# Efficient Control Discretization Based on Turnpike Theory for Dynamic Optimization

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Pertinent Background

#### Turnpike in Optimal Control

## 3. Proposed Adaptive Control Discretization Approach

#### Semi-Uniform Adaptive Control Discretization

#### A Variant Formulation

## 4. Results and Discussion

#### 4.1. Example 1

#### 4.2. Example 2

^{3}or less. The optimization variables are the inlet flowrate ${F}^{in}$ and the cooling power ${P}_{c}$, which are allowed to vary within $[0,40]$ m

^{3}h

^{−1}and $[0,4000]$ kJ h

^{−1}, respectively. With the goal of maximizing the production of B, the dynamic optimization problem can be formulated as:

^{3}h

^{−1}, ${P}_{c,ss}^{*}=3040.6$ kJ h

^{−1}, $({C}_{\mathrm{A},ss}^{*},{C}_{\mathrm{B},ss}^{*},{C}_{\mathrm{C},ss}^{*},{C}_{\mathrm{D},ss}^{*})=(2944.7,977.9,486.2,345.6)$ mol m

^{−3}, $({T}_{ss}^{*},{T}_{c,ss}^{*})=(110,106.5)$ °C, and ${V}_{ss}^{*}=1.8$ m

^{3}.

#### Shorter Time Horizon

#### 4.3. Example 3

^{−1}, $({C}_{\mathrm{A},ss}^{*},{C}_{\mathrm{B},ss}^{*},{C}_{\mathrm{P},ss}^{*},{C}_{\mathrm{I},ss}^{*})=(1.69,0.43,0.82,0.13)$ mol L

^{−1}, and ${V}_{ss}^{*}=0.03$ L.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**An optimal control or state trajectory with two transient phases and a turnpike in between. The green solid line represents the (globally) optimal steady state ${u}_{ss}^{*}$ or ${x}_{ss}^{*}$, and the red dashed lines denote its $\u03f5$-neighborhood.

**Figure 2.**Schematic of an optimal control trajectory resulting from a nonuniform discretization. The dashed line illustrates the ideal optimal trajectory that might be obtained without discretization.

**Figure 3.**Schematic of the semi-uniform control discretization in the real time domain t (

**left**) and the transformed time domain ${t}^{\prime}$ (

**right**). The three transient and turnpike phases are delineated by the blue dashed lines.

**Figure 4.**Adaptive semi-uniform discretization with suboptimal (

**left**) and optimal (

**right**) solution for ${\tau}_{i}$s. The dashed line trajectory shows the ideal optimal control as a reference.

**Figure 5.**Optimal trajectories for Problem (37) in case of a uniform discretization with $N=5$ (

**left**) and $N=60$ (

**right**) epochs.

**Figure 6.**Optimal trajectories for Problem (37) in case of the proposed semi-uniform discretization approach with $N=5$ epochs (both the main and variant formulations) in the ${t}^{\prime}$ (

**left**) and t (

**right**) domains.

**Figure 7.**Optimal trajectories for Problem (40) in the case of a uniform discretization with $N=7$ (

**left**) and $N=60$ (

**right**) epochs.

**Figure 8.**Optimal trajectories for Problem (40) in the case of the main (

**left**) and the variant (

**right**) formulations of the proposed approach with $N=7$ epochs.

**Figure 9.**Optimal trajectories for Problem (40) with a short time horizon solved using a uniform discretization and $N=5$ epochs.

**Figure 10.**Optimal trajectories for Problem (40) with a short time horizon solved using the proposed approach and $N=5$ epochs.

**Figure 11.**Optimal trajectories for Problem (44) in the case of the uniform (

**left**) and proposed (

**right**) discretization methods both with $N=5$ epochs.

**Figure 12.**Optimal trajectories for Problem (44) in the case of the uniform discretization with $N=21$ (

**left**) and the nonuniform discretization with $N=3$ (

**right**) epochs.

**Table 1.**Optimal objective values, number of iterations, and CPU times for different discretization strategies in Problem (37). ${n}_{p}$ denotes the total number of optimization variables for each strategy.

Discretization | N | ${\mathit{n}}_{\mathit{p}}$ | ${\mathcal{J}}^{*}$ | Iter | CPU (s) |
---|---|---|---|---|---|

Uniform | 5 | 5 | 9.32 | 13 | 0.13 |

Uniform | 60 | 60 | 2.45 | 103 | 8.02 |

Nonuniform | 5 | 9 | not converged | >1000 | >11 |

Proposed | 5 (${N}_{s}=2,{N}_{d}=2$) | 7 | 2.45 | 38 | 0.79 |

Proposed (variant) | 5 (${N}_{s}=2,{N}_{d}=2$) | 7 | 2.45 | 33 | 0.89 |

**Table 2.**Constants for the Van de Vusse reactor model (38).

${k}_{0,1}$ | 1.29 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{12}$ | h^{−1} | $\alpha $ | 30.828 | h^{−1} |

${k}_{0,2}$ | 9.04 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{6}$ | m^{3} (mol h)^{−1} | $\beta $ | 86.688 | h^{−1} |

${E}_{1}$ | 9758.3 | K | $\gamma $ | 0.1 | K kJ^{−1} |

${E}_{2}$ | 8560 | K | $\delta $ | 3.52 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ | m^{3} K kJ^{−1} |

$\Delta {H}_{\mathrm{AB}}$ | 4.2 | kJ mol^{−1} | $\eta $ | 30 | m^{3/2} h^{−1} |

$\Delta {H}_{\mathrm{BC}}$ | −11 | kJ mol^{−1} | ${T}_{in}$ | 104.9 | °C |

$\Delta {H}_{\mathrm{AD}}$ | −41.85 | kJ mol^{−1} | ${C}_{\mathrm{A}}^{in}$ | 5.10 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$ | mol m^{−3} |

**Table 3.**Optimal objective values, number of iterations, and CPU times for different discretization strategies in Problem (40). ${n}_{p}$ denotes the total number of optimization variables for each strategy.

Discretization | N | ${\mathit{n}}_{\mathit{p}}$ | ${\mathcal{J}}^{*}$ | Iter | CPU (s) |
---|---|---|---|---|---|

Uniform | 7 | 14 | −3.34 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$ | 251 | 14.7 |

Uniform | 60 | 120 | −3.84 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$ | 227 | 798 |

Uniform | 70 | 140 | failed | 207 | 645 |

Uniform | 80 | 160 | failed | 295 | 1160 |

Nonuniform | 7 | 19 | failed | 1212 | 134 |

Proposed | 7 (${N}_{s}=3,{N}_{d}=3$) | 15 | −3.87 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$ | 800 | 75.2 |

Proposed (variant) | 7 (${N}_{s}=3,{N}_{d}=3$) | 15 | −3.87 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{5}$ | 210 | 21.4 |

**Table 4.**Optimal objective values, number of iterations, and CPU times for different discretization strategies in Problem (40) for the case of a short time horizon.

Discretization | N | ${\mathit{n}}_{\mathit{p}}$ | ${\mathcal{J}}^{*}$ | Iter | CPU (s) |
---|---|---|---|---|---|

Uniform | 5 | 10 | 3987 | 44 | 1.65 |

Uniform | 7 | 14 | 4058 | 53 | 2.29 |

Proposed | 5 (${N}_{s}=2,{N}_{d}=2$) | 11 | 4009 | 69 | 2.77 |

Proposed (variant) | 5 (${N}_{s}=2,{N}_{d}=2$) | 11 | 4009 | 53 | 2.33 |

**Table 5.**Constants and initial conditions for the model (42).

Parameters | Initial Conditions | ||||
---|---|---|---|---|---|

${k}_{1}$ | 0.8 | L (mol min)^{−1} | ${C}_{\mathrm{A}0}$ | 0 | mol L^{−1} |

${k}_{2}$ | 0.5 | L (mol min)^{−1} | ${C}_{\mathrm{B}0}$ | 0 | mol L^{−1} |

${C}_{\mathrm{A}}^{in}$ | 5 | mol L^{−1} | ${C}_{\mathrm{P}0}$ | 0 | mol L^{−1} |

${C}_{\mathrm{B}}^{in}$ | 3 | mol L^{−1} | ${C}_{\mathrm{I}0}$ | 0 | mol L^{−1} |

$\alpha $ | 0.119 | L^{1/2} min^{−1} | ${V}_{0}$ | 0.001 | L |

**Table 6.**Optimal objective values, number of iterations, and CPU times for different discretization strategies in Problem (44). ${n}_{p}$ denotes the total number of optimization variables for each strategy.

Discretization | N | ${\mathit{n}}_{\mathit{p}}$ | ${\mathcal{J}}^{*}$ | Iter | CPU (s) |
---|---|---|---|---|---|

Uniform | 5 | 10 | 0.66 | 31 | 1.3 |

Uniform | 14 | 28 | 0.734 | 23 | 1.26 |

Uniform | 20 | 40 | 0.739 | 27 | 2.38 |

Uniform | 21 | 42 | 0.741 | 34 | 3.53 |

Nonuniform | 2 | 4 | 0.731 | 35 | 0.59 |

Nonuniform | 3 | 7 | 0.742 | 145 | 3.28 |

Proposed | 5 (${N}_{s}=2,{N}_{d}=2$) | 11 | 0.741 | 29 | 1.83 |

Proposed (variant) | 5 (${N}_{s}=2,{N}_{d}=2$) | 11 | 0.741 | 24 | 1.18 |

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**MDPI and ACS Style**

Sahlodin, A.M.; Barton, P.I.
Efficient Control Discretization Based on Turnpike Theory for Dynamic Optimization. *Processes* **2017**, *5*, 85.
https://doi.org/10.3390/pr5040085

**AMA Style**

Sahlodin AM, Barton PI.
Efficient Control Discretization Based on Turnpike Theory for Dynamic Optimization. *Processes*. 2017; 5(4):85.
https://doi.org/10.3390/pr5040085

**Chicago/Turabian Style**

Sahlodin, Ali M., and Paul I. Barton.
2017. "Efficient Control Discretization Based on Turnpike Theory for Dynamic Optimization" *Processes* 5, no. 4: 85.
https://doi.org/10.3390/pr5040085