# The Model Order Reduction Method as an Effective Way to Implement GPC Controller for Multidimensional Objects

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## Abstract

**:**

## 1. Introduction

## 2. GPC Problem Formulation

#### 2.1. Predictive Controller Formulation

#### 2.2. Construction of GPC Algorithm

## 3. Process Identification

## 4. Example Models

#### 4.1. Two Input-Two Output Model

#### 4.2. Transformation to GPC Internal Model Format

#### 4.3. Three Input-Three Output Model

## 5. Model Order Reduction Method

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

GPC | Generalized Predictive Control |

MPC | Model Predictive Control |

PID | Proportional, Integral and Derivative |

MIMO | Multiple-Input Multiple-Output |

SISO | Single-Input Single-Output |

DCS | Distributed Control system |

PLC | Programmable Logic Controllers |

FPGA | Field Programmable Gate Arrays |

MV | Manipulated Variable |

CV | Controlled Variable |

DV | Disturbance Variable |

CRHPC | Constrained Receding Horizon Predictive Control |

SIORHC | Stabilizing Input Output Receding Horizon Control |

QP | Quadratic Programming |

CPU | Central Processing Unit |

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**Figure 3.**Step response of the 6th order model after reduction to a common denominator simulated for 900 time-steps.

**Figure 4.**GPC with the 6th order internal model—closed loop simulation of 300 time-steps, dashed line is the set point, and the solid line is the CV.

**Figure 5.**GPC with the 6th order internal model—closed loop simulation of 300 time-steps, trend for MVs signals.

**Figure 6.**Step response for T

_{1}, T

_{2}, and T

_{3}transfer functions simulated for 900 time-steps.

**Figure 7.**Step response of the 9th order model after reduction to common denominator simulated for 900 time-steps.

**Figure 8.**GPC with the 9th order internal model—closed loop simulation of 300 time-steps; dashed line is a set point, solid is a CV, and control parameters as: control horizon ${N}_{u}=20$, prediction horizon $N=400$, and the R matrix = identity matrix.

**Figure 9.**GPC with the 9th order internal model—closed loop simulation of 300 time-steps, trend for MVs signals. Controller parameters: control horizon ${N}_{u}=20$, prediction horizon $N=400$, and the R matrix = identity matrix.

**Figure 10.**Comparison of original and T

_{1}and after order reduction (black line)—simulation of 900 time-steps.

**Figure 11.**Comparison of original and T

_{2}and after order reduction (black line)—simulation of 900 time-steps.

**Figure 12.**Comparison of original and T

_{3}and after order reduction (black line)—simulation of 900 time-steps.

**Figure 13.**Comparison of the 6th and 9th order models—simulation of 900 time-steps (solid line—9th order model, dashed line—6th order model—model after order reduction).

**Figure 14.**GPC with reduced internal model—closed loop simulation of 300 time-steps where the dashed line is the set point and the solid line is the CV. Controller parameters: control horizon ${N}_{u}=20$, prediction horizon $N=400$, and the R matrix = identity matrix.

**Figure 15.**GPC with reduced internal model closed loop simulation of 300 time-steps, trend for MVs signals (u

_{1}, u

_{2}and u

_{3}). Controller parameters: control horizon ${N}_{u}=20$, prediction horizon $N=400$, and the R matrix = identity matrix.

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**MDPI and ACS Style**

Plamowski, S.; Kephart, R.W.
The Model Order Reduction Method as an Effective Way to Implement GPC Controller for Multidimensional Objects. *Algorithms* **2020**, *13*, 178.
https://doi.org/10.3390/a13080178

**AMA Style**

Plamowski S, Kephart RW.
The Model Order Reduction Method as an Effective Way to Implement GPC Controller for Multidimensional Objects. *Algorithms*. 2020; 13(8):178.
https://doi.org/10.3390/a13080178

**Chicago/Turabian Style**

Plamowski, Sebastian, and Richard W Kephart.
2020. "The Model Order Reduction Method as an Effective Way to Implement GPC Controller for Multidimensional Objects" *Algorithms* 13, no. 8: 178.
https://doi.org/10.3390/a13080178