# Constrained LQR Design Using Interior-Point Arc-Search Method for Convex Quadratic Programming with Box Constraints

**Authors:** Yaguang Yang  
**arXiv:** [1304.4685](https://arxiv.org/abs/1304.4685)  
**Date:** 2013-04-17  
**Summary page:** https://www.summarizepaper.com/en/paper/1304.4685v1/

## Key points

- Classical LQR design method widely successful in engineering designs
- Modifications needed for actuator saturation, known as constrained LQR design
- Constrained LQR design for discrete systems formulated as convex quadratic programming problem with box constraints
- Authors propose efficient algorithm using interior-point method
- Algorithm requires initial interior point available in this specific problem
- Algorithm takes advantage of special structure of box constraints and incorporates arc-search technique
- Proposed algorithm is polynomial and has best-known complexity bound for convex quadratic programming
- Algorithm implemented in MATLAB with example application to constrained LQR design
- Algorithm can be adapted for model predictive control, showcasing versatility and potential for broader applications.

## Summary

The classical LQR design method has been widely successful in engineering designs. However, in some cases, modifications are needed due to actuator saturation. This modified problem is known as the constrained LQR design. For discrete systems, the constrained LQR design problem can be formulated as a convex quadratic programming problem with box constraints. In this study, the authors propose an efficient algorithm for solving the constrained LQR design problem using the interior-point method. The key advantage of this method is that it requires an initial interior point which is available in this specific problem but not in general convex quadratic programming problems. The algorithm takes advantage of the special structure of the box constraints and incorporates a recently introduced arc-search technique for the interior-point algorithm. The authors prove that their proposed algorithm is polynomial and has the best-known complexity bound for convex quadratic programming. To demonstrate its effectiveness and efficiency, they implement the algorithm in MATLAB and provide an example of its application to constrained LQR design. Furthermore, they highlight that their proposed algorithm can easily be adapted for model predictive control, showcasing its versatility and potential for broader applications. Overall, this research presents a refined approach to address actuator saturation in engineering designs through the constrained LQR design framework. The proposed algorithm offers improved efficiency and complexity bounds compared to existing methods and can be readily applied to various control scenarios including model predictive control.