On mean square boundedness of stochastic linear systems with bounded controls

Abstract

We start by summarizing the state of the art in stabilization of stochastic linear systems with bounded inputs and highlight remaining open problems. We then report two new results concerning mean-square boundedness of a linear system with additive stochastic noise. The first states that, given any nonzero bound on the controls, it is possible to construct a policy with bounded memory requirements that renders a marginally stable stabilizable system mean-square bounded in closed-loop. The second states that it is not possible to ensure mean-square boundedness in closed-loop with a bounded control policy for systems affected by unbounded noise and having at least one eigenvalue outside the unit circle

Stabilizing Stochastic Predictive Control Under Bernoulli Dropouts

“…To deal with stochastic systems, we shall impose a constraint in the underlying optimal control problem such that it is (globally) feasible and the closed-loop system is stable in a precise sense. Such constraint embedding for stability has appeared before in the literature concerning deterministic model predictive control in [18] and stochastic model predictive control in [21], [22]. The control channel is modelled as an erasure channel with dropouts occurring according to an i.i.d.…”

Stochastic Model Predictive Control: An Overview and Perspectives for Future Research

“…Lyapunov stability of the matrix A implies that all eigenvalues of A lie in the closed unit disc, and those with magnitude one have equal algebraic and geometric multiplicities (see [80] for a detailed analysis on mean-square boundedness of stochastic linear systems). In [51], the stochastic OCP is augmented with a negative drift condition defined in terms of a stability constraint to render the states of the closed-loop system mean-square bounded.…”

Resource efficient stochastic predictive control under packet dropouts

“…(R1) It is known that linear systems with bounded control cannot be globally stabilised if the system matrix has eigenvalues outside the unit disk: see [24,27] for the corresponding results in the deterministic and stochastic settings, respectively. Therefore, the assumption of Lyapunov stable A is essential for us.…”

“…(R2) The fourth moment bound on the noise is less restrictive than the standard assumption of i.i.d. Gaussian; see [24] for further discussions. Stability results discussed in Section 5 are valid if…”