Dissipativity, inverse optimal control, and stability margins for nonlinear discrete-time stochastic feedback regulators

In this paper, we derive stability margins for optimal and inverse optimal stochastic feedback regulators. Specifically, gain, sector, and disk margin guarantees are obtained for discrete-time nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal controllers that minimise a nonlinear-nonquadratic performance criterion. Furthermore, using the newly developed notion of stochastic dissipativity we derive a return difference inequality to provide connections between stochastic dissipativity and optimality of nonlinear controllers for discrete-time stochastic dynamical systems. In particular, using extended Kalman-Yakubovich-Popov conditions characterising stochastic dissipativity we show that our optimal feedback control law satisfies a return difference inequality predicated on a difference operator of a controlled Markov dispersion process and is stochastically dissipative with respect to a specific quadratic supply rate.