Generalised polynomial chaos expansion approaches to approximate stochastic model predictive control†

This paper studies the model predictive control of dynamic systems subject to stochastic parametric uncertainty due to plant/model mismatches and exogenous disturbance that corresponds to uncertain circumstance in operation of the system. Model and disturbance uncertainties are ubiquitous in any mathematical models of system and control theory. Parametric uncertainty propagation or quantification is approximated using a spectral method called polynomial chaos expansion and exogenous disturbance is assumed to be an additive Gaussian random process. With Gaussian approximation of resulting solution trajectory of a stochastic differential equation using polynomial chaos expansion, we solve convex finite-horizon model predictive control problems that are amenable to online computation of a stochastically robust control policy over the time-horizon. The proposed approach to chance-constrained model predictive control provides an explicit way to handle a stochastic system model in the presence of both model uncertainty and exogenous disturbances. Probabilistic constraints are replaced by convex deterministic constraints that approximate the probabilistic violations with a user-defined confidence level.

Stochastic linear Model Predictive Control with chance constraints – A review

Wind energy has been drawing considerable attention in recent years. However, due to the random nature of wind and high failure rate of wind energy conversion systems (WECSs), how to implement fault-tolerant WECS control is becoming a significant issue. This paper addresses the fault-tolerant control problem of a WECS with a probable actuator fault. A new stochastic model predictive control (SMPC) fault-tolerant controller with the Conditional Value at Risk (CVaR) objective function is proposed in this paper. First, the Markov jump linear model is used to describe the WECS dynamics, which are affected by many stochastic factors, like the wind. The Markov jump linear model can precisely model the random WECS properties. Second, the scenario-based SMPC is used as the controller to address the control problem of the WECS. With this controller, all the possible realizations of the disturbance in prediction horizon are enumerated by scenario trees so that an uncertain SMPC problem can be transformed into a deterministic model predictive control (MPC) problem. Finally, the CVaR object function is adopted to improve the fault-tolerant control performance of the SMPC controller. CVaR can provide a balance between the performance and random failure risks of the system. The Min-Max performance index is introduced to compare the fault-tolerant control performance with the proposed controller. The comparison results show that the proposed method has better fault-tolerant control performance.

Parametric uncertainty assessment in hydrological modeling using the generalized polynomial chaos expansion

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

Stochastic model predictive control (SMPC) with probabilistic path constraints remains an open and challenging issue due to the intractability of probabilistic path constraints, let alone guarantee rigorous satisfaction. For the SMPC problem, the greatest difficulty lies in how to solve a finite time domain optimal control problem in the corresponding sampling step while ensuring that the probabilistic path constraints are rigorously satisfied. Aiming at the above problem, this article presents a new method for SMPC with guaranteed rigorous satisfaction of probabilistic path constraints. Specifically, first, to convert thorny probabilistic path constraints into computationally solvable and equivalent deterministic constraints, a transformation technique is proposed by utilizing a designed ancillary feedback controller and the information on the probabilistic distribution of disturbances. Then, a probabilistic path-constrained dynamic event-triggered SMPC (DETSMPC) algorithm is further designed where not only control input can be obtained but also the probabilistic path constraints can be rigorously satisfied over the entire time domain. In addition, to mitigate redundant computation consumption, a novel event-triggering sampling scheme with two dynamic thresholds is proposed by the virtue of prediction horizon, real-time sampled data, and previous predicted and triggered states. Furthermore, it is proved that the proposed algorithm is feasible for the overall probabilistic path-constrained closed-loop DETSMPC system. Moreover, the convergence in probability of the closed-loop system is presented. Finally, a numerical example is provided to illustrate the effectiveness of the proposed methods.

This study focuses on demonstrating the effectiveness and efficiency of the Stochastic Sequential Model Predictive Control (MPC) framework within the context of the Hjartdøla hydropower system. Multistage MPC, while effective in managing uncertainty, poses challenges due to its high computational demands and complex optimal control problems, particularly in applications requiring long-term forecasting, such as hydropower systems. Through a comparative simulation study with multistage MPC, this paper highlights the superior feasibility and computational speed of the Stochastic Sequential MPC framework. This work contributes to the broader understanding of MPC applications in hydropower systems.

This work investigates the event‐driven model predictive control problem for perturbed nonlinear systems with input and state constraints. Firstly, a new event‐driven triggering condition, considering the gradient and accumulation of the error between the optimal and actual states at adjacent moments with a state‐dependent relative threshold is designed with a guaranteed positive inter‐event time to avoid Zeno behaviour. Secondly, an error gradient and accumulation type event‐driven model predictive control framework is designed based on the new triggering condition and the dual‐mode strategy, for effectively reducing the calculation and communication burden of the model predictive control controller while ensuring desired control performance. Furthermore, the feasibility and the input‐to‐state practical stability (ISpS) of gradient and accumulation‐type event‐driven model predictive control are rigorously guaranteed theoretically. At last, the simulation results are provided to validate the this algorithm.

The importance of model predictive control derives primarily from its industrial success. Theory has contributed to the development of model predictive control mainly in its discovery of conditions that ensure closed-loop stability. Indeed, there is now a rich collection of of useful and interesting results on stability that we summarize because of their relevance to uncertainty, the main topic of this address. While stability is, not the only topic of concern to theoreticians and practitioners alike, progress on other fronts, such as robustness to model error, state estimation error and exogenous disturbances has been slow; we discuss the issues for interesting reasons. The main challenge facing model predictive control is the development of techniques for reducing the inherent complexities in feedback model predictive control.

In this paper, we propose a stochastic model predictive control (MPC) algorithm for linear distributed discrete-time systems affected by unbounded additive Gaussian disturbances, which are subject to local probabilistic constraints. Probabilistic constraints are treated with the concept of probabilistic reachable sets, which are an analogy to robust reachable sets for robust MPC. We present a method which decouples the global covariance matrix into a block diagonal upper bound. Together with the decomposition of the centralized problem, we define local probabilistic invariant sets as terminal regions, where we additionally derive a condition that gives us a probabilistic guarantee of invariance. We demonstrate our approach on an example, highlighting the closed-loop performance and constraint satisfaction compared to a centralized scheme.

This paper proposes a solution method for the discrete-time long-term dynamic portfolio optimization problem with state and asset allocation constraints. We use the ideas of Model Predictive Control (MPC) to solve the constrained stochastic control problem. MPC is a solution technique which was developed to solve constrained optimal control problems for deterministic control applications. MPC solves the optimal control problem with a receding horizon where a series of consecutive open-loop optimal control problems is solved. The aim of this paper is to develop an MPC approach to the problem of long-term portfolio optimization when the expected returns of the risky assets are modeled using a factor model based on stochastic Gaussian processes. We prove that MPC is a suboptimal control strategy for stochastic systems which uses the new information advantageously and thus is better than the pure optimal open-loop control. For the open-loop optimal control optimization, we derive the conditional portfolio distribution and the corresponding conditional portfolio mean and variance. The mean and the variance depend on future decision about the asset allocation. For the dynamic portfolio optimization problem, we consider constraints on the asset allocation as well as probabilistic constraints on the attainable values of the portfolio wealth. We discuss two different objectives, a classical mean–variance objective and the objective to maximize the probability of exceeding a predetermined value of the portfolio. The dynamic portfolio optimization problem is stated, and the solution via MPC is explained in detail. The results are then illustrated in a case study.

Model predictive control has been recognized as one of the essential solutions to achieve considerable energy savings in buildings. However, its performance on a building zone level can be inferior to a well-tuned conventional controller, especially in situations with constant energy prices and conservative comfort constraints. Optimization problem in the background has to be chosen to guarantee recursive feasibility and considerable energy savings without compromising the users comfort at the same time. This paper gives a novel formulation of the model predictive temperature control problem in buildings and its fair comparison with conventional controllers with the same level of flexibility allowed in zone temperature control. All controllers are tested for a system with seasonal heating and cooling, which is the most common case in real applications. It is shown that the introduced formulation leads to the model predictive controller that significantly outperforms conventional controllers both in energy consumption and users comfort.

The domain of attraction is central to stability and recursive feasibility of model predictive control (MPC). For stochastic linear systems, this paper addresses the problem of enlarging the domain of attraction of stochastic MPC (SMPC) for offset-free tracking. The proposed SMPC strategy relies on using an artificial steady-state target to enlarge the domain of attraction while ensuring recursive feasibility. The key advantage of the proposed strategy is that it alleviates the computation of an additional robust control invariant set to greatly simplify the SMPC design. Furthermore, the proposed SMPC strategy for offset-free tracking circumvents feasibility loss due to target changes since the feasible region is independent of the desired steady-state target. The SMPC strategy is demonstrated on a benchmark DC-DC converter case study, where its performance is compared to that of standard SMPC for regulation based on tighter constraints and a recently proposed SMPC strategy with a first-step constraint.

Chapter 14 - Fast stochastic model predictive control of end-to-end continuous pharmaceutical manufacturing

Stochastic Nonlinear Model Predictive Control with Active Model Discrimination: A Closed-Loop Fault Diagnosis Application