An algebraic criterion of absolute stochastic stability of nonlinear discrete-time systems with multiplicative white noise (Q1119238)

The author considers the nonlinear discrete-time stochastic system \[ x(k+1)=Ax(k)+Bf(y(k))+\sum^{N}_{i=1}Q_ ix(k)v_ i(k)+\sum^{m}_{i=1}g_ ih^ T_ if(y(k))w_ i(k) \] \[ y(k)=- Cx(k),k=0,1,... \] where x(k) is an n-dimensional state vector; y(k) is an m-dimensional output vector of the system; f(y) is a nonlinear m- dimensional vector function, \(v_ i(k)\) and \(w_ i(k)\) are independent discrete white noise with unit variances; A,B,C and \(Q_ i\) are constant matrices, \(g_ i\), \(h_ i\) are constant vectors. It is assumed that the matrix A is stable, that the components of the vector f are \(f_ i(y)=f_ i(y_ i),f_ i(0)=0,i=1,2,...,m\), and that they satisfy the constraints \(0\leq f_ i(y_ i)y_ i\leq K_ iy^ 2_ i,i=1,2,...,m\). In the first part of the paper the author has derived sufficient conditions of exponential stability in the mean-square for the system expressed in algebraic form. In the second part an example of the analysis of a digital system taking into account rounding noise has been presented.