scholarly journals Existence and uniqueness of constrained globally optimal feedback controls in a linear-quadratic framework

2006 ◽  
Vol 2006 ◽  
pp. 1-22 ◽  
Author(s):  
Yashan Xu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Existence And Uniqueness ◽  
Closed Loop ◽  
Synthesis Method ◽  
Open Loop ◽  
Linear Quadratic ◽  
Feedback Controls ◽  
Loop Control

A constrained closed-loop optimal control problem is considered in a linear-quadratic framework. To solve the problem, a special type open-loop optimal control problem and a standard open-loop optimal control problem are introduced and carefully studied, via which the existence and uniqueness of the globally optimal closed-loop control is established by a synthesis method.

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 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 Implementation Aspects Regarding Closed-Loop Control Systems Using Evolutionary Algorithms

Inventions ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 53
Author(s):  
Viorel Minzu ◽  
Saïd Riahi ◽  
Eugen Rusu
Keyword(s):  
Optimal Control ◽  
Evolutionary Algorithms ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Closed Loop ◽  
Batch Reactor ◽  
General Structure ◽  
Computational Effort ◽  
Loop Control ◽  
The Impact

When an optimal control problem requires an important computational effort, a metaheuristic algorithm (MA) can be a useful approach. An MA is conceived to solve a specific optimal control problem having a characteristic objective function. This algorithm solely yields only an optimal offline solution. The desideratum to have a closed-loop implementation can be fulfilled through a supplementary “tool”, the Receding Horizon Control (RHC) structure. This paper addresses a particular case and integrates Evolutionary Algorithms into the RHC structure. The main objective is to propose a general harmonization between the Controller of the closed loop and the Evolutionary Algorithm. Some details concerning the implementation of the closed loop and Controller are described. The impact of the RHC’s prediction technique upon the control sequences’ encoding is also analyzed. Two general structure Controllers are proposed, one of them conceived to cope with restrictive time constraints. Practical ideas have been illustrated through a case study: the well-known optimal control of a fed-batch reactor for ethanol production. This time, our implementation achieves a closed-loop solution. The results from the programs and simulation series validate the Controllers, EAs, and the closed-loop structure. Generally speaking, the association between RHC and EA can be a realistic solution to optimal process control.

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.

scholarly journals Prediction error's minimization through a controller based on a Metaheuristic Algorithm

2020 ◽  
Vol 43 (3) ◽  
pp. 5-11
Author(s):  
Viorel Mînzu
Keyword(s):  
Optimal Control ◽  
Theoretical Analysis ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Predictive Control ◽  
Closed Loop ◽  
Open Loop ◽  
The Other ◽  
Metaheuristic Algorithm ◽  
Prediction Technique

A Metaheuristic Algorithm (MA) can be a realistic method to solve a given Optimal Control Problem (OCP), but the result is an open-loop solution. If the Metaheuristic Algorithm is integrated within the Model Predictive Control (MPC) structure, a closed-loop solution can be achieved. The controller works using a prediction technique and prediction error's minimization. On the other side, the MA optimizes (minimizes or maximizes) the OCP's objective function. The controller is faced with two optimization tasks. This paper proves through theoretical analysis and simulations that the prediction error's minimization is implicitly accomplished.

On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

2020 ◽  
Vol 26 ◽  
pp. 41
Author(s):  
Tianxiao Wang
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Time Inconsistency ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

scholarly journals Convergence results for an averaged LQR problem with applications to reinforcement learning

2021 ◽  
Author(s):  
Andrea Pesare ◽  
Michele Palladino ◽  
Maurizio Falcone
Keyword(s):  
Optimal Control ◽  
Reinforcement Learning ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Current System ◽  
Numerical Test ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Lqr Problem ◽  
Quadratic Optimal Control

AbstractIn this paper, we will deal with a linear quadratic optimal control problem with unknown dynamics. As a modeling assumption, we will suppose that the knowledge that an agent has on the current system is represented by a probability distribution $$\pi $$ π on the space of matrices. Furthermore, we will assume that such a probability measure is opportunely updated to take into account the increased experience that the agent obtains while exploring the environment, approximating with increasing accuracy the underlying dynamics. Under these assumptions, we will show that the optimal control obtained by solving the “average” linear quadratic optimal control problem with respect to a certain $$\pi $$ π converges to the optimal control driven related to the linear quadratic optimal control problem governed by the actual, underlying dynamics. This approach is closely related to model-based reinforcement learning algorithms where prior and posterior probability distributions describing the knowledge on the uncertain system are recursively updated. In the last section, we will show a numerical test that confirms the theoretical results.

scholarly journals Two Inverse Problems Solution by Feedback Tracking Control

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 137
Author(s):  
Vladimir Turetsky
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Feedback Linearization ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Linear Quadratic Tracking ◽  
Ill Posed ◽  
Quadratic Optimal Control ◽  
Quadratic Tracking

Two inverse ill-posed problems are considered. The first problem is an input restoration of a linear system. The second one is a restoration of time-dependent coefficients of a linear ordinary differential equation. Both problems are reformulated as auxiliary optimal control problems with regularizing cost functional. For the coefficients restoration problem, two control models are proposed. In the first model, the control coefficients are approximated by the output and the estimates of its derivatives. This model yields an approximating linear-quadratic optimal control problem having a known explicit solution. The derivatives are also obtained as auxiliary linear-quadratic tracking controls. The second control model is accurate and leads to a bilinear-quadratic optimal control problem. The latter is tackled in two ways: by an iterative procedure and by a feedback linearization. Simulation results show that a bilinear model provides more accurate coefficients estimates.

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