Keywords: Economic Predictive Control, Stability and Recursive Feasibility
Abstract: The analysis of convergence and stability conditions for Non-linear Model Predictive Control (NMPC) schemes has seen significant progress during the last decades. Numerous results in the literature state textit{sufficient} conditions for convergence and/or stability of NMPC. Yet, textit{necessary and sufficient} conditions are---to the best of the author's knowledge---not known and, given the variety of choices for MPC design, seemingly appear to be difficult.

This paper analyzes the convergence of NMPC of continuous-time systems in time-invariant settings. We introduce the receding-horizon Hamiltonian---i.e., the value of the optimal control Hamiltonian along the sequence of OCPs---as a novel tool for stability/convergence analysis. Based on the assumption of strict dissipativity, we prove necessary and sufficient conditions for asymptotic convergence of the closed loop to the optimal steady state. Numerical results obtained for the Van de Vusse reactor illustrate the proposed conditions.

Keywords: Process Control, Output Feedback Predictive Control, Stability and Recursive Feasibility
Abstract: Predictive functional control (PFC) is the simplest model-based algorithm, equipped with the attributes of a fully fledged predictive controller but at the cost and complexity threshold of a standard PID regulator. It has proven benefits in controlling stable SISO dynamic systems, but similarly to its competitor PID, it loses efficacy when a challenging application is introduced. In this paper, we present a modified PFC approach, especially tailored for open-loop unstable processes, using pre-stabilisation to efficiently control the undesirable dynamics at hand. This is essentially a two-stage design scheme with implications for PFC tuning and constraint handling. The proposal, nevertheless, is straightforward and intuitive, and provides improved closed-loop control in the presence of external perturbations against the standard PFC, and significantly better performance overall compared to the common PID algorithm, as demonstrated in a numerical case-study.

Keywords: Stability and Recursive Feasibility
Abstract: The stability analysis of NMPC schemes has seen significant progress in recent years, which includes schemes with and without terminal ingredients such as penalties and constraints. In the context of economic MPC, turnpike properties, which are closely related to dissipativity properties of the underlying optimal control problem, have enabled novel insights on stability conditions. In the present note, we show that turnpike properties naturally enable to bound the required stabilizing horizon length in MPC. The main idea is to define the turnpike with respect to a level-set of the terminal penalty. This way we derive a bound on the stabilizing horizon which guarantees that a terminal constraint is satisfied without being explicitly stated in the underlying optimal control problem. A numerical example indicates that if the terminal set is not too small, then the horizon bound is not overly conservative.

Keywords: Stability and Recursive Feasibility, Output Feedback Predictive Control, Learning and Predictive Control
Abstract: Stabilizing conditions for nonlinear predictive control typically rely on standard Lyapunov functions and thus require a monotonically decreasing cost function. These conditions cannot certify stability of predictive controllers in the presence of non--monotonic cost functions. In this paper we develop new dissipativity-based stabilizing conditions for nonlinear predictive control that allow for non--monotonic cost functions. Firstly, we establish that dissipation inequalities with a cyclically negative supply imply asymptotic stability. Secondly, we show that closed-loop trajectories generated by predictive control satisfy a fundamental dissipation inequality. This enables dissipativity-based stabilizing conditions that do not require a special terminal cost and apply to both model-based and data-driven predictive control algorithms.

Keywords: Stability and Recursive Feasibility, Tracking and Path Following Predictive Control
Abstract: In this paper, we provide a stability and performance analysis of model predictive control (MPC) schemes based on finite-tail costs. We study the MPC formulation originally proposed by Magni et al. (2001) wherein the standard terminal penalty is replaced by a finite-horizon cost of some stabilizing control law. In order to analyse the closed loop, we leverage the more recent technical machinery developed for MPC without terminal ingredients. For a specified set of initial conditions, we obtain sufficient conditions for stability and a performance bound in dependence of the prediction horizon and the extended horizon used for the terminal penalty. The main practical benefit of the considered finite-tail cost MPC formulation is the simpler offline design in combination with typically significantly less restrictive bounds on the prediction horizon to ensure stability. We demonstrate the benefits of the considered MPC formulation using the classical example of a four tank system.

Keywords: Hybrid Model Predictive Control, Optimization and Model Predictive Control
Abstract: We propose an efficient algorithm for the optimal control problems (OCPs) of nonlinear switched systems that optimizes the control input and switching instants simultaneously for a given switching sequence. We consider the switching instants as the optimization variables and formulate the OCP based on the direct multiple shooting method. We derive a linear equation to be solved in Newton’s method and propose a Riccati recursion algorithm to solve the linear equation efficiently. The computational time of the proposed method scales linearly with respect to the number of time stages of the horizon as the standard Riccati recursion. Numerical experiments show that the proposed method converges with a significantly shorter computational time than the conventional methods.

Keywords: Optimization and Model Predictive Control
Abstract: We study a property of dynamic optimization (DO) problems (as those encountered in model predictive control and moving horizon estimation) that is known as exponential decay of sensitivity (EDS). This property indicates that the sensitivity of the solution at stage i against a data perturbation at stage j decays exponentially with |i-j|. Building upon our previous results, we show that EDS holds under uniform boundedness of the Lagrangian Hessian, a uniform second order sufficiency condition (uSOSC), and a uniform linear independence constraint qualification (uLICQ). Furthermore, we prove that uSOSC and uLICQ can be obtained under uniform controllability and observability. Hence, we have that uniform controllability and observability imply EDS. These results provide insights into how perturbations propagate along the horizon and enable the development of approximation and solution schemes. We illustrate the developments with numerical examples.

Keywords: Real-Time Implementation of Model Predictive Control, Optimization and Model Predictive Control
Abstract: This paper investigates how various caching strategies can reduce the computational effort of the active set method (ASM) applied to solve constrained model predictive control problems with quadratic objective function and linear constraints. Specifically, we show that during closed-loop operation, the active set method often re-visits the same combination of active constraints while searching for optimal control inputs by factoring Karush-Kuhn-Tucker (KKT) systems. By storing the factors of the corresponding KKT system in a cache, these repetitive calculations can be simplified to a mere cache search and evaluation of the appropriate factors. Since the cache memory is typically fairly restricted, the efficiency of the scheme depends on how well the cache space can be utilized. In particular, when the cache is fully utilized, and a new element needs to be stored, the cache replacement policy needs to determine which element should be removed from the cache to make space for the new one. In the paper, we scrutinize various cache replacement policies and how well they work as a function of the cache size. The results show that by using a cache of modest size, the number of computational operations performed by the ASM can be reduced by up to 80%, thus significantly accelerating the implementation of model predictive control.

Keywords: Optimization and Model Predictive Control, Distributed Model Predictive Control
Abstract: In this paper we present a distributed second-order augmented Lagrangian method for distributed model predictive control. We distribute the computations for search direction, step size, and termination criteria over what is known as the clique tree of the problem and calculate each of them using message passing. The algorithm converges to its centralized counterpart and it requires fewer communications between sub-systems as compared to algorithms such as the alternating direction method of multipliers. Results from a simulation study confirm the efficiency of the framework.

Keywords: Optimization and Model Predictive Control
Abstract: Certain feasible optimal control problems feature properties that make them infeasible upon discretization with conventional direct transcription schemes. We present and discuss examples of such problems, supported with numerical experiments, and show that they can be solved successfully using a penalty-discretization method.

Keywords: Automotive
Abstract: Objective of the vehicle thermal management system is to achieve precise coolant temperature control that ensures operation of the combustion engine in the region with the maximum thermal efficiency. In this study it is achieved by a combined effect of coolant temperature control algorithm that is operating an advanced electronic valve with one input and three outputs that balances distribution of the coolant flow between the heat source and sink components. The objective is to present an application of a nonlinear model predictive control (NMPC) approach for coolant temperature control. It is a simulation study to quantify benefits of precise coolant temperature control with the possible inclusion of preview information.

Keywords: Energy Efficient Buildings, Hybrid Model Predictive Control
Abstract: We address the problem of operating up- and down-time restricted components with activation thresholds using linear predictive control instead of hybrid predictive control. Accounting for operational requirements of this kind usually requires optimal control problems with discrete variables, which are computationally demanding. As a workaround, run-time requirements are often ignored in the control design and enforced independently from the optimal input value. We present conditions and an algorithm that ensures minimum run-time requirements in linear MPC. The control performance for a heating application is compared to linear MPC with rule-based post-processing and hybrid MPC. The results show that our approach ensures the up-and down-time requirements at a high control performance and is significantly faster than hybrid MPC.

Keywords: Fluid Dynamics, Process Control
Abstract: This paper describes the development of a control system for an industrial heating application. In this process a moving substrate is passing through a heating zone with variable speed. Heat is applied by hot air to the substrate with the air flow rate being the manipulated variable. The aim is to control the substrate’s temperature at a specific location after passing the heating zone. First, a model is derived for a point attached to the moving substrate. This is modified to reflect the temperature of the moving substrate at the specified location. In order to regulate the temperature a nonlinear model predictive control approach is applied using an implicit Euler scheme to integrate the model and an augmented gradient based optimization approach. The performance of the controller has been validated both by simulations and experiments on the physical plant. The respective results are presented in this paper.

Keywords: Process Control, Scheduling and Manufacturing, Power Electronics
Abstract: This contribution presents the application of nonlinear model predictive control to the Vertical Gradient Freeze crystal growth process. Due to the time-varying spatial extent of the crystal and melt during growth, this process is characterised by two coupled free boundary problems that form a so called two-phase Stefan problem which is of nonlinear nature. To apply model predictive control to this process, a simplified, spatially distributed representation of the system is derived and transferred into a spatially lumped form by means of the finite element method. For this model, a nonlinear control problem is formulated, that takes process limitations into account and tries to satisfy different quality objectives by formulating demands on the systems spatiotemproal temperature distribution. This provides the foundation for the presented predictive control design. Finally, the approximated model and the controller are verified for different real-world scenarios that include model errors and parameter uncertainties.

Keywords: Optimization and Model Predictive Control

Keywords: Stability and Recursive Feasibility
Abstract: In this paper, we propose a suboptimal moving horizon estimator for nonlinear systems. For the stability analysis we transfer the “feasibility-implies-stability/robustness” paradigm from model predictive control to the context of moving horizon estimation in the following sense: Using a suitably defined, feasible candidate solution based on the trajectory of an auxiliary observer, robust stability of the proposed suboptimal estimator is inherited independently of the horizon length and even if no optimization is performed.

Keywords: Real-Time Implementation of Model Predictive Control, Optimization and Model Predictive Control, Healthcare
Abstract: Model Predictive Control (MPC) is a successful control methodology, which is applied to increasingly complex systems. However, real-time feasibility of MPC can be challenging for complex systems, certainly when an (extremely) large number of constraints have to be adhered to. For such scenarios with a large number of state constraints, this paper proposes two novel MPC schemes for general nonlinear systems, which we call constraint-adaptive MPC. These novel schemes dynamically select at each time step a (varying) set of constraints that are included in the on-line optimization problem. Carefully selecting the included constraints can significantly reduce, as we will demonstrate, the computational complexity with often only a slight impact on the closed-loop performance. Although not all (state) constraints are imposed in the on-line optimization, the schemes still guarantee recursive feasibility and constraint satisfaction. A numerical case study illustrates the proposed MPC schemes and demonstrates the achieved computation time improvements exceeding two orders of magnitude without loss of performance.

Keywords: Optimization and Model Predictive Control

Keywords: Optimization and Model Predictive Control
Abstract: 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.

Keywords: Biological Systems, Healthcare
Abstract: We present an approach for state and parameter estimation in retinal laser treatment by a novel setup where both measurement and heating is performed by a single laser. In this medical application, the temperature that is induced by the laser in the patient’s eye is critical for a successful and safe treatment. To this end, we pursue a model-based approach using a model given by a heat diffusion equation on a cylindrical domain, where the source term is given by the absorbed laser power. The model is parametric in the sense that it involves an absorption coefficient, which depends on the treatment spot and plays a central role in the input-output behavior of the system. After discretization, we apply a particularly suited parametric model order reduction to ensure real-time tractability while retaining parameter dependence. We augment known state estimation techniques, i.e., extended Kalman filtering and moving horizon estimation, with parameter estimation to estimate the absorption coefficient and the current state of the system. Eventually, we show first results for simulated and experimental data from porcine eyes. We find that, regarding convergence speed, the moving horizon estimation slightly outperforms the extended Kalman filter on measurement data in terms of parameter and state estimation, however, on simulated data the results are very similar.

Keywords: Explicit Model Predictive Control, Output Feedback Predictive Control
Abstract: Designing predictive controllers for systems with computationally limited embedded hardware, e.g. for autonomous vehicles, requires solving an optimization problem in real-time taking the vehicle dynamics and constraints into account. Furthermore, often the controller needs to be available in explicit form for verification and validation purposes and should only exploit the available output measurement. We propose an approximation of a special nonlinear model predictive output-feedback formulation considering the infinite-horizon case. The main idea is to offline derive an approximated explicit solution of the underlying Hamilton–Jacobi–Bellman equation. The resulting feedback control law is polynomial in terms of the measurements and estimated parameters. Therefore, the online evaluation can be efficiently implemented. The optimal control law is parameterized in terms of the varying parameters which can be updated/learned online. We provide a proof of convergence and existence of the explicit solution and compare the proposed approximated nonlinear controller to a finite-horizon nonlinear model predictive controller considering the control of a quadcopter subject to wind disturbances.

Keywords: Learning and Predictive Control, Output Feedback Predictive Control, Stability and Recursive Feasibility
Abstract: We present a model predictive control (MPC) scheme to control linear time-invariant systems using only measured input-output data and no model knowledge. The scheme includes a terminal cost and a terminal set constraint on an extended state containing past input-output values. We provide an explicit design procedure for the corresponding terminal ingredients that only uses measured input-output data. Further, we prove that the MPC scheme based on these terminal ingredients exponentially stabilizes the desired setpoint in closed loop. Finally, we illustrate the advantages over existing data-driven MPC approaches with a numerical example.

Keywords: Optimization and Model Predictive Control, Real-Time Implementation of Model Predictive Control
Abstract: This paper presents an approach to design nonlinear control policies for the class of systems with nonlinear steady-state characteristics while assuming constant dynamic behavior. We present an approach where the nonlinear steady-state characteristics is approximated with a nonlinear function. Such a model then serves as the design model for the nonlinear model predictive control strategy. The validity of such a control approach is experimentally implemented on a laboratory scale device.

Keywords: Output Feedback Predictive Control, Optimization and Model Predictive Control, Real-Time Implementation of Model Predictive Control
Abstract: In this paper, we develop an efficient algorithmic implementation of output feedback control of discrete-time linear systems with state and input constraints that is based on model predictive control (MPC) and moving horizon estimation (MHE). We present an iteration scheme which combines previously proposed proximity-based MHE and relaxed barrier function based MPC algorithms in a certainty equivalence output feedback fashion. The resulting overall closed-loop system is shown to converge to the origin under mild assumptions and simulation examples are used to demonstrate the advantages of the proposed approach.

Keywords: Process Control, Output Feedback Predictive Control, Learning and Predictive Control
Abstract: For many complex processes, it is desirable to use a nonlinear model in the MPC design, and the recently proposed Dynamic Response Surface Methodology (DRSM) is capable of accurately modeling nonlinear continuous processes over semi-infinite time horizons. We exploit the DRSM to identify nonlinear data-driven dynamic models that are used in an NMPC. We demonstrate the ability and effectiveness of the DRSM data-driven model to be used as the prediction model for a nonlinear MPC regulator. This DRSM model is efficiently used to solve a non-equally-spaced finite-horizon optimal control problem so that the number of decision variables is reduced. The proposed DRSM-based NMPC is tested on a representative nonlinear process, an isothermal CSTR in which a second-order irreversible reaction is taking place. It is shown that the obtained quadratic data-driven model accurately represents the open-loop process dynamics and that DRSM-based NMPC is an effective data-driven implementation of nonlinear MPC.