A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Volume 5 Issue 4
Jul.  2018

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 11.8, Top 4% (SCI Q1)
    CiteScore: 17.6, Top 3% (Q1)
    Google Scholar h5-index: 77, TOP 5
Turn off MathJax
Article Contents
Lantao Xie, Lei Xie, Hongye Su and Jingdai Wang, "Polyhedral Feasible Set Computation of MPC-Based Optimal Control Problems," IEEE/CAA J. Autom. Sinica, vol. 5, no. 4, pp. 765-770, July 2018. doi: 10.1109/JAS.2018.7511126
Citation: Lantao Xie, Lei Xie, Hongye Su and Jingdai Wang, "Polyhedral Feasible Set Computation of MPC-Based Optimal Control Problems," IEEE/CAA J. Autom. Sinica, vol. 5, no. 4, pp. 765-770, July 2018. doi: 10.1109/JAS.2018.7511126

Polyhedral Feasible Set Computation of MPC-Based Optimal Control Problems

doi: 10.1109/JAS.2018.7511126
Funds:

the Natural Science Foundation of Zhejiang Province LR17F030002

the Science Fund for Creative Research Groups of the National Natural Science Foundation of China 61621002

More Information
  • Feasible sets play an important role in model predictive control (MPC) optimal control problems (OCPs). This paper proposes a multi-parametric programming-based algorithm to compute the feasible set for OCP derived from MPC-based algorithms involving both spectrahedron (represented by linear matrix inequalities) and polyhedral (represented by a set of inequalities) constraints. According to the geometrical meaning of the inner product of vectors, the maximum length of the projection vector from the feasible set to a unit spherical coordinates vector is computed and the optimal solution has been proved to be one of the vertices of the feasible set. After computing the vertices, the convex hull of these vertices is determined which equals the feasible set. The simulation results show that the proposed method is especially efficient for low dimensional feasible set computation and avoids the non-unicity problem of optimizers as well as the memory consumption problem that encountered by projection algorithms.

     

  • loading
  • [1]
    D. Q. Mayne, "Model predictive control: recent developments and future promise, " Automatica, vol. 50, no. 12, pp. 2967-2986, Dec. 2014. http://www.sciencedirect.com/science/article/pii/S0005109814005160
    [2]
    A. Mesbah, "Stochastic model predictive control: an overview and perspectives for future research, " IEEE Control Syst., vol. 36, no. 6, pp. 30-44, Dec. 2016. http://ieeexplore.ieee.org/document/7740982/
    [3]
    F. Scibilia, S. Olaru, and M. Hovd, "On feasible sets for MPC and their approximations, " Automatica, vol. 47, no. 1, pp. 133-139, Jan. 2011. http://dl.acm.org/citation.cfm?id=1923254
    [4]
    L. T. Xie, L. Xie, and H. Y. Su, "A comparative study on algorithms of robust and stochastic MPC for uncertain systems, " Acta Autom. Sinica, vol. 43, no. 6, pp. 969-992, 2017. https://www.researchgate.net/publication/318860024_A_Comparative_Study_on_Algorithms_of_Robust_and_Stochastic_MPC_for_Uncertain_Systems
    [5]
    M. Lorenzen, F. Dabbene, R. Tempo, and F. Allgöwer, "Constraint-tightening and stability in stochastic model predictive control, " IEEE Trans. Autom. Control, vol. 62, no. 7, pp. 3165-3177, Jul. 2017. http://arxiv.org/abs/1511.03488
    [6]
    S. V. Raković, B. Kouvaritakis, M. Cannon, C. Panos, and R. Findeisen, "Parameterized tube model predictive control, " IEEE Trans. Autom. Control, vol. 57, no. 11, pp. 2746-2761, Nov. 2012. http://www.sciencedirect.com/science/article/pii/S0005109815005646
    [7]
    L. T. Xie and L. Xie, Linear mismatched model based offset-free MPC for nonlinear constrained cstr systems with stochastic and deterministic disturbances. Control Engineering Practice, 2018.
    [8]
    J. Löfberg, "Minimax approaches to robust model predictive control, " Ph. D. dissertation, Linköping University, Linköping, Sweden, 2003. https://www.researchgate.net/publication/228869742_Minmax_approaches_to_robust_model_predictive_control
    [9]
    C. N. Jones, E. C. Kerrigan, and J. M. Maciejowski, "On polyhedral projection and parametric programming, " J. Optim. Theory Appl., vol. 138, no. 2, pp. 207-220, Aug. 2008. http://www.ams.org/mathscinet-getitem?mr=2414994
    [10]
    S. V. Raković, B. Kouvaritakis, R. Findeisen, and M. Cannon, "Homothetic tube model predictive control, " Automatica, vol. 48, no. 8, pp. 1631-1638, Aug. 2012. http://www.sciencedirect.com/science/article/pii/S0005109812001768
    [11]
    F. Borrelli, A. Bemporad, and M. Morari, "Geometric algorithm for multiparametric linear programming, " J. Optim. Theory Appl., vol. 118, no. 3, pp. 515-540, Sep. 2003. doi: 10.1023/B%3AJOTA.0000004869.66331.5c
    [12]
    J. Spjotvold, P. Tondel, and T. A. Johansen, "Continuous selection and unique polyhedral representation of solutions to convex parametric quadratic programs, " J. Optim. Theory Appl., 134, no. 2, pp. 177-189, Aug. 2007. doi: 10.1007/s10957-007-9215-z
    [13]
    J. A. Primbs and C. H. Sung, "Stochastic receding horizon control of constrained linear systems with state and control multiplicative noise, " IEEE Trans. Autom. Control, vol. 54, no. 2, pp. 221-230, Feb. 2009. http://www.researchgate.net/publication/4265686_Stochastic_Receding_Horizon_Control_of_Constrained_Linear_Systems_with_State_and_Control_Multiplicative_Noise
    [14]
    M. Cannon, B. Kouvaritakis, S. V. Rakovic, and Q. F. Cheng, "Stochastic tubes in model predictive control with probabilistic constraints, " IEEE Trans. Autom. Control, vol. 56, no. 1, pp. 194-200, Jan. 2011. http://ieeexplore.ieee.org/document/5599849
    [15]
    A. Bemporad and M. Morari, "Robust model predictive control: A survey, " in Robustness in Identification and Control, A. Garulli and A. Tesi, Eds. London, UK: Springer, 1999, pp. 207-226. http://www.springerlink.com/index/n8734746605j8447.pdf
    [16]
    G. Blekherman, P. A. Parrilo, and R. R. Thomas, Semidefinite Optimization and Convex Algebraic Geometry. Philadelphia, Pennsylvania, USA: SIAM, 2012.
    [17]
    S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: Cambridge University Press, 2004.
    [18]
    A. Brondsted, An Introduction to Convex Polytopes. New York, USA: Springer-Verlag, 1983.
    [19]
    E. Castillo, A. Cobo, F. Jubete, and R. E. Pruneda, Orthogonal Sets and Polar Methods in Linear Algebra: Applications to Matrix Calculations, Systems of Equations, Inequalities, and Linear Programming. New York, USA: John Wiley & Sons, 2011.
    [20]
    K. Kellner, T. Theobald, and C. Trabandt, "Containment problems for polytopes and spectrahedra, " SIAM J. Optim., vol. 23, no. 2, pp. 1000-1020, May 2013. http://www.oalib.com/paper/3951878
    [21]
    M. V. Ramana, "Polyhedra, spectrahedra, and semidefinite programming, " in Topics in Semidefinite and Interior-Point Methods, Fields Institute Communications, Providence, RI, USA, vol. 18, pp. 27-38, 1997. http://www.ams.org/mathscinet-getitem?mr=1607318
    [22]
    A. Bhardwaj, P. Rostalski, and R. Sanyal, "Deciding polyhedrality of spectrahedra, " SIAM J. Optim., vol. 25, no. 3, pp. 1873-1884, Sep. 2015. http://arxiv.org/abs/1102.4367
    [23]
    D. Q. Mayne, M. M. Seron, and S. V. Raković, "Robust model predictive control of constrained linear systems with bounded disturbances, " Automatica, vol. 41, no. 2, pp. 219-224, Feb. 2005. http://ogma.newcastle.edu.au/vital/access/manager/Repository/uon:361?exact=creator%3A
    [24]
    D. Limon, I. Alvarado, T. Alamo, and E. F. Camacho, "Robust tube-based MPC for tracking of constrained linear systems with additive disturbances, " J. Process Control, vol. 20, no. 3, pp. 248-260, Mar. 2010. http://www.sciencedirect.com/science/article/pii/S0959152409002169
    [25]
    S. N. Čhernikov, "Contraction of finite systems of linear inequalities, " Dokl. Akad. Nauk SSSR, vol. 152, pp. 1075-1078, 1963. http://www.mathnet.ru/eng/dan28687
    [26]
    Michael Grant, Stephen Boyd, and Y. Y. Ye, "CVX users guide, " 2009.
    [27]
    M. Herceg, M. Kvasnica, C. N. Jones, and M. Morari, "Multi-parametric toolbox 3. 0, " in 2013 European Control Conf. (ECC), Zurich, Switzerland, pp. 502-510.

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Figures(5)

    Article Metrics

    Article views (1393) PDF downloads(205) Cited by()

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return