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Active Disturbance Rejection Control of a Coupled-Tank System.

1. Introduction

In process control, the liquid level control in multiple connected tanks performed by controlling the liquid flow is a typical nonlinear control problem present in many industrial processes. A mathematical model of the plant is required to design a controller to maintain a constant level in such tanks. The mathematical model of the controlled plant can be obtained by two techniques: analytical and experimental. The mathematical models for a coupled-tank system are obtained by applying the laws of energy conservation, mass conservation, etc. The mathematical models obtained by means of analytical designs are generally complex and most often contain nonlinear dependencies of variables. The need to estimate system uncertainties using input and output is a fundamental problem of the control theory. Different methodologies were proposed to solve this problem. Among them are the backstepping control strategy [1], the adaptive fuzzy proportional-integral controller [2], the neurofuzzy-sliding mode controller [3], the hybrid fuzzy inference system that uses artificial hydrocarbon networks at the defuzzification step or the so-called fuzzy-molecular control [4], and second-order sliding mode controllers (SMC) [5]. A hybrid system that combines the advantages of the robustness of the fractional control and the SMC [6] and a digital proportional integral controller [7] were also proposed. Moreover, an observed-state feedback controller via eigenvalue assignment and linear-quadratic-Gaussian control were designed in discrete-time and implemented by an industrial controller (i.e., programmable logic controller) [8]. Several other researchers reported model-based controllers as the development of an optimal PID controller for controlling the desired liquid level using the particle swarm optimization (PSO) algorithm for optimizing the PID controller parameters [9]. A static sliding mode control scheme was proposed for the system [10], and two different dynamic sliding mode control schemes were proposed to reduce the chattering problem associated with the static sliding mode control scheme.

The active disturbance rejection control (ADRC) [11] is a method that does not require a complete mathematical description of the system. The basic idea for this method comprises the use of an extended observer coupled with a feedback controller in the closed-loop control. The observer estimates all states of the system, uncertainties, and external disturbances (total uncertainty). The total uncertainty is considered as an extended state of the system. If the estimation of the observer is accurate, the system to be controlled is converted to a simpler model because the total uncertainty is canceled in real time. The ADRC method has been successfully applied to several practical problems [12-14].

The current study aims to apply the ADRC method to regulate the liquid level in the second tank of a coupled-tank system. In addition, this study attempts to reduce the tuning parameters of the ADRC, use the time-varying parameters of the observer and the controller, and optimize the parameters of the observer and the controller using the integral absolute error (IAE) as the cost function and the genetic algorithm as the optimization method. To the best of the author's knowledge, this is the first study to apply the ADRC method to the problem of the coupled-tank system.

The remainder of this paper is structured as follows: Section 2 derives a mathematical model of the coupled-tank system and introduces the ADRC method; Section 3 describes the ADRC method used herein and discusses the simulation results; and Section 4 concludes this paper.

2. Methods

2.1. Mathematical Modeling of the Coupled-Tank System. Figure 1 shows a schematic of the coupled-tank system, which consisted of two connected tanks. A pump supplied water into the first tank (q). The second tank was filled from the first tank via a connecting pipe ([q.sub.1]). An outlet was located at the bottom of the second tank to change the output flow [q.sub.2]. The mathematical model of the coupled-tank system is nonlinear.

We derive the following equation by applying the flow balance equation for tanks 1 and 2 [10]:

[dh.sub.1]/dt = 1/A (q - [q.sub.1])

[dh.sub.2]/dt = 1/A ([q.aub.1] - [q.sub.2]) (1)

In (1), [q.aub.1] and [q.aub.2]are defined as follows [10]:

[q.aub.1] = [a.aub.1] [square root of (2g([h.aub.1] - [h.sub.2]))] for [h.sub.1] > [h.sub.2]

[q.aub.2] = [a.sub.2] [square root of (2g[h.sub.2])] for [h.sub.2] > 0 (2)

where [h.sub.1] and [h.sub.2] are the water level in tanks 1 and 2, respectively; q is the inlet flow rate; [q.sub.1] is the flow rate from tanks 1 to 2; A is the cross section area for both tanks; [a.sub.1] is the area of the pipe connecting the two tanks; [a.sub.2] is the area of the outlet; and g is the constant of gravity. The system can be considered as a single input-single output system (SISO) if the inlet flow q is selected as the input and the liquid level [h.sub.2] in the second tank is selected as the output. The dynamic model of the coupled tanks is described by the following equation [10]:

[mathematical expression not reproducible] (3)

Parameters [k.sub.1] and [k.sub.2] are defined as follows:

[k.sub.1] = [a.sub.1] [square root of (2g)]/A

[k.sub.2] = [a.sub.2] [square root of (2g)]/A (4)

Note that q is always positive, which means that the pump can pump water into the tank (q [greater than or equal to] 0). At equilibrium, for the constant water level set point, the derivatives with regard to the water levels in the two tanks must be zero, such that the following condition can be written:

d[h.sub.1]/dt = d[h.sub.2]/dt = 0 (5)

Therefore, the following algebraic relationship holds when (3) is used in (5):

[mathematical expression not reproducible] (6)

The equilibrium flow rate q can be calculated as follows:

q = -[Ak.sub.1]sign([h.sub.1] - [h.sub.2]) [square root of ([absolute value of ([h.sub.1] -[h.sub.2]))] (7)

In the case of coupled tanks, the inequality [h.sub.1] [greater than or equal to] [h.sub.2] holds in every operating point, which implies that the terms [k.sub.1] sign([h.sub.1] - [h.sub.2]) [greater than or equal to] 0. The dynamic model can then be written as

[dh.sub.1]/dt = -[k.sub.1] [square root of ([h.sub.1] - h.sub.2])] + [k.sub.2] [square root of ([h.sub.2])]

[dh.sub.2]/dt = -[k.sub.1] [square root of ([h.sub.1] - h.sub.2])] - 2[k.sub.2] [square root of ([h.sub.2])] + 1/A u (8)

Using the following transformation,

[x.sub.1] = [h.sub.2]

[x.sub.2] = -[k.sub.1] [square root of ([h.sub.1])] + [k.sub.2] [square root of ([h.sub.1] - [h.sub.2])] (9)

Eq. (8) can be written as

[[??].sub.1] = [x.sub.2]

[[??].sub.2] = f (x, t) + g(x, t) u

y = [x.sub.1] (10)

Accordingly, f(x, t) and g(x, t) in (10) have the following form:

[mathematical expression not reproducible] (11)

2.2. Active Disturbance Rejection Control. The ADRC method is explained on the second-order SISO dynamical system of the following form:

[[??].sub.1] = [x.sub.2]

[[??].sub.2] = f(x, t) + d(t) + g(x, t)u(t)

y = [x.sub.1] (12)

where u(t) and y(t) are the system input and output, respectively The nonlinear function f(x, t) is the internal dynamics of the system, and d(t) is the external disturbance. Taking the estimation value of g(x, t) as [b.sub.0], (15) can be rewritten as follows:

[mathematical expression not reproducible] (13)

where the state variables [x.sub.1] and [x.sub.2] are the system states, and [x.sub.3] = f is added as an additional state representing the total disturbance. The states of (13) are estimated using an extended state observer (ESO). The main advantage of an ESO is that it can estimate the total uncertainties without knowledge of the system's mathematical model. The ESO treats the total uncertainties as a new state. An ESO for the second-order system is constructed as follows [15,16]:

[mathematical expression not reproducible] (14)

The time-varying function R(t) has the following form:

R(t) = [R.sub.0] 1 - [e.sup.-at]/1 + [e.sup.-at] (15)

The parameter [[alpha].sub.1] in (21) can be determined, such that the characteristic polynomial

[lambda] (s) = [s.sup.3] + [[alpha].sub.1][s.sup.2] + [[alpha].sub.2]s + [[alpha].sub.3] (16)

is Hurwitz.

If the observer tuning procedure is adequate, the observer states converge to the system states [[??].sub.1] [right arrow] [x.sub.1], [[??].sub.2] [right arrow] [x.sub.2], and [[??].sub.3] [right arrow] [x.sub.3] in finite time.

The control objective is to cancel the total disturbance while satisfying the tracking task. The total disturbance is rejected with the system input signal:

u (t) = -[[??].sub.3] (t) [u.sub.0] (t)/[b.sub.0] (17)

where [u.sub.0] (t) is a control signal from a feedback controller. Substituting (17) in (13) and assuming an accurate estimation of the total disturbance, the controlled system transforms to a double integrator:

[mathematical expression not reproducible] (18)

A double integrator can be controlled with any classical controller design. The following control law can be obtained if a linear proportional and derivative controller is used:

[mathematical expression not reproducible] (19)

where r(t) and [??](t) are the reference signal and its derivative, respectively; and [[??].sub.1] (t) and [[??].sub.2](t) are the estimated states of the plant. One possible method to simplify the controller tuning is to set

[k.sub.d] = R (t)

and [k.sub.p = [R.sup.2] (t)/4. (20)

Figure 2 shows the block diagram of the ADRC closed-loop system.

3. Results and Discussion

Table 1 lists the numerical values of the parameters of the coupled-tank system [10].

The range of the pump flow rate was limited between [u.sub.min] = 0 and [u.sub.max] = 50 [[cm.sup.3]/s].

The Methods clearly showed that the parameters of the closed-loop control using ADRC are [[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3], [R.sub.0], [b.sub.0], and a.

The Hurwitz characteristic polynomial is selected as follows with the poles -4.4848, -0.2576 + 2.5735i, -0.2576 - 2.5735i:

[lambda] (s) = [s.sup.3] + 5[s.sup.2] + 9s + 30 (21)

Accordingly, [R.sub.0], [b.sub.0], and a were obtained using a genetic algorithm optimization method with the objective of minimizing the IAE defined as follows:

IAE = [[integral].sup.t.sub.0] [absolute value of (e)] dt (22)

The optimum parameters obtained are [R.sub.0] = 78.14, [b.sub.0] = 0.30, and a = 0.99.

Figure 3 shows the regulation performance of the controller for a desired level of 6 cm and confirms that the controllers successfully regulated the water level. Figure 4 depicts the control signal of the ADRC. Figure 5 presents the ESO performance in estimating the system states. The observer accurately estimated the states. The errors e1(t) = [x.sub.1] - [[??].sub.1], [e.sub.2](t) = [x.sub.2](t)-[[??].sub.2](t) and [e.sub.3](t) = [x.sub.3](t)-[[??].sub.3](t) converged to zero in less than 1 s.

The following performance measures were introduced to facilitate a comparison with the other control methods: the settling time defined as the time taken until the output finally settles within 2% of the steady state value; the rise time, [T.sub.r], defined as the time taken by the output to change from 10% to 90% of its final value; and, in addition to the IAE, the integral squared error (ISE) and the time weighted absolute error (ITAE) computed as follows:

[mathematical expression not reproducible] (23)

ITAE = [[integral].sup.t.sub.0] t [absolute value of (e)] dt (24)

The performance of the ADRC was then compared with that of the SMC method reported in [10]. Table 2 presents the rise time, settling time, and error indices (i.e., IAE, ISE, and ITAE) for the design in [10] and the ADRC method. The table clearly shows that the ADRC outperforms the other designs in all performance measures. The response of the system controlled by the ADRC took 58.2 s to settle, whereas that in the design in [10] took 113 s. The rise time of the output response in the ADRC controller was 46 s, whereas that of the SMC was 52 s. The ADRC method resulted in a 48% smaller settling time than that in [10]. Moreover, the rise time was 27% smaller than that in [10]. The IAE, ISE, and ITAE were 41%, 50%, and 56% smaller than those in [10].

As a second test, we tested the ADRC in a tracking test. The set point tracking test consisted of successively changing the set point during the operation (Figure 6). The set point change was performed at 200 s by a magnitude of 6 cm height in the water level. Consequently, the ADRC method accurately tracked the set point changes in the water level. The same parameters were used for the ADRC for the tracking experiment.

As a third test, we checked the ADRC performance against the input disturbance. An external flow rate of 60 cm3/s that started at 150 s and ended at 200 s was applied. Figure 7 illustrates the closed-loop response of the ADRC control and shows how fast the controller response was to the disturbance and corrected it.

4. Conclusions

In this study, the ADRC approach was successfully implemented with the design tested by a simulation to control the water level in the second tank of a coupled-tank system. The effectiveness of the ADRC method was verified through computer simulations. The results showed that this control method can control a nonlinear system at all possible operating points. The designed ADRC achieved the desired transient response with small rise and settling times. The advantages of the ADRC are as follows: (a) easiness and simplicity in design; (b) nonrequirement of a mathematical model of the plant; and (c) robustness against uncertainty and disturbance. Further work is anticipated in the practical implementation of the proposed ADRC technique.

https://doi.org/10.1155/2018/7494085

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that they have no conflicts of interest.

References

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Fayiz Abukhadra

Faculty of Engineering, King Abdulaziz University, Rabigh, Saudi Arabia

Correspondence should be addressed to Fayiz Abukhadra; fabukhadra@kau.edu.sa

Received 15 April 2018; Accepted 26 June 2018; Published 8 July 2018

Academic Editor: Kamran Iqbal

Caption: Figure 1: Schematic diagram of the coupled-tank system.

Caption: Figure 2: Block diagram of the ADRC for second-order system.

Caption: Figure 3: Response of the system for the 6 cm desired level.

Caption: Figure 4: Performance of the ESO in estimating the system states.

Caption: Figure 5: Control signal of the ADRC for the 6 cm desired level.

Caption: Figure 6: Set point tracking performance of the system.

Caption: Figure 7: Closed-loop response of the system against the input disturbance.
Table 1: Characteristic of the coupled-tank system.

Gravitational rate g                       981 cm/[s.sup.2]
Cross-sectional area of both tanks         208.2 [cm.sup.2]
Area of the connecting pipe [a.sub.12]      0.58 [cm.sup.2]
Area of the outlet [a.sub.2]                0.3 [cm.sup.2]

Table 2: Comparison of the performance index measures.

                     Method
Performance
measure          SMC [10]   ADRC    Improvement [%]

Settling time     113.1     58.2           48
Rise time (s)      52.1     37.8           27
IAE               114.5     6786           41
ISE               254.8     126.8          50
ITAE               2690     1190           56
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Title Annotation:Research Article
Author:Abukhadra, Fayiz
Publication:Journal of Engineering
Date:Jan 1, 2018
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