Optimal Energy-Efficient Programmed Control of Distributed Parameter Systems
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ROL IN SYSTEMS WITH DISTRIBUTED PARAMETERS
Optimal Energy-Efficient Programmed Control of Distributed Parameter Systems Yu. E. Pleshivtsevaa and E. Ya. Rapoporta,* a
Samara State Technical University, Samara, 443001 Russia *e-mail: [email protected]
Received October 7, 2019; revised January 15, 2020; accepted January 27, 2020
Abstract—A constructive solution of the optimal energy-efficient programmed control problem for distributed parameter systems with a given-precision uniform approximation of the space distribution of the controlled variable with respect to the desired state is proposed. The computational algorithm developed below involves a special-form preliminary parametrization procedure for control actions on finite-dimensional subsets of the terminal values of conjugate variables in the boundary-value problem of Pontryagin’s maximum principle, in combination with the subsequent reduction to a semi-infinite optimization problem, which is solved with respect to the requisite parameter vector using the alternance method suggested earlier. An example of optimal energy-efficient control of transient heat conduction, which is of independent interest, is given. DOI: 10.1134/S1064230720030120
INTRODUCTION When solving optimal control problems for dynamic systems in the classical two-point state-space formulation with the fixed-end trajectories of a controlled system, one faces well-known difficulties that further increase for the higher orders of the differential equations of the system’s model and become fundamental for the infinite-dimensional distributed parameter systems (DPSs) [1–3]. For many DPSs, either such problems have no solutions due to their uncontrollability for the typical terminal states of the system (caused by the latter’s inconsistency with the common boundary conditions of the corresponding initialboundary value problems), or the resulting solutions belong to the class of technically non-implementable control actions [1–5]. In such a situation, a constructive approach to the optimal control problems of DPSs consists in the transition to a solvable control problem with implementable control actions and the given target set of the resulting space distributions of the controlled variable; this set must adhere to the achievable tolerances ε on the deviations from the system’s terminal state specified in the initial two-point scheme, which are common for engineering applications and are evaluated in a uniform metric in compliance with standard engineering requirements [5–7]. After the preliminary sequential parametrization procedure of control actions using the analytical optimality conditions, the optimal control problems for DPSs in the formulation described above are reduced to a parametric semi-infinite optimization problem; a technically implementable solution of this problem can be obtained using a constructive approach based on the alternance method [5–7]. In this paper, the constructive approach is adopted for obtaining an algorithmically precise solution of a particular optimal ene
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