scholarly journals Use of semidefinite programming for solving the LQR problem subject to rectangular descriptor systems

2010 ◽  
Vol 20 (4) ◽  
pp. 655-664 ◽  
Author(s):  
Keyword(s):  
Semidefinite Programming ◽  
Linear Quadratic Regulator ◽  
Descriptor Systems ◽  
Programming Approach ◽  
State Control ◽  
Linear Quadratic ◽  
Control Pair ◽  
Elegant Method ◽  
Lqr Problem ◽  
Primal Dual

Use of semidefinite programming for solving the LQR problem subject to rectangular descriptor systemsThis paper deals with the Linear Quadratic Regulator (LQR) problem subject to descriptor systems for which the semidefinite programming approach is used as a solution. We propose a new sufficient condition in terms of primal dual semidefinite programming for the existence of the optimal state-control pair of the problem considered. The results show that semidefinite programming is an elegant method to solve the problem under consideration. Numerical examples are given to illustrate the results.

Extension and Simplification of Salukvadze’s Solution to the Deterministic Nonhomogeneous LQR Problem

1998 ◽  
Vol 123 (1) ◽  
pp. 146-149
Author(s):  
R. D. Hampton ◽  
C. R. Knospe ◽  
M. A. Townsend
Keyword(s):  
Dynamic Systems ◽  
Variational Calculus ◽  
Linear Quadratic Regulator ◽  
Linear Quadratic ◽  
Matrix Methods ◽  
Lqr Problem ◽  
Asme Journal ◽  
Analytic Design ◽  
And Control ◽  
Measurement And Control

In a previous paper (Hampton, R. D., et al., 1996, “A Practical Solution to the Deterministic Nonhomogeneous LQR Problem,” ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 118, pp. 354–360.) the authors presented a solution to the nonhomogeneous linear-quadratic-regulator (LQR) problem, for the case of known, deterministic, persistent (“non-dwindling”) disturbances. The authors used variational calculus and state-transition-matrix methods to produce an optimal matric solution, for bounded determinist forcing terms. A restricted version of this problem (treating dwindling disturbances) was evidently first investigated by Salukvadze, M. E., 1962, “Analytic Design of Regulators (Constant Disturbance),” Automation and Remote Control, Vol. 22, No. 10, Mar., pp. 1147–1155, using a differential-equations approach. The present paper uses Salukvadze’s approach to extend his work to the case of non-dwindling disturbances, with cross-weightings between state- and control vectors, and pursues the solution to the same form reported previously in Hampton et al.

Perturbation Analysis of the LMI-Based Continuous-time Linear Quadratic Regulator Problem for Descriptor Systems

2016 ◽  
Vol 14 (1) ◽  
pp. 13-20
Author(s):  
A. Yonchev
Keyword(s):  
Control Problem ◽  
Perturbation Analysis ◽  
Continuous Time ◽  
Linear Quadratic Regulator ◽  
Descriptor Systems ◽  
Linear Matrix ◽  
Linear Perturbation ◽  
Linear Quadratic ◽  
Perturbation Bounds ◽  
Lqr Control

Abstract This paper considers an approach to perform perturbation analysis of linear quadratic regulator (LQR) control problem for continuous-time descriptor systems. The investigated control problem is based on solving LMIs (Linear Matrix Inequalities) and applying Lyapunov functions. The paper is concerned with obtaining linear perturbation bounds for the continuous-time LQR control problem for descriptor systems. The computed perturbation bounds can be used to study the effect of perturbations in system and controller on feasibility and performance of the considered control problem. A numerical example is also presented in the paper.

scholarly journals Gain scheduling LQI controller design for LPV descriptor systems and motion control of two-link flexible joint robot manipulator

2018 ◽  
Vol 8 (2) ◽  
pp. 201-207 ◽  
Author(s):  
Yusuf Altun
Keyword(s):  
State Space ◽  
Motion Control ◽  
Controller Design ◽  
Linear Quadratic Regulator ◽  
Robotic Manipulator ◽  
Gain Scheduling ◽  
Descriptor Systems ◽  
Natural State ◽  
Linear Quadratic ◽  
Flexible Joint

This paper proposes a gain scheduling linear quadratic integral (LQI) servo controller design, which is derived from linear quadratic regulator (LQR) optimal control, for non-singular linear parameter varying (LPV) descriptor systems. It is assumed that state space matrices are non-singular since many mechanical systems do not have any non-singular matrices such as the natural state space forms of robotic manipulator, pendulum and suspension systems. A controller design is difficult for the systems due to rational LPV case. Therefore, the proposed gain scheduling controller is designed without the difficulty. Accordingly, the motion control design is implemented for two-link flexible joint robotic manipulator. Finally, the control system simulation is performed to prove the applicability and performance.

Polynomial-Chaos-Based Numerical Method for the LQR Problem With Uncertain Parameters in the Formulation

2010 ◽  
Author(s):  
Emmanuel Blanchard ◽  
Corina Sandu ◽  
Adrian Sandu
Keyword(s):  
Numerical Method ◽  
Polynomial Chaos ◽  
Linear Quadratic Regulator ◽  
Mean Value ◽  
Optimal Controller ◽  
Linear Quadratic ◽  
Case Scenario ◽  
Worst Case ◽  
Lqr Problem ◽  
And Performance

This paper proposes a polynomial chaos based numerical method providing an optimal controller for the linear-quadratic regulator (LQR) problem when the parameters in the formulation are uncertain, i.e., a controller minimizing the mean value of the LQR cost function obtained for a certain distribution of the uncertainties which is assumed to be known. The LQR problem is written as an optimality problem using Lagrange multipliers in an extended form associated with the polynomial chaos framework, and an iterative algorithm converges to the optimal answer. The algorithm is applied to a simple example for which the answer is already known. Polynomial chaos based methods have the advantage of being computationally much more efficient than Monte Carlo simulations. The Linear-Quadratic Regulator controller is not very well adapted to robust design, and the optimal controller does not guarantee a minimum performance or even stability for the worst case scenario. Stability robustness and performance robustness in the presence of uncertainties are therefore not guaranteed. However, this is a first step aimed at designing more judicious controllers if combined with other techniques in the future. The next logical step would be to extend this numerical method to H2 and then H-infinity problems.

A Practical Solution to the Deterministic Nonhomogeneous LQR Problem

1996 ◽  
Vol 118 (2) ◽  
pp. 354-359 ◽  
Author(s):  
R. D. Hampton ◽  
C. R. Knospe ◽  
M. A. Townsend
Keyword(s):  
Linear Quadratic Regulator ◽  
Linear Quadratic ◽  
State Equations ◽  
Integral Term ◽  
Gain Matrix ◽  
Lqr Problem ◽  
Full State ◽  
Controller Performance ◽  
Lqr Controller ◽  
Duhamel Integral

A linear-quadratic-regulator-based (LQR) controller originates from a homogeneous set of state-space equations, and consists of a matrix of constant feedback gains. If the state equations are made nonhomogeneous by adding a vector of deterministic forcing terms, the standard LQR solution is no longer optimal. The present paper develops a matrix solution to this augmented (nonhomogeneous) LQR problem. The solution form consists of constant-gain feedback of the full-state vector, summed with a matrix preview (Duhamel integral) term. A practical and usable approximation is presented for the optimal preview term, having the form of a constant preview gain matrix. An example shows the improvement obtainable in controller performance with the use of this preview gain matrix, for exponentially decaying disturbances with a range of time constants.

Stochastic Linear-Quadratic Control via Primal-Dual Semidefinite Programming

SIAM Review ◽  
2004 ◽  
Vol 46 (1) ◽  
pp. 87-111 ◽  
Author(s):  
David D. Yao ◽  
Shuzhong Zhang ◽  
Xun Yu Zhou
Keyword(s):  
Semidefinite Programming ◽  
Linear Quadratic Control ◽  
Linear Quadratic ◽  
Primal Dual

scholarly journals Uncertain DC-DC Zeta Converter Control in Convex Polytope Model Based on LMI Approach

2018 ◽  
Vol 9 (2) ◽  
pp. 829
Author(s):  
Hafez Sarkawi ◽  
Yoshito Ohta
Keyword(s):  
Model Simulation ◽  
Convex Polytope ◽  
Linear Quadratic Regulator ◽  
Input Voltage ◽  
Linear Matrix ◽  
Linear Quadratic ◽  
Lqr Problem ◽  
Switch Mode ◽  
Value Model ◽  
Polytope Model

<span>A dc-dc zeta converter is a switch mode dc-dc converter that can either step-up or step-down dc input voltage. In order to regulate the dc output voltage, a control subsystem needs to be deployed for the dc-dc zeta converter. This paper presents the dc-dc zeta converter control. Unlike conventional dc-dc zeta converter control which produces a controller based on the nominal value model, we propose a convex polytope model of the dc-dc zeta converter which takes into account parameter uncertainty. A linear matrix inequality (LMI) is formulated based on the linear quadratic regulator (LQR) problem to find the state-feedback controller for the convex polytope model. Simulation results are presented to compare the control performance between the conventional LQR and the proposed LMI based controller on the dc-dc zeta converter. Furthermore, the reduction technique of the convex polytope is proposed and its effect is investigated.</span>

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