# An efficient cost reduction procedure for bounded-control LQR problems

## Abstract

A novel approach has been developed for approximating the solution to the constrained LQR problem, based on updating the final state and costate of a related regular problem, and on slightly shifting the switching times (the instants when the control meets the constraints). The main result is the expression of a suboptimal control in feedback form using the solution of some compatible Riccati equation. The gradient method is applied to reduce the cost via explicit algebraic formula for its partial derivatives with respect to the hidden final state/costate of the related regular problem and to the switching times. The numerical method is termed efficient because it does not involve integrations of states or cost trajectories, and reduces to its minimum the dimension of the unknown parameters at the final condition. All the relevant objects are calculated from a few auxiliary matrices, which are computed only once. The scheme is here applied to two case studies whose optimal solutions are known. The first example is a two-dimensional model of the ‘cheapest stop of a train’ problem. The second one refers to the temperature control of a metallic strip leaving a multi-stand rolling mill, a problem with a high-dimensional state.

- Authors:

- Instituto de Desarrollo Tecnológico para la Industria Química (INTEC, UNL-CONICET) (Argentina)

- Publication Date:

- OSTI Identifier:
- 22769353

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; MATHEMATICAL SOLUTIONS; RICCATI EQUATION; TEMPERATURE CONTROL; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Costanza, Vicente, Rivadeneira, Pablo S., and Múnera, John A. Gómez.
```*An efficient cost reduction procedure for bounded-control LQR problems*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0393-X.

```
Costanza, Vicente, Rivadeneira, Pablo S., & Múnera, John A. Gómez.
```*An efficient cost reduction procedure for bounded-control LQR problems*. United States. doi:10.1007/S40314-016-0393-X.

```
Costanza, Vicente, Rivadeneira, Pablo S., and Múnera, John A. Gómez. Tue .
"An efficient cost reduction procedure for bounded-control LQR problems". United States. doi:10.1007/S40314-016-0393-X.
```

```
@article{osti_22769353,
```

title = {An efficient cost reduction procedure for bounded-control LQR problems},

author = {Costanza, Vicente and Rivadeneira, Pablo S. and Múnera, John A. Gómez},

abstractNote = {A novel approach has been developed for approximating the solution to the constrained LQR problem, based on updating the final state and costate of a related regular problem, and on slightly shifting the switching times (the instants when the control meets the constraints). The main result is the expression of a suboptimal control in feedback form using the solution of some compatible Riccati equation. The gradient method is applied to reduce the cost via explicit algebraic formula for its partial derivatives with respect to the hidden final state/costate of the related regular problem and to the switching times. The numerical method is termed efficient because it does not involve integrations of states or cost trajectories, and reduces to its minimum the dimension of the unknown parameters at the final condition. All the relevant objects are calculated from a few auxiliary matrices, which are computed only once. The scheme is here applied to two case studies whose optimal solutions are known. The first example is a two-dimensional model of the ‘cheapest stop of a train’ problem. The second one refers to the temperature control of a metallic strip leaving a multi-stand rolling mill, a problem with a high-dimensional state.},

doi = {10.1007/S40314-016-0393-X},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}