Vol. 46 No. 2 SCIENCE IN CHINA (Series F) April 2003 Design of satisfaction output feedback controls for stochastic nonlinear systems under quadratic tracking risk-sensitive index LIU Yungang ( ) 1 , ZHANG Jifeng ( ) 1 & PAN Zigang ( ) 2 1. Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China; 2. Dept. of Electrical and Computer Engineering and Computer Science, Univ. of Cincinnati, USA Correspondence should be addressed to Zhang Jifeng (jif@control.iss.ac.cn) Received June 17, 2002 Abstract In this paper, the design problem of satisfaction output feedback controls for stochastic nonlinear systems in strict feedback form under long-term tracking risk-sensitive index is investigated. The index function adopted here is of quadratic form usually encountered in practice, rather than of quartic one used to beg the essential difficulty on controller design and performance analysis of the closed-loop systems. For any given risk-sensitive parameter and desired index value, by using the integrator backstepping method, an output feedback control is constructively designed so that the closed-loop system is bounded in probability and the risk-sensitive index is upper bounded by the desired value. Keywords: integrator backstepping, nonlinear system, stochastic disturbance, risk-sensitive in- dex, output feedback. The design of global stabilization controls for nonlinear systems has been a research topic under intensive investigation. After the celebrated characterization of the feedback linearizable systems [1] , a breakthrough is achieved with the introduction of the integrator backstepping de- sign methodology [2] , which provides a general constructive tool for designing global stabilization controls for nonlinear systems in or feedback equivalent to strict-feedback (SF) form. Since the early 1990s, a series of research results on SF systems have been obtained (e.g. refs. [3—10] and the references therein). Stochastic risk-sensitive (RS) control is more general than H ∞ and H 2 control. It is closely related to differential game problems [11-19] . For example, when disturbance vanishes, the large deviation limit of the stochastic RS control is nothing but a deterministic differential game problem. These connections have stimulated the research on stochastic RS controls. A big progress on control design of SF stochastic nonlinear systems has been made recently [20-22] . Under the assumption that the disturbance vector field vanishes at the origin, refs. [20, 21] study the problem of designing a control to asymptotically stabilize the closed-loop systems in probability. Ref. [20] considers full state-feedback control design, while ref. [21] considers output-feedback control design. In ref. [22], with a quadratic regulation RS index, design of satisfaction state-feedback control is studied, and the assumption that the disturbance