# (491a) Real-Time Optimization with Changing Active Constraints Solved through Decentralized Feedback Control

#### AIChE Annual Meeting

#### 2023

#### 2023 AIChE Annual Meeting

#### Computing and Systems Technology Division

#### Predictive Control and Optimization

#### Sunday, November 5, 2023 - 3:30pm to 3:48pm

One of the main challenges that such approaches face is the presence of changing active constraints (Jäschke et al., 2017), which may drastically change the operation mode of the system. For example, if a constraint becomes active due to a disturbance, not taking its control into account leads to infeasible operation. Similarly, if a constraint becomes optimally inactive, the control of such constraint should be given up. For a given set of active constraints, if the cost gradient is measured, it is known that the control of the active constraints together with the control of a projection of the cost gradient over the nullspace of the active constraintsâ€™ gradient leads to optimal operation (JÃ¤schke and Skogestad, 2012). However, the existence of several active constraint regions may deem necessary the use of several independent control structures, each of them being able to provide near-optimal operation for their respective region of design. The switching between such control structures would also become an issue since the lack of a feedback-based switching strategy could lead to improper operation.

In practical applications of process systems, logic elements have been extensively used for reconfiguring control loops, which is often needed for attaining optimal operation (Reyes-Lúa et al., 2018). In particular, selectors have been successfully applied as a tool for automatic detection of active constraint switching for single input systems (Krishnamoorthy and Skogestad, 2020). In this work, we extend this analysis for multivariable systems, proposing a simple framework for decentralized optimal operation under changing active constraints, with the use of PID controllers and selectors. This shifts a bigger part of the RTO problem to the control layer, since the switching is to be optimally attained with the selector blocks.

The main idea in the framework proposal is to design a generic decentralized control structure over every possible active constraint region of the steady-state optimization problem, where the control objectives can be locally changed to account for region switches. It is proposed that the logic for switching between controllers can be performed using selectors, such that the control structure is kept simple. If we assume that the number of constraints is smaller than the number of available degrees of freedom, and the gradient matrix of the constraints with relation to the inputs is always full row rank, it can be proven that we can propose a decentralized set of CVs as described in Figure 1. The proof of optimality follows similar steps to the one in Jäschke and Skogestad (2012), but we proceed to specify an ideal pairing between CVs such that the handling of region switching is done in a decentralized fashion, avoiding changes in the rest of the control structure. The control structure resulting of this proposal is illustrated in Figure 2.

The proposed framework is applied to a simulated case of optimal operation of the Williams-Otto reactor (Williams and Otto, 1960), which has two degrees of freedom and two constraints. The active constraints change according to the disturbance values as shown in Figure 3. The simulations were performed in CasADi (Andersson et al., 2019), which allowed for the calculation of analytic gradients through automatic differentiation. The projection matrices that result from the framework, which depend on the gradient of the constraints, are assumed constant, resulting in a simple control structure. In the closed-loop simulations, the system was disturbed through all possible active constraint regions, in order to check for steady-state economic performance, and the corresponding results are shown in Figure 4. Due to the assumption of the cost gradient being a directly available measurement, operation of the system at the fully unconstrained region was verified to be optimal at steady state. In addition, as the constraints are also measured, the same logic can be applied to the operation at the fully constrained region. In the remaining partly constrained regions, the system did not converge exactly to the steady-state optimum but operated with low economic loss, which is attributed to the use of constant projection matrices. The switches in control objectives are automatically performed by the selectors, preventing constraint violation, and controlling the unconstrained degrees of freedom when the corresponding constraints are not violated.

The control structure resulting from the framework proposed in this work makes use of selectors as simple elements for switching between operating regions. These elements are frequently used in practice for coordinating conflicting control objectives. We show in this work how these elements can be used for the optimal operation of systems under changing active constraints in a systematic manner. The min/max nature of the selectors is ultimately related to the nature of the constraint with respect to the input (Krishnamoorthy and Skogestad, 2020).

The approach proposed in this work is based on the analysis of the partly constrained regions, where we find gradient projections that can be optimally controlled. These projected gradients are also the optimal CVs of other regions, which minimizes the number of control loops that are necessary to account for all regions. Moreover, the proposed switching operates independently for each plant input, which means that the detection of each constraint is done independently, and feasible operation is safely achieved. However, the reconfiguring of CVs done by the selectors may significantly change the interactions between loops, and therefore careful tuning of the controllers is necessary, such that a good performance is achieved regardless of which loops are active.

In the present work, we propose a framework for decentralized optimal operation under changing active constraints, applicable to a class of multivariable problems. Even though the approach is based on the linearization of the constraints, and therefore the quality of the linearization plays a relevant role in the economic performance, the strategy proved to be successful in a nonlinear case study, which encourages its use in other relevant problems of process systems engineering, especially when the gains from the inputs to the constraints do not change greatly in the operating range. The use of adaptive cost gradient projections would also be beneficial for improving economic performance.

**References**

J.A. Andersson, J. Gillis, G. Horn, J.B. Rawlings, M. Diehl, 2019. CasADi: a software framework for nonlinear optimization and optimal control. Mathematical Programming Computation, 11, pp.1-36.

J. JÃ¤schke, Y. Cao, V. Kariwala, 2017. Self-optimizing controlâ€“A survey. Annual Reviews in Control, 43, pp.199-223.

J. JÃ¤schke, S. Skogestad, 2012. Optimal controlled variables for polynomial systems. Journal of Process Control, 22, 1, pp.167-179.

D. Krishnamoorthy, E. Jahanshahi, S. Skogestad, 2018. Feedback real-time optimization strategy using a novel steady-state gradient estimate and transient measurements. Industrial & Engineering Chemistry Research, 58, 1, pp.207-216.

D. Krishnamoorthy, S. Skogestad, 2020. Systematic design of active constraint switching using selectors. Computers & Chemical Engineering, 143, p.107106.

A. Reyes-LÃºa, C. ZoticÄƒ, S. Skogestad, 2018. Optimal operation with changing active constraint regions using classical advanced control. IFAC-PapersOnLine, 51, 18, pp.440-445.

S. Skogestad, 2000. Plantwide control: The search for the self-optimizing control structure. Journal of process control, 10, 5, pp.487-507.

T.J. Williams, R.E. Otto, 1960. A generalized chemical processing model for the investigation of computer control. Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics, 79, 5, pp.458-473.