# Control Systems with Tomographic Sensors—A Review

^{*}

## Abstract

**:**

## 1. Introduction

## 2. General Characteristics of Tomography-Based Control

#### 2.1. Implications for Selection of Control Method

**x**(t) is an n-dimensional state vector

**u**(t) is a vector of input variables,

**y**(t) is a vector of output variables,

**A**,

**B**,

**C**and

**D**are matrices of appropriate dimensions. The number of first-order equations n is finite, and it is called the order of this model.

**A**,

**B**,

**C**,

**D**would be infinite-dimensional operators on Hilbert spaces and not matrices [22]. Such models are popular in textbooks on the mathematical theory of infinite-dimensional systems. However, their relationship to physics is indirect and their practical applicability is more than limited. Nevertheless, they are closely related to partial differential equations (PDEs) and can be derived from them [22]. For this reason, it is better to work with models in the form of PDEs that can be derived from the underlying physics of the process.

_{j}(t) is the steam jacket temperature, u is the heated fluid velocity through the tube, and τ is a positive constant. The steam temperature is considered to be spatially uniform for simplicity.

**x**is composed of temperatures T(1,t) to T(N,t), input vector

**u**includes T(0,t) and T

_{j}(t), and output from the system is T(N,t), the set of Equation (5) can be written in matrix form (1).

#### 2.2. Tomography Data Processing

**x**stands for a vector of internal state variables (e.g., concentration distributions), and

**y**is the vector of the measured outputs. Vectors

**ξ**and

**η**are noises: process noise and measurement noise, respectively. These vectors are not included in the continuous version of the state space Equation (1). They are added to the discretized version because these noises are the standard way the model uncertainties and measurement errors are considered in Kalman filtering and LQ control [45]. System input (manipulated variables, disturbances) is denoted with

**u**and k stands for the discrete time.

## 3. Control Techniques, Methods, and Applications

#### 3.1. Control Based on Distributed Models

#### 3.1.1. Concentration Distribution Control

**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

**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.

_{in}(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.

**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.

_{I}) 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

**V**

_{j}are the voltage measurements, and

**U**

_{j}maps the conductivity to the measured voltages. Vector

**v**

_{j}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

**J**is the Jacobian of the function

**U***(

**c**(k)) evaluated at the operating point c

_{0}.

**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.

**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

**y**

_{ref}at the output boundary. For this reason, the structure of the controller had to be changed to

**y**

_{ref}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].

#### 3.1.2. Microwave Drying Process

#### 3.1.3. Inline Fluid Separation Process

#### 3.1.4. Issues with Control Based on Early-Lumped PDE Models

_{i}(x)

**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

_{0}) and velocity (U

_{0}), while output variables are particle moisture (x

_{p}) and temperature (T

_{p}). 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.

_{R}and c

_{i}are real functions and J

_{0}is the Bessel function of the order zero, and λ

_{i}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.

_{R}and c

_{i}.

_{0}corresponding to a set of particle moisture x

_{p}values for a specified desired permittivity shape and identified coefficients in model (22). The relationship between x

_{p}and optimal U

_{0 opt}resulting from this optimization was then approximated by a third-order polynomial

#### 3.3. Experimental Approaches: Identified Models, Empirical Controller Tuning

#### 3.3.1. Control of a Wurster Fluidized Bed

#### 3.3.2. Control of Microwave Drying

_{1}is the power applied to magnetrons, u

_{2}is the average input moisture, and y is the average permittivity change measured by the ECT sensor corresponding to the change of average output moisture. The input moisture was calculated on the basis of weighting the sheets of wet foams on a digital scale and the known weight of dry foam sheets.

#### 3.3.3. Control of Continuous Casting of Metals

#### 3.3.4. Control of Hydrocyclone Separators

#### 3.4. Knowledge-Based Control, Fuzzy Logic and Artificial Intelligence Approaches

_{A}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.

## 4. Discussion and Future Research Directions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ECT | Electrical Capacitance Tomography |

EIT | Electrical Impedance Tomography |

FEM | Finite Element Method |

LQ | Linear Quadratic |

ODE | Ordinary Differential Equation |

PDE | Partial Differential Equation |

PID | Proportional Integral Derivative |

MIMO | Multi Input Multi Output |

MPC | Model Predictive Control |

SISO | Single Input Single Output |

TRL | Technology Readiness Level |

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**Figure 1.**(

**a**) Flow patterns in the mold; (

**b**) parametrization of the jet flow using a line with a variable angle.

**Figure 5.**(

**a**) Schematic sketch of the fluidized bed dryer and its instrumentation (

**b**) structure of its control-oriented mathematical model.

**Figure 10.**Defuzzification example resulting in a single numerical value of manipulated variable (heater power).

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Hlava, J.; Abouelazayem, S.
Control Systems with Tomographic Sensors—A Review. *Sensors* **2022**, *22*, 2847.
https://doi.org/10.3390/s22082847

**AMA Style**

Hlava J, Abouelazayem S.
Control Systems with Tomographic Sensors—A Review. *Sensors*. 2022; 22(8):2847.
https://doi.org/10.3390/s22082847

**Chicago/Turabian Style**

Hlava, Jaroslav, and Shereen Abouelazayem.
2022. "Control Systems with Tomographic Sensors—A Review" *Sensors* 22, no. 8: 2847.
https://doi.org/10.3390/s22082847