# Methods and Tools for Robust Optimal Control of Batch Chromatographic Separation Processes

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## Abstract

**:**

## 1. Introduction

#### 1.1. Solvent Composition Trajectory Elution Strategies

#### 1.2. PDE-Constrained Dynamic Optimization

#### 1.3. Nonlinear Robust Optimization of Uncertain Dynamic Systems

#### 1.4. Aim and Scope

- i)
- To formulate a nominal open-loop optimal control problem and its robust counterpart, including penalty on the difference on the zero-order hold control discretization, to find optimal control trajectories that remain feasible for all perturbations from the uncertainty set.
- ii)
- To develop a numerically efficient framework for simulation of a Lyapunov differential equation, enabling the computation of the robustified inequality constraint margin in the counterpart problem.
- iii)
- To assess the robustness of general elution trajectories, for both batch elution and cyclic-steady-state operation, and to benchmark with that of the conventional liner trajectories.

#### 1.5. Outline of the Paper

## 2. Process Description

**Figure 1.**Simplified P&ID of the HPLC system. Elementary system component description: (1) buffer mixing unit, (2) high-pressure switching valve unit, (3) HPLC separation column, (4) UV detector, and (5) fractionating valve.

**Table 1.**HPLC system component design parameters and HIC column specifics. The adsorption capacity and the self-association parameters listed are equal for all insulin variants $\alpha \in \{A,B,C\}$.

Parameter | Description | Value | Unit |
---|---|---|---|

$\dot{Q}$ | volumetric flow rate | $6.0\times {10}^{-5}$ | ${\mathrm{m}}^{3}\xb7{\mathrm{s}}^{-1}$ |

$\Delta {t}_{\mathrm{load}}$ | sample load duration | $1.0\times {10}^{-2}$ | s |

${V}_{\mathrm{mix}}$ | buffer mixing unit volume | $2.0\times {10}^{-7}$ | ${\mathrm{m}}^{3}$ |

${L}_{c}$ | column length | $1.0\times {10}^{-1}$ | m |

${D}_{c}$ | column diameter | $1.0\times {10}^{-2}$ | m |

${D}_{p}$ | particle diameter | $3.5\times {10}^{-5}$ | m |

${\epsilon}_{c}$ | interstitial porosity of the column | $3.26\times {10}^{-1}$ | − |

${\epsilon}_{p}$ | apparent particle porosity | $7.54\times {10}^{-1}$ | − |

${\epsilon}_{t}$ | total porosity of the column | ${\epsilon}_{c}+(1-{\epsilon}_{c}){\epsilon}_{p}$ | − |

${\mathcal{D}}_{\mathrm{app}}$^{a} | apparent dispersion coefficient | $\left({v}_{\mathrm{int}}{D}_{p}\right){\mathrm{Pe}}^{-1}$ | ${\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ |

${v}_{\mathrm{int}}$ | interstitial velocity | $4\dot{Q}{\left({D}_{c}^{2}\pi {\epsilon}_{t}\right)}^{-1}$ | $\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

${k}_{\mathrm{kin}}$ | kinetic rate constant | $3.0\times {10}^{-1}$ | ${\mathrm{s}}^{-1}$ |

${K}_{\mathrm{eq}}$ | self-association equilibrium constant | $7.56$ | − |

ν | stoichiometric constant | $4.82\times {10}^{1}$ | − |

${q}_{\mathrm{max}}$ | adsorption capacity | $1.0\times {10}^{2}$ | $\mathrm{kg}\xb7{\mathrm{m}}^{-3}$ |

^{a}Pe is assumed to be constant and equal to $0.50$, i.e., molecular diffusion is neglected [55].

#### 2.1. Control Signal Uncertainties

## 3. Mathematical Modeling

**Table 2.**Kinetic parameters of the Langmuir self-association adsorption model Equation (2). The feed composition, ${c}_{\mathrm{load},\alpha}$, is given in fractional weights for each insulin component and the total feed concentration is $2.0\times {10}^{1}$ $(\mathrm{moL}\xb7{\mathrm{m}}^{-3})$.

Component | α | ${H}_{0,\alpha}\phantom{\rule{3.33333pt}{0ex}}(-)$ | ${\gamma}_{\alpha}\phantom{\rule{3.33333pt}{0ex}}({\mathrm{m}}^{3}\xb7{\mathrm{kmoL}}^{-1})$ | ${c}_{\mathrm{load},\alpha}\phantom{\rule{3.33333pt}{0ex}}(\mathrm{wt}\phantom{\rule{3.33333pt}{0ex}}\%)$ |
---|---|---|---|---|

Insulin aspart | A | $1.14$ | $2.30$ | $33.3$ |

desB30 insulin | B | $1.51$ | $2.39$ | $33.3$ |

Insulin methyl ester | C | $1.78$ | $2.63$ | $33.3$ |

#### 3.1. Spatial Discretization

## 4. Problem Formulation

- i)
- Optimal control strategy I is exclusively concerned with the batch elution operation mode, where the limit cycle criteria are relaxed and the upper fractionating interval endpoint bounds the temporal horizon, i.e., ${t}_{f}={\tau}_{0}+\Delta \tau $.
- ii)
- Optimal control strategy II is concerned with the comprehensive CSS operation mode, where the temporal horizon includes the column regeneration and re-equilibration modes.

#### 4.1. Cyclic-Steady-State Criteria Formulation

#### 4.2. Open-loop Optimal Control Problem Formulation

#### 4.3. Robust Counterpart Problem Formulation

**P**, givings us ${n}_{x}({n}_{x}+1)/2$ extra states. The result is the following approximate robust counterpart of Equation (11).

**P**is a linear approximation of the covariance matrix of the state x. Consequently, the covariance matrix associated with the output function h is approximated by $\mathbf{C}\left(t\right)\mathbf{P}\left(t\right){\mathbf{C}}^{T}\left(t\right)$. While the discussed approximate robust approach is computationally tractable for medium-scale ODE-constrained DOPs, it is not for our original PDE-constrained DOP. For these reasons, the robust counterpart problem is cast in the frame of bilevel optimal control where the upper level concerns forward simulation of the Lyapunov differential equation, and the nominal NLP augmented with the robustified inequality constraint margin is considered in the lower level:

## 5. Methods and Tools

#### 5.1. The JModelica.org Toolchain

#### 5.1.1. Optimization

**C**omputer

**a**lgebra

**s**ystem with

**A**utomatic

**Di**fferentaion) is an open-source, low-level tool for efficiently computing derivatives using AD and is tailored for dynamic optimization. The user encodes the dynamic optimization problem using Modelica and Optimica. The JModelica.org compiler then transfers a symbolic representation of the problem to CasADi Interface [78], after performing symbolic transformations such as index reduction. CasADi Interface serves as a three-way, symbolic Python interface between the optimization problem, user, and numerical algorithm. The main numerical algorithm for dynamic optimization in JModelica.org is based on direct, local collocation and is described in Section 5.2. The algorithm is implemented using CasADi to transcribe the DOP into an NLP. First- and second-order derivatives of the NLP functions can then be conveniently and efficiently computed using CasADi’s algorithmic differentiation while preserving sparsity. The NLP is then finally solved by the open-source, interior-point method IPOPT [43].

**Figure 2.**The dynamic optimization framework of the JModelica.org platform. The problem is formulated using Modelica and Optimica, which is then transferred to CasADi Interface by the JModelica.org compiler. CasADi Interface is a symbolic interface between the compiler, user, and collocation algorithm. After collocation, the problem is finally solved by IPOPT.

#### 5.1.2. Simulation

**Figure 3.**The simulation capabilities of the JModelica.org platform. An FMU is loaded using the package PyFMI, which in turn connects to the simulation package Assimulo which provides access to the solvers. The work-flow is controlled through a user-defined script which additionally allows for providing extra equations to the problem, in this paper the Lyapunov equations.

#### 5.2. Transcription of the Dynamic Optimization Problem

#### 5.3. Simulation Setup and Coupling of the Lyapunov Equation

**Figure 4.**The sparsity pattern of the Jacobian of the spatially discretized, with ${n}_{v}=20$, and ODE transformed system.

## 6. Results and Discussion

Solutions to the Open-loop Optimal | Solutions to the Robust Counterpart | |||||
---|---|---|---|---|---|---|

Control Problem (Equation (11)) | Problem (Equation (19)) | |||||

${Y}_{B}\left({t}_{f}\right)$ | ${X}_{B}\left({t}_{f}\right)$ | $\Gamma \sqrt{\mathbf{C}\left({t}_{f}\right)\mathbf{P}\left({t}_{f}\right){\mathbf{C}}^{T}\left({t}_{f}\right)}$ | ${Y}_{B}\left({t}_{f}\right)$ | ${X}_{B}\left({t}_{f}\right))$ | $\Gamma \sqrt{\mathbf{C}\left({t}_{f}\right)\mathbf{P}\left({t}_{f}\right){\mathbf{C}}^{T}\left({t}_{f}\right)}$ | |

$(\%)$ | $(\%)$ | $(\%)$ | $(\%)$ | $(\%)$ | $(\%)$ | |

$\mathrm{R}=0.00$ | $71.98$ | $97.50$ | $1.30$ | $63.06$ | $98.22$ | $0.72$ |

$\mathrm{R}=0.50$ | $71.23$ | $97.50$ | $1.26$ | $62.27$ | $98.21$ | $0.71$ |

$\mathrm{R}=5.00$ | $70.31$ | $97.50$ | $1.12$ | $61.73$ | $98.19$ | $0.69$ |

${u}_{\mathrm{lin}}$^{a} | $67.41$ | $97.50$ | $1.33$ | $56.08$ | $98.37$ | $0.87$ |

^{a}Benchmarking with the conventional linear elution trajectory, ${u}_{\mathrm{lin}}$, governed by Equation (30).

Solutions to the Open-loop Optimal | Solutions to the Robust Counterpart | |||||
---|---|---|---|---|---|---|

Control Problem (Equation (11)) | Problem (Equation (19)) | |||||

${Y}_{B}\left({t}_{f}\right)$ | ${X}_{B}\left({t}_{f}\right)$ | $\Gamma \sqrt{\mathbf{C}\left({t}_{f}\right)\mathbf{P}\left({t}_{f}\right){\mathbf{C}}^{T}\left({t}_{f}\right)}$ | ${Y}_{B}\left({t}_{f}\right)$ | ${X}_{B}\left({t}_{f}\right))$ | ||

$(\%)$ | $(\%)$ | $(\%)$ | $(\%)$ | $(\%)$ | $(\%)$ | |

$\mathrm{R}=0.00$ | $70.26$ | $97.50$ | $1.15$ | $61.53$ | $98.15$ | $0.65$ |

$\mathrm{R}=0.05$ | $69.95$ | $97.50$ | $1.10$ | $61.10$ | $98.14$ | $0.64$ |

$\mathrm{R}=0.50$ | $62.88$ | $97.50$ | $0.85$ | $55.25$ | $98.06$ | $0.56$ |

$\mathrm{R}=1.00$ | $59.50$ | $97.50$ | $0.79$ | $51.82$ | $98.03$ | $0.53$ |

#### 6.1. Optimal Control Strategy I—Nominal and Robustified Elution Trajectories in the Batch Elution Operation Mode

**Figure 6.**Optimal state and control trajectories, where ${c}_{\alpha}(t,{z}_{f})$ and $\forall \alpha \in \{A,B,C\}$ is normalized with $[0.75,0.75,1.5]\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}(\mathrm{moL}\xb7{\mathrm{m}}^{-3})$, for ${X}_{B,L}=9.75\times {10}^{-1}\phantom{\rule{3.33333pt}{0ex}}(-)$ and $\mathrm{R}\in [0.0,5.0]$. Markers indicate the solution at the Radau collocation points and solid and dashed lines the corresponding simulated response. The shaded areas indicate the pooling interval endpoints, $[{\tau}_{0},{\tau}_{0}+\Delta \tau ]$, and those of the initial load and wash and the terminal re-equilibration.

**Figure 7.**Target component purity, ${X}_{B}\left(t\right)$, (dashed lines) and associated back-off term, $\Gamma \sqrt{\mathbf{C}\left(t\right)\mathbf{P}\left(t\right){\mathbf{C}}^{T}\left(t\right)}$, (solid lines) as a function of time for the nominal elution trajectories depicted in Figure 6.

**Figure 8.**State and control trajectories generated with the set of stochastic control signals, $\tilde{u}$, sampled from a uniform distribution with the uncertainty level ${\sigma}_{w}=5.0\%$. Markers indicate the nominal optimal solution at the Radau collocation points and solid and dashed lines the corresponding simulated response.

**Figure 9.**Probability distribution of the variability in ${X}_{B}\left({t}_{f}\right)$ generated with the set of stochastic control signals, $\tilde{u}$, sampled from a uniform distribution with the uncertainty level ${\sigma}_{w}=5.0\%$. The dashed lines indicate the lower purity constraint, ${X}_{B,L}$, and the deterministic worst-case purity, ${X}_{B,L}-\Gamma \sqrt{\mathit{C}\left({t}_{f}\right)\mathit{P}\left({t}_{f}\right){\mathit{C}}^{T}\left({t}_{f}\right)}$. The solid line indicates the cumulative probability, ${\Phi}_{{\sigma}_{w}^{2}}\left({X}_{B}\left({t}_{f}\right)\right)$.

**Figure 10.**Nominal and robustified state and control trajectories. Markers indicate the robustified solution at the Radau collocation points and solid and dashed lines the corresponding simulated response. The light colored state and control trajectories indicate the nominal solution.

#### 6.2. Optimal Control Strategy II—Nominal and Robustified Elution Trajectories in the Cyclic-Steady-State Operation Mode

**Figure 11.**State and control trajectories generated for ${X}_{B,L}=9.75\times {10}^{-1}\phantom{\rule{3.33333pt}{0ex}}(-)$ and $\mathrm{R}\in [0.0,1.0]$. The set of stochastic control signals, $\tilde{u}$, was sampled from a uniform distribution with the uncertainty level ${\sigma}_{w}=5.0\%$. Markers indicate nominal state and control trajectories that conform to the cyclic-steady state criteria Equation (6) over the temporal horizon $[{t}_{0},{t}_{f}]$ at the Radau collocation points and solid and dashed lines the corresponding simulated response.

**Figure 12.**Target component purity, ${X}_{B}\left(t\right)$, (dashed lines) and associated back-off term, $\Gamma \sqrt{\mathbf{C}\left(t\right)\mathbf{P}\left(t\right){\mathbf{C}}^{T}\left(t\right)}$, (solid lines) as a function of time for the nominal elution trajectories depicted in Figure 11.

**Figure 13.**Probability distribution of the variability in ${X}_{B}\left({t}_{f}\right)$ generated with the set of stochastic control signals, $\tilde{u}$, sampled from a uniform distribution with the uncertainty level ${\sigma}_{w}=5.0\%$. The dashed lines indicate the lower purity constraint, ${X}_{B,L}$, and the deterministic worst-case purity, ${X}_{B,L}-\Gamma \sqrt{\mathit{C}\left({t}_{f}\right)\mathit{P}\left({t}_{f}\right){\mathit{C}}^{T}\left({t}_{f}\right)}$. The solid line indicates the cumulative probability, ${\Phi}_{{\sigma}_{w}^{2}}\left({X}_{B}\left({t}_{f}\right)\right)$.

**Figure 14.**Nominal and robust cyclic-steady-state elution profiles for $\mathrm{R}\in [0.0,1.0]$. Markers indicate the robustified solution at the Radau collocation points and solid and dashed lines the corresponding simulated response. The light colored state and control trajectories indicate the nominal solution.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Holmqvist, A.; Andersson, C.; Magnusson, F.; Åkesson, J.
Methods and Tools for Robust Optimal Control of Batch Chromatographic Separation Processes. *Processes* **2015**, *3*, 568-606.
https://doi.org/10.3390/pr3030568

**AMA Style**

Holmqvist A, Andersson C, Magnusson F, Åkesson J.
Methods and Tools for Robust Optimal Control of Batch Chromatographic Separation Processes. *Processes*. 2015; 3(3):568-606.
https://doi.org/10.3390/pr3030568

**Chicago/Turabian Style**

Holmqvist, Anders, Christian Andersson, Fredrik Magnusson, and Johan Åkesson.
2015. "Methods and Tools for Robust Optimal Control of Batch Chromatographic Separation Processes" *Processes* 3, no. 3: 568-606.
https://doi.org/10.3390/pr3030568