Stabilization of stochastic nonlinear systems driven by noise of unknown covariance
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- 1 August 2001
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Automatic Control
- Vol. 46 (8) , 1237-1253
- https://doi.org/10.1109/9.940927
Abstract
This paper poses and solves a new problem of stochastic (nonlinear) disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise. This is a natural stochastic counterpart of the problem of input-to-state stabilization in the sense of Sontag (1989). Our development starts with a set of new global stochastic Lyapunov theorems. For an exemplary class of stochastic strict-feedback systems with vanishing nonlinearities, where the equilibrium is preserved in the presence of noise, we develop an adaptive stabilization scheme (based on tuning functions) that requires no a priori knowledge of a bound on the covariance. Next, we introduce a control Lyapunov function formula for stochastic disturbance attenuation. Finally, we address optimality and solve a differential game problem with the control and the noise covariance as opposing players; for strict-feedback systems the resulting Isaacs equation has a closed-form solution.Keywords
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