3.1. Control Based on Distributed Models
As has already been stated, tomographic sensors enable distributed sensing. As a result, it is pretty natural to combine them with control approaches that exploit the distributed parameters nature of the controlled process. For this reason, we will start this treatment with this class of approaches. A very important category of processes with distributed parameters is the processes that include single-phase or multiphase flows. Applications of tomography-based control to flow processes have been reported since the very beginnings of industrial tomography.
3.1.1. Concentration Distribution Control
A very basic control task related to flow control is the control of the concentration distribution of a substance in a fluid flow along a straight pipe with a circular cross-section. The concentration profile is controlled by injecting strong concentrate into the flow through one or more injectors. Their flow rates are considered manipulated variables. The control action is based on concentration measurements performed by an electrical impedance tomography system located downstream from the injection point. The structure of this process is shown in
Figure 3.
From one viewpoint, this is a relatively simple, specific process. However, as far as the structure of its mathematical model is concerned, this process reflects the general nature of many fluid flow processes, including liquid/liquid and solid/liquid mixing. Thus, it can be viewed as a paradigmatic controlled process for tomography-based control. For this reason, its control was addressed in many papers, most notably in several papers and a PhD thesis written mainly by S. Duncan, A. Kaasinen, and their co-authors [
46,
47,
48,
49,
50,
51]. Therefore, we will describe this process in some detail. A mathematical model of this process has been given several times in slightly different variants in the papers cited above. The following equations are based mainly on [
51].
The concentration of the injected substance can be described by a PDE model—convection–diffusion equation
where
ξ is the vector of space coordinates within the domain of interest Ω (finite segment of a pipe). This domain is generally three-dimensional, but it can be two-dimensional if it is a straight pipe with circular symmetry. The concentration distribution is denoted as
c(
ξ,
t) where
ξ ∈ Ω and
t ≥ 0,
v(
ξ) is the velocity field and
κ(
ξ) is the diffusion coefficient. The source term
q(
x,
t) describes the injection of substance B into the flow of substance A. If there are multiple injectors, this term can be expressed as
where
u(
t) is the vector of manipulated variables and
Λ is a linear map that depends on the geometry of the injection and the number of injectors
K.
Initial and boundary conditions can be specified as follows
Here cin(t) is the time-varying concentration at the input boundary, i.e., it is generally assumed that even the fluid entering the pipe can contain a fraction of the substance injected by the injectors. This variable is a disturbance from the viewpoint of feedback control.
PDE model is further space-discretized to obtain a finite-dimensional model. This can be done using FEM (e.g., [
47,
48,
49,
50,
51,
52]) or finite difference approximation in the direction of the pipe centerline [
49,
50]. Regardless of the method used, the result is a high order state-space model that can be written in a time-discretized form as
where
c(
k) is the space and time-discretized concentration, which plays the role of a state vector,
u(
k) is the vector of manipulated variables defined in Equation (9), disturbance
d(
k) is input concentration defined in Equation (10), and
w(
k) is a stochastic process representing modeling uncertainties, random components of input signals and other stochastic influences. Assuming that the velocity field and the diffusion coefficient in Equation (8) are not time-varying, the matrices in Equation (13) are constant, i.e., they describe a linear time-invariant finite-dimensional system. It can be used instead of the PDE model (8) in control design. That means that this approach is a sort of early lumping design.
Concentrations are measured using electrical impedance tomography. Depending on the properties of the fluids, either conductivity or permittivity measurements (i.e., electrical resistance or capacitance tomography) are used. Conductivity measurements (i.e., a conductive fluid consisting of components with different conductivities) will be assumed here. However, the main principles would be the same if permittivity measurements were considered.
Electrical tomography does not provide simple and direct information about concentrations. The available measurements are voltages between different pairs of electrodes produced by varying current injection patterns (
j = 1, …,
NI) applied to the electrodes. These voltages depend on the conductivity distribution
σ within the pipe, and this conductivity distribution depends on the concentration of injected substance
c. Voltage measurements for one current injection pattern can be expressed by a compound formula
where
Vj are the voltage measurements, and
Uj maps the conductivity to the measured voltages. Vector
vj represents additive measurement noise. If the controlled process is slow enough, the conductivities can be considered constant during the whole frame, where the full set of all current injection patterns is used. In this case, all measurements and conductivity mappings can be combined into one vector and one mapping [
52]. Equation (9) can then be written as
All variables have the same discrete time, i.e., the time necessary for performing a full set of current injections must be significantly shorter than the controller sampling period. However, these stationarity assumptions are often not satisfied in the real-time control. This non-stationarity can be handled in several ways. It is possible to use a substantially reduced set of current injection patterns or even one pattern optimized in such a way that it provides as much information as possible [
53]. The observation equation has then the same form as Equation (15), but it uses just one measurement or a small number of measurements. As a result of this, the concentrations must be computed on the basis of a smaller amount of data. Alternatively, this observation equation can be considered time-varying where the mappings are different each discrete time
k depending on which current injection pattern
j is used.
Equation (15) can be kept in its nonlinear form, or it can be linearized and written in deviation variables around an appropriately selected operating point
where
J is the Jacobian of the function
U*(
c(
k)) evaluated at the operating point
c0.
Equations (13) and (16) or (15) are then of the same form as Equation (6) or (7). They can be used to design a state estimator using either a linear version of the Kalman filter or some of its nonlinear versions, e.g., an extended Kalman filter. State estimation was originally proposed as an approach to solving the inverse problem and obtaining the concentration (or other) data from voltage measurements where the dynamic Equation (13) was used in order to bring additional information and make the whole procedure feasible, especially in the non-stationary cases, where observation Equation (15) includes just a limited amount of data.
This means the original motivation was mainly data reconstruction. However, this setting is very suitable for control. Equations (13) and (15) or (16) provide a good starting point for control methods based on state-space models. It is just necessary to consider that although observation Equations (15) and (16) can be regarded as output equations of a state-space model, they provide measured outputs that are not controlled variables. The controlled variables are concentrations at a selected region. Most naturally, the output boundary can be considered. This can be described by an additional output equation
Matrix
C can be zeros and ones matrix that selects just the concentrations in the region of interest as controlled outputs. This form of output equation can also cover other cases of control objectives, e.g., an average concentration in a chosen region of interest can be the output variable.
We described this concentration control process in some detail because it corresponds well with the fundamental characteristics of tomographic sensors. Corresponding to the distributed sensing achievable by tomography, there is a distributed parameter model of the process. This model is handled using the early lumping approach, i.e., it is converted into a high order state-space model, which can then be used as a starting point for several controller design approaches.
Early papers by S. Duncan [
49,
50] used PI controller. This controller is suitable for SISO systems. In accordance with this, only one injector was used, and the average concentration at a particular pipe point was the controlled variable. The control was tested in simulation only. Control performance was rather poor, marked by oscillatory responses to set-point changes. These oscillations were caused by the significant delay because of the flow from the injector to the measuring point. It is well known from the theory of time-delay systems that these oscillations can be eliminated either by decreasing the proportional gain or by adding the Smith predictor structure to the PI controller [
54]. However, neither of these approaches was used in [
28,
29].
Since a state-space model is available, it is possible to use advanced model-based controllers instead of PI control. This was done in several papers [
46,
47,
48,
51]. The control approach of choice was the Linear Quadratic (LQ) controller. This controller is based on quadratically optimal state feedback in the form
where
x(
k) is a state vector, i.e., it is the vector of concentrations
c(
k) if model (8) is the controlled plant.
K(
k) is an optimal feedback gain matrix calculated using a discrete-time matrix Riccati equation [
55]. Equations (13), (16), and (17) provide a standard setting for the design and straightforward application of LQ controller. It was only necessary to take into account that the LQ controller in its basic form is a regulator, i.e., its objective is to drive states and system output to zero. In this case, it is not an appropriate objective because the control objective is to obtain a specified nonzero reference concentration
yref at the output boundary. For this reason, the structure of the controller had to be changed to
where
and
are steady-state values corresponding to the state, where the reference concentration
yref is achieved. This modification introduces feedforward action into the controller. Set-point enters the controller indirectly as a variable on the basis of which the steady-state values are computed. The controller described by Equation (19) is used by all of the papers [
46,
47,
48,
51]. The state vector (concentrations), which is input to the controller (19), is estimated either using a Kalman filter based on a linearized observation equation [
46] or using an extended Kalman filter based on a nonlinear observation equation [
51].
This control application of tomography may seem to be a fairly nice example of the early lumping approach applied to a control of a distributed parameter system connected with distributed sensing. It includes a well-established mathematical model as well as an advanced control method. Unfortunately, this concept of tomography-based control may also be a blind alley. The controlled process considered here is relatively simple, and it was the subject of several papers published over more than one decade. Nevertheless, this concept has never gone beyond simulation testing. None of the papers gives more than simulated responses. Moreover, it also seems that this research line found only a very limited continuation in the last decade.
3.1.2. Microwave Drying Process
Looking to the development following 2013 when [
51] was published, almost no papers follow this concept and attempt to transfer it into other controlled processes. The most notable exception is the paper by Hosseini et al. [
56] published in 2020. This paper is focused on the control of a microwave drying process. In microwave drying, a porous dielectric material with high moisture content is placed inside a cavity and exposed to microwave sources (magnetrons). In paper [
56], the control objective is specified in the following way: “The objective is to reach as homogeneous moisture distribution as possible inside the porous material and the average moisture of the material should follow the desired moisture level.” Power levels of microwave sources are used as manipulated variables. The output moisture is measured using electrical capacitance tomography, i.e., using a sensor whose behavior is similar to the sensor described by Equations (14) to (16) but with permittivity instead of conductivity.
Industrial tomography is the key enabling technology for feedback control of this process because it is the only technology that can measure the moisture distribution inside the material. Alternative sensor technologies can provide only surface-related information which is not enough for efficient control. For this reason, paper [
56] is also likely to be the first paper to treat the question of feedback control of microwave drying. It should also be noted that efficient feedback control would be a very important innovation in the case of microwave drying because if the control of microwave sources is not based on the measurements taken inside the material, there is a danger of hot-spot formation [
57]. This danger is particularly high in the case of drying porous polymer foams, where it can lead not only to low-quality processing but also to foam ignition danger [
58].
The control design concept used in [
56] is similar to the concept described in the preceding section for the concentration distribution control process. The microwave drying process was modeled using a parabolic PDE model. This model was space discretized; the LQ controller was designed and tested in simulation. Since the control objective is to reach a nonzero set-point, a modified version of LQ control similar to the controller (19) had to be used. The simulation results presented in [
56] are promising. However, this paper does not go beyond simulation, and no follow-up papers going further to implementation have been published.
3.1.3. Inline Fluid Separation Process
A similar controller design methodology based on the concept of space-discretized PDE models was also applied to control of gas–liquid inline swirl separator. This control design was described in recent (2020) papers [
59,
60]. The structure of the gas–liquid inline swirl separator process is outlined in
Figure 4. The central component of this process is a vaned swirl element generating the swirling flow. This swirling motion should generate a core of the lighter phase, which is then captured by the central pickup tube, while the heavier phase should be captured by the outer tube. In order to improve this phase separation, the flows through the tubes are controlled by two valves. These valves are used as manipulated variables.
Conventional control of this process is explained in a recent paper [
61]. It is inherently indirect and based on the so-called pressure drop ratio, i.e., the ratio of the pressure differences between the pressure in the inlet tube and the two outlet tubes. It is rightly stated in [
61] that such an indirect control approach can reduce the efficiency of these devices significantly and result in violations of the environmental requirements. The authors also propose some extensions and improvements to the control based on the pressure drop ratio. However, these extensions are just variations of the same basic structure, e.g., they add feedforward action or make the setpoint of the ratio controller variable. On the contrary, the structure shown in
Figure 4 allows significantly more direct control because it includes several tomographic sensors.
Void fraction sensors in the outlet tubes can be either ERT or wire-mesh sensors, and they measure cross-sectional volumetric gas fractions in the outlet flows. From the viewpoint of the control system, all these variables can be used as controlled variables. The ERT sensor monitors the gas core created by the swirl element. This measurement can be used both as a controlled variable and as a measured disturbance. The wire mesh sensor in the bottom measures the gas fraction and gas velocity of the incoming flow. In this way, it provides some kind of advanced information for the controller. It can be used as a measured disturbance for controller feedforward action.
The underlying concept used by paper [
59] is the same as was described in the previous sections on microwave drying and concentration distribution control. The starting point is a simplified PDE model of the flows inside the separator process. This model is space discretized using the finite volume method. The resulting finite-dimensional model is then linearized. The result of this procedure is a high-order state-space model of the form (1). Unlike previous sections, this paper uses not LQ but MPC approach for controller design. However, this MPC controller is developed in a really minimalistic form. Although the paper introduces the control structure shown in
Figure 4, this structure is rather an ideal concept that should be achieved in future development. The really developed MPC controller is much more modest. It does not use the information from the inlet wire-mesh sensor. Rather, the ERT sensor measurements serve as a measured disturbance, while the measurements from the void fraction sensors are controlled variables. The controller is unconstrained, and its cost function considers only squared differences between measured void fractions and perfect separation conditions, i.e., increments of manipulated variables are not included in the cost function.
Although the controller was tested in simulations, these simulation tests are still at a very early stage. At this moment, it can be considered a very interesting concept. However, only future development can show its real strength. The open question is not so much the use of MPC but whether the space-discretized PDE model is really the most suitable option. Although [
59] uses this version of the model, in [
61] the authors consider using either identified black box/grey box models or neural network models.
3.1.4. Issues with Control Based on Early-Lumped PDE Models
Papers [
56,
57,
58,
59] are the only papers using the concept of early lumped PDE models for control with tomographic sensors published after 2013. Trying to analyze the reasons for this somewhat discouraging situation, at least two principal issues can be identified.
The first issue to consider is the use of linear quadratic control. This control method was originally developed for aerospace applications where modeling the uncertainties by white or colored noise disturbances such as in Equations (6) and (7) is more or less adequate. However, it is generally known that this is not an adequate model for industrial process control and that the application of the LQ approach to industrial control problems is not really a success story [
62]. Moreover, the version of set-point tracking described by controller (14) is sensitive to model inaccuracy. If the steady-state values are calculated using an inaccurate model, there is no way to compensate for this, and steady-state control error appears. A more robust version of linear quadratic set-point tracking can be obtained by introducing the integral of control error to the controller [
55], but this has never been used in the context described above.
Generally speaking, the importance of linear quadratic control is mainly historical. LQ control was the first control method based on quadratic optimality criterion and state-space models, but it has never found significant industrial acceptance. On the other hand, model predictive control (MPC) [
63] is related to LQ control in many important respects. It also usually uses a quadratic criterion of optimality, and it is based on a model of the process. LQ control is sometimes referred to as the zeroth generation MPC because of these similarities. However, despite these similarities, MPC is much more suitable for industrial use than LQ control. In particular, MPC features a unique ability to handle constraints, a wide range of acceptable models (state-space models, step-response models but also e.g., fuzzy models), and suitability for control of multivariable processes. Nonzero set-point tracking can also be achieved in a straightforward way, and no awkward modifications such as in the case of LQ control are necessary. For these reasons, MPC has found wide industrial acceptance. Nowadays, it is the standard method of choice for control tasks where PID control is unable to achieve desired control performance.
Although the idea to apply MPC in a context similar to the application of linear quadratic control described above may seem obvious, there are just two attempts in this direction. One attempt done in [
59] has already been described, and it is rather a concept than a fully-fledged MPC application. The second attempt is the paper by Sbarbaro and Vergara [
64] published in 2015. This paper uses electrical impedance tomography modeled by Equations (14) and (15) while the controlled plant is a generic discrete-time state-space model with a linear dynamic equation and nonlinear output (observation) equation
This structure corresponds well with the model of the concentration distribution control process described by Equations (18) and (20) as well as with the space-discretized model of the microwave drying process. Similar correspondence could be found with other processes where tomographic sensors can be applied. That means the MPC design proposed there could be fairly general. However, the MPC design in [
64] does not go too far. The paper includes just standard textbook equations of analytical unconstrained MPC controller combined with a nonlinear observer. The resulting control design is tested using a simple numerical example.
The second issue is the dimensionality of the model. PDE models are space-discretized using the finite element method (FEM) or finite differences method in the above-mentioned papers. This is an appropriate approach for numerical simulation, but it does not provide good control-oriented models. For instance, the order of the state-space model used in [
56] is 3670. This extremely high order is likely to result in high computational demands and ill-conditioned computations. Since the processes considered are described by parabolic PDE, it might be better to use the well-known fact that parabolic systems have a spectral gap between finite-dimensional slow and infinite-dimensional stable, fast modes [
65]. This enables an accurate approximation of such PDEs using relatively low-order models based on slow modes. Such models can be derived using spectral methods or the Karhunen–Loève method. For instance, paper [
66] applies these methods to distributed parameters processes with convection-diffusion phenomena and compares the quality of resulting low order models. In this paper, an acceptable approximation is always achieved with model order less than ten, which is by order of several magnitudes less than the order of models resulting from FEM.
Using low order models derived from parabolic PDE models on the basis of spectral methods or the Karhunen–Loève method is an open research direction for further development of tomography-based control. The use of these methods and their advantages are not as straightforward as they might seem at first sight. All approximation methods can be interpreted in such a way that the spatiotemporal PDE variable
y(
x,
t) is expanded by a set of spatial basis functions Φ
i(
x)
where the infinite upper limit is replaced by a sufficiently high finite
n. The low order of the approximated model in spectral methods can be achieved because the spatial basis functions are global, i.e., they are nonzero in the whole domain of interest. On the other hand, FEM can be understood as a method with local spatial basis functions (nonzero only within one element). As a result of this, high
n is necessary for a good approximation.
However, this high n simplifies the form of the output equation. If the model output has to correspond with tomography measurements, (i.e., with measurements distributed within some subdomain such as, e.g., the output boundary), it is enough to take the values of the elements located in the subdomain of interest, if FEM is used. We obtain a simple output equation such as Equation (17) where C is zeros and ones matrix. On the contrary, the construction of the output equation becomes more complicated if any method with global spatial basis functions is used. Despite this problem, methods with global spatial basis functions are worthy of research attention because control designs based on models with thousands of state variables are unlikely to find industrial implementation.
3.2. Control Based on Lumped Parameters Dynamical Models with a Static Model of Distributed Variables
Although no published paper has ever used spectral and related approximation approaches in connection with tomography, several papers use methods that are different in concept but similar in their results. These are the papers by Villegas, Duncan, Wang, and others focused on the control of batch fluidized bed dryers [
67,
68,
69,
70]. A schematic sketch of this dryer and its instrumentation is shown in
Figure 5a.
Such dryers are used in a wide range of industries (food, pharmaceutical, chemical). In these dryers, solid particles are transformed into a fluid-like state by forcing a gas (typically air) to flow in an upward direction through the bed. One of their main advantages is that the high turbulence arising in the bed provides high heat and mass transfer, as well as a high degree of mixing of the solids and gases within the bed. However, this turbulence may also result in non-uniform distribution of the solid phase (powder) inside the bed and local variations of moisture and other physical properties of the powder. This is the motivation for introducing a tomographic sensor that can observe these local variations inside the dryer.
There is actually a significant degree of similarity between this process and the concentration distribution control process shown in
Figure 3. Both of them are fluid flow processes. Electrical tomography is used to measure a distributed controlled variable. However, the modeling approach is different. The authors of [
69] use a dynamic model built from the very beginning as a simplified lumped parameters model based on mass and energy balances of the solid, gas, and bubble phases. Its input variables are inlet air temperature (
T0) and velocity (
U0), while output variables are particle moisture (
xp) and temperature (
Tp). Although there are two input variables, it was shown by a sensitivity analysis published in [
46] that the inlet air velocity has a much more significant influence on the performance of a fluidized bed dryer than the inlet air temperature. For this reason, it is enough to use this velocity as the only manipulated variable while the inlet air temperature can be kept constant.
The distributed model describes the variation of permittivity and thereby also the distribution of the particle moisture over the cross-section of the dryer. It is described by an algebraic expression
where
R is the radius of the dryer,
r ∈ <0,
R>,
fR and
ci are real functions and
J0 is the Bessel function of the order zero, and
λi the
i-th positive zero of this function. The structure of the mathematical model combining lumped dynamical model and (22) is shown in
Figure 5b.
The choice of Bessel functions in the permittivity distribution model (22) is motivated by their suitability for cylindrical coordinates. Model (22) is not a first-principles model but a suitable model structure whose components are to be obtained by identification from experimental data. This was done in [
69] using the data from a small laboratory scale dryer Sherwood M501. Good approximation was obtained with
N = 4 and bivariate polynomials up to second order in the positions of
fR and
ci.
It should be noted that if the modeling started from a PDE model and some method with global spatial basis functions were used for space discretization, there would be similarities between the resulting model structure and the structure shown in
Figure 5b. Again, it would be a relatively low order lumped parameter dynamical model with a static output function. This output function could be linear if it directly described the output variable of interest. However, it would be nonlinear similarly to model (22) if it described variables such as permittivity or conductivity. This is because of the nonlinearity of the relationship between variables such as concentration or moisture on the one hand and conductivity or permittivity on the other hand.
The control objective was to achieve the desired moisture distribution over the cross-section of the dryer. This desired moisture distribution can be converted into a corresponding desired permittivity shape as expressed by the summation term in model (22). The authors of [
69,
70] used offline optimization. They calculated a set of optimal values of
U0 corresponding to a set of particle moisture
xp values for a specified desired permittivity shape and identified coefficients in model (22). The relationship between
xp and optimal
U0 opt resulting from this optimization was then approximated by a third-order polynomial
Putting together the descriptions somewhat scattered between [
44] and [
45], the resulting control structure can be visualized using the scheme in
Figure 6.
The fluidized bed dryer used for experiments was a laboratory-scale dryer Model 501 produced by Sherwood Scientific. Technical details of this dryer are given by the manufacturer in [
71]. This device incorporates an air pump, heating coil, and temperature measurement (with control and timer) in its basic configuration. Microprocessor control of airflow, inlet air temperature, and drying period is available. The device configuration used in the experiments included an optional moisture probe, and it was further equipped with an ECT system with a data acquisition rate of 120 frames per second [
69]. This instrumentation corresponds to
Figure 5a.
A standard humidity probe measures just the humidity of the outlet air and not the moisture content in the particles being dried. These particles are scattered over the whole cross-section of the dryer. As a result, their moisture content cannot be measured by a single-point probe, but electrical tomography must be used instead. On-line evaluation of the average particle moisture is based on ECT images reconstructed using Landweber iteration [
72]. Therefore, this control is basically a simple cascade control loop with a single controlled variable in the master loop. However, the tomography sensor evaluating the whole cross-section of the dryer is necessary to obtain the value of this single controlled variable.
The master controller in the structure shown in
Figure 6 and described by polynomial (23) is a purely static term. It can be understood as a nonlinear version of the proportional controller. The airspeed, which is the manipulated variable of this controller, must be limited. In particular, its lower bound is important to ensure that the speed will never fall below minimum fluidization velocity.
The control objective is to achieve an appropriate moisture distribution, which is translated into achieving the corresponding reference permittivity shape. It is important to note that there is no reference input to the controller, but the control objective is expressed indirectly by the values of coefficients in approximation polynomial (23). If the desired shape changes, repeated off-line optimization must be performed, and the values of coefficients in polynomial (23) must be updated accordingly.
Regarding the relationship between the desired and real permittivity shape, the controller can be classified as a feedforward controller whose robustness is inherently limited. If there is any change in the behavior of the dryer, the functions in the permittivity distribution model (23) should be updated. However, there is no way for the controller to learn about this and a mismatch between desired and real permittivity shapes can appear. It is evident that there are many ways to improve the control. The availability of lumped dynamical model and electrical tomography sensor and other instrumentation shown in
Figure 5 gives many opportunities for control improvement. However, this is an open research direction. At this moment, it seems that the research line documented in [
67,
68,
69,
70,
71] has not found further continuation, at least as far as the modeling and control approaches are considered.
3.4. Knowledge-Based Control, Fuzzy Logic and Artificial Intelligence Approaches
As can be seen from the previous sections, in many cases, control with tomographic sensors is essentially real-time quality control, and its objectives are rather qualitative than quantitative. For instance, we need to achieve a good quality product from continuous casting or good separation efficiency. On the other hand, standard control assumes a specified set-point and feedback control loop that aims to have controlled variable(s) equal to set-point(s). Qualitative requirements must be first translated into standard control engineering terms and control objectives. In this regard, there is no difference between conventional PID control and advanced model-based control approaches such as MPC.
On the contrary, approaches based on fuzzy logic, artificial intelligence, and similar methods can be less rigid in their structure and specifications of objectives. Moreover, they can be designed and implemented without accurate mathematical models. This point is crucial. It is evident that even very simplified mathematical models of most of the processes considered here are very complicated.
Fuzzy control may be briefly introduced as an example of a control approach that can be used even if the process model is unknown or so complicated that controller design based on it would be totally impractical. Just the main ideas can be outlined in this paper. However, there are many good textbooks that can be used for further reference, e.g., recent titles [
88,
89].
It is well known that even very challenging processes can be controlled by human operators using their experience and expert knowledge. For instance, automatic systems cannot safely control aircraft takeoff and landing, but human operators (pilots) can handle even these flight phases. A fuzzy control system can be understood as a system implementing such expertise of a human operator. This expertise is not represented by differential equations or controller parameters but rather by situation/action rules. The operators are experts in operating their processes, and they know what to do under various circumstances. However, their knowledge is not expressed in precise rules such as, e.g., if the temperature is above 80 °C, set the heater power to 1 kW. Their rules are rather qualitative and imprecise. If we consider a really simple control task such as temperature control, we can say that the operator knows what to do, e.g., if the temperature is decreasing fast or if it becomes too high, while the term too high does not simply mean just higher than a single specific threshold.
Fuzzy rules are expressed in the same way as the rules in the minds of human operators. That means they are articulated using qualitative linguistic variables. Examples of typical fuzzy rules can be defined in the following way:
If the temperature is high and increasing, then reduce heater power fast.
If the temperature is high and constant, then reduce heater power slowly.
When fuzzy control is used for standard control tasks, it must be further considered that a fuzzy controller is similar to a human expert operator also in the sense that it looks at exact measured data, and if it performs a control action, this action is precisely specified. The measured temperature is a single numerical value, and the power of an electric heater can always be set only at one specific (crisp) value. It cannot be just low or high. For this reason, the fuzzy inference mechanism consisting of qualitative fuzzy rules must be connected to the real world by operations called fuzzification and defuzzification. This structure is shown in
Figure 8.
The input to the fuzzification block is the measured values of controlled variables, e.g., temperature. Fuzzification means conversion to qualitative variables. It is done using the so-called membership functions, which convert the measurement into a degree of membership. An example of such a membership function is shown in
Figure 9.
Assuming that the measured temperature is, e.g., 80 °C, this temperature is normal to a degree of 0.2, high to a degree of 0.8, and low to a degree of 0, where 1 represents full membership and 0 no membership. This expresses that in the considered control task, the human expert would regard 80 °C to be well above normal operating temperature. That means this temperature is definitely not low, rather high than normal, but on the other hand, the temperature could become even higher. Membership functions conceived in this way correspond to qualitative human reasoning where no strict boundaries exist between qualitative notions such as normal, high, too high, etc. The membership function is based on simple “expert” knowledge in this example. Defining these functions and inference rules in more complex tasks may not always be easy. For this reason, fuzzy control is also sometimes connected to other artificial intelligence approaches, especially with neural networks into neuro-fuzzy control [
90].
The output from the fuzzy inference mechanism combines all fuzzy rules together, and it is again a qualitative variable described by its membership function. Obviously, this function is not constant, but as the controlled variable(s) vary, fuzzy inference yields different results, and this output membership function also varies. This output must be defuzzified, i.e., converted to an exact numerical value that can be sent to the physical actuator, (e.g., heater). Defuzzification can be done in several ways. The most widely used method is the so-called centroid defuzzification which calculates the center of gravity (COG) of the membership function corresponding to the output variable along the x-axis. This defuzzification approach can be expressed by the formula
where
μA is the output membership function defined in the range from
a to
b. An example of this defuzzification method for one specific output membership function is shown in
Figure 10.
The example with temperature control is a simple standard control task where precise temperature measurement must be fuzzified. An output from a tomographic sensor often gives directly qualitative information, e.g., a flow pattern. Although various flow patterns are clearly different if fully developed, the transitions between them may often be gradual and hence naturally fuzzy. For this reason, fuzzy control may be even more suitable for complex tomography-based control tasks than for simple standard control. Therefore, it is not surprising that control approaches from this category have been considered since the very beginnings of process tomography. An important application area has been pneumatic conveying, i.e., a process where an air (or other inert gas) stream is used as a transportation medium to transport various granular solids and dry powders. The most common variant of this process is the dilute phase pneumatic conveying where the conveyed particles are uniformly suspended in the gas stream.
This process has many advantages in terms of routing flexibility and low maintenance costs. However, an appropriate flow regime with a homogeneously dispersed flow must be continuously maintained. Otherwise, several issues may arise such as discrete plugs of material, rolling dunes with a possible high increase in pressure and a blockage, and unstable flow with violent pressure surges resulting in increased plant wear and product degradation. For this reason, standard uncontrolled pneumatic conveying processes can normally be used only for the specific solid materials for which they were designed, and airspeed must be rather high to avoid potential blockage problems. This high speed increases power demands and makes this process less energy efficient than it could be [
91].
This is a motivation for closed-loop control. It is a non-standard control task. The control objective is to maintain a suitable flow pattern. There is no numerical-valued controlled variable with a specified set-point. Following [
92,
93], it is possible to identify several flow regimes besides the desired dilute phase flow achieved at a higher airspeed. If the airspeed becomes smaller, it reaches a point (so-called saltation velocity) where an abrupt change from dilute to dense phase flow occurs and the solid particles begin to settle out. This is marked by saltating flow and dune formation. If the speed is further decreased, so-called dune flow and plug flow can exist, and a significant danger of pipe blockage occurs.
The control of pneumatic conveying was treated by several research groups. The measurement system was always an ECT sensor, but different ECT data processing and control approaches were used. An earlier paper [
92] by Deloughry, Ponnapalli, and others attempts to interpret the task of keeping the desirable flow regime as a standard control task with numerical-valued variables. It considers an experimental pneumatic conveying process where polyethylene nibs are transported. The tomographic image of the pipe cross-section has a total of 816 pixels, and the number of pixels that contain these polyethylene nibs (pixel density) can be used as a measure of the current level of sedimentation (dune formation) inside the pipe. This allows the use of a conventional PID controller whose set-point is set to a sufficiently small number to prevent dune formation. This objective was achieved with set-point 20, while set-point values higher than 50 usually resulted in pipe blockage. A substantial part of this paper discusses the influence of the PID controller parameters on the controller’s ability to prevent dune formation and blockage for different set-point values.
In the end, PID control was not found to be an adequate approach, and later the authors of [
92] proposed a neural network inverse model controller [
93]. ECT image was evaluated in the same way, i.e., by counting the pixels. The authors of [
93] used the procedure of inverse plant identification [
94]. That means they performed a wide range of experiments with the laboratory-scale pneumatic conveying process around the critical saltation speed. Data obtained from these experiments were first used to train a neural network modeling the process in forward configuration, i.e., current and past values of air blower speed and past values of pixel density were inputs, while the current value of pixel density was output.
The purpose of the forward model was to find suitable neural network parameters (number of hidden neurons and past values). The network with the parameters that turned out to be the best for modeling the process (6 hidden neurons and 13 past values) was then trained in the inverse configuration where the current value of the blower speed (i.e., manipulated variable) is the output. Since the output of this model is the manipulated variable, and the set-point can be fed into the current pixel-density input, this neural inverse model can be understood and used as a controller. These model structures are illustrated in
Figure 11.
The paper concludes that this controller was able to clear the dunes automatically while maintaining the air velocity at a minimum value necessary to keep the flow homogeneous. In this respect, it was able to outperform the PI/PID controller. This neural controller responds in some way to changes in flow patterns, but this response is implicit and hidden in the internal structure of the network. There is no direct and explicit identification of these regimes.
An alternative approach is followed by Williams, Owens, and others in [
91]. It uses an explicit classification of flow regimes. The ECT data are processed to obtain the void fraction, i.e., a signal that can be considered complementary to the pixel density, which evaluated the number of occupied pixels. Both magnitude and frequency of changes of void fraction signal are important. Classification of flow regimes is performed by a neural network. The authors then develop an idea of a two-level control strategy. Low-level control with a very fast response does not use ECT measurements. Only flow rate and pressure measurements are used to quickly prevent any blocking tendencies.
High-level control is based on fuzzy logic, and it uses flow classification from neural networks and data from other sensors to keep the desired flow regime and retune the low-level controller. The classification of flow regimes seems to be really tested and working. On the contrary, the control structure proposed by the authors is rather a concept, which is proposed, but not really implemented. This concept does not seem to be ever used for control of the pneumatic conveying process. Papers [
93,
94] are later (2001 and 2007), and they use a different approach without even citing [
91] (1999). Despite this, the approach taken in [
91] turned out to be pretty fruitful and even paradigmatic for controlling other processes. Its main principle can be characterized as the classification of the selected phenomena (in this case, flow patterns) in the tomography data as a first stage and fuzzy or other rule-based systems as a second stage in the control strategy.
This concept is used in [
95] to control oil separators based on ECT images. The ECT images are classified using the principal component analysis approach. The results from this classification are then used as input to a controller based on expert rules. This controller was applied to a laboratory-scale separator process. In
Section 2 of the present paper, we have already mentioned paper [
40] where the classification of flow patterns was done using both neural networks and cascades of Support Vector Machines. These classified patterns were then used as inputs to model-free adaptive controllers based on deep belief networks. Model-free adaptive control [
96] is a control approach that might be very suitable for use with tomography because, as we could see, the models are notoriously difficult to obtain. It is hard to tell how this approach is applied in [
40] because the paper focuses mostly on the classification of flow patterns while the control itself is only outlined. Nevertheless, it is easy to see that the paper again follows the same structure: flow patterns classification using artificial intelligence methods as a first stage and fairly non-classical controller as a second stage.
A similar structure is followed also by [
97], where the focus is on control of the polymer extrusion molding process. ECT tomography is used as the only sensor that can measure the internal temperature in a cross-section of a polymer extrusion process. In this way, ECT enables feedback control of this process normally operated by trial and error. The melt temperature field is obtained from the reconstructed ECT image measured. The difference between the measured and reference temperature fields is then used as input information for a knowledge-based control system. This system uses process variables such as temperature in different sections along the barrel and the screw rotating speed as manipulated variables. That means there is an implicit cascade control structure where this knowledge-based controller is the master controller calculating set-points for slave controllers in the polymer extruder. Similar to most of the previous papers cited in this section, this knowledge-based controller is a proposed concept but was neither implemented nor tested.