Construction of the time-optimal bounded control for linear discrete-time systems based on the method of superellipsoidal approximation. (English. Russian original) Zbl 1530.93248
Autom. Remote Control 84, No. 9, 924-946 (2023); translation from Avtom. Telemekh. 2023, No. 9, 37-67 (2023).
Summary: The speed-in-action problem for a linear discrete-time system with bounded control is considered. In the case of superellipsoidal constraints on the control, the optimal control process is constructed explicitly on the basis of the discrete maximum principle. The problem of calculating the initial conditions for an adjoint system is reduced to solving a system of algebraic equations. The algorithm for generating a guaranteeing solution based on the superellipsoidal approximation method is proposed for systems with general convex control constraints. The procedure of superellipsoidal approximation is reduced to solving a number of convex programming problems. Examples are given.
MSC:
| 93C55 | Discrete-time control/observation systems |
| 93C05 | Linear systems in control theory |
| 93B25 | Algebraic methods |
| 49J05 | Existence theories for free problems in one independent variable |
Keywords:
linear discrete-time systems; speed-in-action problem; maximum principle; superellipse; ellipsoidal approximationsReferences:
| [1] | Pontryagin, L.S., Boltyansky, V.G., Gamkrelidze, R.V., and Mishchenko, B.F., Matematicheskaya teoriya optimal’nykh protsessov (Mathematical Theory of Optimal Processes), Moscow: Nauka, 1969. · Zbl 0179.14001 |
| [2] | Boltyanskii, V.G., Optimal’noe upravlenie diskretnymi sistemami (Optimal Control of Discrete Systems), Moscow: Nauka, 1973. · Zbl 0268.49017 |
| [3] | Propoi, A.I., Elementy teorii optimal’nykh diskretnykh protsessov (Elements of the Theory of Optimal Discrete Processes), Moscow: Nauka, 1973. |
| [4] | Ibragimov, D. N.; Sirotin, A. N., On the Problem of Operation Speed for the Class of Linear Infinite-Dimensional Discrete-Time Systems with Bounded Control, Autom. Remote Control, 78, 1731-1756 (2017) · Zbl 1387.93108 · doi:10.1134/S0005117917100010 |
| [5] | Bellman, R., Dinamicheskoe programmirovanie (Dynamic Programming), Moscow: Inostrannaya Literatura, 1960. |
| [6] | Ibragimov, D. N.; Novozhilin, N. M.; Portseva, E. Yu., On Sufficient Optimality Conditions for a Guaranteed Control in the Speed Problem for a Linear Time-Varying Discrete-Time System with Bounded Control, Autom. Remote Control, 82, 2076-2096 (2021) · Zbl 1483.93361 · doi:10.1134/S000511792112002X |
| [7] | Kamenev, G.K., Chislennoe issledovanie effektivnosti metodov poliedral’noi approksimatsii vypuklykh tel (Numerical Study of the Efficiency of Polyhedral Approximation Methods for Convex Bodies), Moscow: Vychisl. Tsentr Ross. Akad. Nauk, 2010. |
| [8] | Ibragimov, D.N., The Minimum-Time Correction of the Satellite’s Orbit, Elektron. Zh. Tr. MAI, 2017, no. 94. http://trudymai.ru/published.php |
| [9] | Kurzhanskiy, A.; Varaiya, P., Ellipsoidal Techniques for Reachability Analysis of Discrete-Time Linear Systems, IEEE Transactions on Automatic Control, 52, 26-38 (2007) · Zbl 1366.93039 · doi:10.1109/TAC.2006.887900 |
| [10] | Chernous’ko, F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov (Phase State Estimation of Dynamic Systems. The Ellipsoid Method), Moscow: Nauka, 1988. · Zbl 1124.93300 |
| [11] | Gridgeman, N. T., Lame Ovals, The Mathematical Gazette, 54, 31-37 (1970) · Zbl 0189.22901 · doi:10.2307/3613154 |
| [12] | Tobler, W. R., The Hyperelliptical and Other New Pseudo Cylindrical Equal Area Map Projections, J. Geophys. Res., 78, 1753-1759 (1973) · doi:10.1029/JB078i011p01753 |
| [13] | Shi, P. J.; Huang, J. G.; Hui, C.; Grissino-Mayer, H. D.; Tardif, J. C.; Zhai, L. H.; Wang, F. S.; Li, B. L., Capturing Spiral Radial Growth of Conifers Using the Superellipse to Model Tree-Ring Geometric Shape, Frontiers in Plant Science, 6, 1-13 (2015) · doi:10.3389/fpls.2015.00856 |
| [14] | Gielis, J., A Generic Geometric Transformation That Unifies a Wide Range of Natural and Abstract Shapes, Amer. J. Botany, 90, 333-338 (2003) · doi:10.3732/ajb.90.3.333 |
| [15] | Maximidis, R.T., Caratelli, D., Toso, G., and Smolders, B.A., Analysis of a Novel Class of Waveguiding Structures Suitable for Reactively Loaded Antenna Array Design, Doklady TUSUR, 2017, no. 1, pp. 10-13. doi:10.21293/1818-0442-2017-20-1-09-13 |
| [16] | Zolotenkova, M.K. and Egorov, V.V., Development and Analysis of Ultrasound Registrating and Performing Rodent Vocalization Device, IEEE-EDM, 2022, pp. 506-509. doi:10.1109/EDM55285.2022.9855056 |
| [17] | Sadowski, A. J., Geometric Properties for the Design of Unusual Member Cross-Sections in Bending, Engineering Structures, 33, 1850-1854 (2011) · doi:10.1016/j.engstruct.2011.01.026 |
| [18] | Tobler, W. R., Superquadrics and Angle-Preserving Transformations, IEEE-CGA, 1, 11-23 (1981) · doi:10.1109/MCG.1981.1673799 |
| [19] | Desoer, C. A.; Wing, J., The Minimal Time Regulator Problem for Linear Sampled-Data Systems: General Theory, J. Franklin Inst., 272, 208-228 (1961) · Zbl 0161.15002 · doi:10.1016/0016-0032(61)90784-0 |
| [20] | Lin, W.-S., Time-Optimal Control Strategy for Saturating Linear Discrete Systems, Int. J. Control, 43, 1343-1351 (1986) · Zbl 0586.93020 · doi:10.1080/00207178608933543 |
| [21] | Moroz, A. I., Synthesis of Time-Optimal Control for Linear Discrete Objects of the Third Order, Autom. Remote Control, 25, 193-206 (1965) |
| [22] | Krasnoshchechenko, V.I., Simplex Method for Solving the Brachistochroneproblem at State and Control Constraints, Inzhenernyi Zhurnal: Nauka i Innovatsii, 2014, no. 6. http://engjournal.ru/catalog/it/asu/1252.html |
| [23] | Cazanova, L.A., Stability of optimal synthesis in the time-optimality problem, Izvestiya Vuzov, Matematika, 2002, no. 2, pp. 46-57. · Zbl 1096.49523 |
| [24] | Rockafellar, R., Vypuklyi analiz (Convex Analysis), Moscow: Mir, 1973. · Zbl 0251.90035 |
| [25] | Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Fizmatlit, 2012. |
| [26] | Berendakova, A. V.; Ibragimov, D. N., About the Method for Constructing External Estimates of the Limit 0-Controllability Set for the Linear Discrete-Time System with Bounded Control, Autom. Remote Control, 84, 83-104 (2023) · Zbl 1521.93012 · doi:10.1134/S0005117923020030 |
| [27] | Ibragimov, D. N., On the Optimal Speed Problem for the Class of Linear Autonomous Infinite-Dimensional Discrete-Time Systems with Bounded Control and Degenerate Operator, Autom. Remote Control, 80, 393-412 (2019) · Zbl 1431.93040 · doi:10.1134/S0005117919030019 |
| [28] | Ostrowski, A.M., Reshenie uravnenii i sistem uravnenii (Solution of Equations and Systems of Equations), Moscow: Izdatel’stvo Inostrannoi Literatury, 1963. · Zbl 0126.32004 |
| [29] | Landau, L. D.; Lifshitz, E. M., Mekhanika (Mechanics) (1988), Moscow: Nauka, Moscow |
| [30] | Horn, R. and Johnson, C., Matrichnyi analiz (Matrix Analysis), Moscow: Mir, 1989. |
| [31] | Ashmanov, S.A. and Timohov, S.V., Teoriya optimizatsii v zadachakh i uprazhneniyakh (Optimizationtheory in Problems and Exercises), Moscow: Nauka, 1991. · Zbl 0729.90060 |
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