Stabilization of stochastic nonlinear systems driven by noise of unknown covariance
H. Deng, M. Krstic, R. J. Williams
This paper poses and solves a new problem of stochastic (nonlinear)
disturbance attenuation where the task is to make
the system solution bounded (in expectation, with appropriate nonlinear
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. 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
(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.
In IEEE Transactions on Automatic Control, 46 (2001), 1237-1253.
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