# LMI-Based Multi-model Predictive Control of an Industrial C3/C4 Splitter

## Authors

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DOI: 10.1007/s40313-013-0050-1

- Cite this article as:
- Capron, B.D.O. & Odloak, D. J Control Autom Electr Syst (2013) 24: 420. doi:10.1007/s40313-013-0050-1

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## Abstract

In this paper, the robust Model Predictive Control (MPC) of systems with model uncertainty is addressed. The robust approach usually involves the inclusion of nonlinear constraints to the optimization problem upon which the controller is based. At each time step the sequence of control actions is then calculated through the resolution of a NonLinear Programming problem, which can be too computer demanding for high dimension systems. Here, the conventional Multi-model Predictive Control (MMPC) problem is re-casted as an LMI-based problem that can be solved with a lower computational cost. The conventional and LMI-based robust controllers’ performances and computational costs are compared through simulations of the control of an industrial C3/C4 splitter.

### Keywords

Model predictive controlLinear matrix inequalityMulti-model uncertaintyRobust control## 1 Introduction

Model Predictive Control (MPC) has known over the past decades a tremendous development. Originally designed and developed for power plants and oil refineries, this advanced control technology can now be found in many sectors such as chemical, food processing, automotive and aerospace industries (Qin and Badgwell 2003), and in medical research (Lee and Bequette 2009). Based on a model representation of the system to be controlled, MPC basically consists in calculating at every time step the sequence of inputs that optimizes the predicted behavior of the system subject to restrictions on the inputs and the outputs. The first calculated control move is implemented and the optimization problem is solved again at the next time step.

MPC is usually based on a single linear model of the system. However, chemical systems often exhibit highly nonlinear behavior. Consequently, as the system commonly works at different operating points, a controller based on a single linear model may not produce an efficient control of the plant. One way to circumvent this problem is to work with the Multi-model Predictive Control (MMPC) (Porfírio et al. 2003), for which a discrete set of plant models corresponding to different operating points of the system is considered. In that case, an objective function is defined for each model of the set and the multi-model predictive controller results from minimizing the worst case objective function.

Such a controller has already been implemented in Porfírio et al. (2003) on a real industrial C3/C4 splitter, where it actually showed to have a much better performance than the standard MPC formulation. A disadvantage of the MMPC compared to the conventional MPC is the computational burden involved in the solution of the optimization problem that defines the MMPC, which can be prohibitive for high dimension systems. In this case, the LMI techniques that have been developed over the past decades may be of interest as they allow a significant reduction of the computational complexity of the optimization problem (Gahinet et al. 1995). These techniques were first applied for MPC by Kothare et al. (1996) whose work has opened the way to several developments over the years including the recent works of Cuzzola et al. (2002), Alamo et al. (2008), Cychoswski and O’mahony (2010), Ding (2010), Li and Xi (2010), Falugi et al. (2010a, b).

The number of variables of the control problem is much larger than in the conventional MPC. If the control problem is solved on-line, the computational effort may be prohibitive for moderate to large systems.

The way that the input constraints are implemented tends to be conservative, which impacts on the controller domain of attraction and performance.

## 2 System Representation

In the system studied in this work, the uncertainties concentrate on matrices \(B,F\)and \(\varPsi \), so that the discrete set \(\Omega \) of possible plants can be defined as \(\Omega =\left\{ {\Theta _1,\cdots ,\Theta _L } \right\} \) where each \(\Theta _\mathrm{n} \) corresponds to a particular plant: \(\varTheta _{n} =\left( {B,F,\varPsi } \right) _{n},\,n=1,\ldots ,L\). Let us also assume that the true plant is designated as \(\Theta _T \) and that the current estimated state corresponds to the true plant state.

## 3 Conventional NLP-Based Multi-model Predictive Control with Zone Control and Input Target

It is assumed in this work that a Real-Time Optimization (RTO) algorithm lies at the top of the control structure and defines optimum targets for some inputs and outputs of the system while the remaining inputs and outputs have to remain inside predefined boundaries, characterizing the so-called zone control strategy (Maciejowski 2002).

Let us note that the third term of the right hand side of (4) is included to force the inputs to reach their corresponding targets.

## 4 LMI-Based Multi-model Predictive Control for Zone Control and Input Target Strategy

Applying the Schur complement to the nonlinear inequality constraints represented by (11), problem P1 can be easily turned into the following LMI optimization problem:

One should note that the Schur complement states that problem P1 and problem P2 are equivalent. As a result, the controllers based on both problems are expected to have similar performances.

## 5 Application of the LMI-Based MMPC to the C3/C4 Splitter

The system considered in this work is the C3/C4 splitter studied in Porfírio et al. (2003) where more details can be found. The controlled outputs are the percentage of propane (C3) in the bottom stream \((y_1 )\) and the temperature of the first stage of the distillation column top section (\(y_2 )\). The manipulated inputs are the reflux flow rate to the top of the column (\(u_1 )\) and the flow rate of hot oil to the reboiler (\(u_2 )\). The feed flow rate (\(u_3 )\) and the temperature of the hot oil stream (\(u_4 )\) are two measured disturbances of the system.

Transfer functions models coefficients of the C3/C4

\(\varvec{\Theta }_{1}\) | \(\varvec{\Theta }_{2}\) | \(\varvec{\Theta }_{3}\) | \(\varvec{\Theta }_{4}\) | \(\varvec{\Theta }_{5}\) | \(\varvec{\Theta }_{6}\) | |
---|---|---|---|---|---|---|

\(\mathbf{b}_\mathbf{1,1,0}\) | 0.1094e\(-\)4 | 0.4220e\(-\)3 | 0.1532e\(-\)2 | 0.4884e\(-\)3 | 0.5647e\(-\)3 | 0.5656e\(-\)3 |

\(\mathbf{b}_\mathbf{1,2,0}\) | \(-\)0.3824e\(-\)4 | \(-\)1.4050e\(-\)4 | \(-\)0.7811e\(-\)3 | \(-\)0.1862e\(-\)3 | \(-\)0.4780e\(-\)3 | \(-\)0.1452e\(-\)2 |

\(\mathbf{b}_\mathbf{1,3,0}\) | 0.5668e\(-\)2 | 0.0521e\(-\)3 | 0.7698e\(-\)3 | 0.2710e\(-\)3 | 0.6786e\(-\)3 | 0.1125e\(-\)3 |

\(\mathbf{b}_\mathbf{1,4,0}\) | \(-\)0.2174e\(-\)3 | \(-\)0.4740e\(-\)3 | \(-\)0.7701e\(-\)2 | \(-\)0.1781e\(-\)2 | \(-\)0.6481e\(-\)2 | \(-\)0.1642e\(-\)1 |

\(\mathbf{b}_\mathbf{2,1,0}\) | \(-\)0.1116e\(-\)3 | \(-\)0.0063 | \(-0.0008\) | \(-0.0025\) | \(-0.0021\) | \(-0.001235\) |

\(\mathbf{b}_\mathbf{2,2,0}\) | 0.0070 | 0.0045 | 0.0089 | 0.0029 | 0.0081 | 0.0020 |

\(\mathbf{b}_\mathbf{2,3,0}\) | \(-0.0015\) | \(-0.0012\) | \(-0.0009\) | \(-0.0018\) | \(-\)0.7616e\(-\)3 | \(-0.0003829\) |

\(\mathbf{b}_\mathbf{2,4,0}\) | 0.1044 | 0.0575 | 0.0877 | 0.0281 | 0.0766 | 0.006970 |

\(\mathbf{b}_\mathbf{1,1,1}\) | 0.4227e\(-\)4 | \(-\)0.2722e\(-\)3 | \(-\)0.0860e\(-\)2 | \(-\)0.1107e\(-\)3 | \(-\)0.3536e\(-\)3 | \(-\)0.2218e\(-\)3 |

\(\mathbf{b}_\mathbf{1,2,1}\) | \(-\)1.2055e\(-\)4 | \(-\)2.1828e\(-\)4 | \(-\)0.3770e\(-\)3 | \(-\)0.1763e\(-\)3 | \(-\)0.1427e\(-\)3 | 0.7413e\(-\)4 |

\(\mathbf{b}_\mathbf{1,3,1}\) | \(-\)0.0945e\(-\)2 | 0.150e\(-\)3 | 0.8929e\(-\)3 | 0.1854e\(-\)3 | \(-\)0.1149e\(-\)3 | \(-\)0.1138e\(-\)3 |

\(\mathbf{b}_\mathbf{1,4,1}\) | \(-\)0.6856e\(-\)3 | \(-\)0.7365e\(-\)3 | \(-\)0.3716e\(-\)2 | \(-\)0.1686e\(-\)2 | \(-\)0.1935e\(-\)3 | \(-\)0.01324 |

\(\mathbf{b}_\mathbf{2,1,1}\) | \(-\)0.0873e\(-\)3 | \(-\)0.0034 | \(-\)0.0034 | \(-\)0.0039 | \(-\)0.0019 | \(-\)0.001135 |

\(\mathbf{b}_\mathbf{2,2,1}\) | 0.0013 | 0.0002 | 0.0064 | 0.0055 | 0.0053 | \(-0.0003\) |

\(\mathbf{b}_\mathbf{2,3,1}\) | \(-\)0.0004 | \(-\)0.0012 | \(-\)0.0012 | \(-\)0.0004 | \(-\)0.5323e\(-\)3 | 0.0001715 |

\(\mathbf{b}_\mathbf{2,4,1}\) | 0.0194 | 0.0020 | 0.0634 | 0.0527 | 0.0725 | 0.001998 |

\(\mathbf{a}_\mathbf{1,1,1}\) | 0.01090 | 1.6602 | 1.1913 | 0.9881 | 0.8165 | 3.4948 |

\(\mathbf{a}_\mathbf{1,2,1}\) | 0.1342 | 0.1322 | 0.3402 | 0.2605 | 0.3417 | 2.6987 |

\(\mathbf{a}_\mathbf{1,3,1}\) | 0.3975 | 0.2097 | 0.5633 | 0.5111 | 0.9003 | 0.4559 |

\(\mathbf{a}_\mathbf{1,4,1}\) | 0.1342 | 0.1322 | 0.3402 | 0.2605 | 0.3417 | 1.4380 |

\(\mathbf{a}_\mathbf{2,1,1}\) | 0.1317 | 2.0724 | 0.4017 | 0.8868 | 1.1676 | 1.6280 |

\(\mathbf{a}_\mathbf{2,2,1}\) | 2.2605 | 0.8352 | 1.8959 | 0.8602 | 2.4190 | 2.4298 |

\(\mathbf{a}_\mathbf{2,3,1}\) | 1.9247 | 0.8328 | 0.8844 | 1.8877 | 0.9818 | 0.2295 |

\(\mathbf{a}_\mathbf{2,4,1}\) | 2.2605 | 0.8352 | 1.8959 | 0.8602 | 1.6731 | 0.1731 |

\(\mathbf{a}_\mathbf{1,1,2}\) | 0.0243 | 0.2525 | 0.0912 | 0.0646 | 0.0809 | 0.5902 |

\(\mathbf{a}_\mathbf{1,2,2}\) | 0.0111 | 0.0117 | 0.0181 | 0.0091 | 0.0259 | 0.4023 |

\(\mathbf{a}_\mathbf{1,3,2}\) | 0.0850 | 0.0279 | 0.0359 | 0.0265 | 0.0712 | 0.03227 |

\(\mathbf{a}_\mathbf{1,4,2}\) | 0.0111 | 0.0117 | 0.0181 | 0.0091 | 0.0259 | 0.1840 |

\(\mathbf{a}_\mathbf{2,1,2}\) | 0.0073 | 0.2428 | 0.0365 | 0.0840 | 0.1069 | 0.09852 |

\(\mathbf{a}_\mathbf{2,2,2}\) | 0.1366 | 0.0812 | 0.1946 | 0.0392 | 0.1761 | 0.06510 |

\(\mathbf{a}_\mathbf{2,3,2}\) | 0.1353 | 0.0693 | 0.0799 | 0.0669 | 0.0549 | 0.02632 |

\(\mathbf{a}_\mathbf{2,4,2}\) | 0.1366 | 0.0812 | 0.1946 | 0.0392 | 0.1242 | 0.01045 |

As in Porfírio et al. (2003), the time delays \(\theta _{i,j}(\Theta _n)\) are all equal to 1.

For the implementation simulated here, the MATLAB LMI Toolbox routine “mincx” was used to solve Problem P2 and the MATLAB Optimization Toolbox with active-set algorithm of the routine “fmincon” was used to solve Problem P1.

In order to provide a fair comparison of the performance and computation cost of the two controllers, the convergence parameters of routines *mincx* and *fmincon* were selected in such a way that the stopping criterion that is first reached will always be the relative accuracy of the optimal objective, which was set to \(10^{-10}\).

## 6 Conclusion

In this work, it is presented an LMI formulation of the multi-model predictive control (MMPC) of an industrial C3/C4 splitter. Compared to the classical NLP-formulation proposed in Porfírio et al. (2003), the LMI approach allows to a significant reduction in the computational time without penalizing the controller performance. For high dimension systems, this method then may constitute a very interesting alternative to the conventional NLP-based MMPC.