  This paper presents a new paradigm to stabilize uncertain stochastic linear
systems. Herein, second moment polytopic (SMP) systems are proposed that
generalize systems with both uncertainty and randomness. The SMP systems are
characterized by second moments of the stochastic system matrices and the
uncertain parameters. Further, a fundamental theory for guaranteeing stability
of the SMP systems is established. It is challenging to analyze the SMP systems
owing to both the uncertainty and randomness. An idea to overcome this
difficulty is to expand the SMP systems and exclude the randomness. Because the
expanded systems contain only the uncertainty, their stability can be analyzed
via robust stability theory. The stability of the expanded systems is
equivalent to statistical stability of the SMP systems. These facts provide
sufficient conditions for the stability of the SMP systems as linear matrix
inequalities (MIs). In controller design for the SMP systems, the linear MIs
reduce to cubic MIs whose solutions correspond to feedback gains. The cubic MIs
are transformed into simpler quadratic MIs that can be solved using
optimization techniques. Moreover, solving such non-convex MIs is relaxed into
the iteration of a convex optimization. Solutions to the iterative optimization
provide feedback gains that stabilize the SMP systems. As demonstrated here,
the SMP systems represent linear dynamics with uncertain mean and covariance
and other existing systems such as independently identically distributed
dynamics and random polytopes. Finally, a numerical simulation shows the
effectiveness of the proposed method.
