In model identification, especially in industrial applications, one of the key steps is model refinement. This step is traditionally completed after the model is obtained through linear least squares estimation without constraints. After estimating the parameters of the model based on data, model refinement helps ensure that the model is appropriate for the desired goal. Model refinement involves such considerations: forcing material balance, forcing known steady-state gains and gain ratios, removing null models, forcing the exact co-linearity, and repairing ill-conditioned sub-matrices, consistency of steady-state models, and dynamic consistency relationships. As mentioned, these corrections are usually done manually. Proving a means to impose these constraints directly in the identification optimization problem instead of as a post-processing step has the potential to improve a variety of Model Predictive Control (MPC) applications. Applying prior knowledge and model consistency relationships leads to handling ill-conditioning for the purpose of better control performance [1, 2]. The other advantage of incorporating constraints in the identification step is to achieve smaller variance for parameter estimates possibly with less testing [1, 3].

In [2], Thorpe et al. incorporate different constraints such as dead time, settling time, and gains or gains rations for FIR or subspace models. There is a wide range of other possibilities for considering constraints for the identification procedure. Darby et al. in [1], summarized possible improvements in different steps of performing MPC. For example, consistency of steady-state models [4], and dynamic consistency relationships [5] can be imposed during the model identification step.

In this work, the goal of model identification is to achieve the minimum value for a function of error between the system response and the model while satisfying several constraints. The model should satisfy certain conditions so that it reflects known aspects of the system in order to perform well in closed-loop. In most traditional identification methods, these conditions are not imposed on the model identification problem. By imposing constraints, the model identification of Multi-Input Multi-Output (MIMO) systems along with discretization in time or space becomes a large-scale nonlinear constrained optimization problem.

Among different ways of formulating an optimization problem, the open-equation format has found more attention recently. Writing the equations in residual form allows the differential term to be expressed in implicit form [6]. This strategy has the ability to solve problems with a large number of variables and equations simultaneously without nested convergence loops [7,8]. Although the open-equation method still has some challenges to solve very large-scale problems, it has the potential to be efficient for constrained model identification. This work investigates the effect of imposing different constraints on the model identification step, and shows the capabilities and limitations of the open-equation nonlinear solver to solve the estimation problem.

**References:**

1. Darby, M. L., Nikolaou, M., MPC: Current practice and challenges, Control Engineering Practice, 2012, 20:328-342.

2. Thorpe, P. J., Nicholson, D, Osta, S., Harmse M., Constrained Model Identification: Use Knowledge of the process to get better models from less data, AIChE, 2010.

3. Soderstrom, T., and Stoica, P., System identification, Prentice Hall, 1989.

4. Skogestad, S., Consistency of Steady-State Models Using Insight about Extensive Variables, Ind. Eng. Chem. Res. 1991, 30:654-661.

5. Eskinat, E., Dynamic consistency relations for process modeling. AIChE Journal, 2003, 49(8): 2224–2227.

6. Hedengren, J. D., Asgharzadeh Shishavan, R., Powell, K. M., and Edgar, T. F., Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Computers & Chemical Engineering, 2014, 70:133-148.

7. AspenTech, Getting Started Using Equation Oriented Modelling, 2011.

8. Meixell, M. D., Gochenour, B. and Chen, C., Industrial Applications of Plant-Wide Equation-Oriented Process Modeling, Advances in Chemical Engineering, 2011, 40: 119–152.

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