scholarly journals A New Method for Determining the Degree of Controllability of State Variables for the LQR Problem Using the Duality Theorem

2020 ◽  
Vol 10 (15) ◽  
pp. 5234 ◽  
Author(s):  
Lifei Zhang ◽  
Konstantin Avenirovich Neusypin ◽  
Maria Sergeevna Selezneva
Keyword(s):  
Duality Theorem ◽  
Optimal Estimation ◽  
New Method ◽  
Model Parameters ◽  
Control Performance ◽  
State Variables ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Controllability Gramian ◽  
Lqr Problem

The control performance of a dynamic system can be checked by the degree of controllability. In this work, we present a new method for determining the degree of observability of state variables for the linear quadratic optimal estimation (LQE) problem. We carried out the calculation of the degree of controllability for the linear quadratic optimal control (LQR) problem using a duality theorem. Compared with the traditional measures of controllability such as determinant, trace, and maximal eigenvalue of the inverse controllability Gramian, the proposed degree of controllability was developed for each state variable and takes into account both the controllability Gramian and the cost function. The new method is convenient to apply to LQR problem. In the numerical simulation, we determined the influence of the model parameters on the degree of controllability. Besides that, we analyzed the degree of controllability, which gives an insight into the relationship between the system model design and the control performance.

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 A New Approach for Solving Optimal Nonlinear Control Problems Using Decriminalization and Rationalized Haar Functions

2011 ◽  
Vol 1 ◽  
pp. 387-394 ◽  
Author(s):  
Zhen Yu Han ◽  
Shu Rong Li
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Nonlinear Optimal Control ◽  
Control Problems ◽  
State Variables ◽  
Linear Quadratic ◽  
Control Variables ◽  
Linear Quadratic Optimal Control ◽  
Haar Functions ◽  
And Control

This paper presents a numerical method based on quasilinearization and rationalized Haar functions for solving nonlinear optimal control problems including terminal state constraints, state and control inequality constraints. The optimal control problem is converted into a sequence of quadratic programming problems. The rationalized Haar functions with unknown coefficients are used to approximate the control variables and the derivative of the state variables. By adding artificial controls, the number of state and control variables is equal. Then the quasilinearization method is used to change the nonlinear optimal control problems with a sequence of constrained linear-quadratic optimal control problems. To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.

scholarly journals Reducing the model-data misfit in a marine ecosystem model using periodic parameters and Linear Quadratic Optimal Control

2012 ◽  
Vol 9 (8) ◽  
pp. 10207-10239 ◽  
Author(s):  
M. El Jarbi ◽  
J. Rückelt ◽  
T. Slawig ◽  
A. Oschlies
Keyword(s):  
Optimal Control ◽  
Ocean Circulation ◽  
Marine Ecosystem ◽  
Circulation Model ◽  
Ecosystem Model ◽  
Model Parameters ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Marine Ecosystem Model ◽  
Quadratic Optimal Control

Abstract. This paper presents the application of the Linear Quadratic Optimal Control (LQOC) method for a parameter optimization problem in a marine ecosystem model. The ecosystem model simulates the distribution of nitrogen, phytoplankton, zooplankton and detritus in a water column with temperature and turbulent diffusivity profiles taken from a three-dimensional ocean circulation model. We present the linearization method which is based on the available observations. The linearization is necessary to apply the LQOC method on the nonlinear system of state equations. We show the form of the linearized time-variant problems and the resulting two algebraic Riccati Equations. By using the LQOC method, we are able to introduce temporally varying periodic model parameters and to significantly improve – compared to the use of constant parameters – the fit of the model output to given observational data.

LINEAR OPTIMAL CONTROL MODEL FOR FELLING THE OIL PALM TREES

2017 ◽  
Vol 79 (4) ◽  
Author(s):  
Noryanti Nasir ◽  
Mohd Ismail Abd Aziz ◽  
Akbar Banitalebi
Keyword(s):  
Optimal Control ◽  
Palm Oil ◽  
Oil Palm ◽  
Minimum Cost ◽  
Real Data ◽  
Control Model ◽  
Optimal Feedback Control ◽  
State Variables ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control

The increases of operational felling cost have prompted the oil palm industry to look at the current practices. The felling activity is considered as the main aspects to improve and maintain palm oil production through the provision of effective and agronomic practices. To support this success and achieve minimum cost of operation, this study aims to develop a time-invariant linear quadratic optimal control model for controlling the felling and harvest rate of the oil palm plantation. The proposed model involves two state variables which are biomass and crude oil. The optimal parameters for the model are estimated using a set of real data collected from Malaysian Palm Oil Board (MPOB). The study analyzes the solution of the resulting control problem within a limited time frame of 30 years and the results provide an optimal feedback control for the felling and harvest rates.

A Method for Optimal Step Adjustment for a Class of Linear Quadratic Optimal Control Problems

1993 ◽  
Vol 115 (3) ◽  
pp. 535-538 ◽  
Author(s):  
Stelios C. A. Thomopoulos ◽  
Ioannis N. M. Papadakis
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
New Method ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Gradient Algorithms ◽  
Quadratic Optimal Control

In this paper, a new method for the computation of the optimal step in gradient algorithms for a class of linear quadratic optimal control problems is presented. This method improves the convergence of the gradient algorithms and is optimal, i.e., it outperforms any other step searching scheme in the above class of problems.

scholarly journals Reducing the model-data misfit in a marine ecosystem model using periodic parameters and linear quadratic optimal control

2013 ◽  
Vol 10 (2) ◽  
pp. 1169-1182 ◽  
Author(s):  
M. El Jarbi ◽  
J. Rückelt ◽  
T. Slawig ◽  
A. Oschlies
Keyword(s):  
Optimal Control ◽  
Marine Ecosystem ◽  
Ecosystem Model ◽  
Original Model ◽  
Model Parameters ◽  
Model Data ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Marine Ecosystem Model ◽  
Quadratic Optimal Control

Abstract. This paper presents the application of the Linear Quadratic Optimal Control (LQOC) method to a parameter optimization problem for a one-dimensional marine ecosystem model of NPZD (N for dissolved inorganic nitrogen, P for phytoplankton, Z for zooplankton and D for detritus) type. This ecosystem model, developed by Oschlies and Garcon, simulates the distribution of nitrogen, phytoplankton, zooplankton and detritus in a water column and is driven by ocean circulation data. The LQOC method is used to introduce annually periodic model parameters in a linearized version of the model. We show that the obtained version of the model gives a significant reduction of the model-data misfit, compared to the one obtained for the original model with optimized constant parameters. The found inner-annual variability of the optimized parameters provides hints for improvement of the original model. We use the obtained optimal periodic parameters also in validation and prediction experiments with the original non-linear version of the model. In both cases, the results are significantly better than those obtained with optimized constant parameters.

scholarly journals A periodic linear–quadratic controller for suppressing rotor-blade vibration

2019 ◽  
Vol 25 (17) ◽  
pp. 2351-2364 ◽  
Author(s):  
J. F. Camino ◽  
I. F. Santos
Keyword(s):  
Rotor Blade ◽  
Linear Time ◽  
Linear Quadratic Regulator ◽  
Time Varying ◽  
Linear Quadratic ◽  
Blade Vibration ◽  
Linear Quadratic Optimal Control ◽  
Riccati Differential Equation ◽  
Periodic Linear ◽  
Lqr Problem

This paper presents an active control strategy, based on a time-varying linear–quadratic optimal control problem, to attenuate the tip vibration of a two-dimensional coupled rotor-blade system whose dynamics is periodic. First, a periodic full-state feedback controller based on the linear–quadratic regulator (LQR) problem is designed. If all the states are not available for feedback, then an optimal periodic time-varying estimator, using the Kalman–Bucy filter, is computed. Both the Kalman filter gain and the LQR gain are obtained as the solution of a periodic Riccati differential equation (PRDE). Together, these gains provide the observer-based linear–quadratic–Gaussian (LQG) controller. An algorithm to solve the PRDE is also presented. Both controller designs ensure closed-loop stability and performance for the linear time-varying rotor-blade equation of motion. Numerical simulations show that the LQR and the LQG controllers are able to significantly attenuate the rotor-blade tip vibration.

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.

Nearly Exact Bayesian Estimation of Non-linear No-Arbitrage Term-Structure Models*

2021 ◽  
Author(s):  
Marcello Pericoli ◽  
Marco Taboga
Keyword(s):  
Bayesian Estimation ◽  
Term Structure ◽  
Broad Class ◽  
Model Parameters ◽  
State Variables ◽  
Computationally Efficient ◽  
Macroeconomic Factors ◽  
No Arbitrage ◽  
Term Structure Models ◽  
Non Linear

Abstract We propose a general method for the Bayesian estimation of a very broad class of non-linear no-arbitrage term-structure models. The main innovation we introduce is a computationally efficient method, based on deep learning techniques, for approximating no-arbitrage model-implied bond yields to any desired degree of accuracy. Once the pricing function is approximated, the posterior distribution of model parameters and unobservable state variables can be estimated by standard Markov Chain Monte Carlo methods. As an illustrative example, we apply the proposed techniques to the estimation of a shadow-rate model with a time-varying lower bound and unspanned macroeconomic factors.

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