Uniform optimization of controlled systems with distributed parameters

È. Ya. Rapoport

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 3, pp. 419-445.

Résumé

A constructive method is proposed for solving a spatiotemporal control problem in systems with distributed parabolic parameters under the conditions of the given accuracy of uniform approximation of the final state of a plant to the required spatial distribution of the controlled variable. The developed approach is based on the previously designed alternance method for constructing the parameterizable programmed control algorithms, which extended the results of the theory of nonlinear Chebyshev approximations to a wide range of optimization problems and uses the fundamental laws of the subject area. It is shown that in linear quadratic problem optimization the equations of optimal controllers with autonomous modal controls in the open domain of their definition and taking into account restrictions on the nature of the spatial distribution of the control actions specified by the conditions of technical implementation are reduced to linear feedback algorithms for the measured state of the plant with nonstationary transmission coefficients and the given dependence on the spatial arguments of the controlled value. The results obtained are extended to the problem of searching for time-invariant spatially distributed controls, considered as the desired design solutions for a plant.

Détail

Keywords: uniform metric, systems with distributed parameters, alternance method.

Bibliographie

[1] Rapoport E. Ya., Optimal'noe upravlenie sistemami s raspredelennymi parametrami [Optimal Control of Systems with Distributed Parameters], Vyssh. shk., Moscow, 2009, 677 pp. (In Russian)

[2] Rapoport E. Ya., Analiz i sintez sistem avtomaticheskogo upravleniya s raspredelennymi parametrami [Analysis and Synthesis of Automatic Control Systems with Distributed Parameters], Vyssh. shk., Moscow, 2005, 292 pp. (In Russian)

[3] Rapoport E. Ya., Al'ternansnyi metod v prikladnykh zadachakh optimizatsii [Alternance Method in Applied Optimization Problems], Nauka, Moscow, 2000, 336 pp. (In Russian)

[4] Rapoport E. Ya., Optimizatsiia protsessov induktsionnogo nagreva metalla [Optimization of the Induction Heating of Metal Process], Metallurgiia, Moscow, 1993, 279 pp. (In Russian)

[5] Rapoport E. Ya., Pleshivceva Yu. E., Metody polubeskonechnoi optimizatsii v prikladnykh zadachakh upravleniia sistemami s raspredelennymi parametrami Methods of Semi-Infinite Optimization in Applied Problems of Control of Systems with Distributed Parameters, Nauka, Moscow, 2021, 286 pp. (In Russian)

[6] Vladimirov V. S., Uravneniia matematicheskoi fiziki [Equations of Mathematical Physics], Nauka, Moscow, 1981, 512 pp. (In Russian)

[7] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki [Boundary Value Problems of Mathematical Physics], Nauka, Moscow, 1973, 407 pp. (In Russian)

[8] Polyanin A. D., Spravochnik po lineinym uravneniiam matematicheskoi fiziki [Handbook on Linear Equations of Mathematical Physics], Fizmatlit, Moscow, 2001, 576 pp. (In Russian)

[9] Rapoport E. Ya., “Open-loop controllability of multidimensional linear systems with distributed parameters”, J. Comput. Syst. Sci. Int., 54:2 (2015), 184–201 | DOI | DOI | Zbl

[10] Koshlyakov N. S., Smirnov M. M., Gliner E. B., Differential Equations of Mathematical Physics, North-Holland Publ., Amsterdam, 1964, xvi+701 pp. | Zbl

[11] Valeev G. K., Zhautykov O. A., Beskonecnye sistemy differencial'nyh uravnenii [Infinite Systems of Differential Equations], Nauka, Alma–Ata, 1974, 415 pp. (In Russian) | Zbl

[12] Koval' V. A., Spektral'nyi metod analiza i sinteza raspredelennykh upravliaemykh sistem Spectral Method of Analysis and Synthesis of Distributed Control Systems, Saratov State Techn. Univ., Saratov, 1997, 192 pp. (In Russian)

[13] Egorov Yu. V., “Necessary conditions for optimal control in Banach spaces”, Mat. Sb. (N.S.), 64(106):1 (1964), 79–101 (In Russian) | MR | Zbl

[14] Fedorenko R. P., Priblizhennoe reshenie zadach optimal'nogo upravleniia [Approximate Solution of Optimal Control Problems], Nauka, Moscow, 1978, 488 pp. (In Russian)

[15] Moiseev N. N., Elementy teorii optimal'nykh sistem [Elements of the Theory of Optimal Systems], Nauka, Moscow, 1975, 526 pp. (In Russian)

[16] Vasil'ev F. P., Metody optimizatsii [Optimization Methods], Faktorial Press, Moscow, 2002, 824 pp. (In Russian)

[17] Pleshivtseva Yu. E., Rapoport E. Ya., “The successive parameterization method of control actions in boundary value optimal control problems for distributed parameter systems”, J. Comput. Syst. Sci. Int., 48:3 (2009), 351–362 | DOI | Zbl

[18] Rapoport E. Ya., Pleshivtseva Yu. E., Optimal Control of Induction Heating Processes, CRC Press, Boca Raton, 2007, 349 pp. | DOI

[19] Rapoport E. Ya., Pleshivceva Yu. E., Optimal'noe upravlenie temperaturnymi rezhimami induktsionnogo nagreva [Optimal Control of Induction Heating Temperature], Nauka, Moscow, 2012, 309 pp. (In Russian)

[20] Rapoport E. Ya., Pleshivtseva Yu. E., “Optimal control of induction heating of metals prior to warm and hot forming”, Induction Heating and Heat Treatment, v. 4C, AMS Handbook, 2014, 366–401 | DOI

[21] Pleshivtseva Yu. E., Rapoport E. Yu., “Optimal energy-efficient programmed control of distributed parameter systems”, J. Comput. Syst. Sci. Int., 59:4 (2020), 518–532 | DOI | DOI | Zbl

[22] Rapoport E. Ya., “Analytical design of the optimal controllers in linear-quadratic problems of controlling systems with distributed parameters under uniform estimates of target sets”, J. Comput. Syst. Sci. Int., 60:3 (2021), 364–378 | DOI | DOI | Zbl

[23] Rapoport E. Ya., Pleshivtseva Yu. E., “Optimal energy-efficient control in distributed parameter systems”, Optoelectron. Instrument. Proc., 57:4 (2021), 345–355 | DOI | DOI

[24] Pleshivtseva Yu. E., Rapoport E. Yu., “Spatiotemporal control of systems with distributed parameters in linear-quadratic optimization problems with uniform estimates of target sets”, J. Comput. Syst. Sci. Int., 61:4 (2022), 523–538 | DOI | DOI

[25] Butkovskii A. G., Teoriia optimal'nogo upravleniia sistemami s raspredelennymi parametrami [Optimal Control Theory for Systems with Distributed Parameters], Nauka, Moscow, 1965, 474 pp. (In Russian)

[26] Egorov A. I., Optimal'noe upravlenie teplovymi i diffuzionnymi protsessami [Optimal Control of Thermal and Diffusion Processes], Nauka, Moscow, 1978, 500 pp. (In Russian)

[27] Sirazetdinov T. K., Optimizatsiia sistem s raspredelennymi parametrami [Optimization Systems with Distributed Parameters], Nauka, Moscow, 1977, 480 pp. (In Russian)

[28] Ray W. H., Advanced Process Control, McGraw-Hill, New York, 1981, xiii+376 pp.

[29] Degtiarev G. L., Sirazetdinov T. K., Teoreticheskie osnovy optimal'nogo upravleniia uprugimi kosmicheskimi apparatami [Theoretical Foundations of Optimal Control of Elastic Space Vehicles], Mashinostroenie, Moscow, 1986, 214 pp. (In Russian)

[30] Letov A. M., Matematicheskaia teoriia protsessov upravleniia [Mathematical Theory of Control Processes], Nauka, Moscow, 1981, 255 pp. (In Russian)

[31] Letov A. M., Dinamika poleta i upravleniia [Flight and Control Dynamics], Nauka, Moscow, 1969, 360 pp. (In Russian)