The Synthesis of the Proportional‐Differential Regulators for the Systems with Fixed Ends of Trajectories Under Two‐Sided Constraints on Control Values

Murzabekov, Z. N.

doi:10.1002/asjc.1063

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…Итак, будем искать оптимальное управление в форме обратной связи ( , ) в задаче (1)-(3). Для решения используем принцип расширения [7,8], который состоит в том, что исходная задача оптимального управления с ограничениями сводится к задаче без ограничений и при этом решение новой задачи является одновременно и решением первоначальной задачи [9][10][11][12][13]. Для этого задачу (1)-(3) заменяем задачей без ограничений с использованием множителей Лагранжа 1 ( ), 2 ( ), 3 ( ), ( , ).…”

Section: теоретические результатыunclassified

“…Лемма. Пусть выполнены условия: 1) Пара { ( ), ( )} ∈ Δ( 0 , , 0 ); 2) Существуют 1 ( ) ≥ 0, 2 ( ) ≥ 0 такие, что на оптимальной паре ( ̃ ( ), ̃ ( )) ∈ Δ( 0 , , 0 ), которая доставляет минимальное значение функционалу (11), выполняются условия…”

Section: имеет местоunclassified

“…и, таким образом, утверждение леммы имеет место. Для определения пары { ̃ ( ), ̃ ( )}, минимизирующей функционал (11), необходимо найти управление ( ) и определить множители ( , ) = + ( , ), 1 ( ), 2 ( ) так, чтобы при каждом фиксированном ∈ ( 0 , ) подынтегральная функция ( , , ) в ( 11) достигала наименьшего значения среди ( , ) ∈ Δ( 0 , , 0 ).…”

Section: имеет местоunclassified

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Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

Dmitriev

Murzabekov

Mirzakhmedova

2021

Model. anal. inf. sist.

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For a continuous nonlinear control system on a finite time interval with control constraints, where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position, we consider the construction of a feedback according to the Kalman algorithm. For this, the solution of an auxiliary optimal control problem with a quadratic functional is used by analogy with the SDRE approach.Since this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems, where all coefficient matrices in differential equations and criteria can contain state variables, then on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations, with state-dependent coefficient matrices. This circumstance, due to the nonlinearity of the system, in comparison with the Kalman algorithm for linear-quadratic problems, significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V. F. Krotov and developed by V. I. Gurman and allows not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. The article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, depending on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.

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Section: теоретические результатыunclassified

Section: имеет местоunclassified

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Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

Dmitriev

Murzabekov

Mirzakhmedova

2021

Model. anal. inf. sist.

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“…A method based on the Lagrange multipliers of special type [7] is used to solve the optimal control problem (1)-(4).…”

Section: Problem Statementmentioning

confidence: 99%

Design of PI regulators for dynamic systems with constrained control and fixed endpoints of trajectories

Murzabekov

2018

Filomat

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The problem of optimal control for time-varying linear systems with fixed endpoints of trajectories is considered. A corresponding quadratic objective functional depends on the control, the state of the object and on its integral. New technique of designing the PI controller for the automatic control systems with box constraints on values of control is proposed. The problem is solved by using Lagrange multipliers of a special type.

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Construction of Control with Constraints for Nonlinear Systems with Coefficients Depending on the Control Object State

Murzabekov

Mirzakhmedova

2020

J Math Sci

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The Synthesis of the Proportional‐Differential Regulators for the Systems with Fixed Ends of Trajectories Under Two‐Sided Constraints on Control Values

Cited by 5 publications

References 11 publications

Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

Design of PI regulators for dynamic systems with constrained control and fixed endpoints of trajectories

Construction of Control with Constraints for Nonlinear Systems with Coefficients Depending on the Control Object State

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