Enhancing Control of Dynamic Flow Systems Through Innovator Controller Design and Parametric Polynomial Modeling

This research employs parametric polynomial techniques to determine the parameters of a dynamic fluid control system. The process of system identification involves constructing a mathematical model of a dynamic system using measured data in the time or frequency domain. The approach involves fitting a polynomial function to the input-output data, with the polynomial parameters representing the unknown parameters of the system. The objective is to estimate these parameters in order to control the system effectively. In this study, aortic, femoral, iliac, carotid, and coronary artery signals were utilized in modeling and testing studies of the flow system during parametric model studies, as the system forms the basis for hemodynamic research. The Autoregressive model with external input (ARX), Autoregressive moving average model with external input (ARMAX), Box-Jenkins (BJ), Output-Error (OE), and State Space Model (SSM) parametric models were utilized in the modeling process, and the transfer function of the most successful parametric model was calculated in the mathematical performance analysis. Flow control devices such as the AC motor and centrifugal pump were employed. The transfer function that exhibited the most successful performance was used in observer design as the Luenberger Controller. An innovative closed-loop control system was achieved using the Luenberger structure.


INTRODUCTION
In human physiology, the cardiovascular system is a set of organs consisting of the heart and vascular system that provides the transport of necessary substances between organs and systems and the removal of unnecessary substances with the help of blood [1].The circulation of blood in the body is provided by the heart.
All of the blood flow in the arteries and veins depends on the contraction and relaxation of the heart [1].
Cardiovascular diseases are the leading cause of physical disability and death in the world [2].In developing countries cardiovascular diseases appear to be the major cause of death [3].
Numerical and experimental studies are carried out to see the causes and underlying dynamic effects of cardiovascular diseases.These studies can generally be grouped under the title of hemodynamic studies.There are many numerical and experimental studies investigating blood flow and its effects in cardiovascular systems.
Numerical studies: It is done by mathematical modeling of the chemical and vascular geometric data of the blood in computer environment [3][4][5][6][7].Simulation times increase depending on the number of parametric criteria used in numerical studies.In reality, for an experiment that requires a few minutes of data recording, simulation studies can take hours depending on the effects of the parameters.The results obtained still do not include all parametric data made on the real experimental system.Due to the nature of experimental systems, the fact that they contain all data and allow real-time data recording provides great advantages over numerical studies.Even though experimental and numerical studies have advantages and disadvantages compared to each other, blood flow simulation studies are carried out in an experimental and numerical field that support each other [3][4][5][6][7].
With the discovery of MR imaging techniques that make non-invasive measurements possible, blood flow data (velocity and flow) in the human body have begun to be measured.Thanks to these data obtained, in vitro studies have become popular and the number of studies in this field has increased [8][9][10].Although blood flow velocities in the arteries can be measured by non-invasive methods, long-term in-vivo studies on real patients cannot be considered ethically correct [8][9][10] and its implementation in practice may cause harm to patient health.
For this reason, the best solution is to simulate blood flow velocities from real patients on experimental systems and conduct related studies [8][9][10].
In all these studies given in the literature, no performance criterion is given regarding the success of the desired blood flow rate (flow rate) on the system.For example, in a study carried out on the experimental system, the average of the 15 pulsatile blood flow rates and comparison with the desired blood flow rate was made visually on the graph [20].All experiments were carried out with open loop control.During the experiments, the amplitude or frequency information of the measured signal and the reference signal were matched manually by the user.
Experiment system consists of piston pump, gear pump, back pressure valve, two servo motors with 600W and 400W power, computer, data collector card control system.The fact that the installed system can be easily controlled via MATLAB or Labview is given as an advantage in the study [20].
In all of these studies, it is seen that the system control is performed with an open loop.Open loop control work has been tried to be done either with a mechanical system or with user manual control.Pulsatile flows realized with open loop were created in experimental systems with sinusoidal flow signals instead of real blood flow characteristics [9], [12], [18].By calculating the amount of change in the flow rate of the simulated sinusoidal signal, the amount of change in the upper and lower levels was calculated, and the results were tried to be reached, considering that the desired graphical match was made.In the literature review, it was seen that the closed loop (PI) control technique was used in one study [20].The system is driven by a combination of a fixed speed pump and valve.However, the performance of the system in obtaining the desired flow rate was not calculated, instead, its success in matching graphically was discussed visually.There is also a second study given in the literature, new PID control approaches are presented, and closed-loop performances are given by the authors of the article [8].In this study, a more efficient closed-loop control structure is proposed by the authors.It imposes more burden than traditional PID control structures in terms of mathematical model and preliminary studies.On the other hand, it ensures that the reactions of the system are predictable and controllable.
In our study, control-based closed-loop control was implemented with the Luenberger observer structure.
Modeling studies of the flow control system are carried out with parametric polynomials.The performances of the parametric polynomials used in revealing the transfer function of the system were also evaluated mathematically.
In the study, AC motor and centrifugal pump were used as flow control devices.In addition, mathematical analyzes in which the performances between the arterial reference r(n) signal and the y(n) signal generated by the system in the flow meters are calculated are given in the study.Thus, the work I have carried out constitutes a resource for the creation of an in vitro system for hemodynamic studies and eliminates the debate about the results of hemodynamic studies.

EXPERIMENTAL SYSTEM
The diagram and photograph of the system where the experiments were carried out are given in Figure 1 as a and b, respectively.The AC motor used in the experimental system can rotate at 2900 rpm in response to the 220V, 1.2A and 50Hz standard supply of Fasco company.The AC motor transmits the liquid stored in the reservoir tank to the system with the help of a centrifugal pump.Centrifugal pump can only create one-way flow.The speed control of the AC motor is provided by the drive inverter.G110 model of Siemens company was used as inverter.The driver creates a rotation speed of 0-2900 rpm in response to the 0-10V dc signal sent over the DAQ card and therefore the MATLAB/Simulink [28].AC Motor, Centrifugal Pump and Inverter are given in Figure 2. The flow control system and the structure of the devices in the system are given in Figure 2. MF624 DAQ card belonging to Humusoft company was used as a controller.The DAQ card has 8 analog inputs and 4 analog outputs.There are also 8 digital inputs, 8 digital outputs, internal time circuit, Analog/Digital and Digital/Analog converters [29].The DAQ card has a tool compatible with MATLAB/Simulink and can perform real-time data read-write tasks.By reading the flow rate of the water-glycerin mixture on the DAQ card system via the Flow meter, the AC motor can control the speed of the pump via the inverter.The DAQ card is capable of real-time 100Hz signal read-write, compatible with MATLAB/Simulink.Humusoft MF624 DAQ card is introduced on MATLAB program and can be used with Real Time Toolbox via Simulink Block.The 0-10V signal is sent to the inverter to drive the AC motor via the DAQ card.A flow rate of 100 ml/sec occurs in the system against each volt value of the AC motor.Flow information is read from flow meters.Calibrations were made for these measurements.

PARAMETRIC POLYNOMIAL MODELS
System identification can be defined as the mathematical modeling of a dynamic system using measured statistical data in time or frequency space.System diagnostic studies of flow control devices in the experimental system can be done using parametric polynomials and state space model.In the system diagnosis process, it is necessary to plan the experimental system and collect data, to find the models mathematically, to estimate the unknown system parameters from the experimental data, and to test the validity of the found model [25,26].System diagnostic studies: The data obtained by recording the test signals applied to the flow control devices and the flow signals formed on the system were processed in Matlab program tools.Parametric system diagnostic polynomials, one of the Matlab system diagnostic tools, were used to derive the mathematical model of the integrity of the system with each flow control device.
Parametric polynomials propose mathematical models using data recorded in discrete time space.
Dynamic systems with the given diagnostic block; can be expressed with a linear polynomial model and can be written as in (1).
While y(n) is the system output, r(n) is the system input, k is the unit time delay in the system, e(n) is the noise affecting the system.H(q) is transfer function of the stochastic part of system.G(q) is the transfer function of the deterministic part of the system.Parametric polynomials of A(q), B(q), C(q), D(q) and F(q) belonging to G(q) and H(q) transfer functions are given in equations ( 2) and (3) [30].

B(q)
Parametric polynomials can be arrange depending on the A(q), B(q), C(q), D(q) and F(q) from equations (2) and ( 3).The polynomials of the deterministic and stochastic parts for parametric polynomials are given in Table 1.

FLUID MECHANICS
In fluid mechanics, Reynolds and Womersley dimensionless numbers are taken into account for the generation of arterial blood flow signals in the in vitro system.Reynolds number and Womersley number are two important dimensionless parameters in fluid mechanics.Reynolds number (Re) is a ratio of inertial forces to viscous forces and describes the nature of fluid flow, whether it is laminar or turbulent in equation ( 4) [35,36].
V; average speed (m/s), D; diameter (m), ρ; density (kg/m3), μ; dynamic viscosity (kg/m.s)value, when the units are written instead on the equation, it is seen that the Reynolds number is unitless.
On the other hand, Womersley number (Wo) is a dimensionless parameter that characterizes pulsatile flow in a circular tube.It is defined as Wo in equation ( 5).The number used to relate the frequency of the flow between two different viscous systems in systems with pulsatile flow [35,36].Re and Wo dimensionless numbers [37].

SYSTEM IDENTIFICATION
The artery signals r(n) were used as the input signal for the system identification studies.The r(n) signal  3 and Table 4.The transfer functions obtained for arterial signals are given in Table 3 and Table 4.The arrangement of the coefficients of the OE and BJ parametric polynomials causes the same transfer functions to be obtained within the two polynomials.Because for the parametric polynomials OE and BJ, the parameters B(q), F(q) are multiplied by the same coefficient.Also, since the parameters C(q) and D(q) of BJ are multiplied by 0. The coefficients can be seen in Table 2. To find out how successfully the transfer functions represent the system, numerical studies were carried out and the r(n) artery signals were applied as an input to the numerically obtained transfer functions.The performance analysis was made by comparing the output signals produced by the transfer functions with the output signal produced by the real experimental system.Equation ( 6) was used to calculate the performance analysis.
The output signal yh of the model is calculated.The flow signal y recorded from magnetic flow meters during the experiment and their performance were compared with the F function.
shows the Euclidean and y shows the mean value of output signal.In the studies, it was aimed to reach the lowest order transfer function against the highest success.The results of the study are given in Table 5.
Performance analyzes of the models of Transfer Functions were made according to the Equation ( 4).
Parametric models of each arterial signal were tested both in its own signal and from other arterial signals.
Performance results are given in Table 5.The arterial blood flow signals can be obtained as the input signal r(n), the experimental system output signal y(n), and the signal yh(n) formed by the mathematical model.The model with the highest performance among the transfer functions will be used in the observer design and control studies.
When the performances of the transfer functions of Parametric polynomials are examined, OE, BJ transfer functions have the highest mean value.The OE, BJ transfer functions are able to model the y(n) signal, which is the test system signal output, with an average success rate of about 73%.It is seen that the transfer functions with 88% performance are of the 2nd order and their coefficients are quite close to each other.On the other hand, SSM transfer functions have the lowest performance average value with 61%.Mathematically, the model to be included in the Simulink control system is a 2nd order transfer function, which is also a reason for preference in terms of control response.
The transfer function of the OE and BJ parametric polynomials of the Illiac artery was chosen as a mathematical model, since it has the highest performance value of 88.11%.Continuous time and discrete time models of the transfer function are respectively given in equation ( 7) and ( 8). = -9,889.10 - s 2 + 3,956.10 - s 2 + 0.02141s + 3,663.10 - (7) Using the transfer function obtained with the illiac artery signals given in ( 5) and ( 6), the output signals of the other arteries and their performance were calculated according to equation ( 4).The performance results of this calculation are given in Table 6.Using the equations given in ( 5) and ( 6), the r(n), y(n) and yh(n) signals of all arteries are plotted again on   On the other hand, studies on obtaining arterial signals in this area with closed loop and high-performance control belong to the authors [8].In their studies, different closed-loop control studies based on the model were presented.In this study, closed loop control will be realized with the observer design.

LUENBERGER OBSERVER DESIGN
The mathematical models calculated for the centrifugal pump with the AC motor were used in the design of the Luenberger Observer controller.It includes general measurement and sensing techniques as well as modelbased reconstruction process and parallel and feedback connections [26], [31].It provides an effective and live method for data extraction that is not accessible to physical sensors in the observer detection process.For this reason, it is a structure that is also called virtual sensor and uses the form of state space equations ( 9) to describe all states in order to formulate this principle.
̇=  +   =  +  (9) Full state feedback or pole placement [32] method is used in the observer design.In practice, the observer is realized by assigning the current poles of the system to predetermined places so that the system will react in a shorter time and will not cause overshoot.In order to reassign the current poles of the system, linear simulation [33] studies have been carried out on the computer.State space model parameters (A, B, C, D) were used in simulation studies.The control of the characteristic response of the system depends on the location of the poles, and the poles directly correspond to the eigenvalues of the system and are the roots of the characteristic equation.
It can be calculated using functions of eigenvalues with MATLAB.The state space model given in Equation (9) was used in the studies as a block.The closed loop poles of the model of the system directly affect the time response such as rise time and transient oscillations.Therefore, the time response and transition oscillations are optimized by reassigning the closed-loop poles using the state space model.
In order not to interfere with the state equations of the system and the variables to be used during the design of the observer, it is necessary to calculate the observer gain L, with the estimated state variable ŷ estimated output.
The feedback gain K and the state gain u=-K are added to the block as in Figure 6.Observer error would be e =-x.
The observer state space equations given in Figure 6 can be written as follows.
In the observer design, linear simulation experiments were carried out to ensure the desired response time and stability of the poles on the system and eqautions are given in (10).
Simulation studies give preliminary information about the success of the selected poles before experimenting in the real system.In the simulation results, it is decided whether the poles are at the appropriate points by looking at the response time and rise amplitudes of the system.It is preferred that the response time be short and the response amplitude low.Due to the presence of a vascular model of the coronary artery on the experimental system, Luenberger observer studies were continued with the coronary artery flow rate signal.The most successful transfer function in modeling the coronary artery signal of the system was selected among the calculated models, which is given in equation (11).
As a result of the preferred poles, the error performance analyzes of the system were made by using the control blocks in the experimental system.By calculating the error performances of the reference signal r and the response y signals of the system, it is decided whether the selected poles are successful on the system.
The transfer function is given in equation ( 12) by transforming equation ( 9) into state space model.By calculating the pole values of the state space equation, the state space equations were arranged for the Luenberger observer control design.The poles of the system of equation ( 11) or ( 12) are calculated as P1=(-0.7635+ 0.9217i) and P2=(-0.7635-0.9217i).As a result of the simulations, arranging the poles as P11=-5 and P22 = -10 provides control gain as shown in Figure 6.The reaction time of the system is shortened with new poles.As seen in Figure 6,  ̂̇ response time is better than ̇ response.State space equations arranged with new coefficients are given in equation (13).With the new pole assignment, the Luenberger structure of the system is arranged as given in Figure 7.

Figure 1
Figure 1.a.The schematic of the experimental system.b.Experimental system picture.

Figure 2 .
Figure 2. Matlab/Simulink, DAQ Card, Siemens Inverter, AC Motor and Centrifugal Pump cylinder radius, ω; angular frequency, υ; kinematic viscosity and υ=μ/ρ values.According to the Reynolds and Womersley dimensionless numbers, hemodynamic simulation studies were calculated by the authors for the in vitro experimental system.In accordance with the theorems, the arterial signals are expressed as r(n)[34].Arterial blood flow signals can be obtained by MRI clinical methods, giving the opportunity to research on in vitro systems with

Table 2 .Figure 3 .
Figure 3. Open Loop Control of Aortic (a), Femoral (b), Illiac (c), Carotid (d) and Coronary (e) Artery Signals r(n) and y(n) The mathematical values calculated by parametric polynomials are translated into transfer functions in the s domain.The transfer functions obtained for arterial signals are given inTable 3 and Table 4.The transfer functions

Figure 4 .
Figure 4. r(n), y(n) and yh(n) signals of Aorta (a), Femoral (b), Illiac (c), Carotid (d) and Coronary (e) Arteries.The transfer function modeling the arterial signals with the best performance is the model given in equation (5).This transfer function can model the signals of the aorta, femoral and iliac arteries with successful performance.However, in modeling coronary and carotid signals, transfer functions using their own arterial signals have better performance individually.While the coronary artery signal can be modeled with the arx model with a success rate of 81.49%, the carotid can be modeled successfully with 80.51% of BJ, OE transfer functions.r(n), y(n) and yh(n) graphs of carotid and coronary artery signals with their transfer functions are given in Figure 5.

Figure 5 .
Figure 5. r(n), y(n) and yh(n) signals of Coronary (a) %81.49 and Carotid (b) %80.51 Arteries with their own transfer functions.The low flow rate of the coronary artery signal and the complexity of the carotid artery signal make it difficult to model these arterial signals with high performance.It is clear in the study that the performance successes of these two difficult signals are above 80% with their own transfer function models.

Figure 6 .
Figure 6.Observer block diagram with equations of state

Figure 6 .
Figure 6.Poles of ̇ and Poles of  ̂̇ Simulation Response

Table 3 .
Transfer functions of arterial signals belonging to parametric models

Table 4 .
Transfer functions of arterial signals belonging to parametric models

Table 6 .
Illiac Transfer Function Performance