Simultaneous Design and Control of Chemical Plants: A Robust Modelling Approach
Ricardez Sandoval, Luis Alberto
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This research work presents a new methodology for the simultaneous design and control of chemical processes. One of the most computationally demanding tasks in the integration of process control and process design is the search for worst case scenarios that result in maximal output variability or in process variables being at their constraint limits. The key idea in the current work is to find these worst scenarios by using tools borrowed from robust control theory. To apply these tools, the closed-loop dynamic behaviour of the process to be designed is represented as a robust model. Accordingly, the process is mathematically described by a nominal linear model with uncertain model parameters that vary within identified ranges of values. These robust models, obtained from closed-loop identification, are used in the present method to test the robust stability of the process and to estimate bounds on the worst deviations in process variables in response to external disturbances. The first approach proposed to integrate process design and process control made use of robust tools that are based on the Quadratic Lyapunov Function (QLF). These tests require the identification of an uncertain state space model that is used to evaluate the process asymptotic stability and to estimate a bound (γ) on the random-mean squares (RMS) gain of the model output variability. This last bound is used to assess the worst-case process variability and to evaluate bounds on the deviations in process variables that are to be kept within constraints. Then, these robustness tests are embedded within an optimization problem that seeks for the optimal design and controller tuning parameters that minimize a user-specified cost function. Since the value of γ is a bound on one standard deviation of the model output variability, larger multiples of this value, e.g. 2γ, 3γ, were used to provide more realistic bounds on the worst deviations in process variables. This methodology (γ-based) was applied to the simultaneous design and control of a mixing tank process. Although this approach resulted in conservative designs, it posed a nonlinear constrained optimization problem that required less computational effort than that required by a Dynamic Programming approach which had been the main method previously reported in the literature. While the γ-based robust performance criterion provides a random-mean squares measure of the variability, it does not provide information on the worst possible deviation. In order to search for the worst deviation, the present work proposed a new robust variability measure based on the Structured Singular Value (SSV) analysis, also known as the μ-analysis. The calculation of this measure also returns the critical time-dependent profile in the disturbance that generates the maximum model output error. This robust measure is based on robust finite impulse response (FIR) closed-loop models that are directly identified from simulations of the full nonlinear dynamic model of the process. As in the γ-based approach, the simultaneous design and control of the mixing tank problem was considered using this new μ-based methodology. Comparisons between the γ-based and the μ-based strategies were discussed. Also, the computational time required to assess the worst-case process variability by the proposed μ-based method was compared to that required by a Dynamic Programming approach. Similarly, the expected computational burden required by this new μ-based robust variability measure to estimate the worst-case variability for large-scale processes was assessed. The results show that this new robust variability tool is computationally efficient and it can be potentially implemented to achieve the simultaneous design and control of chemical plants. Finally, the Structured Singular Value-based (μ-based) methodology was used to perform the simultaneous design and control of the Tennessee Eastman (TE) process. Although this chemical process has been widely studied in the Process Systems Engineering (PSE) area, the integration of design and control of this process has not been previously studied. The problem is challenging since it is open-loop unstable and exhibits a highly nonlinear dynamic behaviour. To assess the contributions of different sections of the TE plant to the overall costs, two optimization scenarios were considered. The first scenario considered only the reactor’s section of the TE process whereas the second scenario analyzed the complete TE plant. To study the interactions between design and control in the reactor’s section of the plant, the effect of different parameters on the resulting design and control schemes were analyzed. For this scenario, an alternative calculation of the variability was considered whereby this variability was obtained from numerical simulations of the worst disturbance instead of using the analytical μ-based bound. Comparisons between the analytical bound based strategy and the simulation based strategy were discussed. Additionally, a comparison of the computational effort required by the present solution strategy and that required by a Dynamic Programming based approach was conducted. Subsequently, the topic of parameter uncertainty was investigated. Specifically, uncertainty in the reaction rate coefficient was considered in the analysis of the TE problem. Accordingly, the optimization problem was expanded to account for a set of different values of the reaction rate constant. Due to the complexity associated with the second scenario, the effect of uncertainty in the reaction constant was only studied for the first scenario corresponding to the optimization of the reactor section. The results obtained from this research project show that Dynamic Programming requires a CPU time that is almost two orders of magnitude larger than that required by the methodology proposed here. Likewise, the consideration of uncertainty in a physical parameter within the analysis, such as the reaction rate constant in the Tennessee Eastman problem, was shown to dramatically increase the computational load when compared to the case in which there is no process parametric uncertainty in the analysis. In general, the integration of design and control within the analysis resulted in a plant that is more economically attractive than that specified by solely optimizing the controllers but leaving the design of the different units fixed. This result is particularly relevant for this research work since it justifies the need for conducting simultaneous process design and control of chemical processes. Although the application of the robust tools resulted in conservative designs, the method has been shown to be an efficient computational tool for simultaneous design and control of chemical plants.