Abstract
This work investigates an optimal control problem for a class of stochastic differential bilinear systems, affected by a persistent disturbance provided by a nonlinear stochastic exogenous system (nonlinear drift and multiplicative state noise). The optimal control problem aims at minimizing the average value of a standard quadratic-cost functional on a finite horizon. It has been supposed that neither the state of the system nor the state of the exosystem is directly measurable (incomplete information case). The approach is based on the Carleman embedding, which allows to approximate the nonlinear stochastic exosystem in the form of a bilinear system (linear drift and multiplicative noise) with respect to an extended state that includes the state Kronecker powers up to a chosen degree. This way the stochastic optimal control problem may be restated in a bilinear setting and the optimal solution is provided among all the affine transformations of the measurements. The present work is a nontrivial extension of previous work of the authors, where the Carleman approach was exploited in a framework where only additive noises had been conceived for the state and for the exosystem. Numerical simulations support theoretical results by showing the improvements in the regulator performances by increasing the order of the approximation.
Highlights
IntroductionConsider the following stochastic differential system described by means of the Itoformalism: dx(t) Ax(t)dt + Bu(t)dt + Lz(t)dt b
Consider the following stochastic differential system described by means of the Itoformalism: dx(t) Ax(t)dt + Bu(t)dt + Lz(t)dt b+ Mi + Nix(t)dWxi (t), x(0) x0, (1)h dz(t) φ(z(t))dt + Fj + Djz(t)dWzj (t), j 1 z(0) z0, (2)where φ : Rm ⟼ Rm is a smooth nonlinear map
Is work investigates an optimal control problem for a class of stochastic differential bilinear systems, affected by a persistent disturbance provided by a nonlinear stochastic exogenous system. e optimal control problem aims at minimizing the average value of a standard quadratic-cost functional on a finite horizon
Summary
Consider the following stochastic differential system described by means of the Itoformalism: dx(t) Ax(t)dt + Bu(t)dt + Lz(t)dt b. To the best of our knowledge, the only reference that deals with stochastic optimal control problems in a nonlinear framework with incomplete knowledge of the state is [10], though nonlinearities are restricted only to the diffusion term where the noise affects the state dynamics in a multiplicative fashion; the state drift and the output equation providing noisy measurements are both linear. E Carleman approach consists in the embedding of the original nonlinear differential stochastic system onto an infinite-dimensional system whose state accounts for the Kronecker powers of any order of the original state With respect to such a state, the dynamics can be written in a bilinear fashion (linear drift and multiplicative noise), and the ]-degree Carleman approximation is achieved by truncating the higher-than-] Kronecker powers. A similar approach can be found in [26] in a discrete-time framework, and in [27] in a continuous-time framework, where only additive noises had been conceived for the state and for the exosystem. e present note is a nontrivial extension of [27], since multiplicative noises are considered both in the system dynamics and in the exosystem one
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