In this note, we study the problem of output-feedback control design for a class of strict feedback stochastic nonlinear systems. Under an infinite-horizon risk-sensitive cost criterion, the controller designed can guarantee an arbitrary small long-term average cost for arbitrary risk-sensitivity parameter and achieve boundedness in probability for the closed-loop system, using the integrator backstepping methodology. Furthermore, the controller preserves the equilibrium at the origin of the nonlinear system.

研究了一类严格反馈随机非线性系统的输出反馈设计问题。在无限时区风险灵敏度指标下，应用积分反推（integratorbackstepping）技术，设计了控制器。所设计的控制器能够保障对任意风险灵敏度系数具有任意小的指标，并且闭环系统为概率意义下有界的。特别地，所设计的控制器还能保证控制器的平衡条件。仿真例子验证了理论结果的正确性。

考虑了一类随机非线性系统的鲁棒自适应控制问题。采用Itod随机微分方程描述系统，进而在概率意义下研究系统的鲁棒稳定性。应用积分反推（backstepping）方法，系统地给出了设计状态反馈及输出反馈鲁棒自适应控制器的方法。同时构造出了适当形式的四次型的自适应控制Lyapunov函数（CLF）。

This paper addresses the design problem of practical (or satisfaction) output-feedback controls for stochastic strict-feedback nonlinear systems in observer canonical form with stable zerodynamics under long-term average tracking risk-sensitive cost criteria. The cost function adopted here is of the quadratic-integral type usually encountered in practice, rather than the quartic-integral one used to avoid difficulty in control design and performance analysis of the closed-loop system. A sequence of coordinate diffeomorphisms is introduced to separate the zero-dynamics from the entire system, so that the transformed system has an appropriate form suitable for integrator backstepping design. For any given risk-sensitivity parameter and desired cost value, by using the integrator backstepping methodology, an output-feedback control is constructively designed such that (a) the closed-loop system is bounded in probability and (b) the long-term average risk-sensitive cost is upper bounded by the desired value. In addition, this paper does not require the uniform boundedness of the gain functions of the system noise. Furthermore, an example is given to show the effectiveness of the theory.

In this paper, we investigate the stabilization control design problem of nonlinear stochastic SISO systems in strictfeedback form. By introducing a novel reduced-order observer, an output-feedback-based control is constructively designed, which renders the closed-loop system asymptotically stable in the large when the nonlinearities and stochastic disturbance equal zero at the equilibrium point of the open-loop system, and bounded in probability, otherwise. Besides, the obtained controller preserves the equilibrium point of the open-loop nonlinear system.