LMI-based Robust MPC Design

Contact information: Juraj Oravec


Brief introduction of the Institute of Information Engineering, Automation and Mathematics (IAM) [link]. Introduced the possibilities to study at IAM by ERASMUS+ Program [link].


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Part I: Alternative Robust MPC Design


Robust MPC is an advanced control strategy to optimize control performance subject to the constraints of the system inputs/outputs and in the presence of bounded disturbance. Several alternative approaches of online robust MPC design based on LMI formulation are formulated. The proposed alternative approaches are based on existing approaches. Suitable properties of these approaches are adopted to reduce the conservativeness of the quadratic stability condition, to reduce the conservativeness of control inputs constraints evaluation, and to minimize a computational effort. Therefore, these alternative robust MPC methods may be considered as a tool for overcoming some of the robust MPC design obstacles. Simulation case studies are proposed to demonstrate the effectiveness of the proposed strategies.


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Part II: Toolbox


Software MUP represents an efficient and user-friendly MATLAB-based toolbox for online robust MPC design in LMI-framework. The toolbox enables designing robust MPC using an all-in-one MATLAB/Simulink block. The advanced users may benefit from designing robust MPC using MATLAB Command-Line-Interface. One of the most valuable features is an advanced feasibility check, i.e., when the optimization problem is found infeasible, then we suggest to the user how to modify the robust MPC design problem to make it feasible. The MUP is dependent on the YALMIP toolbox that reformulates the optimization problem and delegates it to an external SDP solver, e.g., MOSEK or SeDuMi. An illustrative case study of robust MPC design for a real process is proposed to demonstrate the effectiveness of the MUP toolbox.


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Part III: LMI-based Robust MPC Design Exercises


The LMI-based robust MPC design exercises are oriented on the implementation of simple robust MPC for the uncertain system with input and state constraints. Two approaches are considered, i.e., manual implementation and implementation using the MUP toolbox. The manual implementation aims to point out the key ideas of robust MPC design. On the other hand, MUP-based implementation demonstrates the efficacy of the software and enables the user to focus the attention on the robust MPC tuning to improve the closed-loop control performance. This task includes also some hints on how to solve the considered robust MPC design problem.


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The LMI-based robust MPC design exercises are evaluated using MATLAB/Simulink environment. Although there were not observed obstacles by using older releases of MATLAB, the recommended release is R2014b or later.

We strongly recommend to use tbxmanager for MATLAB whenever possible. The install instructions can be found on its homepage. Then, you can download the required toolboxes YALMIP, SEDUMI, and MUP, using tbxmanager via MATLAB Command Window by typing:

tbxmanager install yalmip sedumi mup

To keep the releases up-to-date you should run the commands:

tbxmanager update yalmip sedumi mup

Part IV: Some references

Obviously, there should be a much more extensive list of references focused on LMI-based robust MPC design. Here are mentioned just several publications closely related to the considered topic. The following works (listed subject to the time of publishing) are crucial to cover the necessary theoretical backgrounds, considered software implementation, and selected applications.

  1. S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan (1994): Linear Matrix Inequalities in System and Control Theory. SIAM.

    Chapters 2 and 3 discussed wide classes of process control problems that can be formulated using linear matrix inequalities. Chapter 7 considered State-Feedback Synthesis for the systems in the continuous-time domain.

  2. L. Vandenberghe and S. Boyd (1996): Semidefinite Programming. SIAM, 38, 49-95.

    Principles of formulating and solving SDP problems.

  3. M. V. Kothare, V. Balakrishnan, M. Morari (1996): Robust Constrained Model Predictive Control Using Linear Matrix Inequalities. Automatica 32, 10, 1361-1379.

    Pioneer work of LMI-based robust MPC design. Based on former American Control Conference (1995) paper.

  4. F. Blanchini (1999): Set invariance in control. Automatica, 35, 1747-1767

    Robust positive invariant sets are crucial to guarantee the closed-loop system stability.

  5. A. Bemporad, M. Morari (1999): Robust model predictive control: A survey. Robustness in Identification and Control, Springer London, 207-226.

    Extensive survey on robust MPC.

  6. J. F. Sturm (1999): Using SeDuMi 1.02, A MATLAB Toolbox for Optimization Over Symmetric Cones. 1-30.

    Reference to well-known and freely available MATLAB-based toolbox to solve the SDP problems.

  7. F. A. Cuzzola, J. C. Geromel, M. Morari (2002): An Improved Approach for Constrained Robust Model Predictive Control, Automatica 38, 7, 1183-1189.

    The conservativeness of the quadratic stability condition was reduced by introducing the parameter-dependent Lyapunov function. Note, that robust MPC design did not consider the time-varying uncertainties. This issue was fixed in the paper Mao (2003).

  8. W. J. Mao (2003): Robust Stabilization of Uncertain Time-Varying Discrete Systems and Comments on "An Improved Approach for Constrained Robust Model Predictive Control". Automatica 39, 1109-1112.

    Fixed the issue of PDLF-based robust MPC design subject to the time-varying uncertainties.

  9. Z. Wan, M. V. Kothare (2003): Efficient Robust Constrained Model Predictive Control with a Time Varying Terminal Constraint Set. System & Control Letters 48, 375-383.

    System states have to converge into the origin in each control step. This paper considered the robust MPC design subject to the required value of states convergence just for the nominal system, i.e., system vertices were forced to converge without preset value.

  10. S. Boyd, L. Vandenberghe (2004): Convex Optimization. Cambridge University Press.

    Chapter 11 discussed in detail the theoretical backgrounds of Interior-point methods.

  11. J. Löfberg (2004): YALMIP: A Toolbox for Modeling and Optimization in MATLAB. IEEE International Symposium on Computer Aided Control Systems Design, Taipei, Taiwan, 284-289.

    Basic reference to an advanced MATLAB-based toolbox YALMIP that serves to formulate the complex optimization problems in user-friendly and tractable form. YALMIP was introduced already in 2003 by the author's PhD-thesis and originally was developed to solve especially LMI-based optimization problems.

  12. Y. Y. Cao, Z. Lin (2005): Min-max MPC Algorithm for LPV Systems Subject to Input Saturation. Control Theory and Applications, IEE Proceedings 153, 266-272.

    Conservativeness of constraints on the control inputs was reduced using advanced additional control input saturation.

  13. B. C. Ding, Y. G. Xi, M. T. Cychowski, T. O'Mahoney (2007): Improving Off-line Approach to Robust MPC Based-on Nominal Performance Cost, Automatica, 43, 158-163.

    PDLF-based robust MPC design subject to the nominal system optimization.

  14. Z. Li, Y. Shi, D. Sun, L. Wang (2008): An Improved Constrained Robust Model Predictive Control Algorithm for Linear Systems with Polytopic Uncertainty. Proc. of the 2008 IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics, Xi'an, China, 1272-1277.

    The drawback of the infinity prediction horizon is that the single controller is considered to control an uncertain system forever. Although the state-feedback controller is redesigned in each control step, the results may be conservative. The conservativeness was reduced by designing two controllers, one for the current control step, and the second for the next control steps.

  15. H. Huang, D. Li, Z. Lin, Y. Xi (2011): An Improved Robust Model Predictive Control Design in the Presence of Actuator Saturation. Automatica 47, 861-864.

    Extension of Cao et al. (2005). The robust MPC design tuning parameter was introduced to set the weight of the constrained and non-constrained control action. It enabled to compute more aggressive control action when the control input is not constrained.

  16. G. Blekherman , P. A. Parrilo and R. R. Thomas (2013): Semidefinite Optimization and Convex Algebraic Geometry

    Chapter 2 of the book provides excellent insight into Semidefinite Optimization.

  17. L. Zhang, J. Wang, K. Li (2013): An improved model predictive control for uncertain systems with input saturation. Journal of the Franklin Institute 350, 9, 2757-2768.

    LMI-based robust MPC design considering a set of linear control laws for a finite control horizon.

  18. L. Zhang, J. Wang, K. Li (2013): Min-Max MPC for LPV Systems Subject to Actuator Saturation by a Saturation-dependent Lyapunov Function. Proc. of the 32th Chinese Control Cnference, Xi'an, China, 4087-4092.

    Conservativeness of robust MPC design subject to the constrained control inputs is reduced by saturation-dependent Lyapunov functions.

  19. L. Zhang, J. Wang, B. Wang (2014): A Multi-Step Robust Model Predictive Control Scheme for Polytopic Uncertain Multi-Input Systems. In Proceedings of the 19th World Congress IFAC, Cape Town, South Africa, 8540-8545.

    Speed up of LMI-based robust MPC design by the advanced fixing of the control inputs in the next control step. The approach leads to the suboptimal solution.

  20. M. Bakošová, J. Oravec (2014): Robust Model Predictive Control of a Laboratory Two-Tank System. In American Control Conference, Portland, Oregon, USA, 5242-5247.

    Basic reference to MATLAB-based toolbox MUP for on-line LMI-based robust MPC design. Case study of robust MPC implementation for real process.

  21. D. Q. Mayne (2015): Model predictive control: Recent developments and future promise. Automatica, 50, 2967-2986.

    Current survey on MPC, issues of robust MPC were briefly discussed.

  22. J. Oravec, M. Bakošová (2015): Alternative LMI-based Robust MPC Design Approaches. In Proceedings of the 8th IFAC Symposium on Robust Control Design, Elsevier, Bratislava, Slovak Republic, 180-184

    Various alternative robust MPC design procedures were proposed. These approaches were derived considering the suitable properties of exiting strategies.

  23. J. Oravec, M. Bakošová (2015): Software for Efficient LMI-based Robust MPC Design. In Proceedings of the IEEE International Conference on Process Control, Slovakia, 272-277.

    Introduction to MATLAB-based toolbox MUP for on-line LMI-based robust MPC design.

  24. J. Oravec, M. Bakošová, A. Mészáros, N. Míková (2016): Experimental Investigation of Alternative Robust Model Predictive Control of a Heat Exchanger. Applied Thermal Engineering.

    Control performance analysis of alternative robust MPC design implemented for real process.

  25. J. Oravec, M. Bakošová, M. Trafczynski, A. Vasičkaninová, A. Mészáros, M. Markowski: (2018): Robust model predictive control and PID control of shell-and-tube heat exchangers. Energy.

    Fouling of the heat exchanger network modelled using the parametric uncertainty and the control performance analysis of alternative robust MPC design.

  26. J. Oravec, J. Holaza, M. Horváthová, N. A. Nguyen, M. Kvasnica, M. Bakošová (2019): Convex-lifting-based Robust Control Design Using the Tunable Robust Invariant Sets. European Journal of Control.

    Novel convex-lifting-based robust control strategy and its implementation to laboratory plants.

  27. J. Oravec, M. Bakošová, L. Galčíková, M. Slávik, M. Horváthová, A. Mészáros (2019): Soft-constrained robust model predictive control of a plate heat exchanger: Experimental analysis. Energy.

    Experimental analysis of the energy-efficient control of plate heat exchanger by implementing soft-constraints.

  28. J. Oravec, M. Horváthová, M. Bakošová (2020): Multivariable robust MPC design for neutralisation plant: Experimental analysis. Energy.

    Extensive experimental analysis of the multivariable control of a chemical reactor using a performance tuning.

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Last Update: 2021-07-08