# Model predictive control

Our reserach on MPC covers a large range of topics from data-based to distributed economic MPC.

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, Christian Ebenbauer, Meriem Gharbi, Johannes Köhler, Philipp Köhler, Matthias Lorenzen, Matthias A. Müller, Fabian Pfitz

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

## MPC Research Fields

### 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 F. Allgöwer.

Economic and distributed model predictive control: recent developments in optimization-based control.

SICE Journal of Control, Measurement, and System Integration, vol. 10, no. 2, pp.39-52, March 2017. - M. A. Müller and L. Grüne.

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

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

Robust Economic Model Predictive Control using Stochastic Information.

Automatica, vol. 74, pp. 151-161, 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. 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.

**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:**

- M. A. Müller and F. Allgöwer.

Economic and distributed model predictive control: recent developments in optimization-based control.

SICE Journal of Control, Measurement, and System Integration, vol. 10, no. 2, pp.39-52, March 2017. - 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.

### 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:**

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

Stochastic MPC with Offline Uncertainty Sampling.

Automatica, 2017, 81, 176-183. - Lorenzen, M.; Dabbene, F.; Tempo, R. and Allgöwer, F.

Constraint-Tightening and Stability in Stochastic Model Predictive Control.

IEEE Transactions on Automatic Control, 2017, 62, 3165-3177. - Lorenzen, M.; Müller, M. A. and Allgöwer, F.

Stochastic Model Predictive Control without Terminal Constraints.

International Journal of Robust and Nonlinear Control, 2017. - 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.

**Cooperations:**

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

### 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, Matthias A. Müller

**Publications:**

- Lorenzen, M.; Allgöwer, F. and Cannon, M.

Adaptive Model Predictive Control with Robust Constraint Satisfaction.

Proceedings of the IFAC World Congress, 2017, 3368-3373. - 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. - 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. - 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.

**Cooperations:**

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

### Moving horizon estimation

Moving horizon estimation (MHE) considers the dual problem of optimization-based state estimation. In particular, at each time step, the current state estimate is determined by solving an optimization problem taking a number of past measurements into account. In order to obtain a good state estimate, ideally all past measurements from the initial time up to the current time should be considered, resulting in a so-called full information estimator. Since this is in general computationally intractable, in moving horizon estimation one instead uses only a fixed number N of past measurements.

Research at the IST studies different system theoretic properties of MHE schemes for linear and nonlinear systems. In particular, in one of our recent results robust stability of the estimation error for general nonlinear detectable systems subject to bounded disturbances was established.

**Contact Persons:**Matthias A. Müller

**Publications:**

- F. D. Brunner, M. A. Müller and F. Allgöwer.

Enhancing Output-feedback mpc with Set-valued Moving Horizon Estimation.

IEEE Transactions on Automatic Control, 2018, to appear. -
M. A. Müller.

Nonlinear moving horizon estimation in the presence of bounded disturbances.

Automatica, vol. 79, pp. 306-314, 2017.

### MPC and MHE 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.

Motivated by these promising results, research at the IST investigates the extension of the concept of barrier functions to the framework of constrained moving horizon estimation (MHE). The goal is to potentially develop an efficient algorithm that combines both the barrier function based MPC and MHE in the context of estimation based control.

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

**Publications:**

- M. Gharbi and C. Ebenbauer.

A proximity approach to linear moving horizon estimation.

In Proc. of the 6th IFAC NMPC Conference 2018, pages 649–655, Madison, USA, 2018. - C. Feller and C. Ebenbauer.

Sparsity-exploiting anytime algorithms for model predictive control: A relaxed barrier approach.

IEEE Transactions on Control Systems Technology, 2018. To appear. - C. Feller and C. Ebenbauer.

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

Automatica, 80:328–339, June 2017. - C. Feller and C. Ebenbauer.

Relaxed logarithmic barrier function based model predictive control of linear systems.

IEEE Transactions on Automatic Control, 62:1223–1238, March 2017.

### 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:**

- Lorenzen, M.; Müller, M. A. and Allgöwer, F.

Stochastic Model Predictive Control without Terminal Constraints.

International Journal of Robust and Nonlinear Control, 2017. - M. A. Müller and L. Grüne.

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

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

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.

**Cooperations:**

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

### 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, Matthias A. Müller

### 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:**

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

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

Automatica, 72, 73–83, 2016. - 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. 1581-1587. - 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. 5th IFAC Workshop on Distributed Estimation and Control in Networked Systems (NecSys)*,*Philadelphia, PA, USA, September 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.

Proc. IFAC Conf. Nonlinear Model Predictive Control (NMPC'15), Seville, Spain, September 2015, pp. 117-122.

**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:**

- 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. - 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.

**Cooperations:**

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