On a control problem for a linear system with measurements of a part of phase coordinates

V. I. Maksimov

Trudy Instituta matematiki i mehaniki, Tome 23 (2017) no. 1, pp. 195-205

Résumé

We consider a control problem for a system of linear ordinary differential equations. It is required to design a feedback control procedure under which the velocity of a part of the phase coordinates of the system would track the velocity of a part of the phase coordinates of another system, which is subject to an unknown perturbation. It is assumed that a part of phase coordinates of each of the systems is measured with error at discrete times. We propose a solution algorithm that is stable to informational disturbances and computation errors. The algorithm is based on the extremal shift method known in the theory of guaranteed control. Since it is impossible to apply the “classical” extremal shift due to the incompleteness of the information on the phase coordinates, we propose a modification of this method that employs elements of the dynamic inversion theory. The latter is based on constructions from the theory of ill-posed problems. In the concluding section of the paper, we specify a class of systems nonlinear in the phase coordinates for which the algorithm is applicable.

Détail

Keywords: control, incomplete information, linear systems.

Bibliographie

[1] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR

[2] Krasovskii N.N., Upravlenie dinamicheskoi sistemoi, Nauka, M., 1985, 516 pp.

[3] Osipov Yu.S., Izbrannye trudy, Izd-vo Mosk. un-ta, Moskva, 2009, 656 pp.

[4] Subbotin A.I., Chentsov A.G., Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 1981, 288 pp. | MR

[5] Ushakov V.N., “K zadache postroeniya stabilnykh mostov v differentsialnoi igre sblizheniya-ukloneniya”, Izv. AN SSSR. Tekhn. kibernetika, 1980, no. 4, 29–36 | Zbl

[6] Barabanova N.N., Subbotin A.I., “O klassakh strategii v differentsialnykh igrakh ukloneniya ot vstrechi”, Prikl. matematika i mekhanika, 35:6 (1971), 345–356 | MR

[7] Patsko V.S., Poverkhnosti pereklyucheniya v lineinykh differentsialnykh igrakh, In-t matematiki i mekhaniki UrO RAN, 2004, 80 pp. | Zbl

[8] Maksimov V.I., “Ob otslezhivanii traektorii dinamicheskoi sistemy”, Prikl. matematika i mekhanika, 75:6 (2011), 951–960 | MR | Zbl

[9] Kryazhimskii A.V., Maksimov V.I., “Zadacha resursosberegayuschego slezheniya na beskonechnom promezhutke vremeni”, Differents. uravneniya, 47:7 (2011), 993–1002 | MR

[10] Maksimov V.I., “O kompensatsii vozmuschenii v nelineinykh upravlyaemykh sistemakh”, Izv. RAN. Teoriya i sistemy upravleniya, 2003, no. 2, 62–68 | Zbl

[11] Fagnani F., Maksimov V., Pandolfi L., “A recursive deconvolution approach to disturbance reduction”, IEEE Trans. Aut. Contr., 49:6 (2004), 907–921 | DOI | MR

[12] Maksimov V.I., “Ob otslezhivanii resheniya parabolicheskogo uravneniya”, Izv. vuzov. Matematika, 2012, no. 1, 40–48 | Zbl

[13] Maksimov V.I., “Ob odnom algoritme otslezhivaniya resheniya parabolicheskogo uravneniya na beskonechnom promezhutke vremeni”, Differents. uravneniya, 50:3 (2014), 366–375 | DOI | Zbl

[14] Maksimov V.I., “Ob odnom algoritme upravleniya lineinoi sistemoi pri izmerenii chasti koordinat fazovogo vektora”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:4 (2014), 218–230

[15] Maksimov V.I., “Ob otslezhivanii predpisannogo resheniya nelineinogo raspredelennogo uravneniya vtorogo poryadka”, Differents. uravneniya, 52:1 (2016), 128–131 | DOI | MR | Zbl

[16] Maksimov V.I., “O vychislenii proizvodnoi funktsii, zadannoi netochno, s pomoschyu zakonov obratnoi svyazi”, Tr. MIAN im. V.A. Steklova, 291, 2015, 231–243 | DOI | Zbl