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Over the course of four decades, model predictive control (MPC) has become one of the great success stories in systems and control. It has grown from its native habitat (chemical process control) into all domains of control applications—power and energy systems, mechatronics and robotics, as well as aerospace and aeronautics. Hence, in a modern systems and control curriculum, MPC triggers not so much the question of if it should be taught. In fact, industrial demand for and the continued research potential of MPC suggest that one should rather ask the Aristotelian 5W1H (what, when, where, why, who, and how?) about teaching MPC. This article presents insights into the 5Ws distilled from the results of a survey on teaching MPC conducted in the systems and control community. Moreover, the how is approached through blueprint suggestions for curricula for an undergraduate discrete-time linear-quadratic MPC course and for graduate courses covering the continuous-time nonlinear avenue and the learning-based route.

In scenarios where high-fidelity physical models are either unavailable or are impractical due to their high complexity, data-based models offer a viable solution to obtain the system model necessary for predictive control. However, the accuracy of the predictions obtained by data-based models is limited. We propose to use neural networks with Bayesian last layer to obtain information about the uncertainty of the predictions. This paper demonstrates the use of Bayesian last layer surrogate models in a robust nonlinear model predictive control setting. The nonlinear model predictive control problem is adapted by considering the predicted uncertainty of the surrogate model, which can be efficiently computed using the Bayesian last layer method, in the cost function. The controller thus takes model uncertainty explicitly into account and by its formulation also avoids areas of extrapolation. The proposed method is applied to a mixed-suspension, mixed-product-removal crystallizer and simulation studies show that it outperforms a standard data-based model.

This article considers the $\mathcal {H}_\infty$ static output-feedback control for linear time-invariant uncertain systems with polynomial dependence on probabilistic time-invariant parametric uncertainties. By applying polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system, which characterizes stochastic state uncertainty propagation. A closed-loop polynomial chaos transformation is proposed to derive the closed-loop expanded system. The approach explicitly accounts for the closed-loop dynamics and preserves the $\mathcal {L}_2$-induced gain, which results in smaller transformation errors compared to existing polynomial chaos transformations. The effect of using finite-degree polynomial chaos expansions is first captured by a norm-bounded linear differential inclusion, and then addressed by formulating a robust polynomial chaos based control synthesis problem. This proposed approach avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of truncation errors, which significantly reduces computational complexity. A numerical example illustrates the effectiveness of the proposed approach.

Model predictive control is a powerful advanced control technique to deal with complex nonlinear systems with constraints. Despite recent advances in computing hardware and optimization algorithms, solving the nonconvex optimization problems associated to nonlinear model predictive control in real time can still be intractable. There are several possibilities to alleviate the challenge of computational complexity. An increasingly popular approach is to use neural networks to approximate the mapping between current system state and the corresponding optimal control input, which is implicitly defined by an optimization problem. These approaches typically need a large amount of data. Although the data can be generated offline by sampling the state space and solving many model predictive control problems in advance, it can become intractable especially for increasing system dimensions.

Adversarial attacks on controllers of dynamic systems have become a serious threat to many real-world systems, making methods for fast identification of attacks an indispensable part of autonomous systems. With the increasing use of model-based controllers, it is valid to exploit model knowledge also for attack identification as long as privacy of individual components is maintained. A scalable, model-based method to reveal generic attacks was introduced in our previous work and is further investigated here. It is designed for coupled systems with nonlinear dynamics and monitors certain coupling states. Based on the exchange of local sensitivity information, it approximates the propagation of an attack through the network and solves a sparse optimization problem to identify the attack. We provide a thorough derivation of the approach and analyze the involved approximation errors to prove rigorous guarantees for successful identification. In an extensive numerical case study with the IEEE 30 bus power system, we prove that not only the guarantees apply for a nonlinear, practically relevant example, but the method also identifies attacked buses with very high success rates.

Robust model predictive control (MPC) is an essential tool for control systems under uncertainty as it allows for constraint satisfaction even if disturbances occur. When a system suffers malicious attacks, in contrast to parametric uncertainties or known systems faults, it is difficult to specify tight uncertainty ranges within which possible disturbances lie. In this case, very conservative solutions or even infeasible problems are obtained. To address this issue, we propose an adaptively robust MPC scheme that adjusts the uncertainty ranges to available knowledge about the attackers. To this end, we combine a recently proposed method identifying unknown attacks on nonlinear systems with a multi-stage approach to robust MPC. We illustrate the potential of the method in a numerical case study with a distributed nonlinear system.

Tube-enhanced multi-stage nonlinear model predictive control is a robust control scheme that can handle a wide range of uncertainties with reduced conservatism and manageable computational complexity. In this paper, we elaborate on the flexibility of the approach from an application point of view. We discuss the path to making design decisions to implement the novel scheme systematically. We illustrate the critical steps in the design and implementation of the scheme for an industrial example.

Optimal experimental design for parameter precision attempts to maximize the information content in experimental data for a most effective identification of parametric model. With the recent developments in miniaturization and parallelization of cultivation platforms for high-throughput screening of optimal growth conditions massive amounts of informative data can be generated with few experiments. Increasing the quantity of the data means to increase the number of parameters and experimental design variables which might deteriorate the identifiability and hamper the online computation of optimal inputs. To reduce the problem complexity, in this work, we introduce an auxiliary controller at a lower level that tracks the optimal feeding strategy computed by a high-level optimizer in an online fashion. The hierarchical framework is especially interesting for the operation under constraints. The key aspect of this method are discussed together with an in silicostudy considering parallel glucose limited bacterial fed batch cultivations.

Since the earliest conceptualizations by Lee and Markus, and Propoi in the 1960s, Model Predictive Control (MPC) has become a major success story of systems and control with respect to industrial impact and with respect to continued and wide-spread research interest. The field has evolved from conceptually simple linear-quadratic (convex) settings in discrete and continuous time to nonlinear and distributed settings including hybrid, stochastic, and infinite-dimensional systems. Put differently, essentially the entire spectrum of dynamic systems can be considered in the MPC framework with respect to both—system theoretic analysis and tailored numerics. Moreover, recent developments in machine learning also leverage MPC concepts and learning-based and data-driven MPC have become highly active research areas. However, this evident and continued success renders it increasingly complex to live up to industrial expectations while enabling graduate students for state-of-the-art research in teaching MPC. Hence, this position paper attempts to trigger a discussion on teaching MPC. To lay the basis for a fruitful debate, we subsequently investigate the prospect of covering MPC in undergraduate courses; we comment on teaching textbooks; and we discuss the increasing complexity of research-oriented graduate teaching of MPC.

Reinforcement learning aims to compute optimal control policies with the help of data from closed-loop trajectories. Traditional model-free approaches need huge number of data points to achieve an acceptable performance, rendering them not applicable in most real situations, even if the data can be obtained from a detailed simulator. Model-based reinforcement learning approaches try to leverage model knowledge to drastically reduce the amount of data needed or to enforce important constraints to the closed-loop operation, which is another important drawback of model-free approaches. This paper proposes a novel model-based reinforcement learning approach. The main novelty is the fact that we exploit all the information of a model predictive control (MPC) computing step, and not only the first input that is actually applied to the plant, to efficiently learn a good approximation of the state value function. This approximation can be included into a model predictive control formulation as a terminal cost with a short prediction horizon, achieving a similar performance to an MPC with a very long prediction horizon. Simulation results of a discretized batch bioreactor illustrate the potential of the proposed methodology.

The operation of large-scale water distribution networks (WDNs) is a complex control task due to the size of the problem, the need to consider key operational, quality and safety-related constraints as well as because of the presence of uncertainties. An efficient operation of WDNs can lead to considerable reduction in the energy used to distribute the required amounts of water, leading to significant economic savings. Many model predictive control (MPC) schemes have been proposed in the literature to tackle this control problem. However, finding a control-oriented model that can be used in an optimization framework, which captures nonlinear behavior of the water network and is of a manageable size is a very important challenge faced in practice. We propose the use of a data-based automatic clustering method that clusters similar nodes of the network to reduce the model size and then learn a deep-learning based model of the clustered network. The learned model is used within an economic nonlinear MPC framework. The proposed method leads to a flexible scheme for economic robust nonlinear MPC of large WDNs that can be solved in real time, leads to significant energy savings and is robust to uncertain water demands. The potential of the proposed approach is illustrated by simulation results of a benchmark WDN model.

Inertial measurement units are commonly used in a growing number of application fields to track or capture motions of kinematic chains, such as human limbs, exoskeletons or robotic actuators. A major challenge is the presence of magnetic disturbances that results in unreliable magnetometer readings. Recent research revealed that this problem can be overcome by exploitation of kinematic constraints. While typically each segment of the kinematic chain is equipped with an IMU, a novel approach called sparse inertial motion tracking aims at inferring the complete motion states from measurements of a reduced set of sensors. In the present contribution, we combine the magnetometer-free and the sparse approach for real-time motion tracking of double-hinge joint systems with non-parallel joint axes. Analyzing the observability of the system, we find a condition which assures that the relative orientations between all segments are uniquely determined by a kinematic constraint, which contains only the gyroscope readings. Furthermore, we propose a moving-horizon estimator and validate it in a simulation study of four movements, which differ by their degrees of excitation. The results of this study confirm the theoretical conjectures and demonstrate that magnetometer-free sparse inertial real-time motion tracking is feasible under precise and simple excitation conditions.

In this paper, we consider a stochastic Model Predictive Control able to account for effects of additive stochastic disturbance with unbounded support, and requiring no restrictive assumption on either independence nor Gaussianity. We revisit the rather classical approach based on penalty functions, with the aim of designing a control scheme that meets some given probabilistic specifications. The main difference with previous approaches is that we do not recur to the notion of probabilistic recursive feasibility, and hence we do not consider separately the unfeasible case. In particular, two probabilistic design problems are envisioned. The first randomization problem aims to design offline the constraint set tightening, following an approach inherited from tube-based MPC. For the second probabilistic scheme, a specific probabilistic validation approach is exploited for tuning the penalty parameter, to be selected offline among a finite-family of possible values. The simple algorithm here proposed allows designing a single controller, always guaranteeing feasibility of the online optimization problem. The proposed method is shown to be more computationally tractable than previous schemes. This is due to the fact that the sample complexity for both probabilistic design problems depends on the prediction horizon in a logarithmic way, unlike scenario-based approaches which exhibit linear dependence. The efficacy of the proposed approach is demonstrated with a numerical example.

Developing tools for attack identification in large-scale networked control systems is a research area of increasing significance for the secure and reliable operation of autonomous control systems. Due to scalability limits and privacy issues of individual subsystems, attack identification methods should not rely on global model knowledge. We address systems of interconnected nonlinear subsystems with coupled dynamics or constraints in a distributed control setup. The local controllers share information about the coupling variables of the subsystems and are designed to be robust towards attacks and uncertain influences through neighboring subsystems. We present a scalable hierarchical attack identification method which monitors the evolution of the coupling variables after an attack occurred in some unknown subsystem. Based on the mutual exchange of local sensitivity information among the subsystems, the propagation of the attack through the network is approximated. The propagation equations are used to formulate a quadratic program whose solution determines the attack signal that explains the observed network evolution best. The developed approach is applied to the IEEE 30 bus system to illustrate attack identification in power systems with faulty buses.

Polymerization processes are typically characterized by strongly exothermic reactions with complicated nonlinear kinetics, and they usually require a very precise temperature control to maintain product uniformity. Polymerization reactors are usually operated with a constant monomer feed, while a cascade of PI controllers manipulates the heating and cooling power to keep the reaction temperature within the tolerance limits. Because a robust operation has to be guaranteed for different products under various disturbing influences, for example, fouling in the reactor, oscillations in the supply of the utilities and feed impurities, the monomers are usually slowly dosed into the reactor, leading to large batch times and limiting the process productivity. In this work, a simple and efficient control scheme for the time optimal operation of exothermic semibatch reactors that can be easily implemented in industry is introduced and evaluated by means of simulations using a well-known industrial benchmark problem. The results show that the new control scheme can shorten the reaction period by more than fifty percent without jeopardizing the quality of the temperature control and performs more robustly than a fully model-based controller that employs a fixed nominal process model.

With the growing share of renewable energy sources, the uncertainty in power supply is increasing. In addition to the inherent fluctuations in the renewables, this is due to the threat of deliberate malicious attacks, which may become more prevalent with a growing number of distributed generation units. Also in other safety-critical technology sectors, control systems are becoming more and more decentralized, causing the targets for attackers and thus the risk of attacks to increase. It is thus essential that distributed controllers are robust toward these uncertainties and able to react quickly to disturbances of any kind. To this end, we present novel methods for model-based identification of attacks and combine them with distributed model predictive control to obtain a resilient framework for adaptively robust control. The methodology is specially designed for distributed setups with limited local information due to privacy and security reasons. To demonstrate the efficiency of the method, we introduce a mathematical model for physically coupled microgrids under the uncertain influence of renewable generation and adversarial attacks, and perform numerical experiments, applying the proposed method for microgrid control.

The multi-stage MPC approach models the evolution of state trajectories for different realizations of the uncertainty as a scenario tree and considers the availability of feedback information explicitly in the predictions and in the computation of the control moves. Since the structure of the feedback policy is not restricted, multi-stage MPC is less conservative when compared to approaches that use e.g. affine feedback policies. The advantages come at the cost of rapid growth in problem size with respect to the number of uncertainties and the prediction horizon. In this paper, we propose to use affine control policies for small disturbances and the multi-stage approach for large uncertainties to obtain a better trade-off between optimality and complexity than the existing robust MPC schemes. The proposed scheme mitigates the problem of the increase in problem complexity with respect to the number of uncertainties at the price of only a small loss of performance. We show that the proposed scheme is robustly exponentially stable and demonstrate its advantages for a CSTR example.

Dealing with uncertainties is one of the most challenging issues that prevent nonlinear model predictive control (NMPC) from being a widespread reality. Many different robust schemes have been presented recently, such as multi-stage NMPC, in which the uncertainty is represented as a scenario tree. While multi-stage NMPC achieves promising performance in practice, it suffers from an exponential increase in complexity with the number of uncertainties considered making its real-time application difficult for large case studies. We suggest in this work to use multi-stage NMPC as a generator of data pairs that are used to learn the robust NMPC policy by means of deep neural networks. This choice is motivated by recent practical successes of deep learning and theoretical results that explain the improved representation capabilities of deep networks with respect to shallow networks. We present empirical evidence which shows that the use of deep neural networks with many hidden layers as opposed to shallow networks with only one significantly improves the learning process of a robust NMPC control law. These findings are illustrated with simulation studies of an industrial polymerization reactor.

In this paper we present an implicit dual robust nonlinear model predictive control (NMPC) in the framework of (bounded-error) guaranteed parameter estimation and multi-stage NMPC, which uses a scenario tree to represent propagation of parametric model uncertainty through a dynamic system. The proposed implicit dual control scheme excites the system if the excitation signals improve the overall performance of the controller. A box (over-)approximation of the solution set of guaranteed parameter estimation which encloses the true parameter values is obtained by solving an optimization problem. The proposed approach uses approximations of the future solution sets and updates the scenario tree of the multi-stage NMPC along the prediction horizon accordingly. This gives rise to a bi-level optimization problem, which is solved as a single-level problem by implicitly solving the lower-level problem using its KKT conditions. The advantages of the proposed approach over the standard multi-stage NMPC are demonstrated for a linear and nonlinear (semi-batch reactor) simulation case studies.

The H2-optimal static output feedback control design for linear time-invariant systems with time-invariant probabilistic parametric uncertainties subject to disturbances and noise is formulated in terms of an optimization with a highly sparse bilinear matrix inequality (BMI) constraint. The BMI formulation is derived by employing polynomial chaos expansions to represent the effect of the parametric uncertainties on the H2norm. The effect of using a finite number of terms in the expansions is discussed and addressed by incorporating a robustifying parameter in the BMI formulation. In a case study, the feedback controller converges using a sixth-order expansion, with a substantial improvement in robustness to parametric uncertainty.

Compared to other sectors, industrial biotechnology only employs a limited number of automatic control strategies. The applied control strategies are mainly recipe-based and often do not take directly the high variability and flexibility of biological systems into account. While this provides sharply defined conditions and quality by design guarantees, process are operated significantly below the optimum, with inefficient product yields and limited reproducibility. The information that is available online, the data from previous batches and the additional biological knowledge are not considered, usually because the integration of this information requires the use of control strategies with an adaptive character, which increases the the design complexity and the implementation cost.

We propose an output feedback model predictive control (MPC) scheme that is independent of the estimation method employed. The proposed approach is non-conservative by design as the feedback information subject to estimation errors is explicitly considered in the predictions. The proposed scheme is robust to both the plant-model mismatch and the estimation error and avoids the unnecessary conservatism introduced by other existing output feedback MPC approaches in the literature. We achieve this by making use of all the available information at every time-step and formulate a control policy with the assumption that bounds for the estimation error are available at all times. We show that the proposed scheme is recursively feasible and demonstrate its advantages using a double-integrator example.

Robust optimal control of pharmacokinetic/pharmacodynamic (PK/PD) models allows for optimal drug dosage design under model uncertainties. Typical PK/PD models are highly uncertain and therefore a robust design of the drug dosage is necessary to guarantee that important health-related constraints are satisfied for all the possible values of the uncertainty. This paper shows that the responses of a simple class of PK/PD models are monotonous in the parameters and in the input. This greatly simplifies the design of a scenario-based nonlinear model predictive controller as just two scenarios are necessary to fully bound the system responses. This is illustrated with the drug dosage design for a model of erythropoietin injection, in which we also show the potential benefit of having intermediate measurements during the application of the therapy.

This manuscript studies the monotonicity of multi-step ligand-receptor signalling motifs. Monotonicity with respect to parameters and state perturbations allow not only to exclude periodic solutions, but also to easily bound the responses in cases of bounded perturbations.

Multi-stage Nonlinear Model Predictive Control (NMPC) is a promising strategy for the design of robust NMPC controllers which is based on describing the evolution of the uncertainty as a scenario tree. The scenario tree makes it possible to consider explicitly that the future control inputs can be adapted to the future information (measurements), thus reducing the conservativeness of the robust approach. This paper reviews the multi-stage approach and illustrates its main advantages using a nonlinear CSTR example. We also provide guidelines for possible multi-stage NMPC users that could help to identify the problems where the use of multi-stage NMPC can result in a significant improvement with respect to standard NMPC or other robust NMPC approaches. Finally, we summarize the different modifications that can be done to the multi-stage approach to enhance its performance. The possible enhancements include: improved performance using parameter estimation, rigorous guarantee of constraint satisfaction, and stability guarantees for the case of discrete-valued uncertainties.

Nonlinear Model predictive control (NMPC) is a popular control strategy for highly nonlinear chemical processes. The ability to handle safety and environmental constraints along with the use of an economic objective makes NMPC highly appealing to industries. The performance of NMPC depends strongly on the accuracy of the model. In reality, there always are plant-model mismatch and state estimation errors. Hence the NMPC controller must be robust to uncertainties in the model as well as against estimation errors. Among the several approaches presented in the literature, the scenario-tree based multi-stage NMPC approach is a non-conservative and efficient formulation. In this approach, the evolutions of the plant for different realizations of the uncertainties are considered as different scenarios and the optimization problem is formulated as a multi-stage stochastic programming problem with recourse. In this work, we consider multi-stage output feedback NMPC using the Unscented Kalman Filter (UKF) where the nonlinearities are represented using deterministically chosen sigma points for state estimation. In the control problem, we explicitly consider the UKF estimation equations to predict the future evolution of the system. The proposed approach is illustrated by simulation results of fed-batch chemical reactor with an economic cost function.

We address the problemof controlling interconnected, possibly large-scale systems of systems using distributed model predictive control. We consider the case in which the nonlinear subsystems can be coupled physically and can have shared constraints. The presented approach is based on the transmission of contracts between neighboring subsystems. Contracts are guaranteed sequences of possible future trajectories or trajectory sets of the coupling variables of the subsystems. We derive for the approach sufficient conditions for guaranteeing recursive feasibility and Input-to-State stability. Furthermore, we discuss the case of the so-called Plug & Play operations in which a subsystem in the network is replaced by a new, possibly different, one and when a subsystem is removed or added to the network.

One way to ensure recursive feasibility, stability and performance of Nonlinear Model Predictive Control is the combined use of a terminal region and a terminal cost. However, finding suitable combinations of the terminal cost and terminal region that guarantee closed-loop stability for nonlinear systems is in general challenging. Most existing methods are either based on the linearized system dynamics and a linear feedback, or assume that a control Lyapunov function for the system close to the origin is know. This paper proposes the use of higher order approximations of the optimal feedback and optimal cost of the infinite horizon problem via Al’brekht’Method to determine a suitable terminal region for polynomial systems. To do so, the stability conditions are reformulated in terms of a sumof-squares problem which is iteratively used to determine the terminal region. For a nonlinear chemical reactor example it is shown that the proposed approach leads to a larger terminal region and an improved performance compared to existing approaches.

Nonlinear Model Predictive control (NMPC) is one of the advanced control strategies for multi-dimensional nonlinear systems with constraints. With uncertainties present in the model, robust NMPC strategies are proposed in order to counteract the effects of the uncertainties and have a safe operation of the plant.Multi-stage NMPC offers a non-conservative alternative as it models the feedback information explicitly in the problem formulation by means of a scenario tree. In order to be robust to both the model uncertainties and the estimation error, we formulate a multi-stage output feedback NMPC strategy by creating additional scenarios by sampling the innovations and use observer equations to predict the future evolution of the plant. Since the observers such as the Extended Kalman Filter (EKF) can be used to estimate the uncertain parameters along with the states, the output feedback NMPC strategy is improved to be adaptive with respect to time varying uncertain parameters and the performance of the controller is improved. We demonstrate the advantages of the proposed adaptive scheme using a nonlinear Continuous Stirred Tank Reactor (CSTR) example.

Nonlinear Model Predictive Control (NMPC) employs a plant model to compute a sequence of optimal control inputs for a finite horizon. As, in reality, there always exists a plant-model mismatch and not all states of the plant can be measured, the NMPC scheme must be robust to plant uncertainties and to estimation errors. Different robust NMPC strategies have been proposed to deal with these uncertainties. Among them, a multi-stage NMPC, which is based upon a scenario tree of future plant evolutions, is less conservative compared to worst-case open-loop approaches because the presence of feedback at future sampling instants is explicitly considered. In multi-stage output feedback NMPC, additional scenarios are created by sampling the innovations that are used to estimate the future states of the plant along the scenario tree. In this paper, we refine our previously published approach to include state estimation errors. Moreover we extend the scheme to guarantee robust constraint satisfaction by calculating reachable sets using Taylor models. The method is demonstrated for a nonlinear chemical process example.

In this paper we present a systematic and efficient approach to deal with uncertainty in Nonlinear Model Predictive Control (NMPC). The main idea of the approach is to represent the NMPC setting as a real-time decision problem under uncertainty that is formulated as a multi-stage stochastic problem with recourse, based on a description of the uncertainty by a scenario tree. This formulation explicitly takes into account the fact that new information will be available in the future and thus reduces the conservativeness compared to open-loop worst-case approaches. We show that the proposed multistage NMPC formulation can deal with significant plant-model mismatch as it is usually encountered in the process industry and still satisfies tight constraints for the different values of the uncertain parameters, in contrast to standard NMPC. The use of an economic cost function leads to a superior performance compared to the standard tracking formulation. The potential of the approach is demonstrated for an industrial case study provided by BASF SE in the context of the European Project EMBOCON. The numerical solution of the resulting large optimization problems is implemented using the optimization framework CasADi.

This paper studies the reduction of the conservativeness of robust nonlinear model predictive control (NMPC) via the reduction of the uncertainty range using guaranteed parameter estimation. Optimal dynamic experiment design is formulated in the framework of robust NMPC in order to obtain probing inputs that maximize the information content of the feedback and simultaneously to guarantee the satisfaction of the process constraints. We propose a criterion for optimal experiment design which provides a minimization of parameter uncertainty in the direction of improved performance of the process under robust economic NMPC. A case study from the chemical engineering domain is studied to show the benefits of the proposed approach.