## Works on Model Predictive Control (MPC)

### A New Formulation of Economic Model Predictive Control without terminal constraint

In this paper, it is shown that a simple formulation of Economic Model Predictive Control can be used which possesses two features that are generally viewed as mutually exclusive, namely, a rather short prediction horizon (reachability-compatible) on one side, and the absence of final constraint on the other side. Practical stability at an arbitrarily small neighborhood of the optimal unknown steady- state pair is shown when some design parameters increase. It is also shown that when the system is originated from the time discretization of a continuous-time dynamics, the size of the terminal region can be reduced by decreasing the sampling period for the same design parameter setting. A commonly used example is given to illustrate the results.

M. Alamir and G. Pannocchia. A new formulation of Economic Model Predictive Control without terminal constraint. **Automatica**, Vol. 125, 2021

### On the use of supervised clustering in Stochastic NMPC Design

In this article, a supervised clustering-based heuristic is proposed for the real-time implementation of approximate solu- tions to stochastic nonlinear model predictive control frameworks. The key idea is to update online a low cardinality set of uncertainty vectors to be used in the expression of the stochastic cost and constraints. These vectors are the centers of uncertainty clusters that are built using the optimal control sequences, cost, and con- straints indicators as supervision labels. The use of a moving clus- tering data buffer which accumulates recent past computations enables to reduce the computational burden per sampling period while making available at each period a relevant amount of samples for the clustering task. A relevant example is given to illustrate the contribution and the associated algorithms.

M. Alamir. On the use of supervised clustering in stochastic NMPC design. **IEEE Transactions on automatic control**. Volume 65, Issue 12, pp. 5392-5398, December 2020.

### GPU-Based parameterized NMPC scheme for control of half car vehicle with semi-active suspsension system

In this letter, we propose a black-box compatible simulation-based approach for solving nonlin- ear model predictive control (NMPC) problem via a param- eterized technique to control the vertical dynamics of a half car vehicle equipped with semi-active (SA) suspen- sion system. The method taps the potential of the graphic processing unit (GPU) to simulate the system parallelly for several combinations of control inputs and the optimal input is elicited which minimizes the objective function and satisfies the constraints. The method was tested in MATLAB/Simulink environment by means of simulations and a comparative study was conducted with ACADO- qpOASES NMPC framework. The simulation results display better performance of the proposed approach in terms of computation time, closed loop objective, and constraint satisfaction when juxtaposed to the ACADO-qpOASES NMPC controller.

K. M. M. Rathai, O. Sename and M. Alamir, GPU-based parametrized NMPC scheme for control of half car vehicle with semi-active suspension system. **IEEE Control Systems Letters**, Vol. 3, Number 3, pages 631-636, 2019.

### Fixed-point Based Hierarchical MPC Control Design For a Cryogenic Refrigerator

In this paper, a simple, general and scalable hierarchical control framework is proposed and validated through the interconnection of the Joule-Thomson and the Brayton cycle stages of a cryogenic refrigerator. The proposed framework enables to handle the case of destabilizing interconnections through state and/or control signals (which is the case of the cryogenic refrigerator example). Moreover, it offers the possibility to simply change the behavior of the overall system (depending on the context) by only changing the coordinator problem's parameters without changing the set of local controllers used by subsystems which is a common industrial requirement regarding industrial control architectures. Finally, the proposed scheme enables a smooth operator handover on a specific subsystem and/or actuator. In this paper, a simple, general and scalable hierarchical control framework is proposed and validated through the interconnection of the Joule-Thomson and the Brayton cycle stages of a cryogenic refrigerator. The proposed framework enables to handle the case of destabilizing interconnections through state and/or control signals (which is the case of the cryogenic refrigerator example). Moreover, it offers the possibility to simply change the behavior of the overall system (depending on the context) by only changing the coordinator problem's parameters without changing the set of local controllers used by subsystems which is a common industrial requirement regarding industrial control architectures. Finally, the proposed scheme enables a smooth operator handover on a specific subsystem and/or actuator.

**Reference**

M. Alamir, P. Bonnay, F. Bonne and V. V. Trinh. Fixed-point based hierarchical MPC control design for a cryogenic refrigerator. Journal of Process Control, Vol 58, pages 117-130, 2017.

### Stability proof for nonlinear MPC design using monotonically increasing weighting profile without terminal constraints

In this note, a new formulation of Model Predictive Control (MPC) framework with no stability-related terminal constraint is proposed and its stability is proved under mild standard assumptions. The novelty in the formulation lies in the use of time-varying monotonically increasing stage cost penalty. The main result is that the 0-reachability prediction horizon can always be made stabilizing without any terminal constraints} provided that the increasing rate of the penalty is made sufficiently high. Moreover, it is shown through an illustrative example that the time varying penalty may improve the resulting closed-loop performance computed with the original stage-cost when compared to the traditional MPC formulation with final constraint on the state.

**Reference**

M. Alamir. Stability proof for nonlinear MPC design using monotonically increasing weighting profile withour terminal constraints. Automatica, Vol 87, pp. 455-459, 2018.

M. Alamir. Numerical investigation regarding an MPC scheme with non uniformly weighted stage cost without terminal constraints: Application to the control of real-life cryogenic plant. Proceedings of the IFAC workshop on Nonlinear Model Predicitive Control NMPC2018, Wisconsin, Madison, USA, August, 2018.

### The PDF-MPC Package: A Free-Matlab-Coder package for real-time nonlinear model predictive control.

This paper describes the Parametrized Derivative-Free Model Predictive Control (pdf-mpc) package, a matlab coder-based set of subroutines that enables a model predictive control problem to be defined and solved. the \pac is made available for free download and use through the website of the author.

**Reference**

M. Alamir. The PDF-MPC Package: A Free-Matlab-Coder package for real-time nonlinear model predictive control. 2017. arXiv:1703:08255

### Experimental investgation of control updating period monitoring in industrial PLC-Based fast MPC: Application to the constrained control of a cryogenic refrigerator

In this paper, a complete industrial validation of a recently published scheme for on-line adaptation of the control updating period in Model Predictive Control is proposed. The industrial process that serves in the validation is a cryogenic refrigerator that is used to cool the supra-conductors involved in particle accelerators or experimental nuclear reactors. Two recently predicted features are validated: the first states that it is sometimes better to use less efficient (per iteration) optimizer if the lack of efficiency is over-compensated by an increase in the updating control frequency. The second is that for a given solver, it is worth monitoring the control updating period based on the on-line measured behavior of the cost function.

**Reference**

F. Bonne, M. Alamir and P. Bonnay. Experimental investigation of control updating period monitoring in industrial PLC-based fast MPC: Application to the constrained control of a cryogenic refrigerator. Journal of control theory and technology (2017).

### Contraction-based nonlinear model predictive control formulation without stability-related terminal constraints

Reference

M. Alamir, Contraction-based nonlinear model predictive control formulation without stability-related terminal constraints, Automatica, Vol. 75, pp 288-292, 2017.

### A State-Dependent Updating Period For Certified Real-Time Model Predictive Control

In this paper, a state-dependent control updating period strategy is proposed for use in interrupted implementation of real-time Model Predictive Control (MPC). The strategy can be used as soon as a certification bound is available for the underlying optimization algorithm. Moreover, a new fast-Gradient based certifiable algorithm is defined with the associated certification bounds for convex generally constrained optimization problems.

Reference

M. Alamir, A state-dependent updating period for certified real-time Model Predictive Control, to appear in IEEE Transactions on Automatic Control 2017.

### From Certification of algorithms to certified MPC: The missing links

Deriving certification bounds for optimization algorithms is an active research area in the control community. This is mainly impulsed by the use of on-line optimization algorithms in real-time MPC through limited computation power. However, the way such bounds are then used to derive a convergence certification for MPC frameworks is still not sufficiently mature. This paper contributes in clarifying what are the unavoidable additional ingredients that need to be combined with any algorithm's certification bound in order to derive a relevant certification result for the MPC-based closed-loop performance. Moreover, the paper gives such a general certification result based on these ingredients for any pair of certified algorithm and provably stable ideal MPC formulation. The proposed framework is then instantiated to the particular case of linear MPC and a simple example is given to illustrate the introduced concepts.

**Reference**

M. Alamir, From certification of algorithms to certified MPC: the missing links. Proceedings of the 5th IFAC conference on Nonlinear Model Predictive Control. Seville, Spain 2015. (Invited paper) [download]

### Scalability study for a hierarchical NMPC scheme for resource sharing problems

This paper deals with the computational efficiency evaluation of a hierarchical DMPC (distributed model predictive control) framework for resource sharing problems. The provided DMPC framework is based on a dual decomposition of the centralized open-loop controller which is decomposed into several subproblems and one coordinator problem. At coordinator level the bundle method is used in order to recover the globally optimal solution through an iterative process.

The main focus of this paper is a detailed discussion of the impact of the bundle method’s parametrization on the computational performance of the whole scheme. Additionally a qualitative comparison with a similar scheme based on primal decomposition is provided and some rules of thumb for determining an effective parametrization of the bundle method are established. In the provided simulations the scheme is applied to a large-scale problem of the smart district context. More precisely the centralized optimization problem of a district composed of 1000 buildings sharing a globally limited power resource is able be solved to optimality using our proposed framework in around 3 seconds.

**Reference**

P. Pflaum, M. Alamir, M. Y. Lamoudi. Scalability study for a hierarchical NMPC scheme for resource sharing problems. Proceedings of the European Control Conference, ECC2015, Linz, Austria, 2015. [download]

### Fast NMPC: Some good news and some facts to keep in mind

**Reference**

M. Alamir. Fast NMPC: Some good news and some facts to keep in mind. Semi-plenary talk at the European Control conference (ECC2014), Strasbourg, France.

### A distributed cooperative control scheme with optimal priority assignment and stability assessment

a distributed partially cooperative control framework is proposed for a network of linear inter- connected subsystems. It is assumed that each subsystem in the network possesses its own objective and a corresponding nominal interaction-free state feedback law, the proposed framework enables each subsystem to compute an additional control term in order to help maintaining the integrity of the overall network. As this cooperation-like behavior involves relative priority assignment, a communication aware heuristic is proposed with an associated stability assessment that is based on the closed-loop network matrix’s spectrum monitoring. A nice feature in the proposed solution is that it can be applied to the linearized network model in order to compute the relative priority vector components that can then be used in a nonlinear cooperation scheme. Illustrative examples are used to assess the effectiveness of the proposed scheme including a distributed load frequency problem.

**Reference**

H. Ding, M. Alamir and A. Hably. A distributed cooperative control scheme with optimal priority assignment and stability assessment. Proceeding of the IFAC World Congress, South Africa (2014). [download]

### Fast NMPC: A Reality-Steered Paradigm: Key Properties of Fast NMPC Algorithms

In this paper, the paradigm of fast Nonlinear Model Predictive Control is recalled. Then a fundamental inequality that conditions the closed-loop stability is derived. Based on this inequality, it is shown that the comparison between different algorithms in the context of Fast NMPC must be based not only on the efficiency per iteration but also on the time needed to perform a single iteration. An illustrative example is used to assess the fact that under some circumstances, it is worth using less efficient algorithms (in conventional sense) that correspond to less amount of computation per iteration and this even when perfect model is used and in a disturbance-free context

**Reference**

M. Alamir. "Fast MPC, A reality steered paradigm: Key properties of fast NMPC Algorithms. Proceedings of the European Control Conference (ECC2014), Strasbourg, France (Tutorial sessions of fast NMPC). [download]

### A Pragmatic Story of Model Predictive Control: Self Contained Algorithms and Cases Studies

Nowadays, optimality is a major concern in modern controlled systems, and since optimality generally steers the systems to the boundary of their admissible operational domain, no control designer can afford ignoring Model Predictive Control (MPC). MPC is the only control design methodology that enables systematic handling of constraints and optimality concerns. This book is addressed to under-graduate students in engineering who are interested in control design issues. It can also be used by control researchers or practitioners that are not specialized in MPC but who are willing to acquire a concise and concrete knowledge of this advanced control methodology. Researchers in Robotics, Mechatronics, Process control, Automotive control, Aerospace, Power Systems and so many other application domains where the control plays a crucial role can find in this book answers to their challenging problems. The book covers both linear and nonlinear constrained MPC with many case-studies. All the scripts that are used to produce the results are explicitly and exhaustively given in the book. These scripts can therefore serve as templates to engineers when facing real-life control problems. The book uses the Matlab scientific programming language and shows systematically how to use the Matlab Coder toolbox in order to produce real-time implementable compiled MPC solutions. Several reading advices are also given regarding the state of the art and the major current mainstream research directions on this hot and challenging topic.

**Reference**

M. Alamir, A Pragmatic Story of Model Predictive Control: Self Contained Algorithms and Cases Studies. CreateSpace Independent Publishing Platform. ISBN-10 1489541349. 2013.

### A framework for real-time implementation of low-dimensional parameterized NMPC

In this paper, a novel approach is proposed to implement low-dimensional parameterized Nonlinear Model Predictive Control (NMPC) schemes for systems showing fast dynamics. The proposed scheme is based on distributing the reconstruction of the cost function over the real lifetime of the controlled system. The framework is particularly suitable for NMPC formulations that use low dimensional control parametrization. The concrete example of a Planar Vertical Take-Off and Landing (PVTOL) aircraft stabilization problem is used to illustrate the efficiency of the proposed formulation.

**Reference**

M. Alamir. A Framework For Real-Time Implementation of Low Dimensional Parameterized NMPC. Automatica. Vol 48. pp. 198-204 (2011). [download]

### A Novel Distributed NMPC Control Structure For Partially Cooperative Systems under limited information sharing

In this contribution, a new cooperative control framework is proposed for a network of subsystems sharing limited information and showing destabilizing interconnection. The scheme is based on the exchange of Lyapunov function levels together with associated constraints between neighbors. Moreover, a tunable cooperation index is used by each subsystem in order to define the extent to which it accepts to degrade its own performance index in order to recover the integrity of its neighbors. An interesting feature is the use by each subsystem of a priority vector that enables to introduce hierarchical order of its neighbors leading to a cooperative strategy that can preserve critical nodes of the network. Finally, the scheme assumes no particular structure nor linearity of the involved dynamics. The efficiency of the entire scheme is shown through two examples containing 3 and 12 subsystems respectively.

**Reference**

M. Alamir, A. Hably and H. Ding. A Novel Distributed NMPC Control Structure For Partially Cooperative Systems under limited information sharing. Proceedings of the IFAC World Congress, Milano, Italy, (2011). [download]

### Une méthode du point fixe pour la mise en œuvre de la commande prédictive non linéaire sous contraintes sur des EDP

In this contribution, a new fixed point based iterative scheme is proposed for the im- plementation of constrained Nonlinear Model Predictive Control (NMPC) to dynamic systems that are described by nonlinear Partial Differential Equations (PDE’s). The design method is illustrated through the problem of constrained stabilization of the Kuramoto Sivashinski non- linear PDE.

**Reference**

Alamir M. A Fixed Point Method for the implementation of constrained NMPC to PDE's: Application to the constrained stabilization of the Kuramoto-Sivashinsky nonlinear PDE's. Journal Européen des Systèmes Automatisés (In French), Vol 45/7-10 - 2011 - pp.693-713 (2011). [download]

### Distributed Partially Cooperative NMPC Under Limited Communication and Destabilizing Interconnections

In this paper, a new formulation of distributed and partially cooperative control under limited communication is proposed. Unlike many existing schemes, destabilizing inter- connections are considered and partial load shading-like decisions may be potentially taken in order to maintain the overall system integrity. The example of power systems black-out can be viewed as the targeted context although the present paper gives the general framework regardless any specific application.

**Reference**

M. Alamir. Distributed Partially Cooperative NMPC Under Limited Communication and Destabilizing Interconnections. Proceedings of the IFAC Workshop on Time-Delay Systems, 2009. [download]

### A Framework for Control Updating Period Monitoring In Real-Time NMPC Schemes

In this contribution, a general scheme is proposed that enables the con- trol updating period used in Nonlinear Model Predictive Control (NMPC) scheme to be dynamically optimized. Such a scheme can be of great interest when applying NMPC to systems with fast dynamics. The updating scheme is based on the on-line identification of generic models for both solver efficiency and disturbance effect on the optimal cost behavior. The efficiency of the proposed approach is illustrated on several examples.

**Reference**

Alamir, M. A Framework for Monitoring Control Updating Period in Real-Time NMPC. In International Workshop on Assessment and Future Directions in Nonlinear. In Model Predictive Control. L. Magni, D. Raimondo and F. Allgöwer (Ed) . Lecture Notes in Control and Information Sciences, Springer-Verlag, (2008). [download]

### Stabilization of Nonlinear Systems Using Receding-Horizon Control Schemes: A Parametrized Approach for Fast Systems

**Reference**

Alamir, M.

Stabilization of Nonlinear Systems Using Receding-Horizon Control Schemes: A Parametrized Approach for Fast Systems. Lecture Notes in Control and Information Sciences, Springer, London, ISBN 1-84628-470-8 (2006)

### A low dimensional contractive NMPC scheme for nonlinear systems stabilization: Theoretical framework and numerical investigation on relatively fast systems

In this paper, a new contractive receding horizon scheme is proposed for the stabilization of constrained nonlinear systems. The proposed formulation uses a free finite prediction horizon without explicit use of a contraction stability constraint. Another appealing feature is the fact that the resulting receding horizon control is in pure feedback form unlike existing contractive schemes where open- loop phases or a memorized threshold are used to ensure the contraction property in closed loop. The control scheme is validating on the swing-up and stabilization problem of a simple and a double inverted pendulums.

**Reference**

Alamir, M. A Low Dimensional Contractive NMPC Scheme for Nonlinear Systems Stabilization: Theoretical Framework and Numerical Investigation on Relatively Fast Systems. In R. Findeisen, F. Allgower, L. Biegler (eds.). Assessement and future directions in Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, Springer-Verlag, (2006). [download]

### La Commande Prédictive

**Reference**

Alamir, M. Commande Prédictive Non Linéaire. Dans "La Commande Prédictive", édition Hermès. Coordinateur: D. Dumur. (2006)

### New path-generation based receding-horizon formulation for constrained stabilization of nonlinear systems

In this paper,a new formulation of constrained stabilizing receding-horizon control is proposed.This formulation is based on the use of open-loop steering path generators. The open-loop optimization problem associated to the proposed receding-horizon formulation is scalar in which the optimization variable is the prediction horizon length. Stability is proved in a sampling control scheme. A simple example is given to illustrate the main concepts.

**Reference**

Alamir, M.New path-generation based receding-horizon formulation for constrained stabilization of nonlinear systems. Automatica 40, No.4, 647-652 (2004). [download]

### Solutions of nonlinear optimal and robust control problems via a mixed collocation/DAE's based algorithm

A new algorithm for computing the solutions of nonlinear optimal and robust Hinfinity control problems is proposed. The algorithm is based on the use of the collocation method to transform the PDE's into ODE's. Some convergence results are given and several examples are presented

**Reference**

Alamir, M. Solutions of nonlinear optimal and robust control problems via a mixed collocation/DAE's based algorithm. Automatica 37, No.7, 1109-1115 (2001). [download]

### On the feasibility of finite difference homotopy based algorithm for nonlinear optimal control problems

**Rereference**

Alamir, M. On the feasibility of finite difference homotopy based algorithm for nonlinear optimal control problems. Proceedings of the IFAC World Congress, Bejing, China (1999).

### Optimal control with harmonic rejection of induction machines

In this paper a new algorithm is used to regulate flux and torque in induction machine. The aim is to show the possibility of attenuating the harmonics as early as the phase of the design of the control law while preserving good regulation performances. The control law results from an optimisation algorithm associated to a cost function. By adding some harmonic weighting factor to the regulation term in the cost function, the algorithm carries out a trade-off between regulation performances and harmonic attenuation

**Reference**

Alamir, M. and Balloul, I. Optimal Control with Harmonic Reduction in Induction Machines. In Nonlinear Control Networks (eds). Lecture Notes in Control and Information Sciences, Springer-Verlag (2000). [download]

### Robust constrained control algorithm for general batch processes

In this paper, an implementable algorithm that enables to robustly control batch processes is proposed. The resulting state feedback algorithm is based on repeated on-line solution of constrained open loop min-max problems associated to the worst-case perturbations and/or uncertainties. These solutions are then used in a receding horizon scheme in order to yield a robust state feedback controller. A novel algorithm is proposed for the solution of the open loop constrained minmax problems that is based on chattering control combined with the variable stabilizing penalty approach. Simulations are provided in order to illustrate the effectiveness of the proposed control algorithm.

**Reference**

Alamir, M.; Balloul, I. Robust constrained control algorithm for general batch processes. Int. J. Control 72, No.14, 1271-1287 (1999).

### Numerical Stabilisation of Non-linear Systems: Exact Theory and Approximate Numerical Implementation

In this paper, a theoretical background is presented for the stabilisation of non-linear systems. A numerical implementation is then proposed. The class of systems concerned with the proposed practical approach is quite large and contains all flat systems as a particular subset. The stabilising strategy is based on path generation strategy and avoids the integration of the differential system. The numerical implementation extensively uses the interpolation on a function basis. Two examples of systems known to be hard to stabilise are given to illustrate the proposed algorithm

**Reference**

Alamir, M.; Marchand, N. Numerical stabilisation of non-linear systems: Exact theory and approximate numerical implementation. Eur. J. Control 5, No.1, 87-97 (1999). [download]

### Stability of a Truncated Infinite Constrained Receding Horizon Scheme: the General Discrete Nonlinear Case

Some results concerning the constrained receding horizon formulation with infinite and truncated prediction horizon are presented. In this formulation a distinction is made between prediction and control horizons. The main result is a generalization of a known fact for linear systems, namely that, under certain technical requirements, the asymptotic stability of the above formulation holds for a sufficiently large finite prediction horizon. Existence results are also provided under stabilizability assumptions.

**Reference**

Alamir, M.; Bornard, G. Stability of a truncated infinite constrained receding horizon scheme: The general discrete nonlinear case. Automatica 31, No.9, 1353-1356 (1995). [download]

### On the stability of receding horizon control of nonlinear discrete-time systems

In this paper, we give sufficient conditions that guarantee the existence of the receding horizon control and the stability of the feedback system. The controlled system is a gen- eral nonlinear discrete-time system, Local and global stability are studied separately, giving rise to two sets of sufficient condi- tions, The context is that of regulation and'tracking with inter- nal model control scheme,

**Reference**

Alamir, M.; Bornard, G. On the stability of receding horizon control of nonlinear discrete-time systems. Syst. Control Lett. 23, No.4, 291-296 (1994). [download]

### Sur la stabilité de la commande optimal à horizon glissant des systèmes discrets non linéaires

**Reference**

Alamir, M.; Bornard, G. Sur la stabilité de la commande optimal à horizon glissant des systèmes discrets non linéaires. C. R. Académie des Sciences, série I, 318, 769-773 (1994).