# Local Decay of Residuals in Dual Gradient Method Applied to MPC Studied using Active Set Approach

### Matija Perne, Samo Gerkšič, Boštjan Pregelj

#### Abstract

A dual gradient method is used for solving quadratic programs resulting from a model predictive control problem in real-time control context. Evolution of iterates and residuals throughout iterations of the method is studied. In each iteration, the set of active inequality constraints whose corresponding components of the Lagrange multiplier are non-zero can be defined. It is found that the set of active constraints tends to stay constant through multiple iterations. Observing the decay of residuals for intervals where the set of active constraints is constant leads to interesting findings. For such an interval, the dual residual can be expressed in a base so that its components are decaying independently, uniformly, and at predictable rates. The base and the rates only depend on the system matrices and the set of active constraints. The calculated decay rates match the rates observed in numerical simulations of MPC control, which is shown for the AFTI-16 benchmark example.

Download#### Paper Citation

#### in Harvard Style

Perne M., Gerkšič S. and Pregelj B. (2017). **Local Decay of Residuals in Dual Gradient Method Applied to MPC Studied using Active Set Approach** . In *Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,* ISBN 978-989-758-263-9, pages 54-63. DOI: 10.5220/0006416500540063

#### in Bibtex Style

@conference{icinco17,

author={Matija Perne and Samo Gerkšič and Boštjan Pregelj},

title={Local Decay of Residuals in Dual Gradient Method Applied to MPC Studied using Active Set Approach},

booktitle={Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},

year={2017},

pages={54-63},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006416500540063},

isbn={978-989-758-263-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,

TI - Local Decay of Residuals in Dual Gradient Method Applied to MPC Studied using Active Set Approach

SN - 978-989-758-263-9

AU - Perne M.

AU - Gerkšič S.

AU - Pregelj B.

PY - 2017

SP - 54

EP - 63

DO - 10.5220/0006416500540063