Stochastic Model Predictive Control Algorithm Research Articles

Spatiotemporal learning-based stochastic MPC with applications in aero-engine control

To deal with the issue of model uncertainty, extensive stochastic MPC methods have been investigated, where the uncertainties are typically assumed to follow given statistical distributions. However, in practical scenarios, the statistical properties of model uncertainties may depend on certain hyperparameters varying both temporally and spatially. For instance, in the applications of aero-engine control, different types of uncertainties may occur under different operational conditions, such as flight altitudes and Mach numbers, which are dynamically changing. Traditionally, the stochastic MPC methods may not be able to handle these types of uncertainties directly. Therefore, we propose a spatiotemporal learning-based stochastic model predictive control algorithm to study the stochastic optimal control problem with dynamically changing uncertainties, by constructing spatiotemporal Gaussian processes (GPs) to approximate the uncertainties based on measurement data. Since the spatiotemporal GPs may be difficult to evaluate with long processing time series, we present a state–space representation of GPs to employ computationally efficient Kalman filtering. Then, we derive a computationally tractable control strategy by parameterizing the controller and reformulating the cost and chance constraints, and analyze the corresponding recursive feasibility and closed-loop stability. Finally, the proposed algorithm is applied to the compressor control of the aero-engine, and the comparisons with other MPC controllers are demonstrated to show the effectiveness of our methods.

Stochastic Model Predictive Control With a Safety Guarantee for Automated Driving

Automated vehicles require efficient and safe planning to maneuver in uncertain environments. Largely this uncertainty is caused by other traffic participants, e.g., surrounding vehicles. Future motion of surrounding vehicles is often difficult to predict. Whereas robust control approaches achieve safe, yet conservative motion planning for automated vehicles, Stochastic Model Predictive Control (SMPC) provides efficient planning in the presence of uncertainty. Probabilistic constraints are applied to ensure that the maximal risk remains below a predefined level. However, safety cannot be ensured as probabilistic constraints may be violated, which is not acceptable for automated vehicles. Here, we propose an efficient trajectory planning framework with safety guarantees for automated vehicles. SMPC is applied to obtain efficient vehicle trajectories for a finite horizon. Based on the first optimized SMPC input, a guaranteed safe backup trajectory is planned using reachable sets. This backup is used to overwrite the SMPC input if necessary for safety. Recursive feasibility of the safe SMPC algorithm is proved. Highway simulations show the effectiveness of the proposed method regarding performance and safety.

Online learning stochastic model predictive control of linear uncertain systems

AbstractThis article presents an online learning stochastic model predictive control method for linear uncertain systems with state‐dependent additive uncertainties, where the uncertainty is modeled as Gaussian process. The proposed scheme utilizes the probabilistic reachable sets as time‐varying tubes, which are formulated by forecasting the variance propagation of uncertainty via Gaussian process regression, to embody the chance constraints. The proposed learning based stochastic model predictive control algorithm is designed by refining the active data dictionary to train the Gaussian process model of uncertainty online. In particular, the data points in the active data dictionary are selected from the raw data around the predicted optimal nominal trajectories, which reduces the computational load as well as preserves the control performance. Then the algorithm feasibility and closed‐loop stability of the developed algorithm are analyzed. Finally, the efficacy and superiority over existing methods are verified by simulation studies.

A stochastic MPC scheme for distributed systems with multiplicative uncertainty

This paper presents a Distributed Stochastic Model Predictive Control algorithm for networks of linear systems with multiplicative uncertainties and local chance constraints on the states and control inputs. The chance constraints are approximated via Cantelli’s inequality by means of expected value and covariance. The cooperative control algorithm is based on the distributed Alternating Direction Method of Multipliers, which renders the controller fully distributedly implementable, recursively feasible and ensures point-wise convergence of the states. The aforementioned properties are guaranteed through a properly selected distributed invariant set and distributed terminal constraints for the mean and covariance. The paper closes with an example highlighting the chance constraint satisfaction, numerical properties and scalability of our approach.

Stochastic model predictive control for linear systems with unbounded additive uncertainties

This paper presents two stochastic model predictive control methods for linear time-invariant systems subject to unbounded additive uncertainties. The new methods are developed by formulating the chance constraints into deterministic form, which are treated in analogy with robust constraints, by using the probabilistic reachable set. The first one is the time-varying tube-based stochastic model predictive control algorithm, which is designed by employing the time-varying probabilistic reachable sets as tubes. The second one is the constant tube-based stochastic model predictive control algorithm, which is developed by enforcing a constant tightened constraint in the entire prediction horizon. In addition, the soft constraints are proposed to associate with the state initialization in the algorithms to enhance the feasibility. The algorithm feasibility and closed-loop stability results are provided. The efficacy of the approaches is demonstrated by means of numerical simulations.

Stochastic model predictive control for driver assistance control of intelligent vehicles considering uncertain driving environment

Advanced driver assistance control faces great challenges in cooperating with the nearby vehicles. The assistance controller of an intelligent vehicle has to provide control efforts properly to prevent possible collisions without interfering with the drivers. This paper proposes a novel driver assistance control method for intelligent ground vehicles to cooperate with the nearby vehicles, using the stochastic model predictive control algorithm. The assistance controller is designed to correct the drivers’ steering maneuvers when there is a risk of possible collisions, so that the drivers are not interfered. To enhance the cooperation between the vehicles, the nearby vehicle motion is predicted and included in the assistance controller design. The position uncertainties of the nearby vehicle are considered by the stochastic model predictive control approach via chance constraints. Simulation studies are conducted to validate the proposed control method. The results show that the assistance controller can help the drivers avoid possible collisions with the nearby vehicles and the driving safety can be guaranteed.

Double-layer stochastic model predictive voltage control in active distribution networks with high penetration of renewables

The high penetration of renewable energy into distribution networks poses increasing challenges on voltage control. To address this issue, this paper presents a double-layer stochastic model predictive control algorithm to regulate voltage profile in active distribution networks. In the proposed algorithm, voltage regulation is achieved by coordination of an upper layer controller and a lower layer controller. In the upper layer, the number of operation of mechanical voltage regulation devices, including transformer with on-load tap changer and capacitor banks, is minimized in an hourly timescale. In the lower layer, the controller minimizes the active power curtailments and power losses with a control period of 5 min. The proposed double-layer stochastic model predictive voltage control utilizes not only the reactive power control, but also the active power curtailment to regulate bus voltages. In addition, mechanical voltage regulation devices and distributed generations are controlled in two different timescales. Case studies on a modified IEEE-33 bus system demonstrate that compared with traditional control and two-stage stochastic voltage control, the proposed algorithm can achieve an improvement of 8.05% and 7.43%, respectively.

Distributed Stochastic MPC for Networked Linear Systems With a Multirate Sampling Mechanism

In this article, a distributed stochastic model predictive control (MPC) algorithm with a multirate sampling mechanism is proposed for a networked linear system with multiple dynamic subsystems. A delta operator approach is used for the multiple dynamic subsystems with different sampling periods in the multirate sampling mechanism. Both stochastic disturbances and probabilistic constraints of the multiple dynamic subsystems are satisfied by the distributed stochastic MPC algorithm. Packets dropout are considered by the stochastic MPC algorithm and state predicted errors are compensated. Recursive feasibility and quadratic stability are obtained for the networked linear system under the distributed stochastic MPC algorithm. A numerical example is given to illustrate effectiveness of the proposed algorithm.

Distributed stochastic MPC for systems with parameter uncertainty and disturbances

SummaryA distributed stochastic model predictive control algorithm is proposed for multiple linear subsystems with both parameter uncertainty and stochastic disturbances, which are coupled via probabilistic constraints. To handle the probabilistic constraints, the system dynamics is first decomposed into a nominal part and an uncertain part. The uncertain part is further divided into 2 parts: the first one is constrained to lie in probabilistic tubes that are calculated offline through the use of the probabilistic information on disturbances, whereas the second one is constrained to lie in polytopic tubes whose volumes are optimized online and whose facets' orientations are determined offline. By permitting a single subsystem to optimize at each time step, the probabilistic constraints are then reduced into a set of linear deterministic constraints, and the online optimization problem is transformed into a convex optimization problem that can be performed efficiently. Furthermore, compared to a centralized control scheme, the distributed stochastic model predictive control algorithm only requires message transmissions when a subsystem is optimized, thereby offering greater flexibility in communication. By designing a tailored invariant terminal set for each subsystem, the proposed algorithm can achieve recursive feasibility, which, in turn, ensures closed‐loop stability of the entire system. A numerical example is given to illustrate the efficacy of the algorithm.

An Efficient Model Based Control Algorithm for the Determination of an Optimal Control Policy for a Constrained Stochastic Linear System

In this paper, the authors have proposed an ensemble Kalman filter based stochastic model predictive control algorithm to determine an optimal control policy at every sampling time instant for a constrained stochastic linear system. To determine an optimal control policy for the constrained linear system affected by random disturbances and measurements corrupted by random noise, the authors have minimized the uncertain objective function, subject to uncertain state & output constraints and deterministic input constraints using the quantile based scenario analysis approach. In this work, ensemble Kalman filter is being employed, to generate a recursive estimate of states of the constrained stochastic linear system. The number of scenarios is considered to be equivalent to that of number of sample points used in the ensemble Kalman filter. Each scenario is viewed as one realization of the process noise, measurement noise over the prediction horizon as well as the ith sample point of the state estimate at the beginning of the prediction horizon generated by the ensemble Kalman filter. Simulation studies have been carried out to assess the efficacy of the proposed control scheme on the simulated model of the constrained single-input and single-output linear stochastic system.

Constraint-Tightening and Stability in Stochastic Model Predictive Control

Constraint tightening to non-conservatively guarantee recursive feasibility and stability in Stochastic Model Predictive Control is addressed. Stability and feasibility requirements are considered separately, highlighting the difference between existence of a solution and feasibility of a suitable, a priori known candidate solution. Subsequently, a Stochastic Model Predictive Control algorithm which unifies previous results is derived, leaving the designer the option to balance an increased feasible region against guaranteed bounds on the asymptotic average performance and convergence time. Besides typical performance bounds, under mild assumptions, we prove asymptotic stability in probability of the minimal robust positively invariant set obtained by the unconstrained LQ-optimal controller. A numerical example, demonstrating the efficacy of the proposed approach in comparison with classical, recursively feasible Stochastic MPC and Robust MPC, is provided.

Stochastic MPC with offline uncertainty sampling

For discrete-time linear systems subject to multiplicative disturbance described by random variables, we develop a sampling-based Stochastic Model Predictive Control algorithm. Unlike earlier results employing a scenario approximation, we propose an offline sampling approach in the design phase instead of online scenario generation. The paper highlights the structural difference between online and offline sampling and provides rigorous bounds on the number of samples needed to guarantee chance constraint satisfaction. The approach does not only significantly speed up the online computation, but furthermore allows to suitably tighten the constraints to guarantee robust recursive feasibility when bounds on the uncertain variables are provided. Under mild assumptions, asymptotic stability of the origin can be established.

Model predictive control of linear systems with multiplicative unbounded uncertainty and chance constraints

This paper presents a novel stochastic Model Predictive Control algorithm for linear systems characterized by multiplicative and possibly unbounded model uncertainty. Probabilistic constraints on the states and inputs are considered, and a quadratic cost function is minimized. The stochastic control problem, and in particular the probabilistic constraints, are reformulated in deterministic terms by means of the Cantelli inequality, so that the on-line computational burden of the algorithm is similar to the one of a standard MPC method. The properties of the algorithm, namely the recursive feasibility and the pointwise convergence of the state, are proven by suitably selecting the terminal cost and the constraints on the mean and the variance of the state at the end of the prediction horizon, and by considering as additional optimization variables also the mean and the covariance of the state at the beginning of the prediction horizon. An extension to deal with the case of expectation, rather than probabilistic, constraints is reported. The numerical issues related to the off-line selection of the algorithm’s parameters and its on-line implementation are discussed. Simulation results referred to a system with unbounded uncertainty are shown to compare the performances achievable with probabilistic and expectation constraints.

Model predictive control of linear systems with multiplicative unbounded uncertainty and average constraints

In this paper a novel Stochastic Model Predictive Control algorithm is developed for systems characterized by multiplicative and possibly unbounded model uncertainty, average constraints on the states and inputs, and quadratic cost function. The stochastic control problem, and in particular the average constraints, are reformulated in deterministic terms. The properties of the algorithm, namely the recursive feasibility and the pointwise convergence of the state, are proven by suitably selecting the terminal cost and the constraints on the mean and the variance of the state at the end of the prediction horizon and by considering the mean and the covariance of the state at the beginning of the prediction horizon as additional optimization variables. The numerical issues related to the off-line selection of the algorithm's parameters and its on-line implementation are also discussed in depth. A simulation campaign witnesses the effectiveness of the proposed control scheme.

Formulation of MPC for Multiplicative Stochastic Uncertainty by Multi-step Probabilistic Sets

Abstract This paper designs multi-step probabilistic sets for linear, discrete-time, stochastic systems with unbounded multiplicative noise and probabilistic constraints. Multi-step probabilistic sets strengthen IWPp by bringing more degrees of freedom to optimize the applicable region of finite-step probabilistic constraints, and extending the prediction horizon of IWPp to infinity for infinite-horizon probabilistic constraints. Conditions for multi-step probabilistic sets are then incorporated into a stochastic model predictive control algorithm to satisfy probabilistic constraints. Closed-loop mean-square stability is guaranteed by the algorithm. A numerical example shows the performance of the proposed algorithm.

Optimization of warehousing and transportation costs, in a multi-product multi-level supply chain system, under a stochastic demand

The centralized management approach provides a general view to set a better coordination between the ele- ments of the supply chain, and look for the equilibrium between the stock and the shipped quantities. This paper describes a stochastic model predictive control algorithm to optimize the warehousing cost and the shipping cost on the same time. We consider a multi-level and multi-product supply chain dealing with a stochastic demand. We solve the stochastic model predictive control problem by a dynamic programming to determine the optimal policies to minimize the costs of shipping and warehousing products.