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

Abstract

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.

Estimation and control using sampling‐based Bayesian reinforcement learning

“…Since there are no time-varying stochastic nonlinear MPC algorithms that guarantee feasibility for parameters with Gaussian noise, linear MPC was selected for comparison [21], [24]. Linear MPC acts as a baseline to benchmark the performance between the optimal control and reinforcement learning algorithms.…”

“…Variants of MPC extend to nonlinear systems [15], [16] and account for uncertainty by propagating worstcase outcomes [17]- [19] or using probabilistic constraints [20]. Equations from the unscented transform have been incorporated as MPC constraints in an attempt to improve state estimation [21]- [23]. However, these methods cannot guarantee stability for systems with time-varying parameters following a Gaussian distribution as the possible changes to the system are unbounded and cannot be corrected by an inputconstrained control law [21], [24].…”

Combining Gaussian processes and polynomial chaos expansions for stochastic nonlinear model predictive control

“…PCEs in this context are a scenario-based SNMPC algorithm that uses leastsquares estimation online for every iteration of inputs to approximate the coefficients of an orthogonal polynomial approximation, known as non-intrusive PCE 32 . For polynomial-type systems Galerkin projection is used instead to determine the coefficients, which is called intrusive PCE 33 . Chance constraints can either be given using Chebychev's inequality 34 or applying a MC sampling approximation on the orthogonal polynomials themselves 33 .…”

“…For polynomial-type systems Galerkin projection is used instead to determine the coefficients, which is called intrusive PCE 33 . Chance constraints can either be given using Chebychev's inequality 34 or applying a MC sampling approximation on the orthogonal polynomials themselves 33 . The PCE based SNMPC algorithm has been extended to the case of output feedback in Bradford and Imsland 35 , Bradford et al 36 by combining the approach with a PCE nonlinear state estimator.…”

Stochastic Nonlinear Model Predictive Control with State Estimation by Incorporation of the Unscented Kalman Filter

“…A simple solution to SN-MPC can be found by [11], who linearises the nonlinear system successively and then applies a probabilistic tube method. A popular approach in SNMPC is given by the use of polynomial chaos (PC) expansions, which is a computationally efficient tool for accelerating sampling-based techniques [12]. In this method, implicit mappings between variables/parameters and the states are replaced by orthogonal polynomials.…”