of LPV Models With Uncertain Measurements of the Time-Varying Parameter
,” 47th IEEE Conference on Decision and Control
(CDC
), Cancun, Mexico, Dec. 9–11, pp. 4491
–4496
.2.
Lee
, J.-W.
, 2006
, “On Uniform Stabilization of Discrete-Time Linear Parameter-Varying Control Systems
,” IEEE Trans. Autom. Control
, 51
(10
), pp. 1714
–1721
.3.
Chang
, W. J.
, Chen
, M. W.
, and Ku
, C. C.
, 2013
, “Passive Fuzzy Controller Design for Discrete Ship Steering Systems Via Takagi-Sugeno Fuzzy Model With Multiplicative Noises
,” J. Mar. Sci. Technol.
, 21
(2
), pp. 159
–165
.4.
Rodrigues
, M.
, Hamdi
, H.
, Graiek
, N. B. H.
, and Theilliol
, D.
, 2014
, “Observer-Based Fault Tolerant Control Design for a Class of LPV Descriptor Systems
,” J. Franklin Inst.
, 351
(6
), pp. 3104
–3125
.5.
Qin
, W.
, and Wang
, Q.
, 2007
, “Modeling and Control Design for Performance Management of Web Servers Via an LPV Approach
,” IEEE Trans. Control Syst. Technol.
, 15
(2
), pp. 259
–275
.6.
Prempain
, E.
, Postlethwaite
, I.
, and Benchaib
, A.
, 2002
, “A Linear Parameter Variant H∞ Control Design for an Induction Motor
,” Control Eng. Pract.
, 10
(6
), pp. 633
–644
.7.
Qiu
, W.
, Vittal
, V.
, and Khammash
, M.
, 2004
, “Decentralized Power System Stabilizer Design Using Linear Parameter Varying Approach
,” IEEE Trans. Power Syst.
, 19
(4
), pp. 1951
–1960
.8.
Oliveira
, M. C.
, Bernussou
, J.
, and Geromel
, J. C.
, 1999
, “A New Discrete-Time Robust Stability Condition
,” Syst. Control Lett.
, 37
(4
), pp. 261
–265
.9.
Caigny
, J. D.
, Camino
, J. F.
, Oliveira
, R. C. L. F.
, Peres
, P. L. D.
, and Swevers
, J.
, 2010
, “Gain-Scheduled H2 and H∞ Control of Discrete-Time Polytopic Time-Varying Systems
,” IET Control Theory Appl.
, 4
(3
), pp. 362
–380
.10.
Yu
, J.
, and Sideris
, A.
, 1997
, “H∞ Control With Parametric Lyapunov Functions
,” Syst. Control Lett.
, 30
(2–3
), pp. 57
–69
.11.
Franze
, G.
, Famularo
, D.
, Garone
, E.
, and Casavola
, A.
, 2009
, “Dilated Model Predictive Control Strategy for Linear Parameter-Varying Systems With a Time-Varying Terminal Set
,” IET Control Theory Appl.
, 3
(1
), pp. 110
–120
.12.
Wu
, F.
, and Grigoriadis
, K. M.
, 2001
, “LPV Systems With Parameter-Varying Time Delays: Analysis and Control
,” Automatica
, 37
(2
), pp. 221
–229
.13.
Daafouz
, J.
, and Bernussou
, J.
, 2001
, “Parameter Dependent Lyapunov Functions for Discrete Time Systems With Time Varying Parametric Uncertainties
,” Syst. Control Lett.
, 43
(5
), pp. 355
–359
.14.
Ramo
, D. C. W.
, and Peres
, P. L. D.
, 2001
, “A Less Conservative LMI Condition for the Robust Stability of Discrete-Time Uncertain Systems
,” Syst. Control Lett.
, 43
(5
), pp. 371
–378
.15.
Tóth
, R.
, Heuberger
, P. S. C.
, and Van den Hof
, P. M. J.
, 2010
, “Discretisation of Linear Parameter-Varying State-Space Representations
,” IET Control Theory Appl.
, 4
(10
), pp. 2082
–2096
.16.
Baoyong
, Z.
, and Shengyuan
, X.
, 2009
, “Delay-Dependent Robust H∞ Control for Uncertain Discrete-Time Fuzzy Systems With Time-Varying Delays
,” IEEE Trans. Fuzzy Syst.
, 17
(4
), pp. 809
–823
.17.
Zhang
, L.
, Shi
, P.
, and Basin
, M.
, 2008
, “Robust Stability and Stabilisation of Uncertain Switched Linear Discrete Time-Delay Systems
,” IET Control Theory Appl.
, 2
(7
), pp. 606
–614
.18.
Yang
, R.
, Gao
, H.
, and Shi
, P.
, 2009
, “Novel Robust Stability Criteria for Stochastic Hopfield Neural Networks With Time Delays
,” IEEE Trans. Syst., Man, Cybern.-Part B
, 39
(2
), pp. 467
–474
.19.
Boukas
, E.-K.
, 2006
, “Discrete-Time Systems With Time-Varying Time Delay: Stability and Stabilizability
,” Math. Probl. Eng.
, 2006
, p. 42489
.20.
Ramezanifar
, A.
, Mohammadpour
, J.
, and Grigoriadis
, K. M.
, 2013
, “Sampled-Data Control of Linear Parameter Varying Time-Delay Systems Using State Feedback
,” American Control Conference
(ACC
), Washington, DC, June 17–19, pp. 6847
–6852
.21.
Wu
, L.
, Shi
, P.
, Wang
, C.
, and Gao
, H.
, 2006
, “Delay-Dependent Robust H∞ and L2-L∞ Filtering for LPV Systems With Both Discrete and Distributed Delays
,” IEE Proc. Control Theory Appl.
, 153
(4
), pp. 483
–492
.22.
Ramezanifar
, A.
, Mohammadpour
, J.
, and Grigoriadis
, K.
, 2012
, “Sampled-Data Control of LPV Systems Using Input Delay Approach
,” 51st IEEE Conference on Decision and Control
(CDC
), Maui, HI, Dec. 10–13, pp. 6303
–6308
.23.
Zhou
, S.
, and Zheng
, W. X.
, 2008
, “A Parameter-Dependent Lyapunov Function Based Approach to H∞-Control of LPV Discrete-Time Systems With Delays
,” 47th IEEE Conference on Decision and Control
(CDC
), Cancun, Mexico, Dec. 9–11, pp. 4669
–4674
.24.
Zhu
, X. L.
, and Yang
, G. H.
, 2008
, “Jensen Inequality Approach to Stability Analysis of Discrete-Time Systems With Time-Varying Delay
,” American Control Conference
(ACC
), Seattle, WA, June 11–13, pp. 1644
–1649
.25.
Ma
, S.
, and Boukas
, E. K.
, 2009
, “Stability and Robust Stabilisation for Uncertain Discrete Stochastic Hybrid Singular Systems With Time Delay
,” IET Control Theory Appl
, 3
(9
), pp. 1217
–1225
.26.
Moon
, Y. S.
, Park
, P.
, Kwon
, W. H.
, and Lee
, Y. S.
, 2001
, “Delay-Dependent Robust Stabilization of Uncertain State-Delayed Systems
,” Int. J. Control.
, 74
(14
), pp. 1447
–1455
.27.
Ghaoui
, L. E.
, 1995
, “State-Feedback Control of Systems With Multiplicative Noise Via Linear Matrix Inequalities
,” Syst. Control Lett.
, 24
(3
), pp. 223
–228
.28.
Eli
, G.
, Uri
, S.
, and Isaac
, Y.
, 2005
, H∞ Control and Estimation of State-Multiplicative Linear Systems
, Springer
, London
.29.
Gao
, H.
, Lam
, J.
, and Wang
, Z.
, 2007
, “Discrete Bilinear Stochastic Systems With Time-Varying Delay: Stability Analysis and Control Synthesis
,” Chaos, Solitons Fractals
, 34
(2
), pp. 394
–404
.30.
Xu
, H.
, and Jagannathan
, S.
, 2013
, “Stochastic Optimal Controller Design for Uncertain Nonlinear Networked Control System Via Neuro Dynamic Programming
,” IEEE Trans. Neural Networks Learn. Syst.
, 24
(3
), pp. 471
–484
.31.
Chang
, W. J.
, Jheng
, S. S.
, and Ku
, C. C.
, 2012
, “Fuzzy Control With Robust and Passive Properties for Discrete-Time Takagi–Sugeno Fuzzy Systems With Multiplicative Noises
,” Proc. Inst. Mech. Eng., Part I: J. Syst. Control Eng.
, 226
(4
), pp. 476
–485
.32.
Chang
, W. J.
, Jheng
, S. S.
, and Ku
, C. C.
, 2013
, “Passive Estimated State Feedback Fuzzy Controller Design for Discrete Perturbed Fuzzy Systems With Multiplicative Noises
,” J. Chin. Inst. Eng.
, 36
(6
), pp. 684
–695
.33.
Zhang
, X.
, Ho
, D. W. C.
, and Lu
, G.
, 2004
, “Robust Stabilization and State Estimation for Uncertain Stochastic Discrete-Delay Large-Scale Systems
,” 8th International Conference on Control
, Automation, Robotics and Vision (ICARCV
), Kunming, China, Dec. 6–9, Vol. 1
, pp. 42
–47
.34.
Li
, X.
, 2005
, “Robust H∞ Control for Uncertain Discrete-Time Delayed Stochastic Systems With Saturating Nonlinear Actuators
,” IEEE International Conference on Networking, Sensing and Control (ICNSC
), Tuscon, AZ, Mar. 19–22, pp. 844
–849
.35.
Chang
, W. J.
, Wu
, W. Y.
, and Ku
, C. C.
, 2009
, “Observer-Based H∞ Fuzzy Control Design for Discrete-Time Stochastic T-S Fuzzy Model With Multiplicative Noise
,” IEEE International Conference on Information and Automation
(ICIA
), Zhuhai, China, June 22–24, pp. 1345
–1350
.36.
Boyd
, S.
, Ghaoui
, L. E.
, Feron
, E.
, and Balakrishnan
, V.
, 1994
, Linear Matrix Inequalities in System and Control Theory
, SIAM
, Philadelphia, PA
.37.
Ghaoui
, L. E.
, Oustry
, F.
, and AitRami
, M.
, 1997
, “A Cone Complementarity Linearization Algorithm for Static Output-Feedback and Related Problems
,” IEEE Trans. Autom. Control
, 42
(8
), pp. 1171
–1176
.38.
Tanaka
, K.
, and Sano
, M.
, 1994
, “A Robust Stabilization Problem of Fuzzy Control Systems and Its Application to Backing up Control of a Truck-Trailer
,” IEEE Trans. Fuzzy Syst.
, 2
(2
), pp. 119
–134
.Copyright © 2017 by ASME
You do not currently have access to this content.
