1 Introduction
Plantwide Process Control (PWC) system design is an ongoing research field. In only a few words, this field treats the whole process at once instead of as a collection of individual unit operations. The advantage of this approach is that interactions between process units are considered since the very first analysis, enabling a better decisionmaking through the control design.
A major issue is that real plants are often nonlinear and can have a massive amount of inputs and outputs, i.e., an oil refinery, with thousands of variables involved in highly nonlinear phenomena. Such a high complexity can dim the benefits of linear control methods. First off, obtaining a linear model around an operative point can be computationally costly. Even after getting such a model, there is no guarantee that the model will remain representative of the real operation of the industrial site, since in practice is rather impossible to keep all variables at a specific point. Additionally, some linear representations are strongly illconditioned in the frequency range of interest. As a consequence, most of the analysis and design methods that are usually accepted in the literature are no longer suitable nor trustworthy. These characteristics suggest that nonlinear concepts may be useful to the control design process, at least to some extent.
The methods from PWC are usually presented as a series of steps, from the degreeoffreedom assessment of the plant up to the controller tuning
[6, 7]. Some of these steps aim to gain insight into the plant, such that the final control structure is at least reasonable for the entire plant. In this regard, a key decision for the control structure is the selection of the controlled variables (CV). These are the variables that are fed back into the control system, in order to keep them around desired setpoint values. This work proposes a new kind of application of a chaos characterization to suggest which variables might be critical to control.2 Chaos as a ControlSystem Feature
In the context of nonlinear dynamics, chaos is a phenomenon by which the system output is extremely sensitive to initial conditions [3]. This leads to unpredictability of the system output, an essential feature by which chaos is commonly recognized.
Chaos must be distinguished from random behavior, since chaotic systems are deterministic, not stochastic. The unpredictability of chaotic phenomena is deeply rooted in the relations of the system variables. These interactions amplify any infinitesimal difference between two different initial conditions to produce two completely different system outputs in the long run.
Chaotic systems are a subset of nonlinear systems, meaning that only nonlinear systems are able to present chaotic behavior. What is more, this feature is usually found in strongly nonlinear systems. Some examples are atmospheric dynamics, turbulent flow, buckled elastic structures, mechanical systems with play or backlash, and feedback control devices [3].
As one might expect, it is always better to avoid chaotic modes in controls systems, especially in the context of PWC design. However, this is nearly an impossible task for high dimensional systems, as there is no way to recognize beforehand which variables are chaotic. A suitable workaround is to change the design procedure into one that tackles the most complex behavior of the plant in the first place.
2.1 On Stability and Chaos
Another reason to consider chaos theory in control systems is the close relation between chaos and stability, a fundamental property of control systems theory.
Consider the Lyapunov stability of dynamical systems, which establish that an equilibrium point is stable if solutions that initialize near from remain in a neighborhood of at all times. This concept is commonly used in optimal control and Model Predictive Control, to name a few [8]. This concept of stability will be used throughout the remainder of this work.
In chaos theory, the Lyapunov exponent measures the sensitivity of a given dynamical system to small changes in initial conditions. A natural link between the Lyapunov exponent and stability is given by the sign of the exponent . Whenever the trajectories of solutions starting nearby will eventually diverge, revealing instability in the Lyapunov sense. More precisely, let be an initial condition, a arbitrarily small perturbation in , and the euclidean distance at time between the trajectories starting from and from . With these definitions, the Lyapunov exponent is given by:
(1) 
Thus, at some time the separation of the trajectories is , where . Hence chaotic systems have , while stable nonchaotic systems have . The Fig. 1 shows a typical dissociation of dynamic trajectories for the Lorenz’s model, one of the first studied in the literature.
Typically, a chaotic system will have at least an equilibrium point that is unstable. Therefore, chaoticity implies instability, at least in a local manner. Whenever the plant operative conditions are nearby an unstable equilibrium point, the trajectory complexity will increase arbitrarily. This fact constitutes the cornerstone that supports this exploration.
2.1.1 Estimation of Chaotic Signals
For small and noisy data sets, the verification of chaos behavior is not trivial, let alone the computation of Lyapunov exponents. In either assessment, random variations interfere with the estimation process. Consequently, tests for chaos are scarce in the literature. Recently, a datadriven approach proposed by BenSaïda
[2] admits noisy signals. This is possible due to the remarkable advantage of distinguishing between chaos and randomness. The algorithm to detect chaos in signals has two stages. Primarily, the code computes the largest Lyapunov exponentby using a Jacobian optimization method. Then, a hypothesis test is carried out to assess the presence of chaos in a probabilistic sense. This onetailed test has a null hypothesis given by
, which indicates the presence of chaos. The significance level can be set up to any desired level. The main outputs of the code are the exponent and the value. A small value of casts doubt on the validity of the null hypothesis of chaos.Internally, the algorithm follows a theoretical formulation given by Eckmann and Ruelle [11]. Departing from a scalar time series , a noisy chaotic system can be written as:
(2) 
where represents the added noise, and is a time delay to allow the possibility of skipping samples in the algorithm. The parameter is the embedding dimension or the length of past dependence. Moreover, the system dynamics is unknown, so it is the following mapping:
(3) 
The Jacobian of the former mapping offers a consistent estimation of Lyapunov exponents, even in the presence of noise. However, an approximation of the mapping is required in the first place. Neural networks are suitable for this purpose, since they can approximate any smooth, nonlinear function to any desired accuracy, whenever the complexity of the network is appropriate [9].
A relatively simple neural network architecture is enough for this application, with an mdimensional input layer, a qdimensional hidden layer, and an output layer, which is shown in Fig. 2
. The activation function
governs the propagation of the signal to the next layer. The hyperbolic tangent can handle numerical problems better than other commonly used activation functions, being so a natural choice for this work hereafter.The neural network ”learns” from the inputs and outputs of the chaotic map by adjusting its internal coefficients . Eventually, the network becomes a reasonable approximation of . For this to happen, the noise should be removed from the data. Consequently, an appropriate estimation algorithm is the nonlinear least squares, which minimizes the noise of the system given by , where
represents the coefficients vector that define the neural network.
The triplet accounts for the complexity of the neural network, thus defining the mapping approximation. The time delay should not be too large, since it would cause loss of information from the signal. Conversely, a small value of interferes with the distinction between noise and signal. Additionally, the embedding dimension
should be large enough to capture the chaotic dynamics. A method to derive this parameter is by setting to the largest eigenvalue of
, where is the lagged matrix associated with the time series . Any value of the triplet can lead to unnecessary overhead in the numerical computation if chosen too large.From the mapping it is possible to estimate the Lyapunov exponent through the expression
(4) 
where is an integer associated with the evaluation points, , and stands for the largest eigenvalue of the matrix , with and
(5) 
Additionally, a practical procedure to confirm chaos with high probability is to retrieve positive
for different choices of the triplet . Consequently, a useful approach begins by set a bound on each parameter , then perform the nonlinear least square estimation for different combinations of parameters. As a result, if there is an occurrence of chaos, there will be a set of positive values of . In this case, the final choice is the largest value.Once the Lyapunov exponent is estimated, a hypothesis test is performed to assess the confidence level of the prediction. This is an important step since the noise is not completely removed in the estimation of
, hence there is some degree of uncertainty in the mapping, especially if the signal to noise ratio is low. The test is based on distributional results for the Lyapunov exponent. Such results rely on the hypothesis of
bounded, its Jacobian is also bounded, the noise variablesare independent and identically distributed, and their probability distribution function is bounded. These conditions yield an asymptotically normal distribution through the central limit theorem.
The above describes the basic mechanics of the estimation of Lyapunov exponents. This method was confirmed with several sets of data, both chaotic and stochastic. For additional details refer to [10].
3 The Tennessee Eastman plant
This PWC problem was originally proposed by engineers of the Eastman company as a plantwide control problem challenge for the academic community [13]. It is a petrochemical plant given in the form of an algorithm, so mathematical expressions to describe the system are not available. Additional properties are nonlinearity, strong interaction between variables, openloop instability, and hard operative restrictions over process conditions. The system presents 40 inputs, 50 states, and 41 outputs. The inputs 1 to 12 are related to valves and actuators that it is possible to manipulate to control the plant, while the rest of the inputs represent specific perturbations that any desirable control system should be able to handle. Considering the above features, it is not evident how to choose the CV solely based on process engineering considerations.
The Fig. 3 shows a process flow diagram for the plant. The process involves different units: chemical reactor, condenser, separator, compressor, and stripper. Additionally, it has a recycle stream to increase reactant use efficiency. The process is fed with four gas streams, which introduce the reactants A, C, D, E and a small amount of noncondensible component B. Four simultaneous reactions occur at the reactor, all exothermic. They produce two products of interest G and H, and a byproduct F. The products are taken from the bottom of the stripper. There is another output stream at the top of the separator which serves as a purge.
Each processing unit has its own characteristic dynamics. Particularly, the reactor is openloop unstable, and contains both liquid and vapor phases. The interconnections between the units regarding material streams, momentum transfer, and internal energy, produce a complex behavior for the whole plant. Traditional control system design procedures, based on assembling control subsystems for every unit might fail for complex plants like this one. This is one of the main reasons why plantwide process control is an active research area up to this date.
Even some modern approaches may either fail or underperform against this problem. A distinctive feature of the TE plant is that the linear model is strongly illconditioned, meaning that the condition number of the transfer function matrix is extremely high at every frequency of interest. As result, the use of control design methods for selecting CV based on the linear model can be misleading due to numerical issues. An extensive discussion, analysis, and control system design for this plant is found in my Ph.D. thesis [12].
4 Discussion and Results
Results of the chaos test for the Tennessee Eastman plant output signals are shown in Table 1. The first column reveals the identification symbol for the signal, sorted from 1 to 41, while the second column has designations that resemble the type and location of each variable. For instance, TR is the reactor temperature. The third and the fourth columns present the estimated Lyapunov exponent and value respectively. The first conclusion from this column is that 27 of the output signals present chaotic behavior. However, the degree of certainty varies across the different variables. Notice that some of the Lyapunov exponents are negative. This suggests that in principle, the associated variables would not be chaotic. However, they appear in the table since the value is relatively small, so there is not enough confidence to reject the null hypothesis . On the other hand, some signals have both positive Lyapunov exponents and values close to unity, confirming a chaotic behavior for those signals.
For the particular case of the Tennessee Eastman plant, it is worth mentioning that the confidence in the chaoticity may vary across variables with and . The reason being a combination of two facts. Firstly, there is a limit in the signal length that is achievable from the openloop simulation of the plant, since it is programmed to stop whenever any critical variable surpasses any operative limit. Secondly, not all variables have the same sampling time. The signals with are related to chemical compositions, so they have a higher sampling time compared with the rest of the signals. Consequently, while there are over 600 measures for every output signal , with there are only 11 different numerical values for compositions from to , and 5 different values from to . This leads to less information on the dynamic evolution of the compositions, so conclusions obtained for these signals could be biased by the low volume of actual data acquired. Nonetheless, this fact will be neglected for the remainder of this section, as in real industrial plants there is no shortage of data in general.
Symbol  Abbrev.  Symbol.  Abbrev.  

F2D  4.7226  0.1569  TER  4.7226  0.1150  
F3E  4.7226  0.1569  TEC  4.7226  0.0980  
PR  4.7226  0.1569  CBF6R  0.0081  0.6802  
NR  4.7226  0.0963  CDF6R  0.0014  0.5757  
TR  4.7226  0.1306  CFF6R  0.0014  0.5625  
FP  0.1029  1.0000  CDP  0.2485  0.9609  
TS  4.7226  0.1004  CFP  0.0068  0.7721  
NS  4.7226  0.0558  CGP  0.0150  0.9219  
PS  4.7226  0.1569  CHP  0.0083  0.8271  
ND  4.7226  0.0572  CDFLD  0.0037  0.6589  
PD  4.7226  0.1569  CEFLD  0.0256  0.9990  
TD  4.7226  0.0826  CFFLD  0.3692  1.0000  
FVD  4.7226  0.1478  CGFLD  4.7226  0.0644  
PC  4.7226  0.1551  

All in all, the former results confirm that the plant is a chaotic system, since the variables FP and CFFLD are chaotic almost surely, as shown in Table 1. In fact, most of the process output variables exhibit chaotic behavior to some degree, as it is not possible to reject the null hypothesis for 27 of the 41 signals.
The characterization of chaos enables a simple procedure to prioritize which variables should be controlled, since for real multivariable systems it is not generally possible, nor desirable, to control every output variable. To perform a reasonable selection, the criterion suggested in this work is to control those variables that are more unpredictable from a chaos theory standpoint. It is expected that this selection will increase the stability and performance of the final control system in a natural manner, since the technique aims to reduce the complexity of the overall system. To this objective, a hierarchical list of potential CV is computed. The list follows from sorting the algorithm output in descending order of the product . This product represents a compromise between the degree of chaotic behavior and the certainty about it. Notice that this sorting automatically sets the variables with above those with . However, it is advisable that only variables with should be selected from this list.
It is possible that some processes show no results with , yet there are still variables for which the null hypothesis can not be rejected. In this case, an alternative approach is listing the variables in ascending order of values. This way, the selected CV will be those for which is less probable than . What is more, a combination of the two former criteria can be performed by selecting the first variables from the set of with large values, and then complementing the list with the set of with minimum values.
Symbol  Abbrev.  

CFFLD  0.36928  1.0000  0.36928  
CDP  0.24857  0.90840  0.22580  
FP  0.10293  1.00000  0.10293  
CEFLD  0.02569  0.99908  0.02566  
CGP  0.01500  0.91616  0.01374  
CHP  0.00833  0.82718  0.00689  
CBF6R  0.00815  0.69181  0.00563  
CFP  0.00689  0.77210  0.00532  
CDFLD  0.00377  0.65895  0.00248  
CDF6R  0.00145  0.57573  0.00084  
CFF6R  0.00148  0.56257  0.00083 
Now, recalling that the Tennessee Eastman plant has up to twelve manipulated variables, one might be interested in selecting a similar amount of output variables to control. To reduce the list, variables with either or are filtered out, although this step is not mandatory. According to the above criteria, the set of selected CV is listed in Table 2. According to these results, CFFLD is the first CV to be selected, followed by CDP, FP, and so forth. Notice that variables with higher should be associated with more robust control loops. Consequently, the resulting hierarchy of variables should impact on the controller tuning. In extreme cases, robust control methods should be used for these variables.
5 Conclusions
The former study shows potential in the characterization of chaotic signals for the selection of controlled variables in a control system design problem. This criterion is especially useful for unstable, highly multivariable plants with complex dynamics. These kinds of features arise naturally in industrial plantwide processes.
The criteria proposed can be adapted to a wide variety of levels of outputunpredictability. It is also very simple to compute if the recommended algorithm is used. Additionally, it is a datadriven technique, meaning that it relies uniquely on openloop data from the process, regardless of how complex it is. However, because of the novelty of this work, it is advisable that these criteria should be complemented with additional suitable process considerations, that will be strongly dependent on the process under study.
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