An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations

Arguchintsev, A. V.; Поплевко, Василиса Павловна

doi:10.3390/g12010023

Cited by 6 publications

(5 citation statements)

References 11 publications

(21 reference statements)

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“…Numerical experiments prove effectiveness of the method. This effective algorithm for solving the initialboundary value problem allows to proceed to the implementation of methods for optimal control of flows in columns using variational optimality conditions for such problems [Arguchintsev, 2021].…”

Section: Discussionmentioning

confidence: 99%

Numerical solution of the initial-boundary value problem describing separation processes in a distillation column

Arguchintsev,

Kopylov

2023

CAP

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A modification of the numerical method of characteristics is developed to solve the initial-boundary value problem that arises when modeling the rectification process in the column. The process is described by a system of first-order hyperbolic equations. A specific peculiarity of the model is in the boundary conditions of a special type. At each of the boundaries, boundary conditions are determined from a system of ordinary differential equations, which also includes unknown values of functions on another boundary. A characteristic difference grid is constructed on the base of a linear transformation of a classical rectangular grid. Implicit second-order difference schemes are used, taking into account the features of the problem at the boundaries. The advantage of this approach is in consideration of the specifics of the propagation of perturbations in hyperbolic equations. Numerical implementation of the method was carried out. An illustrative example shows the effectiveness of the proposed modification of the characterization method. This method is a base for further solution of optimal control problems of flows in columns.

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Section: Discussionmentioning

confidence: 99%

Numerical solution of the initial-boundary value problem describing separation processes in a distillation column

Arguchintsev,

Kopylov

2023

CAP

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“…11 and 12 ) constrained with the linearized dynamics of Eq. 6 with costate parameters p(t) used in the Hamiltonian problem formulation leads to time-optimal control ( Flugge-Lotz, 1953 ; Pontryagin et al, 1962 ; Boltyanskii, 1971 ; Sands et al, 2009 ; Sands and Ghadawala, 2011 ; Duprez et al, 2017 ; Heidlauf and Cooper, 2017 ; Baker et al, 2018 ; Sands, 2019 ; Smeresky et al, 2020 ; Arguchintsev and Poplevko, 2021 ; Malecek, 2021 ; Srochko et al, 2021 ).

…”

Section: Methodsmentioning

confidence: 99%

“…This manuscript seeks to extend the notion of tacking nonlinear transport theorem to include time-varying angular velocities. Arguchintsev and Poplevko (2021 ) proposed an optimal control for linear hyperbolic systems of ordinary differential equations by estimating the residuals in terms of the value that characterizes the smallness of the measure of the domain of the needle variation of control. Emphasis was placed on problem formulation by Srochko et al (2021 ), but the focus was parameterizing the cost functional rather than the nonlinear constraint function as done in this work.…”

Section: Introductionmentioning

confidence: 99%

Treatise on Analytic Nonlinear Optimal Guidance and Control Amplification of Strictly Analytic (Non-Numerical) Methods

Sands

2022

Front. Robot. AI

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Optimal control is seen by researchers from a different perspective than that from which the industry practitioners see it. Either type of user can easily become confounded when deciding which manner of optimal control should be used for guidance and control of mechanics. Such optimization methods are useful for autonomous navigation, guidance, and control, but their performance is hampered by noisy multi-sensor technologies and poorly modeled system equations, and real-time on-board utilization is generally computationally burdensome. Some methods proposed here use noisy sensor data to learn the optimal guidance and control solutions in real-time (online), where non-iterative instantiations are preferred to reduce computational burdens. This study aimed to highlight the efficacy and limitations of several common methods for optimizing guidance and control while proposing a few more, where all methods are applied to the full, nonlinear, coupled equations of motion including cross-products of motion from the transport theorem. While the reviewed literature introduces quantitative studies that include parametric uncertainty in nonlinear terms, this article proposes accommodating such uncertainty with time-varying solutions to Hamiltonian systems of equations solved in real-time. Five disparate types of optimum guidance and control algorithms are presented and compared to a classical benchmark. Comparative analysis is based on tracking errors (both states and rates), fuel usage, and computational burden. Real-time optimization with singular switching plus nonlinear transport theorem decoupling is newly introduced and proves superior by matching open-loop solutions to the constrained optimization problem (in terms of state and rate errors and fuel usage), while robustness is validated in the utilization of mixed, noisy state and rate sensors and uniformly varying mass and mass moments of inertia. Compared to benchmark, state-of-the-art methods state tracking errors are reduced one-hundred ten percent. Rate tracking errors are reduced one-hundred thirteen percent. Control utilization (fuel) is reduced eighty-four percent, while computational burden is reduced ten percent, simultaneously, where the proposed methods have no control gains and no linearization.

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“…The reduced problem can be solved using a wide range of efficient methods used for this class of optimal control problems in systems of ODEs. This approach was proposed in [Arguchintsev, 2021] for classic optimal control problems with fixed boundary conditions and without delay. Two symmetric variational conditions were proved.…”

Section: Introductionmentioning

confidence: 99%

Variational optimality condition in control of hyperbolic systems with boundary delay parameters

Arguchintsev

Поплевко

Sinitsyn

2022

CAP

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An optimal control problem of a first-order hyperbolic system is studied, in which a boundary condition at one of the ends is determined from a controlled system of ordinary differential equations with constant state lag. Control functions are bounded and measurable functions. The system of ordinary differential equations at the boundary is linear in state. However the matrix of coefficients depends on control functions. Therefore, the optimality condition of Pontryagin’s maximum principle type in this problem is a necessary, but not a sufficient optimality condition. In this paper, the problem is reduced to an optimal control problem of a special system of ordinary differential equations. The proposed approach is based on the use of an exact formula of the cost functional increment. The reduced problem can be solved using a wide range of effective methods used for optimization problems in systems of ordinary differential equations. Problems of this kind arise when modeling thermal separation processes, suppression of mechanical vibrations in drilling, wave processes and population dynamics.

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An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations

Cited by 6 publications

References 11 publications

Numerical solution of the initial-boundary value problem describing separation processes in a distillation column

Numerical solution of the initial-boundary value problem describing separation processes in a distillation column

Treatise on Analytic Nonlinear Optimal Guidance and Control Amplification of Strictly Analytic (Non-Numerical) Methods

Variational optimality condition in control of hyperbolic systems with boundary delay parameters

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