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There has been an increasing interest in explicit and cheap-to-evaluate control policies that approximate (computationally expensive) control laws such as model predictive control (MPC). However, approximate control policies are subject to approximation errors, leading to asymptotic performance losses. The contribution of this paper is three-fold: (i) a closed-loop training scheme is presented for deep neural network approximation of economic MPC; (ii) an online adaptive correction scheme is presented to account for the performance losses induced by approximation errors; and (iii) an offline performance verification scheme is presented to ensure that the approximate control policy converges to an equilibrium point of the system. The proposed approach is illustrated using a Williams-Otto reactor problem.

The closed-loop performance of model predictive controllers (MPCs) is highly dependent on the choice of prediction models, controller formulation, and tuning parameters. However, prediction models are typically optimized for prediction accuracy, instead of performance, and MPC tuning is typically done manually to satisfy (probabilistic) constraints. In this work, we demonstrate a general approach for automating the tuning of MPC under uncertainty. In particular, we formulate the automated tuning problem as a constrained black-box optimization problem that can be tackled with derivative-free optimization. We rely on a constrained variant of Bayesian optimization to solve the MPC tuning problem that can directly handle noisy and expensive-to-evaluate functions. The benefits of the proposed automated tuning approach are demonstrated on a benchmark continuously stirred tank reactor case study.

Individual chance constraints can be used to systematically seek trade-offs between control performance and constraint violation for a given disturbance description. This paper presents a Stochastic Model Predictive Control (SMPC) approach with adaptive individual chance constraints that relies on online adaptation of the disturbance description using the empirical cumulative distribution (ECDF). Individual chance constraints are ensured by using a suitable worst-case confidence interval derived from the ECDF. The confidence interval may, however, be excessively conservative due to the empirical nature of the ECDF. To reduce this conservatism, the proposed approach accounts for the updated disturbance information, which is sampled online from the one-step ahead prediction error. Hence, the initial ECDF can be obtained from a reduced number of samples since the conservative handling of the chance constraint is continuously mitigated. This will also allow for using simpler models of the stochastic system disturbances. Convergence and recursive feasibility of the proposed adaptive approach are established. A DC-DC converter benchmark problem is used to illustrate the usefulness of the proposed approach.

Atmospheric pressure plasma jets (APPJs) are increasingly used for biomedical applications. Reproducible and effective operation of APPJs hinges on controlling the nonlinear effects of plasma on a target substrate in the face of intrinsic variabilities of the plasma as well as exogenous disturbances. This paper presents a low-memory fast approximate nonlinear model predictive control (NMPC) strategy for an APPJ with prototypical applications in plasma medicine. The NMPC objective is to regulate the delivery of the cumulative thermal effects of plasma to a substrate, while adhering to constraints pertaining to a patient’s safety and comfort. Deep neural networks are used to approximate the implicit NMPC law with a cheap-to-evaluate explicit control law that has low memory requirements. Robust constraint satisfaction is guaranteed by projecting the output of the neural network onto a set that ensures the state stays within an appropriately defined invariant set. Closed-loop simulations and real-time control experiments indicate that the proposed approximate NMPC strategy is effective in handling nonlinear control costs at fast sampling times, while guaranteeing satisfaction of safety-critical system constraints. This work takes a crucial step toward fast NMPC of safety-critical plasma applications using resource-limited embedded control hardware.

Mars life support system models consist of numerous mission-critical, interrelated, and scenario-specific parameters. The large size and involved nature of these models make them computationally expensive, with parameters that are subject to several sources of uncertainty. Accurately characterizing these uncertainties and their impact on overall model predictions is crucial for decision-support and mission optimization. This paper focuses on uncertainty quantification of a model of a space crop cultivation system, which is one of several systems that are required on a long-duration manned Mars mission. The model performs constrained optimization of the equivalent system mass (ESM) metric, which augments shipped mass costs with those of pressurized volume, demanded power, thermal control, and needed crew time. This paper uses surrogate modeling for fast quantification of the effect of probabilistic uncertainty in mission-critical parameters of semi-empirical equations that describe crop growth and equipment operation. This work shows sparse polynomial chaos-Kriging (PCK) yields a computationally cheap-to-evaluate surrogate for the minimum ESM that accounts for probabilistic uncertainty in 86 model parameters. This surrogate model accelerates a global sensitivity analysis that elucidates which crop growth and equipment operation parameters are critical to mission outcome variability. The PCK surrogate model realizes a 100-fold computational speed gain in the estimation of the probability distribution of the minimum ESM.

This paper presents a comparative study between two constraint-tightening approaches for tube-based stochastic nonlinear model predictive control (SNMPC) with and without terminal constraints. A simple constraint-tightening method based on the exponential decay rate of a δ-Lyapunov function is extended to the stochastic setting. This method uses the notion of incremental stabilizability to alleviate the need for offline, but involved computation of terminal constraints. The proposed method is compared to a SNMPC formulation that employs terminal constraints and Lipschitz constant-based constraint tightening. A comparative analysis is presented on a benchmark continuous stirred-tank reactor problem. Practical approximations for computing terminal sets are discussed in the context of this comparison.

Reliable and timely diagnosis of system faults under uncertainties is imperative for safe, reliable, and profitable operation of technical systems. This paper presents an input design method for active fault diagnosis for nonlinear systems that are subject to probabilistic model uncertainty and stochastic disturbances, and are under operational constraints. A computationally efficient sample-based method is presented for joint propagation of model uncertainty and stochastic disturbances using non-intrusive generalized polynomial chaos and unscented transformation. A tractable sample-based distance measure, inspired by the k-nearest neighbors algorithm, is used for fault diagnosis, which seeks to discriminate between probabilistic predictions of the model hypotheses for normal and faulty operation. Simulation results on a benchmark bioreactor case study demonstrate the effectiveness of the proposed input design method for reliable fault diagnosis under uncertainty through online model discrimination.

The majority of work on chance constrained model predictive control (MPC) for stochastic systems adopts the concept of implicit constraint backoffs for handling state chance constraints. However, implicit backoffs cannot be computed analytically in nonlinear MPC (NMPC), so that they must be approximated and, as a result, chance constraints are not guaranteed by the closed-loop system. This paper proposes a strategy for explicit computation of constraint backoffs that allows tightly guaranteeing chance constraints in NMPC for stochastic systems with arbitrary uncertainty distributions. The proposed method relies on uncertainty propagation through closed-loop system dynamics. Thus, this paper also investigates the use of stochastic surrogate models for constructing the distribution of closed-loop quantities of interest such as state constraint functions or any general performance objectives. To this end, an extension of polynomial chaos that can handle arbitrary probability measures is proposed for a class of disturbance models representing plant-model mismatch. The proposed methods are illustrated on a benchmark continuously-stirred tank reactor problem.

This paper addresses the problem of model predictive control with multiple models for nonlinear systems subject to stochastic disturbances. The multiple models can represent various operating conditions such as system faults or failures, or arise from model structure uncertainty. The paper presents a stochastic nonlinear model predictive control (SNMPC) approach with endogenous learning for active discrimination between the competing models based on closed-loop system observations. The system learning is endogenized through explicit inclusion of a model discrimination measure into the stochastic optimal control problem, which facilitates probabilistic discrimination between the predictions of multiple models. The control approach uses a Bayesian estimation algorithm for recursive estimation of the probabilities that represent the degree to which each model predicts the online system observations. The performance of the proposed SNMPC approach with active model discrimination is demonstrated for closed-loop fault diagnosis.

Dynamic simulation of stochastic systems requires uncertainty propagation. Traditional sample-based uncertainty propagation methods are often computationally intractable for online optimization-based estimation and control applications. Generalized polynomial chaos (gPC) is an efficient uncertainty propagation method that has been used for solving various nonlinear estimation and optimal control problems. However, gPC requires knowledge of the exact probability distribution of the random variables, and does not explicitly account for correlations between these random variables. This paper demonstrates the use of arbitrary polynomial chaos (aPC) for propagation of correlated multivariate random variables. aPC constructs orthogonal polynomial basis functions from only the raw moments of the random variables. Thus, aPC can be used for propagation of uncertainties with arbitrary probability distributions, even if their functional forms are unknown. The main contributions of this paper consist of presenting an algorithm for generating a set of orthogonal polynomial basis functions for correlated multivariate random variables and applying the Galerkin projection to compute closed-form expressions for the dynamics of the aPC expansion coefficients. An algorithm is also presented for efficient calculation of inner products between polynomial basis functions needed in the Galerkin projection. The error convergence properties of aPC are investigated and compared to that of gPC and Monte Carlo using a dynamic simulation case study.

This paper presents a robust model predictive control (MPC) approach for offset-free tracking of piece-wise constant references in the presence of bounded deterministic and stochastic disturbances. The system is considered to be linear with two sources of additive bounded uncertainties on the states. The first uncertainty source accounts for unknown, deterministic structural/parametric plant-model mismatch. The second uncertainty source represents stochastic exogenous system disturbances. The proposed deterministic-stochastic robust MPC approach uses estimates of the deterministic model uncertainties to modify the nominal state and input targets. This allows for achieving offset-free tracking of the mean of the controlled variables. A non-conservative constraint tightening procedure is used to handle probabilistic state constraints and hard input constraints in the presence of stochastic uncertainties. The computational complexity of the proposed robust MPC approach is comparable to that of nominal MPC. The closed-loop performance of the proposed robust MPC approach is compared to that of robust tube-based MPC and stochastic MPC in a simulation study.

When the stochastic description of system uncertainties is available, a natural approach to predictive control of uncertain systems involves explicitly accounting for the probabilistic occurrence of uncertainties in the optimal control problem. This work presents a stochastic nonlinear model predictive control (SNMPC) approach for nonlinear systems subject to time-invariant uncertainties as well as additive disturbances. The generalized polynomial chaos (gPC) framework is used to derive a deterministic surrogate for the stochastic optimal control problem. The key contribution of this paper lies in extending the gPC-based SNMPC approach reported in our earlier work to handle stochastic disturbances. This is done via mapping the stochastic disturbances onto the space of the coefficients of polynomial chaos expansions, which enables efficient propagation of stochastic disturbances. A sample-based approach to joint chance constraint handling is employed to fulfill the state constraints in a probabilistic sense. A gPC-based Bayesian parameter estimator is utilized to update the probability distribution of uncertain system parameters at each sampling time. In a simulation case study, the closed-loop performance of the SNMPC approach is demonstrated on an atmospheric-pressure plasma jet that is developed for biomedical applications.

Accurate estimation of parameters is paramount in developing high-fidelity models for complex dynamical systems. Model-based optimal experiment design (OED) approaches enable systematic design of dynamic experiments to generate input-output data sets with high information content for parameter estimation. Standard OED approaches however face two challenges: (i) experiment design under incomplete system information due to unknown true parameters, which usually requires many iterations of OED; (ii) incapability of systematically accounting for the inherent uncertainties of complex systems, which can lead to diminished effectiveness of the designed optimal excitation signal as well as violation of system constraints. This paper presents a robust OED approach for nonlinear systems with arbitrarily-shaped time-invariant probabilistic uncertainties. Polynomial chaos is used for efficient uncertainty propagation. The distinct feature of the robust OED approach is the inclusion of chance constraints to ensure constraint satisfaction in a stochastic setting. The presented approach is demonstrated by optimal experimental design for the JAK-STAT5 signaling pathway that regulates various cellular processes in a biological cell., Comment: Submitted to ADCHEM 2015