A.S. Bortakovskii. Necessary optimality conditions for switching systems ... P. 67-78

We consider an optimal control problem for a switching system whose state vector contains both continuous and discretely varying components. The continuous and the discrete parts of the system are described by differential and recursive equations, respectively. The discrete part switches the operating modes of the continuous part and is itself influenced by the latter. The switching times and their number are not predefined. They are found as a result of optimizing the quality functional of the control process; here processes with instantaneous multiple switchings are not excluded. In deriving the necessary optimality conditions, we use small variations of the controls of the discrete part and variations of the switching times, which are represented by needle variations of the control of the continuous part of the system. The obtained conditions differ from the traditional equations for auxiliary variables due to the presence of instantaneous multiple switchings. The application of the optimality conditions is demonstrated by an academic example.

Keywords: hybrid systems, switching systems, optimal control

Received February 1, 2021

Revised February 22, 2021

Accepted March 1, 2021

Alexandr Sergeevich Bortakovskii, Dr. Phys.-Math. Sci., Prof., Moscow Aviation Institute (National Research University), Moscow, 125993 Russia; National University of Science and Technology MISIS, Moscow, 119049 Russia, e-mail: asbortakov@mail.ru

REFERENCES

1.   Vasiliev S.N., Malikov A.I. Some results on the stability of switchable and hybrid systems. In: Actual problems of continuum mechanics, vol. 1. Kazan’: Folio Publ., 2011, pp. 23–81. ISBN: 978-5-905576-03-4 .

2.   Bortakovskii A.S. Optimizatsiya pereklyuchayushchikh sistem [Optimization of switching systems]. Moscow: Mosk. Aviats. Inst. Publ., 2016, 120 p. ISBN: 978-5-4316-0329-7 .

3.   Branicky M.S., Borkar V.S., Mitter S.K. A unified framework for hybrid control: Model and optimal control theory. IEEE Trans. Automatic Control, 1998, vol. 43, no. 1, pp. 31–45. doi: 10.1109/9.654885 

4.   Brockett R.W. Hybrid models for motion control systems. In: Trentelman H.L., Willems J.C. (eds), Essays on control: perspectives in the theory and its applications, Progress in Systems and Control Theory, vol. 14, Boston: Birkhauser, 1993, pp. 29–53. doi: 10.1007/978-1-4612-0313-1_2 

5.   Miller B.M., Rubinovich E.Ya. Optimizatsiya dinamicheskikh sistem s impul’snymi upravleniyami [Optimization of dynamic systems with impulse controls]. Moscow: Nauka Publ., 2005, 429 p. ISBN: 5-02-033458-8 .

6.   Bortakovskii A.S., Panteleev A.V. Sufficient conditions for optimal control of batch systems. Avtomat. i Telemekh., 1987, no. 7, pp. 57–66 (in Russian).

7.   Bortakovskii A.S. Synthesis of optimal control-systems with a change of the models of motion. J. Comp. Syst. Sci. Internat., 2018, vol. 57, no. 4, pp. 543–560. doi: 10.1134/S1064230718040056 

8.   Sussmann H.J. A maximum principle for hybrid optimal control problems. Proc. of 38th IEEE Conf. on Decision and Control. Phoenix, AZ, USA, 1999, vol. 1, pp. 425–430. doi: 10.1109/CDC.1999.832814 

9.   Hedlund S., Rantzer A. Optimal control of hybrid systems. Proc. 38th IEEE Conference on Decision and Control. Phoenix, AZ, 1999, vol. 4, pp. 3972–3977. doi: 10.1109/CDC.1999.827981 

10.   Dmitruk A.V., Kaganovich A.M. Maximum principle for optimal control problems with intermediate constraints. Comput. Math. Model., 2011, vol. 22, no. 2, pp. 180–215. doi: 10.1007/s10598-011-9096-8 

11.   Ioffe A.D., Tihomirov V.M. Theory of extremal problems. Studies Math. Appl., vol. 6. Amsterdam; N Y; Oxford: North-Holland Publ. Comp., 1979, 460 p. ISBN: 0444851674 . Original Russian text published in Ioffe A.D., Tikhomirov V.M. Teoriya ekstremal’nykh zadach. Moscow: Nauka Publ., 1974, 480 p.

12.   Krotov V.F., Gurman V.I. Metody i zadachi optimal’nogo upravleniya [Methods and problems of optimal control]. Moscow: Nauka Publ., 1973, 446 p.

13.   Fedorenko R.P. Priblizhennoe reshenie zadach optimal’nogo upravleniya [Approximate solution of optimal control problems]. Moscow: Nauka Publ., 1978, 488 p.

14.   Letov A.M. Dinamika poleta i upravlenie [Flight dynamics and control]. Moscow: Nauka Publ., 1973, 390 p.

15.   Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The mathematical theory of optimal processes N Y; London: John Wiley & Sons, 1962, 360 p. ISBN: 0470693819 . Original Russian text published in Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal’nykh protsessov. Moscow: Fizmatgiz Publ., 1961, 391 p.

16.   Vasil’ev F.P. Metody optimizatsii [Optimization methods]. Moscow: Factorial Press, 2002, 824 p. ISBN: 5-88688-056-9 .

Cite this article as: A.S. Bortakovskii. Necessary optimality conditions for switching systems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 2, pp. 67–78.