# General information

Course title: **Advanced topics in control systems** (ATCS)

Instructors: P. Sopasakis and A. Bemporad

Duration: 20 hours

February 2016

Version: Last updated on 13 January 2015.

# Description

In this course we will venture to go through some of the most advanced control schemes whose development has been motivated by problems in process control and economics. The course's main objective will be to bring students in touch with the state of the art in MPC theory and explore various research opportunities that emerge.

We will see how the mature concept of model predictive control can be combined with process economics to yield a unifying framework -- known as economic model predictive control (EMPC) -- for simultaneous control and process optimisation. The EMPC-controlled closed-loop trajectories need not be stable/convergent, but they provide certain performance/cost guarantees for the process. We establish stability conditions for the closed-loop system and study various EMPC formulations and their properties.

Special emphasis will be put on the study of MPC methodologies for uncertain systems. We will discuss various stochastic MPC methodologies and study their closed-loop properties. We will provide a comprehensive theory of Markovian systems for which we will define new notions of stability such as mean square stability, almost sure stability and uniform stability.

# Course topics

### Economic model predictive control

- Process economics coupled with process control
- Examples: control of drinking water networks, dynamic energy pricing and fluctuating demand
- Performance bounds of economic MPC
- Dissipative systems and stability results for EMPC
- Averagely constrained economic MPC
- Optimal operation and average performance
- Generalised terminal constraint for EMPC

### Markovian switching systems

- Stochastic processes and classes of stochastic systems
- Counter-intuitive properties of stochastic systems
- Markovian processes and Markov jump linear systems (MJLS) - definition and properties
- Mean square stability and stability conditions for homogeneous MJLS
- Lyapunov-type conditions for mean square stability of MJLS
- Almost sure stability
- Finite and infinite horizon optimal control of MJLS
- Dynamic programming solution of stochastic optimal control problems
- Mean square stability conditions for Markovian switching systems
- Stochastic MPC for Markovian switching systems

### Stochastic MPC

- Various stochastic MPC formulations
- Stochastic MPC with chance constraints
- Stabilising conditions for systems with iid disturbances
- Risk-averse optimisation

# Prerequisites

Linear algebra & calculus; Linear discrete-time dynamical systems; Model predictive control theory.

# Grading plan

Students will be evaluated by preparing and giving a presentation after the end of the course or solving a set of exercises. You may find detailed instructions in this document (download) where you may also find a few ideas about projects.

# Suggested readings

We recommend the following bibliographic resources:

### Markovian systems - analysis and design

- A must-read on Markov jump linear systems (theory elegantly explained): O.L.V. Costa, M.D. Fragoso and R.P. Marques, Discrete-time Markov Jump Linear Systems, Springer 2005.
- P. Patrinos, P. Sopasakis, H. Sarimveis and A. Bemporad, Stochastic model predictive control for constrained discrete-time Markovian switching systems, Automatica 50, 2504-2514, 2014.

### Stochastic MPC & Stochastic and risk-averse optimal control

- A. Shapiro, D. Dentcheva and A. Ruszczynski, Lectures on stochastic programming: modeling and theory, SIAM and MPS editions, 2009; available online.
- D.P. Bertsekas and S.E. Shreve, Stochastic optimal control: the discrete-time case, Academic Press, New York, 1978.
- D. Bernardini and A. Bemporad, Stabilizing model predictive control of stochastic constrained linear systems, IEEE Trans. Aut. Contr. 57(6), pp. 1468-1480, 2012.
- M. Korda, R. Gondhalekar, J. Cigler and F. Oldewurtel, Strongly Feasible Stochastic Model Predictive Control, In 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011.
- H. Follmer and A. Schied, Stochastic Finance - an introduction in discrete time, Walter de Gruyter, Berlin, 2004.
- B. Kouvaritakis, M. Cannon, S.V. Rakovic and Q. Cheng, Explicit use of probabilistic distributions in linear predictive control, Automatica 46, pp. 1719-1724, 2010.
- P. Krokhmal, M. Zabarankin, and S. Uryasev, "Modeling and optimization of risk," Surveys in Operations Research and Management Science, vol. 16, no. 2, pp. 49 – 66, 2011.

### Economic MPC

- A very good reference paper for EMPC: D. Angeli, R. Amrit and J.B. Rawlings, On average performance and stability of economic model predictive control, IEEE Trans. Aut. Contr. 57(7), pp. 1615-1626, 2012.
- T. Tran, K.-V. Ling and J.M. Maciejowski, An overview of the most recent developments in EMPC, In: 31st International Symposium on Automation and Robotics in Construction and Mining, 2014
- J.B. Rawlings, D. Angeli and C.N. Bates, Fundamentals of economic model predictive control, In 51st IEEE Conf. Decision and Control, Maui, Hawaii, USA, Dec. 10-13 2012
- R. Amrit, Optimizing process economics in model predictive control, PhD. dissertation, Univ. Wisconsin-Madison, 2011
- L. Fagiano and A. Teel, Generalized terminal state constraint for model predictive control, Automatica 49, pp. 2622 - 2631, 2013
- L. Grüne, Economic receding horizon control without terminal constraints, Automatica 49, pp.725 - 734, 2013
- M.A. Müller, D. Angeli and F. Allgöwer, On convergence of averagely constrained economic MPC and necessity of dissipativity for optimal steady-state operation, In: American Control Conference (ACC), Washington, 2013.
- M.A. Müller, D. Angeli and F. Allgöwer, Economic model predictive control with self-tuning terminal cost, European Journal of Control 19, pp. 408-416, 2013.
- J.C. Willems, Dissipative dynamical systems - part I: General theory, Archive for rational mechanics and analysis 45(5), pp. 321-351, 1972.

# Time schedule

Course schedule for 2016:

- February 01, 14:00-16:00
- February 02, 11:00-13:00
- February 03, 14:00-16:00
- February 04, 11:00-13:00
- February 05, 16:00-18:00
- February 08, 11:00-13:00
- February 09, 11:00-13:00
- February 10, 11:00-13:00
- February 11, 11:00-13:00
- February 12, 11:00-13:00