Interior Point Solution of Multilevel Quadratic Programming Problems in Constrained Model Predictive Control Applications

Baker, Rhoda; Swartz, Christopher L.E.

doi:10.1021/ie070270r

Cited by 28 publications

(11 citation statements)

References 35 publications

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“…The resulting DRTO problem can in principle be solved using a sequential solution approach in which optimization iterations and closed‐loop simulation are decoupled. However, in order to avoid potential difficulties with derivative discontinuities due to MPC input inequality constraints, we adopt a simultaneous solution strategy in which the MPC subproblems are replaced by their first‐order KKT optimality conditions, resulting in a single‐level mathematical program with complementarity constraints (MPCC). We illustrate this by considering the following general quadratic programming problem:

\begin{matrix} right \begin{matrix} left min & left \frac{1}{2} z^{T} H boldz + g^{T} boldz \\ left normals . normalt . & left A boldz = boldb \\ left & left boldz \geq bold 0 \\ left & left \end{matrix} & center & left \end{matrix}

for which the KKT optimality conditions are:

\begin{matrix} H z + g - A^{T} λ - μ = 0 \\ A z = b \\ μ_{i} {boldz}_{i} = 0, i \in I \\ z, μ 0 \end{matrix}

…”

Section: Drto Formulationmentioning

confidence: 99%

“…The nonlinear model is used to simulate the real plant dynamics. The objective function provided by Heath et al [31] is used for the DRTO layer of the evaporator process, and aims to minimize the operating cost due to electricity, steam, and cooling water: J = 1.009(F2 + F3) + 96(F1+F3)(T100−T2) + 0.6F200 (23) where:…”

Section: Case Studymentioning

confidence: 99%

“…[19][20][21][22] The closedloop response of a process under constrained MPC has also been considered in operating point back-off and plant design formulations that include dynamic performance considerations. Baker and Swartz [23] propose the reformulation of the MPC optimization subproblems as algebraic equations using the first-order KKT optimality conditions and including them within a single optimization problem that takes the form of a mathematical program with complementarity constraints (MPCC). Lam et al [24] utilize a formulation of this type for reference trajectory optimization for systems in transition.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The utilization of closed‐loop prediction for dynamic real‐time optimization

Jamaludin

Swartz

2017

Can J Chem Eng

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Real‐time optimization (RTO) is a layer within the hierarchical process automation architecture in which economically optimal set‐points are computed for the underlying plant control system. RTO calculations are traditionally based on steady‐state models, but an increasingly global and dynamic marketplace has led to the development of dynamic RTO (DRTO) strategies. Typical DRTO approaches optimize process input trajectories based on the open‐loop response dynamics of the process, with controller set‐point trajectories constructed from the resulting output response. This paper describes recent developments that utilize closed‐loop prediction in the DRTO calculations for MPC regulated processes. A rigorous closed‐loop DRTO problem is formulated as a multilevel dynamic optimization problem due to the inclusion of a sequence of MPC quadratic programming subproblems to generate the closed‐loop response dynamics. A simultaneous solution strategy is applied in which the MPC subproblems are replaced by their equivalent Karush‐Kuhn‐Tucker (KKT) optimality conditions, permitting reformulation of the original problem as a single‐level mathematical program with complementarity constraints (MPCC). Closed‐loop approximation techniques are proposed to reduce the dimension of the DRTO problem while maintaining good closed‐loop prediction accuracy. The performance of the proposed approaches is illustrated using case studies. Conclusions are drawn, and further research directions identified.

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\begin{matrix} right \begin{matrix} left min & left \frac{1}{2} z^{T} H boldz + g^{T} boldz \\ left normals . normalt . & left A boldz = boldb \\ left & left boldz \geq bold 0 \\ left & left \end{matrix} & center & left \end{matrix}

for which the KKT optimality conditions are:

\begin{matrix} H z + g - A^{T} λ - μ = 0 \\ A z = b \\ μ_{i} {boldz}_{i} = 0, i \in I \\ z, μ 0 \end{matrix}

…”

Section: Drto Formulationmentioning

confidence: 99%

Section: Case Studymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The utilization of closed‐loop prediction for dynamic real‐time optimization

Jamaludin

Swartz

2017

Can J Chem Eng

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show abstract

“…It is assumed that only one comonomer (either butene or hexene) can be used at a time. The following complementarity constraints are used to meet this requirement:55 …”

Section: Experimental Designs For the Simplified Polyethylene Mwd Modelmentioning

confidence: 99%

Design of Optimal Sequential Experiments to Improve Model Predictions from a Polyethylene Molecular Weight Distribution Model

Thompson

McAuley

McLellan

2009

Macro Reaction Engineering

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Reliable model predictions require an appropriate model structure and also good parameter estimates. For good parameter estimates to be obtained, it is important that the data used in parameter estimation are informative. Alphabet-optimal experimental designs can be used to ensure that new experiments are as informative as possible. This work presents the development of D-and A-optimal sequential experimental designs for improving parameter precision in a molecular-weight-distribution model for Ziegler-Natta-catalyzed polyethylene. Novel V-optimal designs techniques are developed to improve the precision of model predictions, and anticipated benefits are quantified. Problems with local minima are discussed and comparisons between the optimality criteria are made.

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“…Then, the dynamics are collocated at a specified set of collocation points. By applying pseudospectral methods, the original DOPs are converted into finite dimensional nonlinear programming (NLP) problems, which can be solved by some numerical optimization techniques, such as sequential quadratic programming (SQP) 19, interior point (IP) 20–22, and others 23–25.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Pseudospectral Method for Constrained Dynamic Optimization Problems in Chemical Engineering

Xiao

Liu

2016

Chem Eng & Technol

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An adaptive pseudospectral method with two novel strategies is proposed to solve constrained dynamic optimization problems in chemical engineering. The first strategy is to introduce slope information in designing appropriate subintervals, so that the approach becomes more efficient to track the optimal control profiles. The second strategy is to redistribute the collocation points based on the approximation error, thus ensuring the accuracy of the method. Two constrained dynamic optimization problems with multiple control variables are tested as illustrations, and the proposed approach is compared with other methods. The research results reveal the effectiveness of the proposed method in improving the solution accuracy of dynamic optimization problems.

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Interior Point Solution of Multilevel Quadratic Programming Problems in Constrained Model Predictive Control Applications

Cited by 28 publications

References 35 publications

The utilization of closed‐loop prediction for dynamic real‐time optimization

The utilization of closed‐loop prediction for dynamic real‐time optimization

Design of Optimal Sequential Experiments to Improve Model Predictions from a Polyethylene Molecular Weight Distribution Model

An Adaptive Pseudospectral Method for Constrained Dynamic Optimization Problems in Chemical Engineering

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