Model Predictive Control for Level Control of a Conical Tank

Abstract

Conical tanks have a high application rate in industrial processes, especially in colloidal mills, chemical processes, and food processing. The use of conical tanks presents significant benefits because they contribute to sedimentation and reduce the accumulation of impurities compared to conventional cylindrical tanks. However, level control of a conical tank due to its shape requires advanced strategies to guarantee efficient control. In this research, a model predictive control (MPC) method was designed and implemented for the level control of a conical tank on a laboratory scale. To evaluate the performance of the controller, it was compared with a traditional proportional–integral (PI) controller, and the rise time, settling time, overshoot, and error in the steady state were analyzed when different set point changes were tested. In addition, the system was subjected to disturbances, and the MPC demonstrated better performance in a transient state, as well as smooth and stable action controls that allowed for an increase in the useful life of the actuator. In addition, an interactive graphical interface was developed that allowed a dynamic response in a real plant to be experienced; this provides an academic tool for designing control strategies before implementation in a real process.

1. Introduction

Conical tank systems have high applicability rates in industrial processes; they are mainly used in colloidal mills; in leach extraction in pharmaceutical industries, chemical industries, food processing industries, and oil industries; and in mixing tanks, biodiesel processing, reactors, cider production, alcohol production, vertical rainwater tanks, fertilizers, and chemical storage tanks. Their importance lies in the fact that conical tanks allow for benefits such as sedimentation and the drainage of impurities [1].

Conical tank research is relevant in process control and involves analyzing several control theories after obtaining a mathematical model. Due to their high nonlinearity rates, the study, modeling, and control of conical tanks present a challenge. Due to their shape, they have different dynamics at the point of operation.

The level control of conical tanks is a process with nonlinear dynamics, where applying traditional or linear control strategies does not guarantee an efficient transient state at its level. In the specialized literature, the control of conical tanks has been reported with conventional techniques, such as PID controls, and variants, such as fractional order PID (FOPID), which differs from the traditional control method by the process of obtaining a fractional integrator and derivative and establishing parameters of the order that each one will have; despite this method having a complex structure and being difficult to tune, it, however, guarantees effective stability [2]. On the other hand, advanced control strategies have also been reported in conical tanks, such as fuzzy control algorithms, where establishing a set of rules and relevant membership functions allows for obtaining better performance in tuning parameters such as a fast settling time and low impulses [3], neural control [4], and the use of the SMC rolling horizon control strategy, which consists of establishing the state of the system and extracting it to a sliding surface plane; this type of control strategy guarantees a high degree of robustness in its design, unlike traditional control strategies [5].

Predictive control algorithms based on MPC models have been applied in many industrial processes with nonlinear characteristics—for example, in temperature and level control strategies [6], combined-cycle thermal power plants, biphasic separators [7], correlated temperature and level processes [8], and industrial processes [9]. On the other hand, for model predictive control, there are several methods to obtain the process model—for example, by using an echo state network (ESN), which uses the recursive least squares method applied to the level control strategy for a conical tank [10]. The application of control algorithms in MIMO systems is complex due to the various dynamics of the process. Having a nonlinear process such as a conical tank control strategy requires advanced algorithms and the use of a dynamic matrix controller (DMC) for correction by using dynamic behavior to control the process [11] with no lineal model—for example, in the case of fuzzy logic- and proportional–integral–derivative-based multi-objective optimization of the active suspension system of a 4 × 4 in-wheel, motor-driven electrical vehicle [12], which uses an advanced control algorithm for the control of nonlinear plants.

A conical tank plant, due to its nonlinearity when controlling with a traditional control strategy, tends to have low performance, which is the reason why self-adjustments are performed—for example, the use of an interval type-2 fuzzy logic system with genetic self-adjustment of its membership functions [13]. Another alternative is the use of mathematical modeling to represent a hybrid interactive two-tank system (IHTTS) by analyzing a multivariable system with nonlinearities [14] or an analysis of traditional tuning parameters using the Ziegler–Nichols tuning (ZNT) method, root locus technique (RLT), and (IAE) through comparisons of server responses [15]. Another alternative to improve control is the implementation of advanced control methods such as fuzzy control (FLC) and the use of the Kalman algorithm, which employs fuzzy logic rules to help reduce error in noisy environments [16]. Another alternative is an RTDA controller (robustness, set point tracking, disturbance rejection, aggressiveness), whose main advantage is parameter tuning, thus improving the performance of the closed-loop system [17], or the rolling horizon control strategy for improved robustness to disturbances [18].

Advanced control strategies have demonstrated superiority in processes with nonlinear and multivariable dynamics [5]. In conical tanks, predictive control based on MPC models has also been used, but, in the control process, they perform a model mismatch; that is, the design model does not coincide with the real process, and, within the structure, an estimator parallel to the process occurs. This type of control strategy is known as NMPC [19], and it should be noted that within the literature investigated, it has been implemented in a conical tank plant, but the MPC applied to the same plant has not been developed. However, in one paper, the authors simulated the process, thus showing an idealistic model of a conical tank. Other examples of these are featured in similar research such as “Design and Comparison of Strategies for Level Control in a Nonlinear Tank” [20] and “Real-Time Level Control of Conical Tank and Comparison of Fuzzy and Classical Pid Controller” [21].

The controllers in these works have been validated by simulation, while in this research, control proposals are also implemented and validated experimentally. In addition, a model predictive control (MPC) strategy is proposed to protect actuators and, at the same time, provide good performance.

MPC is an advanced technique that provides a sophisticated way to control complex systems in real-time, considering multiple variables and constraints to achieve desired objectives efficiently and accurately. In this research, a model predictive control (MPC) strategy is designed and implemented for the level control of a conical tank on a laboratory scale. The evaluation performance of the controller is compared with a traditional integral (PI) controller, and the rise time, settling time, overshoot, and steady-state error are analyzed when subjected to different set points. In addition, an interactive interface is developed in LabVIEW software 2021 Sp1 so that students can directly experience how MPC works in real or simulated situations, which would provide them with a deeper and more practical understanding of this advanced technique.

The contributions of this research include (i) an approximate model of a conical tank plant on a laboratory scale; (ii) the design of an interactive interface that allows for the applications of automatic control and advanced control to a conical tank; (iii) and the design, implementation, and validation of an advanced MPC proposal that improves the performance of level control in a conical tank in comparison to classic PI controllers.

2. Description of Conical Tank

This section presents the technical characteristics of the design and implementation of a conical tank for level control on a laboratory scale, as well as the mathematical modeling of the process.

2.1. Design and Implementation of a Conical Tank on a Laboratory Scale

In the structural design procedure of a plant on a laboratory scale, it is relevant to consider the following parameters: the hydraulic and mechanical structure of the plant, as well as the assembly design, programming requirements, and necessary components for the implementation of the conical tank.

Figure 1 shows the implemented experimental plant, which consists of a SITRANS LR140 radar sensor from the SIEMENS brand, a conical tank structure made of steel, manual valves, a display interface, a storage tank, and a centrifugal pump triphasic.

Table 1. Conical tank specifications.

Equipment	Details
Conical Tank	It is designed with stainless-steel material, its height is 80 cm, its upper diameter is 35.5 cm, its lower diameter is 7.64 cm reduced to 2.54 cm, and its thickness is 1.5 mm.
Radar-level sensor	The SITRANS LR140 model from the SIEMENS brand is a type of sensor especially used for conical tanks since it has a wide measurement range and accuracy.
Disturbance valve	The disturbance ball valve is 1 and nickel plated.
Tank fill valves	The nickel-plated valve (1 in.) is used to fill the water.
Pipeline	The pipe for filling and draining the conical tank is made of PVC, which measures 1 in.
Storage tank	For storage, the tank has a height of 34.5 cm, a width of 61 cm, and a surface of 40 cm. This tank is used to drain water to the pump, fill it through the disturbance outlet of the conical tank, and fill it from the pump outlet.

and

Table 2. Implemented conical tank specifications.

Instrument	Material	Dimensions
Conical tank	Stainless steel	80 cm high, largest diameter = 35 cm, smallest radius = 1 in; sensor support, metal structure
Pump	Paolo brand	1/2 HP, 115 V to 230 V, 2.5 Amp, 3400 RPM, Q.MAX = 40 L/m
Electric components	-	Fuses, resistance, electric cable, 220 V to 110 V transformer, I/V converter
Variable frequency drive	SIEMENS	220 V/240 V
Radar level sensor	SIEMENS	SITRANS LR140
24 V Power Supply	SIEMENS	LOGO Power 24 V

show the materials and dimensions of all the elements that comprise the implemented laboratory-scale conical tank plant.

Figure 2 shows the P&ID diagram of the laboratory-scale conical tank plant where its control is carried out through a variable frequency drive that has a single-phase 220 VAC input, generating a three-phase output that is directed towards the three-phase pump, where the manipulated variable is in a frequency range of 0–60 Hz. Additionally, to measure the water level, a SITRANS LR40 radar transmitter (LT) is chosen, with 2 threads, 4–20 mA output, and a current converter to voltage (0–10 V) to be able to obtain output measurements of the level where the controlled variable is the water level in a range of 0–80 cm. The level process has two modes of operation, manual and automatic.

Additionally, in the interactive learning interface in LabVIEW software, a plant is graphically designed to resemble the real process. Similarly, for better organization of PID and MPC visualization, window management is designated. Finally, the numerical and graphical behavior of the process variable, the set points, and the control variable is shown, resulting in a dynamic interface (see Figure 3) where the user can set various set points, read and store data in real-time, and, most importantly, observe the real-time behavior of the conical tank level process.

Figure 4 shows the parameters configured for the PID control strategy of the conical tank in LabVIEW. It is composed of a switch that allows the control process to be changed from automatic to manual. Additionally, it has a parameter block to designate gains, and, at the bottom, the maximum and minimum ranges of control actions are located.

Figure 5 shows the design parameters of the MPC interface, which are composed of a state-space block to locate the plant model, a block to input the real values of the plant, which allows for designating the working range of control actions, and a tuning parameters block in which the prediction horizon, control horizon, and weights are established to give greater importance to either the control action or the errors.

2.2. Modeling of the Conical Tank

To obtain the mathematical model of the conical tank, the most significant characteristics of the system must be considered, such as its inlet and outlet flows, lower radius and upper radius, height, etc. For this reason, to derive the mathematical model and perform level control, the results are shown in

Table 3. Parameters of conical tank.

Parameter	Worth	Description
${\mathrm{R}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	35 cm	Greater upper radius of the cone
${\mathrm{R}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$	2.54 cm	Bottom radius of cone
${\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	80 cm	Maximum cone height
${\mathrm{F}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	-	Maximum flow capacity
$\mathrm{F}\left(\mathrm{t}\right)$	-	Inflow
${\mathrm{F}}_{\mathrm{o}}$	-	Outflow
${\mathrm{K}}_{\mathrm{v}}$	-	Kinematic viscosity
$\mathrm{h}$	-	Liquid level
$\mathrm{r}$	-	Liquid level radius

.

$\mathrm{h}$, $\mathrm{F}\left(\mathrm{t}\right)$, $\mathrm{r}$, are variables, and the rest are constants; with these data, we look at Figure 6.

It begins with the relationship that exists between the variation in the material balance [mass (m)] with the inputs [inflow is a variable, $\mathrm{F}\left(\mathrm{t}\right)]$ and outputs (outflow is not a constant, ${\mathrm{F}}_{\mathrm{o}}$) that directly influence the conical tank, as observed in Equation (1).

$${\displaystyle \frac{\mathrm{d}\mathrm{m}}{\mathrm{d}\mathrm{t}}}=\mathrm{F}\left(\mathrm{t}\right)-{\mathrm{F}}_{\mathrm{o}}$$

(1)

It is considered that the relationship between mass and volume is represented in Equation (2), where $\mathsf{\rho }$ is the density and $\mathbf{V}$ is volume.

$$\mathsf{\rho }={\displaystyle \frac{\mathrm{m}}{\mathrm{V}}}$$

(2)

By clearing the mass, we have the following Equation (3):

$$\mathrm{m}=\mathsf{\rho }\mathrm{V}$$

(3)

And when applying the derivative of each element, considering that the density is a constant, we have Equation (4).

$${\displaystyle \frac{\mathrm{d}\mathrm{m}}{\mathrm{d}\mathrm{t}}}=\mathsf{\rho }{\displaystyle \frac{\mathrm{d}\mathrm{V}}{\mathrm{d}\mathrm{t}}}$$

(4)

The volume of the conical tank is represented according to Equation (5), where r is the radius that depends on the level of the tank (h). By substituting Equation (5) for Equation (4), we obtain Equation (6).

$$\mathrm{V}={\displaystyle \frac{1}{3}}\mathsf{\pi }{\mathrm{r}}^2\mathrm{h}$$

(5)

$${\displaystyle \frac{\mathrm{d}\mathrm{m}}{\mathrm{d}\mathrm{t}}}=\mathsf{\rho }{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\left({\displaystyle \frac{1}{3}}\mathsf{\pi }{\mathrm{r}}^2\mathrm{h}\right)$$

(6)

The radius and the level of the tank present a relationship, which can be approximated as shown in the following Figure 7.

Therefore, we have Equation (7).

$$\mathrm{r}={\displaystyle \frac{{\mathrm{R}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\mathrm{h}$$

(7)

By replacing Equation (7) with Equation (6), we obtain (8).

$${\displaystyle \frac{\mathrm{d}\mathrm{m}}{\mathrm{d}\mathrm{t}}}=\mathsf{\rho }{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\left({\displaystyle \frac{1}{3}}\mathsf{\pi }{\displaystyle \frac{{{\mathrm{R}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2}{{{\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2}}{\mathrm{h}}^3\right)$$

(8)

Applying the derivative corresponding to Equation (9), we obtain the following:

$${\displaystyle \frac{\mathrm{d}\mathrm{m}}{\mathrm{d}\mathrm{t}}}=\mathsf{\rho }\mathsf{\pi }{\displaystyle \frac{{{\mathrm{R}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2}{{{\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2}}{\mathrm{h}}^2\left({\displaystyle \frac{\mathrm{d}\mathrm{h}}{\mathrm{d}\mathrm{t}}}\right)$$

(9)

By replacing Equation (9) with Equation (1) and solving for dh/dt, we obtain (10).

$${\displaystyle \frac{\mathrm{d}\mathrm{h}}{\mathrm{d}\mathrm{t}}}={\displaystyle \frac{\mathrm{F}\left(\mathrm{t}\right)-{\mathrm{F}}_{\mathrm{o}}}{\mathsf{\rho }\mathsf{\pi }\frac{{{\mathrm{R}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2}{{{\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2}{\mathrm{h}}^2}}$$

(10)

Considering that a conical tank has dynamic balance, the output flow is a function of the square root of the height of the water in the tank, as shown in Equation (11).

$${\mathrm{F}}_{\mathrm{o}}={\mathrm{K}}_{\mathrm{v}}\sqrt{\mathrm{h}}$$

(11)

where ${\mathrm{F}}_{\mathrm{o}}$ is the flow, and ${\mathrm{K}}_{\mathrm{v}}$ is the viscosity constant.

Therefore, by substituting Equation (11) for Equation (10), we obtain Equation (12), which represents the mathematical model of the conical tank.

$${\displaystyle \frac{\mathrm{d}\mathrm{h}}{\mathrm{d}\mathrm{t}}}={\displaystyle \frac{\mathrm{F}\left(\mathrm{t}\right)-{\mathrm{K}}_{\mathrm{v}}\sqrt{\mathrm{h}}}{\mathsf{\rho }\mathsf{\pi }\frac{{{\mathrm{R}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2}{{{\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}^2}{\mathrm{h}}^2}}$$

(12)

As can be seen, the result obtained indicates that we have a nonlinear differential equation.

This mathematical model is used for the design of the simulation and interactive interface of the conical tank, as shown in Figure 1. It is important to note that the model is nonlinear concerning time as represents the plant; therefore, this model is not used for the controller design methodology. In this work, the model was previously validated with a real plant.

2.3. Design of Control Algorithms for the Conical Tank

For the design of control algorithms, two strategies are applied: a traditional one based on PI and another advanced one using MPC. The described strategies are implemented in the interactive learning interface of the conical tank and are also evaluated on a real plant at the laboratory scale.

2.3.1. PI Controller Design for the Conical Tank

The control architecture used for the PI algorithm is shown in Figure 8, where the PI controller generates the control actions directed to the frequency converter, which operates from 0 to 60 Hz, with input voltage changes of 0–5 V. In this way, the speed of pumping the liquid into the conical tank is controlled depending on the set point.

The designed PI control algorithm is shown in Equation (13), where $\mathrm{u}\left(\mathrm{t}\right)$ is the control action (0–60 Hz), $\mathrm{e}\left(\mathrm{t}\right)$ is the control error, ${\mathrm{K}}_{\mathrm{p}}$ is the proportional gain, ${\mathrm{T}}_{\mathrm{i}}$ is the integral action time (min), and ${\mathrm{T}}_{\mathrm{d}}$ is the derivative time (min).

$$\mathrm{u}(\mathrm{t})={\mathrm{K}}_{\mathrm{p}}[\mathrm{e}\left(\mathrm{t}\right)+{\displaystyle \frac{1}{{\mathrm{T}}_{\mathrm{i}}}}\int \mathrm{e}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}+{\mathrm{T}}_{\mathrm{d}}{\displaystyle \frac{\mathrm{d}\left(\mathrm{e}\left(\mathrm{t}\right)\right)}{\mathrm{d}\mathrm{t}}}]$$

(13)

To obtain the constants of the reference gains ${\mathrm{K}}_{\mathrm{p}}$, ${\mathrm{T}}_{\mathrm{i}}$, and ${\mathrm{T}}_{\mathrm{d}}$, the HAALMAN tuning technique is used, which is based on a forced approximation model to a first-order system with time delay. To obtain the parameters, an input and output model is designed by taking measurements of the input (frequency) and the output level and using computational tools; the model shown in Equation (14) is obtained, and it is then defined as $\mathsf{\sigma },\mathsf{\tau },\mathsf{\theta }$.

$$\begin{array}{c}\mathrm{G}\left(\mathrm{s}\right)={\displaystyle \frac{\mathsf{\sigma }{\mathrm{e}}^{-\mathsf{\theta }\mathrm{s}}}{1+\mathsf{\tau }\mathrm{s}}}\\ \mathrm{G}\left(\mathrm{s}\right)={\displaystyle \frac{1.6181{\mathrm{e}}^{-3\mathrm{s}}}{1+30.195\mathrm{s}}}\end{array}$$

(14)

The reference gains are obtained; however, as expected, the gains obtained by this technique are reference, this is mainly because this technique is applied to first-order linear systems and the known tank does not have these characteristics, but they are a starting point to then apply fine-tuning. Finally, the profits obtained are as follows: ${\mathrm{K}}_{\mathrm{p}}$ is 4.1468, and ${\mathrm{T}}_{\mathrm{i}}$ is 0.50325 (min).

To obtain the PI control gains, the HAALMAN method is used. This method requires approximation to a first-order model with time delay, whose parameters are used by applying Equation (14). These gains are initialization gains because it is necessary to perform fine-tuning, trial, and error; this is because the technique uses a model that approximates a first-order system, whereas the conical tank has different characteristics.

It is then established that the HAALMAN method results in a better response in aspects such as guaranteeing rapid stability without oscillations, in addition to presenting fewer errors in steady state and over-impulses.

It is important to note that the model (14) is used only to be able to apply the HAALMAN methodology for the tuning of the PI controller gains, where an approximation model is required for first-order systems with time delay because this methodology is designed for linear systems. In our cases, these gains are used as initial conditions to then perform fine-tuning because the conical tank is nonlinear.

2.3.2. Design of MPC Controller for Conical Tank

The control architecture applied to the MPC of the conical tank is described in Figure 9; it consists of a prediction model that generates a vector of the predicted dynamic variable of the process, an optimizer that includes the objective function with its operating constraints, prediction horizon, and control horizon, and ${\mathrm{h}}_{\mathrm{s}\mathrm{p}}$ is a vector of the set point.

The control law is defined by the optimization problem, where the objective function is defined by Equation (15). The control objective seeks to minimize the level of steady-state error and variation control actions. In Equation (15), ${\mathrm{h}}_{\mathrm{s}\mathrm{p}}$ is the vector set point desired level, ${\mathrm{N}}_{\mathrm{p}}$ is the prediction horizon, ${\mathrm{N}}_{\mathrm{c}}$ is the control horizon, $\widehat{\mathrm{h}}$ represents predicted level, $∆\mathrm{u}$ is the variation in the control actions $\mathrm{u}\left(\mathrm{k}\right)-\mathrm{u}(\mathrm{k}-1)$, ${\mathsf{\lambda }}_1$ is the weight for the objective of minimizing the error, and λ₂ is the weight for the objective of minimizing the variation in control actions. If ${\mathsf{\lambda }}_1$ > ${\mathsf{\lambda }}_2$ is faster in the transient response and if in ${\mathsf{\lambda }}_1$ < ${\mathsf{\lambda }}_2$, the control actions are faster, they will be more abrupt to reach the set point; thus, ${\mathsf{\lambda }}_1$ = ${\mathsf{\lambda }}_2$= 1, and k represents the position of each vector of the future set point, the future level of the deposit, and the control action. Furthermore, if $N_p$ > $N_c$, there is a longer prediction time, where the control actions using the sliding method take each sample and recalculate in shorter times. In addition, the MPC algorithm, in the design of its controller, internally uses an internal model (18) that is used to make predictions of future behavior, creating a future scenario.

$$\mathrm{J}\left(\mathrm{k}\right)=\sum _{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{p}}}{\mathsf{\lambda }}_1{\left[{\mathrm{h}}_{\mathrm{s}\mathrm{p}}\left(\mathrm{k}+\mathrm{i}|\mathrm{k}\right)-\widehat{\mathrm{h}}\left(\mathrm{k}+\mathrm{i}|\mathrm{k}\right)\right]}^2+\sum _{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{c}}}{\mathsf{\lambda }}_2{\left[∆\mathrm{u}\left(\mathrm{k}+\mathrm{i}-1\right)\right]}^2$$

(15)

The optimization problem is subject to inequality constraints that apply minimum and maximum limits of the variable to be controlled for the tank level described in Equation (16), where ${\mathrm{H}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is the minimum level, and ${\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum level. The process also presents limits associated with the control action depending on the operating ranges of the actuator. Equation (17) shows the control action constrains, where $u_{min}$ is the minimum control action, and ${\mathrm{u}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum control action.

$${\mathrm{H}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\le \mathrm{h}\left(\mathrm{k}\right)\le {\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(16)

$${\mathrm{u}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\le \mathrm{u}\left(\mathrm{k}\right)\le {\mathrm{u}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(17)

The process limits parameters applying the MPC, and the data of Equation (15) are defined in

Table 4. Parameters and tuning for MPC.

Parameters	Tuning
Prediction horizon ${\mathrm{N}}_{\mathrm{p}}$	80
Control horizon ${\mathrm{N}}_{\mathrm{c}}$	2
Weight for the goal of minimizing error	1
Weight for the objective of minimizing incremental increment $∆\mathrm{u}$	1
Minimum limit for control action ${\mathrm{u}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$	0 Hz
Maximum limit for control action ${\mathrm{u}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	60 Hz
Minimum tank level limit ${\mathrm{H}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$	0 cm
Maximum tank level limit ${\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	80 cm

. It is important to consider that ${\mathrm{u}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathrm{u}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are 0 and 5 V, whose signal is sent to the frequency converter and represents 0–60 Hz.

To design the MPC, the prediction model of the conical tank is defined, represented by a nonlinear state-space model, obtained based on real measurements with a sampling time of 0.1, and described in Equation (18).

Table 5. Matrix A of the state-space model.

A	C1	C2	C3	C4
F1	−0.03351	−0.05374	−0.2222	0.04771
F2	0.1559	−0.2318	−16.16	3.873
F3	0.3667	−1.085	−168	−14.8
F4	−0.08385	0.2712	77.55	−5.012

corresponds to matrix A,

Table 6. Matrix B of the state-space model.

B	C1
F1	0.002093
F2	0.1437
F3	0.8883
F4	−0.5224

to matrix B,

Table 7. Matrix C of the state-space model.

C	C1	C2	C3	C4
F1	−710.6	−3.566	−29.43	5.059

to matrix C, and

Table 8. Matrix D of the state-space model.

D	C1
F1	0

to matrix D.

$$\begin{array}{c}\dot{\mathrm{x}}\left(\mathrm{k}+1\right)=\mathrm{A}\mathrm{x}\left(\mathrm{k}\right)+\mathrm{B}\mathrm{u}\left(\mathrm{k}\right)\\ \mathrm{y}\left(\mathrm{k}\right)=\mathrm{C}\mathrm{x}\left(\mathrm{k}\right)+\mathrm{D}\mathrm{u}\left(\mathrm{k}\right)\end{array}$$

(18)

It is important to note that the state-space model (18) is used only for the MPC controller, generated through the system identification technique based on real measurements.

3. Results

This section presents the comparative results between a traditional PI control algorithm and an MPC controller applied to the level control of a conical tank. The validation is carried out by simulation in the interactive interface and experimentally in a real plant. The process is subjected to set point changes and also to disturbances to evaluate the performance of the level in a transient and permanent state.

3.1. Simulation Results in the Interactive Learning Interface

To evaluate the performance of the controllers implemented in the simulated plant, it is subjected to four ascending and descending set point changes every 1500 s, thus operating at different operating points.

Figure 10a shows the process variable (level) when applying two control algorithms, where the x-axis is the time in (s) and the y-axis is the value of the level in (cm). The evolution can be observed in the level after three set point changes, where the blue signal represents the response when using an MPC, while the red signal corresponds to the level with PI control. The two strategies are compared, and it can be noted that the MPC algorithm presents a shorter establishment time than with a PI; the MPC controller presents a small overshoot, which represents 0.64%.

Another variable to analyze is the control action because the useful life of the controller output depends on it. Figure 10b shows the control actions generated when applying MPC (blue color) and PI control, in red. When comparing the two control signals, it can be seen that with MPC, it is more stable and less aggressive than with a PI control.

Table 9. System simulation results and performance specifications of the level control strategies.

Method	Tr [s]	OS [%]	Ts [s]
PI CONTROL	300	0	430
MPC CONTROL	110	0.64%	367.5

shows a comparison of the performance of the two control algorithms, analyzing rise time (Tr), overshoot (OS), and settling time (Ts). The advanced MPC presents a better performance in terms of establishment time, which is three times lower than the PI control, as well as its rise time. However, it presents a small overshoot, which is acceptable for this process.

In the proposed simulator, the plant is modeled by Equation (12). The PI controller is tuned by the HAALMAN technique using the model (14). It is necessary to use this model because the methodology requires the use of this approximation; also, fine-tuning is necessary. Model (18) is used for the design of the MPC controller. The simulated plant model and the model used for the controller are different; this is an important aspect because it incorporates the modeling error and, therefore, the prediction error in the simulation, which is an aspect that occurs in real life.

3.2. Experimental Results

To evaluate the performance of the controllers implemented in the real plant, it is subjected to three ascending and descending SP changes every 2000 s, thus operating at different operating points.

Figure 11a shows the process variable (level) when applying two control algorithms in the experimental plant at that laboratory scale; the behavior of the plant is shown under the PI control in red and the MPC in blue, and the PI control signal results in control without overshoot. However, it has a longer settling time, unlike the level with MPC, which has a quick settling time, but it is considered that there are over-impulses when applying the advanced control.

Figure 11b shows the control actions, in red for PI control and in blue for MPC, and the control actions are more stable with MPC; therefore, as shown when applying it to real plants, it can be determined that the MPC algorithm allows a longer lifetime, mainly for the peripheral pump, while the PI control presents more oscillatory control actions.

Table 10. System experimental results and performance specifications of the level control strategies.

Method	Tr [s]	OS [%]	Ts [s]
PI Control	622.5	0	745
MPC Control	95	0.9%	306

shows a comparison of the performance of the two experimental control algorithms, analyzing rise time (Tr), overshoot (OS), and settling time (Ts). The advanced MPC presents better performance in terms of establishment time, which is three times lower than with the PI control, as well as its rise time. However, it presents a small overshoot of 0.9% but quickly regains control. The control operation used by the MPC algorithm is the rolling horizon strategy, where, at each step, the predicted control actions are recalculated, and the inputs are updated at each time instant.

When comparing the results with other related works such as [16], where a controller based on fuzzy logic is presented, it can be considered that the fuzzy control fails to stop the oscillations; that is, it cannot completely reach the desired value. Therefore, the control actions will be very abrupt and will risk the useful life of the process. However, MPC stabilizes immediately, and the control actions are smoother than control based on fuzzy logic.

3.3. Experimental Results in the Face of Disturbances

For the analysis of the experimental results against disturbances, a disturbance is performed through the percentage opening of 50% to 60% of the manual valve located at the outlet of the conical tank at 1250 s.

Figure 12a shows the level before disturbances with the PI control (in red) and the MPC (blue); it can be noted that at 1250 [s], when the disturbance occurs, the MPC controller reacts quickly compared to the PID controller, and a similar response occurs to the set point change. Figure 12b shows that the MPC action is more stable in the steady state, which shows better operation of the actuator.

According to

Table 11. System experimental results in the face of disturbances and performance specifications of the level control strategies.

Method	Ts [s]
PI Control	504
MPC Control	333

, advanced MPC recovers quickly from disturbances, achieving fast control compensation. This is because the predictive control predicts future actions and knows the error a priori, which helps to respond quickly to disturbances. On the other hand, the PI controller responds adequately, at a similar time to MPC.

4. Conclusions

In this research, a conical tank was implemented for level control, obtained via design of an interactive learning interface. A comparison of two control algorithms, a traditional PI and an MPC, was performed in a real and simulated process, analyzing rise time (Tr), overshoot (OS), and settling time (Ts).

The efficient interaction of the interactive learning interface made in Lab-VIEW with the real plant was verified, showing similarities in the dynamics of the process both in the real-time animation of the tank level and in the activation of components, as well as the dynamic response of the level. The interface will have a direct impact on student learning since it allows multiple practices to be carried out in a nonlinear conical tank and can be used to investigate new advanced and advanced control methods, being able to put theoretical knowledge into practice.

The use of an advanced MPC algorithm improved the performance of the level control strategy used for the conical tank in terms of transient response and steady state compared to the experimental results, as well as to the traditional PI controller tuned by the HAALMAN method. Furthermore, it can be noted that the MPC presented better performance concerning the rise time, with a reduction of 527.5 [s] as compared to the value obtained for the traditional control strategy, causing the level signal to have a rapid establishment time; however, for this reason, a maximum overshoot of 0.9% was generated. When faced with disturbances, the establishment time of the MPC was less than 171 [s] than in the PI controller, in an MPC controller of 333 [s] and PI controller of 504 [s]. It can be concluded that when using an MPC, the rise and establishment time is faster than with a PI controller. Furthermore, when the set point changes, the control actions are more stable and less aggressive to the actuator.