Journal of Systems Science and Complexity

, Volume 28, Issue 4, pp 813–829 | Cite as

State and output feedback finite-time guaranteed cost control of linear itô stochastic systems

  • Zhiguo YanEmail author
  • Guoshan Zhang
  • Jiankui Wang
  • Weihai Zhang


The problem of guaranteed cost control based on finite-time stability for stochastic system is first investigated in this paper. The motivation of solving this problem arises from an observation that finite/infinite-horizon guaranteed cost control does not consider the transient performance of the closed-loop system, but guaranteed cost control based on finite-time stability involves this practical requirement. In order to explain this problem explicitly, a concept of the stochastic finite-time guaranteed cost control is introduced, and then some new sufficient conditions for the existence of state and output feedback finite-time guaranteed cost controllers are derived, which guarantee finite-times to chastic stability of closed-loop systems and an upper bound of a quadratic cost function. Furthermore, this problem is reduced to a convex optimization problem with matrix inequality constraints and a new solving algorithm is given. Finally, an example is given to illustrate the effectiveness of the proposed method.


Finite-time stability guaranteed cost control matrix inequalities stochastic systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Petersen I R, Ugrinovskii V A, and Savkii A V, Robust Control Design Using H 8 Methods, Springer-Verlag, New York, 2000, 125–184.Google Scholar
  2. [2]
    Mukaidani H, The guaranteed cost control for uncertain nonlinear large-scale stochastic systems via state and static output feedback, Journal of Mathematical Analysis and Applications, 2009, 359: 527–535.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Mukaidani H, Robust guaranteed cost control for uncertain stochastic systems with multiple decision makers, Automatica, 2009, 45(7): 1758–1764zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Zhang J, Shi P, and Qiu J, Non-fragile guaranteed cost control for uncertain stochastic nonlinear time-delay systems, Journal of the Franklin Institute, 2009, 346: 676–690.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Petersen I R, Robust output feedback guaranteed cost control of nonlinear stochastic uncertain systems via an IQC approach, IEEE Trans. on Automatic Control, 2009, 54(6): 1299–1304.CrossRefGoogle Scholar
  6. [6]
    Petersen I R, Guaranteed cost control of stochastic uncertain systems with slope bounded nonlinearities via the use of dynamic multipliers, Automatica, 2011, 47(2): 411–417.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Langton R, Stability and Control of Aircraft Systems: Introduction to Classical Feedback Control, John Wiley and Sons, Ltd., 2006.CrossRefGoogle Scholar
  8. [8]
    Zivanovic M D and Vukobratovic M K, Multi-Arm Cooperating Robots: tiDynamics and Control, 1st edition, Spring-Verlag, 2006.Google Scholar
  9. [9]
    Kamenkov G, On stability of motion over a finite interval of time, Journal of Applied Mathematics and Mechanics, 1953, 17: 529–540.zbMATHMathSciNetGoogle Scholar
  10. [10]
    Lebedev A, The problem of stability in a finite interval of time, Journal of Applied Mathematics and Mechanics, 1954, 18: 75–94.zbMATHGoogle Scholar
  11. [11]
    Lebedev A, On stability of motion during a given interval of time, Journal of Applied Mathematics and Mechanics, 1954, 18: 139–148.zbMATHGoogle Scholar
  12. [12]
    Dorato P, Short time stability in linear time-varying systems, Proceeding of IRE International Convention Record, 1961, 4: 83–87.Google Scholar
  13. [13]
    Weiss L and Infante E F, Finite time stability under perturbing forces and on product spaces, IEEE Trans. on Automatic Control, 1967, 12(1): 54–59.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Amato F, Ariola M, and Dorato P, Finite time control of linear system subject to parametric uncertainties and disturbances, Automatica, 2001, 37(9): 1459–1463.zbMATHCrossRefGoogle Scholar
  15. [15]
    Amato F, Ariola M, and Cosentino C, Finite-time stabilization via dynamic output feedback, Automatica, 2006, 42(2): 337–342.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Amato F and Ariola M, Finite-time control of discrete-time linear systems, IEEE Trans. on Automatic Control, 2005, 50(5): 724–729.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Zhang W and An X, Finite-time control of linear stochastic systems, International Journal of Innovative Computing, Information and Control, 2008, 4: 687–694.Google Scholar
  18. [18]
    Yan Z, Zhang G, and Zhang W, Finite-time stability and stabilization of linear ItÔ stochastic systems with state and control-dependent noise, Asian Journal of Control, 2013, 15(1): 270–281.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Yan Z and Zhang G, Finite-time H8 control for linear stochastic systems, Control and Decision, 2011, 26(8): 1224–1228.MathSciNetGoogle Scholar
  20. [20]
    Yan Z and Zhang G, Finite-time H8 filtering for a class of nonlinear stochastic uncertain systems, Control and Decision, 2012, 29(3): 419–424.Google Scholar
  21. [21]
    Feng J, Wu Z, and Sun J, Finite-time control of linear singular systems with parametric uncertainties and disturbances, Acta Automatica Sinica, 2005, 31(4): 634–637.MathSciNetGoogle Scholar
  22. [22]
    Hasminskii R Z, Stochastic Stability of Differential Equations, Sijtjoff and Nordhoff, Alphen, 1980.CrossRefGoogle Scholar
  23. [23]
    Zhang Q and Zhang W, Properties of storage functions and applications to nonlinear stochastic H8 control, Journal of Systems Science and Complexity, 2011, 24(5): 850–861.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    Bhat S P and Bernstein D S, Finite-time stability of continuous autonomous systems, SIAM Journal on Control and Optimization, 2000, 38(3): 751–766.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    Hong Y, Wang J, and Cheng D, Adaptive finite time control of nonlinear systems with parametric uncertainty, IEEE Trans. on Automatic Control, 2006, 51(5): 858–862.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Yin J, Khoo S, Man Z, and Yu X, Finite-time stability and instability of stochastic nonlinear systems, Automatica, 2011, 47(12): 2671–2677.zbMATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    Oksendal B, Stochastic Differential Equations: An Introduction with Applications, Fifth edition, Springer-Verlag, New York, 2000.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Zhiguo Yan
    • 1
    • 2
    Email author
  • Guoshan Zhang
    • 2
  • Jiankui Wang
    • 2
  • Weihai Zhang
    • 3
  1. 1.School of Electrical Engineering and AutomationQilu University of TechnologyJinanChina
  2. 2.School of Electrical Engineering and AutomationTianjin UniversityTianjinChina
  3. 3.College of Electrical Engineering and AutomationShandong University of Science and TechnologyQingdaoChina

Personalised recommendations