Design of Linear Regulators for Nonlinear Stochastic Systems

Abstract¶

The problem of regulating a nonlinear stochastic system with uncertain parameters and nonlinear noisy measurements is considered. The second-order terms in an expansion about the reference path are viewed as multiplicative disturbances in state and control. The linear estimation/control problem is solved by approximating these as wideband disturbances. The Certainty-Equivalence Principle and the Separation Theorem do not hold. The control and filter gains, found by solving a TPBVP, explicitly hedge against the parameter uncertainties and nonlinearity effects. Numerical results are presented for a combined guidance/navigation system for shuttle entry.

Summary¶

Develops a methodology for designing linear regulators for nonlinear stochastic systems by linearizing about a nominal trajectory and applying LQR synthesis to the resulting linear stochastic model. One of the earliest works identifying key aspects in the design and motivation of optimal control for linear systems with multiplicative noise, including the observation that second-order noise terms show up as state- and input- multiplicative noise, and that the Certainty-Equivalence Principle and the Separation Theorem do not hold in this multiplicative noise setting. These ideas were picked up again by later authors such as Todorov (“A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems” ACC 2005) and Gravell ("Data-driven Control and Planning for Uncertain Complex Systems" UTD 2022).

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