Processes 2023 , 11 , 809
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the Kappa number and cellulose DP and obtain optimal input profiles for appropriate control actions. The controller solves an inverse problem by identifying an input profile that provides the closest output values to the desired set-point value. These inputs can be both constant or time-varying. These inputs are then applied to the first-principles process model, and the optimization problem is solved again for the next set of time instants in a receding horizon fashion. The LSTM-ANN-based feedback control system is designed to regulate both the Kappa number and cellulose DP . The control framework includes an objective function and process constraints as illustrated below:
2 + DP f − DP sp
2
min T , τ , ρ
(16)
κ f − κ sp
s.t. T ( t ) , τ ( t ) , ρ ( t )= 0 for t > τ
(17) (18) (19) (20) (21) (22) (23) (24)
T min ≤ T ( t ) ≤ T max τ min ≤ τ ( t ) ≤ τ max ρ min ≤ ρ ( t ) ≤ ρ max τ ( t )= τ ( t + 1 )
ρ ( t )= ρ ( t + 1 )
t = 0, . . . , τ
{ κ ( t + 1 ) , DP ( t + 1 ) } = f LSTM
T ( t ) , τ ( t ) , ρ ( t ) , κ meas ( t ) , DP meas ( t ))
− ANN (
where κ is the Kappa number, DP is the cellulose degree of polymerization, T is the free- liquor temperature, ρ is the NaOH concentration, and τ is the cooking time of the pulping process. The desired Kappa number is κ sp , and the final obtained Kappa number after the pulping process is κ f . Similarly, the desired cellulose DP is DP sp , and the obtained DP is DP f . As per Equation (17), all three inputs (cooking time, concentration, and temperature) are set to zero after the cooking time is exceeded. The upper and lower bounds of the inputs are described in Equations (18)–(20). Equations (21) and (22) indicate that the cooking time and NaOH concentration remain constant throughout the pulping process. The LSTM-ANN model, as represented in Equation (24), runs at every iteration, calculating the Kappa number and cellulose DP , and optimizing the input profiles. As shown in Equation (24), we include the dependency of the Kappa number and DP obtained from the multiscale model at time t to predict the outputs at the next time step t + 1. Here, we use the inherent nature of LSTM models to predict the future states when we have the true measurement values of the past from the process, denoted by κ meas and DP meas [58–60]. Thus, the input to the LSTM-ANN model includes the current measurement values from the process (i.e., the multiscale model) along with the manipulated inputs. The MPC then solves the optimization problem and obtains the input profiles for the next time instants to apply to the multiscale model. After applying the first input values to the multiscale model, the measurement values of the Kappa number and DP are obtained for the next instant, which is then fed to the LSTM-ANN model along with the other three inputs. This ensures closed-loop operation of the process. The results of this control system will be discussed in the next section. Remark1. The accuracy of the control framework is heavily dependent on the variety and quality of the data used to train the model. Therefore, it is essential to run the model with different sets of input profiles. Additionally, the accuracy of the controller is also highly dependent on the optimizer chosen to solve the optimization problem.
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