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        "description": "# pytope\nPackage with a limited set of operations for polytopes, zonotopes, and invariant sets.\n\nThe currently implemented features include \n* constructing polytopes from inequalities (halfspace representation), vertices, and upper and lower bounds, \n* linear mapping (multiplying a matrix *M* and a polytope *P*: *M P*), \n* the Minkowski sum and Pontryagin difference of polytopes, \n* the intersection of polytopes, and\n* simple plotting. \n\nThe figures below are generated with pytope. \nThe first two illustrate the currently implemented operations (see [demo.py](https://github.com/heirung/pytope/blob/master/pytope/demo.py)); \nthe third uses pytope to plot a robust MPC trajectory (implemented in [CasADi](https://github.com/casadi), combining Figures 1 and 2 from Mayne et al. (2005)) (with a rough approximation of the disturbance sequence); \nthe fourth shows an outer *ε*-approximation of a minimal robust positively invariant (or MRPI) set (computed with pytope), reproducing the example from Rakovi\u0107 et al. (2005).  \n\nMost of pytope is experimental, fragile, largely untested, and buggy.\n\n![Illustration of various polytope operations](https://raw.githubusercontent.com/heirung/pytope/master/docs/various_operations.svg?sanitize=true)  \nFigure: Illustration of various polytope operations.\n\n![The Minkowski sum of two polytopes](https://raw.githubusercontent.com/heirung/pytope/master/docs/minkowski_sum.svg?sanitize=true)  \nFigure: The Minkowski sum of two polytopes.\n\n![Robust MPC trajectory from Mayne et al. (2005)](https://raw.githubusercontent.com/heirung/pytope/master/docs/Mayne_2005.svg?sanitize=true)  \nFigure: Robust MPC trajectory from Mayne et al. (2005), combining Figures 1 and 2.\n\n![Outer *ε*-approximation of a minimal RPI](https://raw.githubusercontent.com/heirung/pytope/master/docs/Rakovic_2005.svg?sanitize=true)  \nFigure: Outer *ε*-approximation of a minimal robust positively invariant (MRPI) set \u2013 a reproduction of the example in Rakovi\u0107 et al. (2005).\n\n### References\n* Mayne, D.Q., Seron, M.M., & Rakovi\u0107, S.V. (2005). \nRobust model predictive control of constrained linear systems with bounded disturbances. \n[*Automatica*, 41(2), 219\u2013224](https://doi.org/10.1016/j.automatica.2004.08.019).\n* Rakovi\u0107, S.V., Kerrigan, E.C., Kouramas, K.I., & Mayne, D.Q. (2005). \nInvariant approximations of the minimal robust positively invariant set. \n[*IEEE Transactions on Automatic Control*, 50(3), 406\u2013410](https://doi.org/10.1109/TAC.2005.843854).\n\n",
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