Linearization techniques

2.12.1 Local Linearization

Linearization methods are categorised into either local or global. Local techniques linearize around a defined operating point and are commonly used because they can be implemented with minimal effort (Hartman, 1963). However, the main drawback for this technique is a reduced accuracy in predicting the behaviour of a nonlinear system at conditions that are far from the operating point used for linearization. This drawback can also be accentuated by the degree of nonlinearity in the dynamics, i.e. the more nonlinear the system dynamics the more

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inaccurate the prediction will be. This can be detrimental to process control given the control decisions are made using the linearized system. Fortunately, global linearization techniques exist which can alleviate the concerns of using a linear model for controlling nonlinear processes.

2.12.2 Global Linearization

Global linearization methods were first applied onto nonlinear systems by Gilbert and Ha (1984). They considered state variable transformations using a state feedback control law. The method transforms a nonlinear input-state process into a linear input-state model. The terminology used here, nonlinear input-state, simply implies the relationship of the process input to states is nonlinear. However, there is a drawback in that a linear input-state model does not guarantee reliable output prediction because the state-output system may also be nonlinear. This drawback led to an alternative feedback linearization framework developed by Kravaris and Chung (1987) for single-input single-output (SISO) systems by considering the input-output. The aim was to develop a framework for global linearization for solving control problems. Applying this linearization to a nonlinear system, a new linear system can be identified as shown in Figure 2-4 (Mesbah et al., 2010). The state feedback linearization (SFL) block contains a control law which uses the states of the nonlinear plant, 𝑥, with the input to the linear system, 𝑣, to obtain the input to the plant, 𝑢. The plant may also be affected by disturbances or uncertainties (𝑑) and the output of the plant is 𝑦. The SFL and nonlinear plant together create a linear system, which can be controlled by MPC. However, the SFL is the states of the plant and the inputs to the plant are within the linear system and are not intuitively accessed directly by the MPC.

Figure 2-4 Linearization schematic of a nonlinear plant using state-feedback linearization

The output from the nonlinear system is mapped to the MPC input 𝑣 using the states and the plant input 𝑢 using Lie derivatives. The transformation means the states are not accessible in

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the MPC because they are inside the linear system, and this is the system that the MPC would control. Therefore, the MPC would find the changes in 𝑣 required to converge output 𝑦 to a reference trajectory.

This linearization technique has been applied to the batch crystallization model and MPC performance has been evaluated by Jansen (2011) and Vissers, Jansen and Weiland, (2011). The output of the system was supersaturation, and the input to the plant was the temperature of the coolant in the jacket. They evaluated the performance of the controller with a growth parameter mismatch and uncertainty between the MPC and process model which mimicked the crystallization system, to establish if MPC was sufficiently robust to control the process. The results show that the MPC is able converge the supersaturation trajectory onto the reference throughout the simulation, but the input profile varies significantly based on the growth parameter and uncertainty. Furthermore, there was an unsuccessful attempt made to calculate constraints on feedback linearization input, 𝑣. For multiple-input multiple-output control with linearization, there have been implementations of MIMO control with SFL outside pharmaceutical crystallization, namely in proton exchange membrane fuel cells (Chang and Chen, 2014) resulting in the successful proportional-integral control of the process. Similarly, there is a MIMO implemented on MSMPR crystallization but again using a PID controller in place of an MPC (Quintana- hernandez et al. , 2012), the control objectives were to stabilize the third moment and crystallization temperature, and both were successfully achieved with the model-free controller. This shows the possibility of extending the SFL to a MIMO system for crystallization.

2.12.3 Methods for Applying Constraints to Global Linearization

If a global linearization is to be used for MPC, an important requirement is the ability to translate the real process constraints to the MPC, so as to not lose an important function of MPC, the ability to accurately model and predict feasible future input moves for a process. In literature, a few successful attempts have been made on applying constraints to a feedback linearization optimization problem. Two techniques introduced by Kurtz and Henson (1996) and further detailed by Kurtz and Henson (1998) discuss how the SFL control law can be used to identify the state-dependent constraints at each interval that the MPC is executed. In the

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first technique, named constant constraint technique (CCT), the constraints are calculated for the next immediate MPC input, 𝑣, using the most recent states, 𝒙,and these constraints are applied on 𝑣 across the entire horizon. The second technique named variable constraint technique (VCT) uses the horizon of 𝑣 from the previous time the MPC was executed to estimate the future states, 𝒙. These future states make it possible to calculate the future inputs into the process (𝑢), using the SFL control law. In both techniques, the constraints on the first input in the horizon will be correct and given that this is the only input from the MPC that is implemented onto the plant, this trade-off has been accepted knowing that the horizon prediction may not be accurate or reliable. However, this is a major drawback for MPC because the important characteristic of MPC is the ability to predict how the trajectory will evolve over the future horizon. Therefore, infeasible horizon prediction is arguably of little to no use.

Kurtz and Henson (1996) also disclose that the method to ensure exact constraints would be to create a constraint calculation strategy which uses the nonlinear system’s real states but this route wasn’t chosen by the authors as it would be relatively computationally inefficient compared to even nonlinear MPC, hindering the real-time capabilities. This claim can now be challenged with advances in computational efficiency, as it may now be possible to form a more computationally demanding constraints calculation strategy whilst also maintaining the ability for real-time operation.

Further advances have since been made by Van Soest, Chu and Mulder (2006) in using a similar technique to CCT and VCT, using the constraint techniques on a MIMO SFL with decoupling. Deng, Becerra and Stobart (2009) also explored the capabilities of constraint implementation using artificial neural networks (ANN), but the technique appears computationally demanding and requires large training sets of data to train the ANN, even then the constraint selection was not reliable. Furthermore, this application was in the aerospace domain and the viability of using ANNs in pharmaceutical crystallization control would require new datasets to be formed.

The most recent advance by Schnelle and Eberhard (2015) introduces a constraint mapping technique for SFL again based on VCT but they introduce the use of recursively calculating the future plant states and using them to dynamically update the VCT constraints technique with

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the constraints on the MPC input. The advantage of this technique is that the constraints will be valid throughout the entire horizon, which means the prediction horizon will be the most accurate of all the previous techniques discussed. However, this method could be improved upon if there was a method to directly constrain the plant states and inputs without the need to convert them into the MPC input constraints.

One solution to implementing plant constraints across the entire prediction horizon could lie in the use of the Sequential Quadratic Programming (SQP) algorithm in optimization. Boggs and Tolle, (1995) introduce the SQP algorithm and state that SQP has the ability to solve nonlinear control problems with nonlinear constraints by using an iterative approach which is capable of converging a feasible optimal solution. This optimization algorithm therefore presents an opportunity to address the problem of handling real plant constraints, by coupling the SFL transformation with the nonlinear state space system into nonlinear constraints function. The SQP algorithm can then be employed to solve the control problem on the state feedback linearized system subject to the nonlinear constraints function.

In summary, this method for linearization poses some complications with bounding and constraints on the real inputs to the nonlinear system, but there is sufficient motivation to explore this area and find a solution. This form of linearization would make it possible to use MPC with a globally linearized model for crystallization and the offline linearization method would only need to be performed once. This is advantageous over local linearization around one or multiple operating points. The ability to linearize a model over its whole design space is an appealing idea for control application.