Three-time-scale difference equations with application to open-loop optimal control problem

1985 ◽  
pp. 120-147
Author(s):  
Desineni S. Naidu ◽  
Ayalasomayajula K. Rao
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Difference Equations ◽  
Time Scale ◽  
Open Loop ◽  
Scale Difference

scholarly journals On linear-quadratic optimal control of implicit difference equations

2018 ◽  
Vol 36 (3) ◽  
pp. 779-833
Author(s):  
Daniel Bankmann ◽  
Matthias Voigt
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Difference Equations ◽  
Matrix Pencil ◽  
Spectral Structure ◽  
Optimal Controls ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Quadratic Optimal Control

Abstract In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyse a discrete-time Lur’e equation and a corresponding Kalman–Yakubovich–Popov (KYP) inequality. We show that solvability of the KYP inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur’e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context.

scholarly journals Prospects on Solving an Optimal Control Problem with Bounded Uncertainties on Parameters using Interval Arithmetics

2021 ◽  
Author(s):  
Etienne Bertin ◽  
Elliot Brendel ◽  
Bruno Hérissé ◽  
Julien Alexandre dit Sandretto ◽  
Alexandre Chapoutot
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Closed Loop ◽  
Minimum Principle ◽  
Open Loop ◽  
Pontryagin Minimum Principle ◽  
Interval Method ◽  
Bounded Uncertainties ◽  
The Given

An interval method based on the Pontryagin Minimum Principle is proposed to enclose the solutions of an optimal control problem with embedded bounded uncertainties. This method is used to compute an enclosure of all optimal trajectories of the problem, as well as open loop and closed loop enclosures meant to enclose a concrete system using an optimal control regulator with inaccurate knowledge of the parameters. The differences in geometry of these enclosures are exposed, as well as some applications. For instance guaranteeing that the given optimal control problem will yield a satisfactory trajectory for any realization of the uncertainties or on the contrary that the problem is unsuitable and needs to be adjusted.

The Optimal Adaptive Sampled-Data Regulator

1974 ◽  
Vol 96 (3) ◽  
pp. 334-340
Author(s):  
R. A. Schlueter ◽  
A. H. Levis
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Closed Loop ◽  
Open Loop ◽  
Sampled Data ◽  
Threshold Values ◽  
Regulator Problem

The optimal control problem for sampled-data processes in which sampling is not triggered by a timer, but occurs when one or more outputs attain preset threshold values is formulated as an adaptive optimal sampled-data regulator problem. Both open loop and closed loop solutions are determined and a comparison of a system’s performance with adaptive and with periodic sampling is presented.

scholarly journals Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems of Markovian regime switching system

2021 ◽  
Author(s):  
Xin Zhang ◽  
Xun Li ◽  
Jie Xiong
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Regime Switching ◽  
Closed Loop ◽  
Open Loop ◽  
Switching System ◽  
Linear Quadratic ◽  
Markovian Regime Switching ◽  
Markovian Regime

This paper investigates the stochastic linear quadratic (LQ, for short) optimal control problem of Markovian regime switching system. The representation of the cost functional for the stochastic LQ optimal control problem of Markovian regime switching system is derived by the technique of It{\^o}'s formula with jumps. For the stochastic LQ optimal control problem of Markovian regime switching system, we establish the equivalence between the open-loop (closed-loop, resp.) solvability and the existence of an adapted solution to the corresponding forward-backward stochastic differential equation with constraint (the existence of a regular solution to Riccati equations). Also, we analyze the interrelationship between the strongly regular solvability of Riccati equations and the uniform convexity of the cost functional. Finally, we present an example which is open-loop solvable but not closed-loop solvable.

scholarly journals An analogue of the linearized maximum principle for the optimal control problem for nonlinear difference equations of fractional order

2021 ◽  
pp. 9-15
Author(s):  
S. T. Alieva ◽  
◽  
Kamil Mansimov ◽  
◽  
◽  
...  
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Fractional Order ◽  
Difference Equations ◽  
Nonlinear Difference Equations

Problem of optimal control of an object, described by a system of nonlinear difference equations of fractional order, is considered. Applying one version of the method of increments, a discrete an-alogue of the linearized maximum principle is established

scholarly journals Parametric Approach to Trajectory Tracking Control of Robot Manipulators

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Shijie Zhang ◽  
Yi Ning
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Trajectory Tracking ◽  
Tracking Control ◽  
Optimal Trajectory ◽  
Open Loop ◽  
Parametric Approach ◽  
Trajectory Tracking Control ◽  
Loop Control

The mathematic description of the trajectory of robot manipulators with the optimal trajectory tracking problem is formulated as an optimal control problem, and a parametric approach is proposed for the optimal trajectory tracking control problem. The optimal control problem is first solved as an open loop optimal control problem by using a time scaling transform and the control parameterization method. Then, by virtue of the relationship between the optimal open loop control and the optimal closed loop control along the optimal trajectory, a practical method is presented to calculate an approximate optimal feedback gain matrix, without having to solve an optimal control problem involving the complex Riccati-like matrix differential equation coupled with the original system dynamics. Simulation results of 2-link robot manipulator are presented to show the effectiveness of the proposed method.

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