# Model Predictive Control

Model predictive control (MPC), also referred to as moving horizon control or receding horizon control, is one of the most successful and most popular advanced control methods. The basic idea of MPC is to predict the future behavior of the controlled system over a finite time horizon and compute an optimal control input that, while ensuring satisfaction of given system constraints, minimizes an a priori defined cost functional. To be more precise, the control input is calculated by solving at each sampling instant a finite horizon open-loop optimal control problem; the first part of the resulting optimal input trajectory is then applied to the system until the next sampling instant, at which the horizon is shifted and the whole procedure is repeated again. MPC is in particular successful due to its ability to explicitly incorporate hard state and input constraints as well as a suitable performance criterion into the controller design.

The research at the IST focuses on the development and the analysis of MPC schemes for which desired closed-loop properties can be established. These include closed-loop stability and performance guarantees, robustness with respect to uncertainties and disturbances and the development of distributed control structures. Another reasearch topic at the IST is the design of efficient MPC methods, in terms of computation as well as communication in the control system.

**Contact Persons:**Frank Allgöwer, Florian Bayer, Christian Ebenbauer, Johannes Köhler, Philipp Köhler, Matthias Lorenzen, Matthias Müller, Fabian Pfitz

###### Please find below all our recent MPC research fields at the Institute for Systems Theory and Automatic Control.

## Robust MPC

In Robust MPC, the goal is to stabilize uncertain systems or systems that are affected by disturbances. The behavior of such systems cannot be exactly predicted. If bounds on the uncertainty are known, however, it is possible to determine bounds on the future system behavior. Feasibility and stability can be established using set-theoretic methods. The drawback of many existing robust predictive control schemes is a high computational load due to the complexity involved in such set-valued predictions as well as conservatism induced by these methods. Research at the IST focusses on finding appropriate and tighter bounds for the uncertain future behavior for both linear and nonlinear systems and the development of efficient interpolation methods based on convex optimization.

**Contact Persons:**Frank Allgöwer, Florian Bayer

**Publications:**

- F. A. Bayer, M. Bürger, and F. Allgöwer.

Discrete-time Incremental ISS: A Framework for Robust NMPC.

In Proc. of the European Control Conference (ECC), Zürich, Switzerland, 2013, pp. 2068-2073. - F. D. Brunner, M. Lazar, and F. Allgöwer.

An Explicit Solution to Constrained Stabilization via Polytopic Tubes.

In Proc. of the 52nd IEEE Conference on Decision and Control (CDC), Florence, Italy, 2013, pp. 7721-7727. - F. D. Brunner, M. Lazar, and F. Allgöwer.

Stabilizing Linear Model Predictive Control: On the Enlargement of the Terminal Set.

International Journal of Robust and Nonlinear Control, 2015, in press. - F. D. Brunner and F. Allgöwer.

Approximate Predictive Control of Polytopic Systems.

In Proc. of the 19th IFAC World Congress, Cape Town, South Africa, August 2014, pp. 11060-11066.

**Cooperations:**

- Mircea Lazar, Dapartment of Electrical Engineering, University of Technology, Eindhoven, The Netherlands

## Economic MPC

In standard MPC formulations, the considered control objective is typically the stabilization of some (given) setpoint or trajectory to be tracked. In contrast, the main focus in economic MPC is on closed-loop performance. This means that some general cost function is employed within the repeatedly solved optimal control problem which need not be positive definite with respect to some setpoint, resulting in the fact that the closed-loop system may not be convergent. The naming and original motivation for economic MPC stem from the process industry, where the general cost function is usually related to the profit to be maximized. Research at the IST studies different system theoretic properties of economic MPC such as a classification of optimal operational regimes or the effect of disturbances on the achievable performance; furthermore, economic MPC algorithms are developed for which desired closed-loop properties can be guaranteed, such as average performance bounds and the satisfaction of average constraints.

**Contact Persons:**Frank Allgöwer, Florian Bayer, Matthias A. Müller

**Publications:**

- M. A. Müller and L. Grüne.

Economic model predictive control without terminal constraints for optimal periodic behavior.

Automatica, vol. 70, pp. 128-139, 2016. - M. A. Müller, D. Angeli, and F. Allgöwer.

On necessity and robustness of dissipativity in economic model predictive control.

IEEE Transactions on Automatic Control, vol. 60, no. 6, pp. 1671-1676, 2015. - F. A. Bayer, M. Lorenzen, M. A. Müller, and F. Allgöwer

Improving Performance in Robust Economic MPC Using Stochastic Information.

Proc. 5th IFAC Conference on Nonlinear Model Predictive Control, Seville, Spain, 2015, pp. 411-416. - F. A. Bayer, M. A. Müller, and F. Allgöwer.

Tube-based Robust Economic Model Predictive Control.

Journal of Process Control, vol. 24, no. 8, pp. 1237-1246, 2014. Special Issue on Economic MPC. - M. A. Müller, D. Angeli, F. Allgöwer, R. Amrit, and J. B. Rawlings.

Convergence in economic model predictive control with average constraints.

Automatica, vol. 50, no. 12, pp. 3100-3111, 2014. - M. A. Müller, D. Angeli, and F. Allgöwer.

Transient average constraints in economic model predictive control.

Automatica, vol. 50, no. 11, pp. 2943-2950, 2014. - M. A. Müller, D. Angeli, and F. Allgöwer.

Economic model predictive control with self-tuning terminal cost.

European Journal of Control, vol. 19, pp. 408-416, 2013. Special Issue for ECC 2013.

**Cooperations:**

- David Angeli, Imperial College, London, UK
- Lars Grüne, University of Bayreuth, Germany
- James B. Rawlings and Rishi Amrit, University of Wisconsin-Madison, USA

## Distributed MPC

In large-scale dynamical systems and networks of cooperating systems, it is often not possible or desirable to control the overall system with one centralized controller. Hence in recent years, the field of distributed MPC has gained significant attention, where each of the subsystems ∑_{i} is locally controlled by an MPC controller and exchanges information about predicted trajectories with its neighbors, in order to ensure satisfaction of coupling constraints and achievement of the overall control objective. At the IST, distributed MPC schemes are developed which besides the classical goal of setpoint stabilization are also suited for more general cooperative control tasks, including for example consensus and synchronization problems.

**Contact Persons:**Frank Allgöwer, Matthias A. Müller, Philipp Köhler

**Publications:**

- P. N. Köhler, M. A. Müller, and F. Allgöwer.

A distributed economic MPC scheme for coordination of self-interested systems.

In Proc. of the American Control Conference, Boston, MA, USA, 2016, pp. 889–894. - M. A. Müller, and F. Allgöwer.

Distributed economic MPC: a framework for cooperative control problems.

In Proc. of the 19th IFAC World Congress, Cape Town, South Africa, 2014, pp.1029-1034. - M. A. Müller, M. Reble, and F. Allgöwer.

Cooperative control of dynamically decoupled systems via distributed model predictive control.

International Journal of Robust and Nonlinear Control, vol. 22, no. 12, pp. 1376-1397, 2012. - M. A. Müller, B. Schürmann, and F. Allgöwer.

Robust cooperative control of dynamically decoupled systems via distributed MPC.

In Proc. of the IFAC Conference on Nonlinear Model Predictive Control, Noordwijkerhout, the Netherlands, 2012, pp. 412-417.

## Unconstrained MPC

The most well-studied MPC approaches with guaranteed stability use a control Lyapunov function as terminal cost. Since the actual calculation of such a function can be difficult, it is desirable to replace this assumption by a less restrictive controllability assumption. For discrete-time systems, the latter assumption has been used in the literature for the stability analysis of so-called unconstrained MPC, i.e., MPC without terminal cost and terminal constraints. At the IST, we have extended these results to continuous-time systems. Starting from this result, we have developed novel alternative MPC formulations based on combinations of the controllability assumption with terminal cost and terminal constraints. Thereby, we have shown connections of our results to previous MPC schemes and and gained insight in the advantages of the different schemes. One of the main contributions is the development of a unifying MPC framework which allows to consider both MPC schemes with terminal cost and terminal constraints as well as unconstrained MPC as limit cases of our framework.

**Contact Persons:**Frank Allgöwer, Matthias A. Müller

**Publications:**

- M. Reble.

Model Predictive Control for Nonlinear Continuous-Time Systems with and without Time-Delays.

Doctoral Thesis, Institute for Systems Theory and Automatic Control, University of Stuttgart, Germany, 2013. - M. Reble and F. Allgöwer.

Unconstrained Model Predictive Control and Suboptimality Estimates for Nonlinear Continuous-Time Systems.

Automatica, Vol. 48, No. 8, pp. 1812-1817, 2012. - K. Worthmann, M. Reble, L. Grüne, and F. Allgöwer.

The Role of Sampling for Stability and Performance in Unconstrained Nonlinear Model Predictive Control.

SIAM Journal on Control and Optimization, Vol. 52, No. 1, pp. 581-605, 2014. - M. Reble, D. E. Quevedo, and F. Allgöwer.

A Unifying Framework for Stability in MPC using a Generalized Integral Terminal Cost.

In Proc. of the American Control Conference (ACC), Montreal, Canada, June 2012, pp. 1211-1216. - M. A. Müller and L. Grüne.

Economic model predictive control without terminal constraints for optimal periodic behavior.

Automatica, vol. 70, pp. 128-139, 2016.

**Cooperations:**

- Lars Grüne, University of Bayreuth
- Karl Worthmann, TU Ilmenau
- Daniel E. Quevedo, University of Paderborn, Germany

## Stochastic MPC

In Stochastic Model Predictive Control, uncertainty in the system description or external disturbances are taken explicitely into account in the design process. In contrast to Robust MPC, where uncertainties are usually assumed to be unknown but bounded and constraints should be satisfied for all possible realizations, this approach assumes stochastic disturbance or parameter variations. This assumptions allows to design controllers which optimize the expected value or other risk measures such as CVaR rather than the worst case. Performance can be further increased by allowing an a priori specified probability of constraint violation. The focus of our research in Stochastic MPC are computational tractable formulations of the constraints and objective as well as ensuring recursive feasibility. To this end deterministic and sample based methods are taken into account.

**Contact Persons:**Frank Allgöwer, Matthias Lorenzen

**Publications:**

- M. Lorenzen, F. Allgöwer, F. Dabbene and R. Tempo.

An Improved Constraint-Tightening Approach for Stochastic MPC.

In Proc. of the American Control Conference (ACC), pages 944-949, Chicago, IL, USA, 2015. - M. Lorenzen, F. Allgöwer, F. Dabbene and R. Tempo.

Scenario-Based Stochastic MPC with Guaranteed Recursive Feasibility.

IEEE Conference on Decision and Control (CDC), 2015. - F.A. Bayer, M. Lorenzen, M.A. Müller and F. Allgöwer.

Improving Performance in Robust Economic MPC Using Stochastic Information.

5th IFAC Conference on Nonlinear Model Predictive Control (NMPC), 2015.

**Cooperations:**

- Roberto Tempo, Fabrizio Dabbene, CNR-IEIIT, Politecnico di Torino, Italy

## MPC Algorithms with Systems Theoretic Guarantees

Conventional MPC schemes usually assume that the required optimal control input can be computed at each sampling instant exactly and in a negligible amount of time. Especially in the context of fast system dynamics or restricted hardware specifications, this idealizing assumption may not be valid anymore, which makes it hard to give stability or constraint satisfaction guarantees when it comes to the practical implementation of MPC. Thus, there is a well-founded need for efficient algorithmic MPC implementations that allow a rapid computation of the optimal control input - or at least of a sufficiently good approximation - while still being able to give guarantees on system theoretic properties of the overall closed-loop system, which now consists of the combined dynamics of both plant and optimization algorithm. Consequently, our research focus lies on the intersection of the areas systems theory, predictive control, and optimization algorithms. In particular, the overall goal is the development and analysis of so-called *barrier function based MPC approaches* in conjunction with suitable, or even specifically tailored, optimization algorithms (both discrete- and continuous-time). By taking the dynamics of the underlying optimization into account, the resulting *MPC algorithms* allow to guarantee properties like closed-loop stability and constraint satisfaction for in principle arbitrary fast system dynamics.

**Contact Persons:**Christian Ebenbauer, Fabian Pfitz, Christian Feller

**Publications:**- C. Feller and C. Ebenbauer.

A barrier function based continuous-time algorithm for linear model predictive control.

In Proc. of the European Control Conference (ECC), Zürich, Switzerland, 2013, pp. 19-26. - C. Feller and C. Ebenbauer.

Ein zeitkontinuierlicher Optimierungsalgorithmus für die modellprädiktive Regelung linearer Systeme.

In 18. Steirisches Seminar über Regelungstechnik und Prozessautomatisierung, Schloss Retzhof, Leibnitz, Austria, 2013, pp. 1-28. - C. Feller and C. Ebenbauer.

Continuous-time linear MPC algorithms based on relaxed logarithmic barrier functions.

19th IFAC World Congress, Cape Town, South Africa, August 2014. - C. Feller, C. Ebenbauer.

Barrier function based linear model predictive control with polytopic terminal sets.

53rd IEEE Conference on Decision and Control, Los Angeles, USA, Dezember 2014. - C. Feller, C. Ebenbauer.

Weight recentered barrier functions and smooth polytopic terminal set formulations for linear model predictive control.

American Control Conference 2015, Chicago, USA, July 2015. - C. Feller, C. Ebenbauer.

Input-to-state stability properties of relaxed barrier function based MPC.

5th IFAC Conference on Nonlinear Model Predictive Control 2015, Seville, Spain, September 2015. - C. Feller, C. Ebenbauer.

Relaxed Logarithmic Barrier Function Based Model Predictive Control of Linear Systems.

IEEE Transactions on Automatic Control, to appear - C. Feller, C. Ebenbauer.

A stabilizing iteration scheme for model predictive control based on relaxed barrier functions.

March 2015, arXiv:1603.04605 [math.OC]

## Machine Learning and Data Mining in MPC

This research project aims at creating advanced MPC schemes via enhancing classical MPC algorithms with methods from the fields of machine learning and data mining. Most existing MPC schemes require generation and processing of large amounts of data which generally remain unexploited to a large extend. By applying machine learning and data mining algorithms, the essence of the appearing data can be extracted and made utilizable to simplify online computations and enhance performance of MPC algorithms. For example, results on parameterizing the predicted input trajectory in MPC based on subspace clustering methods were presented recently. This way, MPC schemes with reduced computational requirements can be formulated.

**Contact Persons:**Frank Allgöwer, Gregor Goebel

## MPC under Communication Constraints

If the cost of communication between sensors, controllers, and actuators in a control system cannot be neglected, the controller design must weigh the amount of necessary communication against the achieved performance. Research at the IST focuses on self-triggered and event-triggered predictive control and estimation methods for disturbed systems with a quantifiable trade-off between closed-loop performance and average sampling frequency.

**Contact Persons:**Frank Allgöwer, Florian Brunner

**Publications:**

- F. D. Brunner, W. P. M. H. Heemels and F. Allgöwer.

Robust Self-Triggered MPC for Constrained Linear Systems.

In Proc. of the European Control Conference (ECC), Strasbourg, France, 2014, pp. 472-477. - E. Aydiner, F. D. Brunner, W. P. M. H. Heemels and F. Allgöwer.

Robust Self-Triggered Model Predictive Control for Constrained Discrete-Time LTI Systems Based on Homothetic Tubes.

In Proc. of the European Control Conference (ECC), Linz, Austria, 2015, pp. 1587-1593. - F. D. Brunner, T. M. P. Gommans, W. P. M. H. Heemels and F. Allgöwer.

Communication Scheduling in Robust Self-Triggered MPC for Linear Discrete-Time Systems

Proc.of the 5th IFAC Workshop on Distributed Estimation and Control in Networked Systems (NecSys), Philadelphia, PA, USA, 2015, pp. 132-137. - F. D. Brunner, W. P. M. H. Heemels and F. Allgöwer.

Robust Event-Triggered MPC for Constrained Linear Discrete-Time Systems with Guaranteed Average Sampling Rate.

In Proc.of the IFAC Conference on Nonlinear Model Predictive Control (NMPC'15), Seville, Spain, 2015, pp. 117-122. - Brunner, F.D., Heemels, W.P.M.H., and Allgöwer, F. (2016).

Robust self-triggered MPC for constrained linear systems: A tube-based approach.

Automatica, 72, 73–83.

**Cooperations:**

- Maurice Heemels, Eindhoven University of Technology, The Netherlands

## MPC Using Reduced Models

Applying model predictive control to high-dimensional systems typically leads to a prohibitive computational complexity. Therefore, reduced order models are employed in many applications. This introduces an approximation error which may deteriorate the closed-loop behavior. At the IST we work on novel model predictive control schemes using a reduced order model for prediction in combination with the error bounding system proposed in [Hasenauer et al, 2012]. By employing the error bound, we achieve design conditions for constraint fulfillment, recursive feasibility and asymptotic stability despite the mismatch between the high-dimensional system and the reduced order model.

**Contact Persons:**Frank Allgöwer, Martin Löhning

**Publications:**

- J. Hasenauer, M. Löhning, M. Khammash, and F. Allgöwer.

Dynamical optimization using reduced order models: A method to guarantee performance.

Journal of Process Control, Vol. 22, No. 8, September 2012, pp. 1490-1501. - M. Löhning, M. Reble, J. Hasenauer, S. Yu, and F. Allgöwer.

Model predictive control using reduced order models: Guaranteed stability for constrained linear systems.

Journal of Process Control, Vol. 24, No. 11, November 2014, pp. 1647-1659.

**Cooperations:**

- Bernard Haasdonk, Institute of Applied Analysis and Numerical Simulation, University of Stuttgart