Stochastic optimal control in the perspective of the Wiener chaos

Huschto, Tony; Säger, Sebastian

doi:10.23919/ecc.2013.6669220

Cited by 6 publications

(5 citation statements)

References 20 publications

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“…In any case, the prediction could be improved by adapting the orthogonal basis as proposed in [51]. Another interesting future extension could be the consideration of time-dependent disturbances (e. g. [40]).…”

Section: Discussionmentioning

confidence: 99%

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Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints

Streif,

Karl,

Mesbah

2014

Preprint

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This paper presents a stochastic model predictive control approach for nonlinear systems subject to time-invariant probabilistic uncertainties in model parameters and initial conditions. The stochastic optimal control problem entails a cost function in terms of expected values and higher moments of the states, and chance constraints that ensure probabilistic constraint satisfaction. The generalized polynomial chaos framework is used to propagate the timeinvariant stochastic uncertainties through the nonlinear system dynamics, and to efficiently sample from the probability densities of the states to approximate the satisfaction probability of the chance constraints. To increase computational efficiency by avoiding excessive sampling, a statistical analysis is proposed to systematically determine a-priori the least conservative constraint tightening required at a given sample size to guarantee a desired feasibility probability of the sample-approximated chance constraint optimization problem. In addition, a method is presented for samplebased approximation of the analytic gradients of the chance constraints, which increases the optimization efficiency significantly. The proposed stochastic nonlinear model predictive control approach is applicable to a broad class of nonlinear systems with the sufficient condition that each term is analytic with respect to the states, and separable with respect to the inputs, states and parameters. The closed-loop performance of the proposed approach is evaluated using the Williams-Otto reactor with seven states, and ten uncertain parameters and initial conditions. The results demonstrate the efficiency of the approach for real-time stochastic model predictive control and its capability to systematically account for probabilistic uncertainties in contrast to a nonlinear model predictive control approaches.

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Section: Discussionmentioning

confidence: 99%

“…See also[23] for a recent review 2. The use of the PC framework for stochastic MPC and optimal control has also been investigated in[35,[38][39][40][41].…”

mentioning

confidence: 99%

Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints

Streif,

Karl,

Mesbah

2014

Preprint

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“…SMPC approaches enable shaping the predicted probability distribution functions (PDFs) of system states and outputs in an optimal manner over a finite prediction horizon (e.g., see [5], [6], [7], [8], [9], and the references therein). In SMPC, chance constraints can be considered in a probabilistic sense to circumvent the inherent conservatism of deterministic worst-case robust MPC approaches.…”

Section: Introductionmentioning

confidence: 99%

Fast stochastic model predictive control of high-dimensional systems

Paulson

Mesbah

Streif

et al. 2014

53rd IEEE Conference on Decision and Control

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Probabilistic uncertainties and constraints are ubiquitous in complex dynamical systems and can lead to severe closed-loop performance degradation. This paper presents a fast algorithm for stochastic model predictive control (SMPC) of high-dimensional stable linear systems with time-invariant probabilistic uncertainties in initial conditions and system parameters. Tools and concepts from polynomial chaos theory and quadratic dynamic matrix control inform the development of an input-output formulation for SMPC with output constraints. Generalized polynomial chaos theory is used to enable efficient uncertainty propagation through the high-dimensional system model. Galerkin projection is used to construct the polynomial chaos expansion for a general class of linear differential algebraic equations (DAEs), so that the SMPC algorithm is applicable to both regular and singular/descriptor systems. The fast SMPC approach is applied for control of an endto-end continuous pharmaceutical manufacturing process with approximately 8000 states. The on-line computational cost of the proposed probabilistic input-output SMPC algorithm is independent of the state dimension and, therefore, alleviates the prohibitive computational costs of control of uncertain systems with large state dimension.

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“…We refer e.g. to [11][12][13]25] for applications of the Wiener chaos expansion in the context of SDEs and to [20][21][22]26] for applications of polynomial expansion in modelling, simulation and filtering of stochastic partial differential equations. The aim of this work is to use the chaos expansion (1.3) to numerically approximate the solution of the SDE (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…where the subset I p,k ⊂ I refers to using orthogonal projections Ψ α only with respect to the first k basis elements (e i ) 1≤i≤k and only up to the pth order Wiener chaos. This method is mostly related to the articles [11,12,22,21]. More specifically, the works [21,22] study the L 2 -error associated with the approximation (1.5), but only for a particular choice of the basis (e i ) i≥1 .…”

Section: Introductionmentioning

confidence: 99%

The asymptotic error of chaos expansion approximations for stochastic differential equations

Huschto¹,

Podolskij²,

Säger³

2019

Modern Stochastics: Theory and Applications

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In this paper we present a numerical scheme for stochastic differential equations based upon the Wiener chaos expansion. The approximation of a square integrable stochastic differential equation is obtained by cutting off the infinite chaos expansion in chaos order and in number of basis elements. We derive an explicit upper bound for the L 2 approximation error associated with our method. The proofs are based upon an application of Malliavin calculus.

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Stochastic optimal control in the perspective of the Wiener chaos

Cited by 6 publications

References 20 publications

Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints

Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints

Fast stochastic model predictive control of high-dimensional systems

The asymptotic error of chaos expansion approximations for stochastic differential equations

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