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December  2018, 38(12): 6149-6162. doi: 10.3934/dcds.2018153

## On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints

 Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaja street, Ekaterinburg, 620990, Russia

Received  September 2017 Revised  December 2017 Published  April 2018

We deal with a problem of target control synthesis for dynamical bilinear discrete-time systems under uncertainties (which describe disturbances, perturbations or unmodelled dynamics) and state constraints. Namely we consider systems with controls that appear not only additively in the right hand sides of the system equations but also in the coefficients of the system. We assume that there are uncertainties of a set-membership kind when we know only the bounding sets of the unknown terms. We presume that we have uncertain terms of two kinds, namely, a parallelotope-bounded additive uncertain term and interval-bounded uncertainties in the coefficients. Moreover the systems are considered under constraints on the state ("under viability constraints"). We continue to develop the method of control synthesis using polyhedral (parallelotope-valued) solvability tubes. The technique for calculation of the mentioned polyhedral tubes by the recurrent relations is presented. Control strategies, which can be constructed on the base of the polyhedral solvability tubes, are proposed. Illustrative examples are considered.

Citation: Elena K. Kostousova. On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6149-6162. doi: 10.3934/dcds.2018153
##### References:
 [1] L. Asselborn, D. Groß and O. Stursberg, Control of uncertain nonlinear systems using ellipsoidal reachability calculus, IFAC Proceedings Volumes (IFAC PapersOnline)(Issue 23), 46 (2013), 50-55. doi: 10.3182/20130904-3-FR-2041.00204. [2] N. Athanasopoulos and G. Bitsoris, Unconstrained and constrained stabilisation of bilinear discrete-time systems using polyhedral Lyapunov functions, Int. J. Control, 83 (2010), 2483-2493. doi: 10.1080/00207179.2010.531396. [3] J. -P. Aubin, A. M. Bayen and P. Saint-Pierre, Viability Theory: New Directions, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-642-16684-6. [4] R. Baier and F. Lempio, Computing Aumann's integral, in Modeling Techniques for Uncertain Systems (Sopron, 1992), (eds. A. B. Kurzhanski and V. M. Veliov), Progr. Systems Control Theory, Birkhäuser, Boston, 18 (1994), 71–92. doi: 10.1007/978-3-642-78787-4_7. [5] B. R. Barmish and J. Sankaran, The propagation of parametric uncertainty via polytopes, IEEE Trans. Automat. Control., 24 (1979), 346-349. doi: 10.1109/TAC.1979.1102011. [6] P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Set-valued numerical analysis for optimal control and differential games, in Stochastic and Differential Games: Theory and Numerical Methods (Birkhäuser, Boston, 1999), Ann. Internat. Soc. Dynam. Games, 4 (1999), 177–247. [7] F. L. Chernousko, State Estimation for Dynamic Systems, CRC Press, Boca Raton, 1994. [8] A. N. Daryin and A. B. Kurzhanski, Parallel algorithm for calculating the invariant sets of high-dimensional linear systems under uncertainty, Comput. Math. Math. Phys., 53 (2013), 34–43, Transl. from Zh. Vychisl. Mat. Mat. Fiz. , 53 (2013), no. 1, 47–57 [Russian]. doi: 10.1134/S096554251301003X. [9] T. Donchev, E. Farkhi and B. S. Mordukhovich, Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces, J. Differential Equations, 243 (2007), 301-328. doi: 10.1016/j.jde.2007.05.011. [10] T. Filippova, Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty, Discrete Contin. Dyn. Syst. 2011, Dynamical Systems, Differential Equations and Applications, 8th AIMS Conference, Suppl. Ⅰ (2011), 410–419. [11] T. F. Filippova, Ellipsoidal estimates of reachable sets for control systems with nonlinear terms, IFAC Proceedings Volumes (IFAC PapersOnline)(Issue 1), 50 (2017), 15355-15360. doi: 10.1016/j.ifacol.2017.08.2460. [12] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three-body problem, J. Differential Equations, 77 (1989), 167-198. doi: 10.1016/0022-0396(89)90161-7. [13] M. I. Gusev, Application of penalty function method to computation of reachable sets for control systems with state constraints, AIP Conf. Proc., 1773 (2016), 050003. doi: 10.1063/1.4964973. [14] L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics, Springer-Verlag, London, 2001. doi: 10.1007/978-1-4471-0249-6. [15] E. K. Kostousova, Control synthesis via parallelotopes: Optimization and parallel computations, Optim. Methods Softw., 14 (2001), 267-310. doi: 10.1080/10556780108805805. [16] E. K. Kostousova, Polyhedral Approximations in Problems of Guaranteed Control and Estimation, Doctoral dissertation, Institute of Nathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, 2005 [Russian]. [17] E. K. Kostousova, On polyhedral estimates in problems of the synthesis of control strategies in linear multistep systems, in Algorithms and Software for Parallel Computations, Ross. Akad. Nauk Ural. Otdel., Inst. Mat. Mekh., Ekaterinburg, 9 (2006), 84–105 [Russian]. [18] E. K. Kostousova, On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty, Discrete Contin. Dyn. Syst. 2011, Dynamical Systems, Differential Equations and Applications, 8th AIMS Conference, Suppl. Ⅱ (2011), 864–873. [19] E. K. Kostousova, On target control synthesis under set-membership uncertainties using polyhedral techniques, in System Modeling and Optimization, 26th IFIP TC 7 Conference, CSMO 2013, IFIP AICT, 443 (2014), 170–180. doi: 10.1007/978-3-662-45504-3_16. [20] E. K. Kostousova, On control synthesis for uncertain dynamical discrete-time systems through polyhedral techniques, Discrete Contin. Dyn. Syst. 2015, Dynamical Systems, Differential Equations and Applications, 10th AIMS Conference, Suppl. (2015), 723–732. doi: 10.3934/proc.2015.0723. [21] E. K. Kostousova, On a polyhedral method for solving problems of control strategy synthesis, Proc. Steklov Inst. Math. , 292 (2016), Suppl. 1, S140–S155, Transl. from Tr. Inst. Mat. Mekh. , 20 (2014), no. 4,153–167. doi: 10.1134/S0081543816020127. [22] E. K. Kostousova, On feedback target control for uncertain discrete-time bilinear systems with state constraints through polyhedral techniques, AIP Conf. Proc., 1895 (2017), 110004. doi: 10.1063/1.5007410. [23] N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3716-7. [24] A. B. Kurzhanski and I. Vályi, Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, 1997. doi: 10.1007/978-1-4612-0277-6. [25] A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes., Theory and Computation (Systems & Control: Foundations & Applications, Book 85), Birkhäuser Basel, 2014. doi: 10.1007/978-3-319-10277-1. [26] A. A. Kurzhanskiy and P. Varaiya, Reach set computation and control synthesis for discrete-time dynamical systems with disturbances, Automatica J. IFAC, 47 (2011), 1414-1426. doi: 10.1016/j.automatica.2011.02.009. [27] J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM Journal of Optimization, 9 (1999), 112-147. doi: 10.1137/S1052623496303470. [28] R. de la Llave, A tutorial on KAM theory, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69 (2001), 175–292. doi: 10.1090/pspum/069/1858536. [29] A. V. Lotov, Method for constructing an external polyhedral estimate of the trajectory tube for a nonlinear dynamic system, Doklady Mathematics, 95 (2017), 95–98, Transl. from Dokl. Akad. Nauk, 472 (2017), no. 1, 18–22 [Russian]. doi: 10.1134/S1064562417010045. [30] K. Martynov, N. Botkin, V. Turova and J. Diepolder, Real-time control of aircraft takeoff in windshear. Part Ⅰ: Aircraft model and control schemes, in 2017 25th Mediterranean Conference on Control and Automation (MED), IEEE Xplore Digital Library, (2017), 277–284. doi: 10.1109/MED.2017.7984131. [31] K. Martynov, N. Botkin, V. Turova and J. Diepolder, Real-time control of aircraft takeoff in windshear. Part Ⅱ: Simulations and model enhancement, in 2017 25th Mediterranean Conference on Control and Automation (MED), IEEE Xplore Digital Library, (2017), 285–290. doi: 10.1109/MED.2017.7984132. [32] R. R. Mohler, Bilinear Control Processes. With Applications to Engineering, Ecology, and Medicine, Academic Press, New York and London, 1973. [33] M. S. Nikol'skii, On controllable variants of Richardson's model in political science, Proc. Steklov Inst. Math., 275 (2011), Suppl.1, S78-S85. doi: 10.1134/S0081543811090070. [34] A. M. Taras'yev, A. A. Uspenskiy and V. N. Ushakov, Approximation schemas and finitedifference operators for constructing generalized solutions of Hamilton-Jacobi equations, J. Comput. Systems Sci. Internat., 33 (1995), no. 6,127–139, Transl. from Izv. Ross. Akad. Nauk Tekhn. Kibernet., (1994), no. 3,173–185 [Russian]. [35] A. Yu. Vazhentsev, On internal ellipsoidal approximations for problems of control synthesis with bounded coordinates, J. Comput. Systems Sci. Internat., 39 (2000), 399-406, Transl. [36] V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions, Systems Control Lett., 13 (1989), 263-269. doi: 10.1016/0167-6911(89)90073-X.

show all references

##### References:
 [1] L. Asselborn, D. Groß and O. Stursberg, Control of uncertain nonlinear systems using ellipsoidal reachability calculus, IFAC Proceedings Volumes (IFAC PapersOnline)(Issue 23), 46 (2013), 50-55. doi: 10.3182/20130904-3-FR-2041.00204. [2] N. Athanasopoulos and G. Bitsoris, Unconstrained and constrained stabilisation of bilinear discrete-time systems using polyhedral Lyapunov functions, Int. J. Control, 83 (2010), 2483-2493. doi: 10.1080/00207179.2010.531396. [3] J. -P. Aubin, A. M. Bayen and P. Saint-Pierre, Viability Theory: New Directions, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-642-16684-6. [4] R. Baier and F. Lempio, Computing Aumann's integral, in Modeling Techniques for Uncertain Systems (Sopron, 1992), (eds. A. B. Kurzhanski and V. M. Veliov), Progr. Systems Control Theory, Birkhäuser, Boston, 18 (1994), 71–92. doi: 10.1007/978-3-642-78787-4_7. [5] B. R. Barmish and J. Sankaran, The propagation of parametric uncertainty via polytopes, IEEE Trans. Automat. Control., 24 (1979), 346-349. doi: 10.1109/TAC.1979.1102011. [6] P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Set-valued numerical analysis for optimal control and differential games, in Stochastic and Differential Games: Theory and Numerical Methods (Birkhäuser, Boston, 1999), Ann. Internat. Soc. Dynam. Games, 4 (1999), 177–247. [7] F. L. Chernousko, State Estimation for Dynamic Systems, CRC Press, Boca Raton, 1994. [8] A. N. Daryin and A. B. Kurzhanski, Parallel algorithm for calculating the invariant sets of high-dimensional linear systems under uncertainty, Comput. Math. Math. Phys., 53 (2013), 34–43, Transl. from Zh. Vychisl. Mat. Mat. Fiz. , 53 (2013), no. 1, 47–57 [Russian]. doi: 10.1134/S096554251301003X. [9] T. Donchev, E. Farkhi and B. S. Mordukhovich, Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces, J. Differential Equations, 243 (2007), 301-328. doi: 10.1016/j.jde.2007.05.011. [10] T. Filippova, Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty, Discrete Contin. Dyn. Syst. 2011, Dynamical Systems, Differential Equations and Applications, 8th AIMS Conference, Suppl. Ⅰ (2011), 410–419. [11] T. F. Filippova, Ellipsoidal estimates of reachable sets for control systems with nonlinear terms, IFAC Proceedings Volumes (IFAC PapersOnline)(Issue 1), 50 (2017), 15355-15360. doi: 10.1016/j.ifacol.2017.08.2460. [12] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three-body problem, J. Differential Equations, 77 (1989), 167-198. doi: 10.1016/0022-0396(89)90161-7. [13] M. I. Gusev, Application of penalty function method to computation of reachable sets for control systems with state constraints, AIP Conf. Proc., 1773 (2016), 050003. doi: 10.1063/1.4964973. [14] L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics, Springer-Verlag, London, 2001. doi: 10.1007/978-1-4471-0249-6. [15] E. K. Kostousova, Control synthesis via parallelotopes: Optimization and parallel computations, Optim. Methods Softw., 14 (2001), 267-310. doi: 10.1080/10556780108805805. [16] E. K. Kostousova, Polyhedral Approximations in Problems of Guaranteed Control and Estimation, Doctoral dissertation, Institute of Nathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, 2005 [Russian]. [17] E. K. Kostousova, On polyhedral estimates in problems of the synthesis of control strategies in linear multistep systems, in Algorithms and Software for Parallel Computations, Ross. Akad. Nauk Ural. Otdel., Inst. Mat. Mekh., Ekaterinburg, 9 (2006), 84–105 [Russian]. [18] E. K. Kostousova, On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty, Discrete Contin. Dyn. Syst. 2011, Dynamical Systems, Differential Equations and Applications, 8th AIMS Conference, Suppl. Ⅱ (2011), 864–873. [19] E. K. Kostousova, On target control synthesis under set-membership uncertainties using polyhedral techniques, in System Modeling and Optimization, 26th IFIP TC 7 Conference, CSMO 2013, IFIP AICT, 443 (2014), 170–180. doi: 10.1007/978-3-662-45504-3_16. [20] E. K. Kostousova, On control synthesis for uncertain dynamical discrete-time systems through polyhedral techniques, Discrete Contin. Dyn. Syst. 2015, Dynamical Systems, Differential Equations and Applications, 10th AIMS Conference, Suppl. (2015), 723–732. doi: 10.3934/proc.2015.0723. [21] E. K. Kostousova, On a polyhedral method for solving problems of control strategy synthesis, Proc. Steklov Inst. Math. , 292 (2016), Suppl. 1, S140–S155, Transl. from Tr. Inst. Mat. Mekh. , 20 (2014), no. 4,153–167. doi: 10.1134/S0081543816020127. [22] E. K. Kostousova, On feedback target control for uncertain discrete-time bilinear systems with state constraints through polyhedral techniques, AIP Conf. Proc., 1895 (2017), 110004. doi: 10.1063/1.5007410. [23] N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3716-7. [24] A. B. Kurzhanski and I. Vályi, Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, 1997. doi: 10.1007/978-1-4612-0277-6. [25] A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes., Theory and Computation (Systems & Control: Foundations & Applications, Book 85), Birkhäuser Basel, 2014. doi: 10.1007/978-3-319-10277-1. [26] A. A. Kurzhanskiy and P. Varaiya, Reach set computation and control synthesis for discrete-time dynamical systems with disturbances, Automatica J. IFAC, 47 (2011), 1414-1426. doi: 10.1016/j.automatica.2011.02.009. [27] J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM Journal of Optimization, 9 (1999), 112-147. doi: 10.1137/S1052623496303470. [28] R. de la Llave, A tutorial on KAM theory, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69 (2001), 175–292. doi: 10.1090/pspum/069/1858536. [29] A. V. Lotov, Method for constructing an external polyhedral estimate of the trajectory tube for a nonlinear dynamic system, Doklady Mathematics, 95 (2017), 95–98, Transl. from Dokl. Akad. Nauk, 472 (2017), no. 1, 18–22 [Russian]. doi: 10.1134/S1064562417010045. [30] K. Martynov, N. Botkin, V. Turova and J. Diepolder, Real-time control of aircraft takeoff in windshear. Part Ⅰ: Aircraft model and control schemes, in 2017 25th Mediterranean Conference on Control and Automation (MED), IEEE Xplore Digital Library, (2017), 277–284. doi: 10.1109/MED.2017.7984131. [31] K. Martynov, N. Botkin, V. Turova and J. Diepolder, Real-time control of aircraft takeoff in windshear. Part Ⅱ: Simulations and model enhancement, in 2017 25th Mediterranean Conference on Control and Automation (MED), IEEE Xplore Digital Library, (2017), 285–290. doi: 10.1109/MED.2017.7984132. [32] R. R. Mohler, Bilinear Control Processes. With Applications to Engineering, Ecology, and Medicine, Academic Press, New York and London, 1973. [33] M. S. Nikol'skii, On controllable variants of Richardson's model in political science, Proc. Steklov Inst. Math., 275 (2011), Suppl.1, S78-S85. doi: 10.1134/S0081543811090070. [34] A. M. Taras'yev, A. A. Uspenskiy and V. N. Ushakov, Approximation schemas and finitedifference operators for constructing generalized solutions of Hamilton-Jacobi equations, J. Comput. Systems Sci. Internat., 33 (1995), no. 6,127–139, Transl. from Izv. Ross. Akad. Nauk Tekhn. Kibernet., (1994), no. 3,173–185 [Russian]. [35] A. Yu. Vazhentsev, On internal ellipsoidal approximations for problems of control synthesis with bounded coordinates, J. Comput. Systems Sci. Internat., 39 (2000), 399-406, Transl. [36] V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions, Systems Control Lett., 13 (1989), 263-269. doi: 10.1016/0167-6911(89)90073-X.
Results of polyhedral synthesis for case (A) using the following controls: only control $u$, only $U$, both $u$ and $U$
Results of polyhedral synthesis with both controls $u$ and $U$ for cases (A), (B, ⅱ), and (B, ⅱ; SC)
Suitable polyhedral tubs $\mathcal{P}^{-}{[\cdot]}$ and corresponding controlled trajectories with both $u$ and $U$ for cases (A) and (B, ⅱ; SC)
Corresponding controls $u$ and $U$ for cases (A) and (B, ⅱ; SC)
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