Implications of dissipativity, inverse optimal control, and stability margins for nonlinear stochastic feedback regulators

Abstract

Summary 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 nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton‐Jacobi‐Bellman controllers that minimize a nonlinear‐nonquadratic performance criterion with cross‐weighting terms. 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 stochastic dynamical systems. In particular, using extended Kalman‐Yakubovich‐Popov conditions characterizing stochastic dissipativity, we show that our optimal feedback control law satisfies a return difference inequality predicated on the infinitesimal generator of a controlled Markov diffusion process if and only if the controller is stochastically dissipative with respect to a specific quadratic supply rate.

Passivity‐based control design frameworks for hybrid nonlinear time‐varying dynamical systems

“…where (A ct , B ct ) and (A ds , B ds ) are assumed to be controllable. Proceeding to the second step, the linear strict passivity-based gain matrix, P(t), can be computed, with ⌢ A ct and ⌢ A ds thus determined, through solving ( 28) and (31) simultaneously in an interacting manner as follows:…”

“…19 Building on these results, stability criteria characterizing Lyapunov, asymptotic, and exponential properties for the feedback interconnection of hybrid nonlinear dissipative dynamical systems were derived in Reference20. The notion of dissipativity has also been extended to a different variety of dynamical systems, including discrete-time nonlinear systems for observer design purposes, [21][22][23] nonnegative and compartmental systems, 24,25 large-scale systems, 26,27 port-controlled Hamiltonian systems, 28,29 and stochastic systems 30,31 to name but a few.…”

Definition and analysis of stability margins for a class of nonlinear systems

“…Similarly, in Reference 5, the authors present a fuzzy control‐based design of a maximum stability margin for NL uncertain systems; however, the results do not illustrate any gain or phase margin model. Stability margins for nonlinear stochastic feedback regulators are discussed in Reference 6. A concept of singular perturbation margin for linear time‐invariant (LTI) is introduced in Reference 7 and is extended for nonlinear time‐invariant (NLTI) systems in Reference 8.…”