On Polyhedral Estimation of Reachable Sets in the “Extended” Space for Discrete-Time Systems with Uncertain Matrices and Integral Constraints

E. K. Kostousova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 1, pp. 141-155

Résumé

The problems of reachability and construction of estimates of reachable sets are considered for discrete-time systems with initially linear structure and uncertainties in the initial conditions, matrices, and additive input actions. The uncertainties are restricted by given parallelepiped-valued, interval, and integral nonquadratic constraints, respectively. The systems under consideration turn out to be of bilinear type due to the uncertainty in the matrices. The reachable sets are considered not only in the original space $\mathbb{R}^{n}$ but also in the “extended” space $\mathbb{R}^{n+1}$, where the last coordinate $\mu$ corresponds to the current reserve of the additive input action. An exact description is given for the reachable sets $\mathcal{Z}[k]$ in the “extended” space using multivalued recurrence relations. Here, the representation of sets in the form of the union of their $\mu$-sections is used, and the recurrence relations include operations with sets; one of the operations (multiplication by an interval matrix) acts on each cross-section independently, and another combines the Minkowski sum and the union over cross-sections. The reachable sets $\mathcal X[k]$ in $\mathbb{R}^{n}$ are determined by the cross-sections of $\mathcal{Z}[k]$ corresponding to $\mu=0$. However, it is usually difficult to calculate $\mathcal{Z}[k]$ exactly from the above relations. Methods are proposed for the construction of parameterized families of external and internal polyhedral estimates of the sets $\mathcal{Z}[k]$ in the form of polytopes of a special type. On this basis, external parallelepiped-valued and internal parallelotope-valued estimates of $\mathcal X[k]$ are constructed. All estimates are found by explicit formulas from systems of recurrence relations.

Détail

Keywords: reachable set, integral constraints, polyhedral estimates, parallelepipeds
Mots-clés : uncertain matrix, parallelotopes.

Bibliographie

[1] Krasovskii N.N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 476 pp.

[2] Kurzhanskii A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 392 pp.

[3] Kurzhanski A.B., Varaiya P., Dynamics and control of trajectory tubes: theory and computation, Systems Control: Foundations Applications, 85, Birkhauser, Basel, 2014, 445 pp. | DOI | MR

[4] Kurzhanski A.B., Daryin A.N., Dynamic programming for impulse feedback and fast controls: The linear systems case, LNCIS, 468, Springer, London, 2020, 275 pp. | DOI | MR | Zbl

[5] Lotov A.V., “Metod postroeniya vneshnei poliedralnoi otsenki trubki traektorii nelineinoi dinamicheskoi sistemy”, Dokl. AN, 472:1 (2017), 18–22 | DOI | Zbl

[6] Guseinov K.G., Ozer O., Akyar E., Ushakov V.N., “The approximation of reachable sets of control systems with integral constraint on controls”, Nonlinear Differential Equations and Appl., 14:1–2 (2007), 57–73 | DOI | MR | Zbl

[7] Kurzhanski A.B., Valyi I., Ellipsoidal calculus for estimation and control, Birkhauser, Boston, 1997, 321 pp. | MR | Zbl

[8] Chernousko F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov, Nauka, M., 1988, 319 pp.

[9] Ananev B.I., Gusev M.I., Filippova T.F., Upravlenie i otsenivanie sostoyanii dinamicheskikh sistem s neopredelennostyu, Izd-vo SO RAN, Novosibirsk, 2018, 193 pp.

[10] Le V.T.H., Stoica C., Alamo T., Camacho E.F., Dumur D., Zonotopes: From guaranteed state-estimation to control, Wiley-ISTE, Croydon, 2013, 150 pp. | DOI

[11] Sharma U., Thangavel S., Gottu Mukkula A.R., Paulen R., “Effective recursive parallelotopic bounding for robust output-feedback control”, IFAC-PapersOnLine, 51:15 (2018), 1032–1037 | DOI

[12] Dreossi T., Dang T., Piazza C., “Reachability computation for polynomial dynamical systems”, Formal Methods in System Design, 50:1 (2017), 1–38 | DOI | Zbl

[13] Kostousova E.K., “O vneshnem poliedralnom otsenivanii mnozhestv dostizhimosti v “rasshirennom” prostranstve dlya lineinykh mnogoshagovykh sistem s integralnymi ogranicheniyami na upravlenie”, Vychisl. tekhnologii, 9:5 (2004), 54–72 | Zbl

[14] Tang W., Wang Z., Shen Y., “Interval Estimation methods for discrete-time linear time-invariant systems”, Systems Control Letters, 123 (2019), 69–74 | DOI | MR | Zbl

[15] Filippova T.F., Matviychuk O.G., “Estimates of reachable sets of control systems with bilinear-quadratic nonlinearities”, Ural Math. J., 1:1 (2015), 45–54 | DOI | Zbl

[16] Mazurenko S.S., “Partial differential equation for evolution of star-shaped reachability domains of differential inclusions”, Set-Valued Var. Anal., 24:2 (2016), 333–354 | DOI | MR | Zbl

[17] Sinyakov V.V., “Metod vychisleniya vneshnikh i vnutrennikh approksimatsii mnozhestv dostizhimosti bilineinykh differentsialnykh sistem”, Differents. uravneniya, 51:8 (2015), 1101–1114 | DOI | MR | Zbl

[18] Kostousova E.K., “O poliedralnykh otsenkakh mnozhestv dostizhimosti mnogoshagovykh sistem s bilineinoi neopredelennostyu”, Avtomatika i telemekhanika, 2011, no. 9, 49–60 | Zbl

[19] Kostousova E.K., “State estimates of bilinear discrete-time systems with integral constraints through polyhedral techniques”, IFAC-PapersOnLine, 51:32 (2018), 245–250 | DOI

[20] Chernousko F.L., Rokityanskii D.Ya., “Ellipsoidal bounds on reachable sets of dynamical systems with matrices subjected to uncertain perturbations”, J. Optim. Theory Appl., 104:1 (2000), 1–19 | DOI | MR | Zbl

[21] Kostousova E.K., “O poliedralnykh otsenkakh mnozhestv dostizhimosti differentsialnykh sistem s bilineinoi neopredelennostyu”, Tr. In-ta matematiki i mekhaniki UrO RAN, 18:4 (2012), 195–210