Sliding Mode Control of Coupled Tank Systems Using Conditional Integrators

Sankata Bhanjan Prusty, Sridhar Seshagiri, Umesh Chandra Pati,, and Kamala Kanta Mahapatra,

Abstract—For the problem of set point regulation of the liquid level in coupled tank systems, we present a continuous sliding mode control (SMC) with a “conditional integrator”, which only provides integral action inside the boundary layer. For a special choice of the controller parameters, our design can be viewed as a PID controller with anti-windup and achieves robust regulation.The proposed controller recovers the transient response performance without control chattering. Both full-state feedback as well as output-feedback designs are presented in this work. Our output-feedback design uses a high-gain observer (HGO) which recovers the performance of a state-feedback design where plant parameters are assumed to be known. We consider both interacting as well as non-interacting tanks and analytical results for stability and transient performance are presented in both the cases. The proposed controller continuous SMC with conditional integrators (CSMCCI) provides superior results in terms of the performance measures as well as performance indices than ideal SMC, continuous SMC (CSMC) and continuous SMC with conventional integrator (CSMCI). Experimental results demonstrate good tracking performance in spite of unmodeled dynamics and disturbances.

I. INTRODUCTION

AN important problem in the process industries is to control liquid level as well as flow between tanks in coupled tank systems (CTS) [1]-[4]. Industrial applications of liquid level control include food processing, filtration, effluent treatment, water purification systems, nuclear power plants, and automatic liquid dispensing and replenishment devices, to name but a few. Several researchers have investigated the problem of liquid level control of which we mention a few below. Traditional adaptive control methods such as gain scheduling, self tuning regulator (STR), and model reference adaptive control (MRAC) designs have been applied to improve the system’s transient performance in [5], [6]. A constrained predictive control algorithm has been presented in[7]. In [8], a new two degree of freedom level control scheme has been developed analytically based on the internal model control for processes with dead time. This method is simple and transparent in design to produce smooth response. A web based laboratory control experiments of coupled tank setup has been discussed in [9]. It has described the impact on both teaching as well as research.

A fractional proportional-integral-derivative (PID) controller has been experimentally validated through a conical tank level in [10]. A new robust optimal decentralized PI controller based on nonlinear optimization for liquid level control in a coupled tank system has been described in [11], where the stability and performance of the controller have been verified by considering multiplicative input and output uncertainties.Intelligent control algorithms, including fuzzy control [12],neural network control [13]-[15] and genetic algorithms (to tune MRAC parameters) [6] have also been applied.

Several sliding mode control (SMC) designs have also appeared in the literature. A “static” and two “dynamic” SMC methods for the coupled tanks system has been proposed by Almitairi and Zribi [16]. A second order sliding mode control(SOSMC) based on the super-twisting algorithm has been presented in [17]. SMC for the quadruple tank process [18]has also been developed by Biswaset al. [19], where a controller is designed based on the feedback linearization and SMC that the authors claim outperforms traditional PI control.A high-gain output feedback design for the quadruple tank process has been described in [20], and an adaptive sliding mode design that replaces the discontinuous term in ideal SMC with an adaptive PD term in [21].

Our current work focuses on set point regulation of the liquid level in two tank systems using a continuous approximation to sliding mode control with conditional integral action based on the results in [22]. Both statefeedback as well as output-feedback designs are considered.The CTS has a relative degree of 2, for which the control can be viewed as a saturated high-gain PID control with antiwindup. This is significant because industrial practice has traditionally relied on such controllers. The SMC controller with conditional integrator (CI) retains the desirable properties of ideal SMC such as robustness to uncertain parameters/matched disturbances, but without control chattering, so that there is less wear and tear in mechanical components such as valve actuators. This is achieved by the conditional integral action, which only requires the boundary layer width to be sufficiently small and can be tuned experimentally, to recover both transient as well as steady-state performance of a benchmark ideal SMC. Another significant contribution is that we provide analytical results for regional/semi-global regulation and performance recovery, along with guidelines for tuning the controller parameters based on a simple/intuitive ideal SMC design. Such results are had to obtain with a general PI/PID design derived from other considerations. The proposed controller continuous SMC with conditional integrators (CSMCCI) has been compared with the ideal SMC, continuous SMC (CSMC) and continuous SMC with conventional integrator (CSMCI) in terms of performance measures and performance indices. Other applications of the conditional integrator design include[23]-[26].

The rest of the paper is organized as follows: the dynamic mathematical model of CTS (interacting and non-interacting)is briefly presented in Section II. The SMC controller with CI is presented and discussed in Section III with analytical results, extracted from [22]. In Section IV, we present experimental results that validate our analytical results and the efficacy of the proposed approach. The work is summarized with some concluding remarks in Section V. A preliminary version of this paper appeared in [27]; this work expands upon[26] in two directions: 1) extending the results to the case of non-interacting systems, and 2) provinding experimental validation of the simulation results therein.

II. DYNAMIC MODEL OF COUPLED TANK SYSTEMS

The schematic diagram of interacting CTS is shown in Fig. 1.In the figure,q1is the inlet flow rate into the Tank 1,q12is the flow rate from Tank 1 to Tank 2 through the valve andq2is the flow rate at the outlet of Tank 2 through a valve. The variablesh1andh2denote liquid levels of Tank 1 and Tank 2 respectively. The valve connecting the two tanks allows water to flow from Tank 1 to Tank 2 and the valve connected at the output of Tank 2 allows water out to a reservoir. The control input to the process is the inlet flow rateq1and the level of liquid in Tank 2h2is the output of the process. The aim of the process is to regulate the inlet flow rateq1so as to keep the level of liquid in Tank 2h2to a desired set point levelH. A Centrifugal pump is provided to supply the water from the reservoir to the first tank.

Using mass balances and Bernoulli’s law, the dynamic model of the interacting two-tank system is as follows ([15]):

Fig. 1. Schematic diagram of interacting coupled tank system.

Defining the diffeomorphic change of variables

We rewrite the system in (1) in standard normal form below

For the non-interacting case, the schematic diagram is as shown in Fig. 2 and the model equations are modified as follows:

The change of variables

transforms the system to normal form as

The control design is discussed in Section III.

Fig. 2. Schematic diagram of non-interacting coupled tank system.

III. CONTROL DESIGN

The control objective is to regulate the liquid levelh2of Tank 2 to a desired reference levelHby manipulating the inlet flow rateq1. While we focus on the case of interacting tanks for definiteness, all of our discussions easily carry over to the non-interacting case as well. Define the error variables

From (3), it is clear that the system is of relative degree 2,with no zero dynamics. We assume that the measured outputs are the tank levelsh1andh2, so that full-state feedback (that uses the erroreor the statex) requires the parametersa1anda2to be known.

A. State-Feedback Design

In ideal SMC design, the sliding surface can be chosen as

wherek1＞ 0. This guarantees that when the motion is restricted tos=0, the errore1converges asymptotically to zero.Taking the derivative of (8) and using (3)1In the non-interacting case, the form of the equations are the same, except for the definitions of f(e) and g(e)., we have

Note thatg(e) is physically bounded away from zero, i.e.,g(e)＞0 8e2 R2. It is standard to design the controluto have two components: a “ nominal control” that cancels known terms in (9) and a “sliding component” that overbounds the uncertain terms, so thatss˙ ＜0 . Accordingly, the controlucan be defined as

and suppose

where sgn(·) is the standard signum function. It is easy to check that this guaranteesfrom which it follows that the output errore1asymptotically converges to zero. If the supremum condition for the choice of ρ(.) is the entire statespace, then the stability results hold “globally” on the entire state-space.

It can be noted that one possible nominal control component design results from the choice

In order to alleviate the chattering problem due to switching nonidealities or unmodeled high frequency dynamics, it is common to replace the discontinuous term sgn(s) with it’s continuous approximation sat(s/μ), where μ＞0 is the width of the boundary layer. However, this only guarantees that the errorebecomesO(μ), but not zero. In order to make the error“small”, μ value must be small, but which could lead to chattering again. In [22], we presented a design that introduces integral action “conditionally” inside the boundary layer with the following 3 important properties

1) It recovers the asymptotic error regulation of ideal SMC but without the transient performance degradation by using a conventional integrator σ ˙=e1, and

2) The inclusion of intgeral action means that we do not have to make the boundary layer width very “small”, just small enough to stabilize the origin. This has the effect of eliminating the trade-off between tracking accuracy and robustness to high-frequency unmodeled dynamics such as actuator dynamics including lag.

3) It does not require that an ideal SMC design (without integral action) can be redone. Our design can be thought of as a “retrofit” for an original ideal SMC design.

Accordingly, we replace the ideal SMC control (12) with the continuous sliding mode controller with conditional integral action below

where σ is the output of the “conditional integrator”

wherek0＞0. From (13) and (14), it is clear that

1) outside the boundary layerso that the integrator state remains “small”.

2) inside the boundary layerwhich indicates thatei=0 at equilibrium.

Consequently, (14) is an integrator equation that produces integral action “conditionally” inside the boundary layer, i.e.,this design gives asymptotic error regulation. This design does not degrade the transient response performance which is same in case of a conventional SMC scheme that utilizes the integratorSincethe controller equations (13) -(14) is considered as a tuned saturated PID controller with anti-windup (see [22], Section VI).

B. Output-Feedback Design

Our state-feedback design uses

which assumes that the parametersa1anda2are known. If that is not the case, the control (13) can be extended to the output-feedback case by replacinge2by its estimateusing the high-gain observer (HGO)

where ? ＞0, and the positive constants α1,α2are selected in such a way that the roots of λ2+α1λ+α2=0 have negative real parts.

To complete the design, we need to specify howk, μ and? are chosen. As previously mentioned, the gainkcan simply be chosen as the largest possible control magnitude, while μ and ?are taken “sufficiently small”, the former to recover the performance of ideal (discontinuous) SMC (without an integrator) and the latter to recover the performance under state-feedback with continuous SMC. Analytical results for stability and performance are given in [22], and are paraphrased non-rigorously below for sake of completeness.

Theorem 1 (Asymptotic Error Regulation):Consider the closed-loop system consisting of the interacting two-tank system model (1) and the output-feedback control (13)-(14)withe2replaced by its estimateobtained using the high gain observer (HGO) (15). Given any compact set Δ ?R2withe(0)2 Δ, there existsk?＞0 and μ?＞0, such that fork≥k?, and μ≤μ?, there exists ??=??(μ)＞0, such that for ? ≤??, all state variables of the closed-loop system are bounded, and

Performance 1:Ifx?(t) be the state of the closed-loop system under the ideal state-feedback SMC (12) andx(t) that with the output-feedback continuous SMC with conditional integrator ( 13)-(14), withx(0)=x?(0). Then, for every τ ＞0,there exists μ?＞0, and for each μ2(0,μ?], there exists ??=??(μ)＞0 such that for ? 2(0,??], kx(t)-x?(t)k ≤τ8t≥0.

IV. RESULTS AND DISCUSSIONS

The controller designed in Section III has been experimentally validated for both interacting as well as noninteracting cases on the four tank system provided by Vi Microsystems, using LabVIEW software and the VDAS 01 data acquisition (DAQ) card of Vi Microsystems. The experimental setup is shown in Fig. 3. Tank 1 and Tank 2 are used for interacting case and Tank 2 and tank 3 are used for non-interacting case.

Fig. 3. Experimental setup of interacting two-tank system.

Two identical differential pressure transmitter (DPT) are used for level measurement. The DPT requires a supply voltage between 10.6 V and 42.4 V DC, and its output is a current between 4 mA and 20 mA. The measured signals are transmitted to the interfacing VDAS 01 DAQ card, and the control algorithm is implemented in LabVIEW with the controller outputs interfaced with a DAQ card, that outputs a current. The DAQ output goes to a current to pressure (I/P)converter, with an output pressure in the range of 3 - 15 psi.This is used to actuate a pneumatic control valve, which in turn regulates the flow of liquid into the tank. The block diagram of system, measurement and control of the interacting two-tank system is shown in Fig. 4, and the technical specifications of the experimental setup is tabulated in Table I.The front panel of interacting two-tank system is shown in Fig. 5, while that of the non-interacting two-tank system is shown in Fig. 6.

Fig. 4. Block diagram of system, measurement and control.

Simulation results for the designs of ideal SMC, continuous SMC with CI (CSMCCI) were presented in [27] with a tabulation of transient performances such as settling time, rise time, % overshoot, and error indices such as integral of absolute error (IAE), integral of squared error (ISE), time integral of absolute error (ITAE) and time integral of squared error (ITSE), and are not repeated here. It was shown that the output-feedback CSMCCI retains the error regulation and robustness to uncertainty/disturbances of ideal state-feedback SMC and it’s transient performance, but not suffering from the problem of chattering.

We present experimental results that expand on the simulation results in [27]. The area of cross-section of Tank 1 and Tank 2 are the same, i. e.,A= 176.625 cm2. The value ofA12is 3.8 cm2and the value ofA2is 2.54 cm2. The value ofgis 981 cm/s2. The desired setpoint of the output level of the interacting two-tank system is chosen asH= 15 cm, and theinput flow rate is constrained as 0 cm3/s ≤u≤ 50 cm3/sec.The controller parameters used in the simulation are chosen to bek1= 0.04 andk= 10 (for the ideal SMC), the boundary layer width is taken as μ=0.1 for the continuous approximation, the gaink0=0.4 (for the conditional integrator), and the HGO parameters (used in output feedback) are α1=1, α2=0.25, and ? =102.

TABLE I TECHNICAL SPECIFICATIONS OF THE EXPERIMENTAL SETUP

Fig. 5. Front panel of interacting two-tank system.

Fig. 6. Front panel of non-interacting two-tank system.

Figs. 7(a) and 7(e) show the experimental results for ideal SMC applied to the interacting two-tank system. It can be observed from the figure that the output response converges to its desired set pointHasymptotically, but there is a lot of chattering in the control signal, which will cause mechanical wear and tear in the valves. Figs. 7(b) and 7(f) show the experimental results of the response using the continuous SMC (CSMC). It is seen that the output converges to anO(μ)neighborhood of the desired value, i.e., there is a non-zero steady-state error; however the control chattering problem has been ameliorated. The experimental results with the continuous SMC with conventional integrator (CSMCI) are shown in Figs. 7(c) and 7(g). It can be observed from the figure that the output response tracks the desired set pointHasymptotically, but has 16% overshoot. Furthermore, as an inclusion of the integrator, the excursions of the sliding variablesoutside the boundary layer are higher, resulting in control chattering. In other words, the controller parameters(from ideal SMC) will need to be retuned to guarantee that the sliding condition is satisfied.

Fig. 7(d) shows the experimental results when the conditional integral design of this paper (continuous SMC with CI (CSMCCI)) is used. From the figure, it is observed that the output response converges to its desired set pointHfaster, and also there is no transient performance degradation.The chattering present in ideal SMC (and also conventional integral control) are absent in the CSMCCI design, as is clear from Fig. 7(h).

The comparison of transient response performance measures such as settling time, rise time, % overshoot, and the error indices (MSE, IAE, ISE, ITAE and ITSE) are tabulated in Table II for the experimental results. From the table, it is observed that the conditional integrator design (CSMCCI) has superior performance as compared to the other methods.

Next, we present results that demonstrate when disturbance enters into the system2This is done by removing water from tank 2 using the outlet hand valve.and/or the system parameter values are not exactly known. The parametersA12andA2are increased by 25% of their nominal values. As previously mentioned, this means we cannot use state-feedback design, since the sliding surface variablesrequiresx2, which is computed from the measured quantitiesh1andh2through the unknown parametersa1anda2. Therefore, we replacesby its estimatewhich in turn uses the estimateofe2=x2obtained using the HGO. For the ideal SMC though, we assume that these values are known, so that we can compare our design to this case.

The ideal SMC case is presented in Figs. 8(a) and 8(e), and it is clear that the controller rejects the disturbance. The CSMC, CSMCI, and CSMCCI designs are shown respectively in Figs. 8(b)-8(d) and Figs. 8(f)-8(h). The responses are as expected, and validate the effectiveness and superiority of the conditional integrator based CSMC design. In fact, it is clear from the figures that the responses are a) practically identical to the nominal case (when the parameters are exactly known)and b) after a small transient, the control rejects the disturbance, i.e., is robust to both unknown parameters as well as external disturbances.

Fig. 7. Experimental results for the response of liquid level in tank 2: (a) Ideal SMC; (b) CSMC; (c) CSMCI; (d) CSMCCI; (e) Control signal for ideal SMC;(f) Control signal for CSMC; (g) Control signal for CSMCI; and (h) Control signal for CSMCCI.

TABLE II EXPERIMENTAL RESULTS FOR PERFORMANCE MEASURES AND PERFORMANCE INDICES

Fig. 8. Experimental results for the response of liquid level in tank 2 in the presence of disturbance: (a) Ideal SMC; (b) CSMC; (c) CSMCI; (d) CSMCCI; (e)Control signal for Ideal SMC; (f) Control signal for CSMC; (g) Control signal for CSMCI; and (h) Control signal for CSMCCI.

For the sake of completeness, we also include experimental results for the non-interacting tank case. Figs. 9 and 10 duplicate the results of the previous two figures, but now for the non-interacting case. It is clear that our controller provides very good performance for this case as well.

Fig. 9. Experimental results for the response of liquid level in tank 2: (a) Ideal SMC; (b) CSMC; (c) CSMCI; (d) CSMCCI; (e) Control signal for Ideal SMC;(f) Control signal for CSMC; (g) Control signal for CSMCI; and (h) Control signal for CSMCCI.

Fig. 10. Experimental results for the response of liquid level in tank 2 in the presence of disturbance: (a) CSMC; (b) SMCS; (c) SMCI; (d) SMCCI; (e) Control signal for CSMC; (f) Control signal for SMCS; (g) Control signal for SMCI; and (h) Control signal for SMCCI.

V. CONCLUSION

This paper presents sliding mode control with conditional integrators to control the level of liquid in coupled tank systems. The integrator is designed in such a way that it gives integral action only conditionally, i.e. it provides integral action only inside the boundary layer. The highlight of the design is the transformation to normal form, and the use of high-gain observer to robustly estimate the derivatives that the controller needs. Unlike several designs in the literature, the controller is very simple, a saturated PID with anti-windup,which is widely used in industrial practice. We provide guidelines to tune the controller parameters to emulate an ideal SMC design, and provide analytical results for error convergence and transient performance. Simulation and experimental results demonstrate that the design recovers the performance robustness to the change in system parameters and disturbances, and hence at the advantage of increased life of the mechanical actuator. Since the design is only for minimum-phase systems, future work will include to extend the design for the non-minimum phase case. Such designs have been presented for general systems in [28].