2016 Indian Control Conference (ICC) Indian Institute of Technology Hyderabad January 4-6, 2016. Hyderabad, India
Sliding Mode Control of Coupled Tanks using Conditional Integrators Sankata B. Prusty, S. Seshagiri, Umesh C. Pati and Kamala K. Mahapatra
that replaces the discontinuous term in ideal SMC with an adaptive PD term in Larguech et al. [14]. b) Our Contribution: Our current work focuses on set point regulation of the liquid level in coupled tank systems, using a continuous approximation to sliding mode control with conditional integral action, based on the results in [15]. Both partial state-feedback as well as output-feedback designs are considered, and we also consider 2 cases: 1) a SISO case with 2 tanks, which is a relative degree two system, for which the control can be viewed as a saturated high-gain PID control with anti-windup, and 2) a MIMO case with quadruple tanks, which has vector relative degree {1, 1}, and where the zero-dynamics can be minimum-phase or non-minimum phase depending on system parameters [11]. Since our emphasis is on application of the results in [15], we are restricted by the results in that paper to the minimumphase case. However, we believe that the results in [16] can be applied to the non-minimum phase case, even though we do not pursue it here. For the minimumphase case that we consider here, the SMC controller can be viewed as a decentralized saturated high-gain PI control with anti-windup, which was also studied in [11], but based on a linearized model. As mentioned/discussed in [15], the SMC controller with conditional integrator retains the desirable properties of ideal SMC such as robust regulation, but without control chattering, and without the transient response degradation as a result of using a conventional integrator in continuous SMC. Analytical results for regional as well as semi-global regulation and performance recovery of ideal state-feedback SMC are provided in [15]. Our earlier remarks that the controllers we present are PI/PID type is significant because industrial practice has traditionally relied on such controllers, and our work provides guidelines for tuning the controller parameters. Simulation results that show good regulation in spite of disturbances, unknown parameters and comparisons with other designs are provided. Other applications of the conditional integrator design include [17]–[20]. c) Organization: The rest of this paper is organized as follows: The dynamic mathematical model of CTS is briefly presented in Section II. The SMC controller with CI is presented and discussed in Section III (this work is extracted from [15]). Section IV shows the simulation results and compares our results with an existing SMC approach. The work is summarized and some concluding remarks made in Section V.
Abstract— This paper is on set point regulation of the liquid level in coupled tanks, using continuous sliding mode control (SMC) with integral action. The integrator action occurs only conditionally inside the boundary layer. This “conditional integrator” (CI) design results in improved transient performance over conventional SMC with integral action. It also retains the robustness to uncertain system parameters and disturbances that ideal SMC exhibits, but without control chattering, which is important, because chattering can cause valve wear and tear. We consider two cases: (i) a SISO system with 2 tanks, and (ii) a MIMO system of quadruple tanks. Our controller happens to be a saturated PID controller with anti-windup structure in the SISO case, and 2 decentralized PI controllers in the MIMO case. This is of particular significance because while the application of nonlinear control strategies such as feedback linearization, adaptive/neural control and nonlinear model predictive control has reached considerable maturity for process systems, industrial practice has traditionally relied on PI/PID controllers. Simulation results show that good tracking performance is achieved, in spite of disturbances and unmodeled dynamics.
I. I NTRODUCTION a) M otivation: An important problem in the process industries is to control liquid level as well as flow between tanks in coupled tank systems (CTS) [1], [2]. Several other researchers have studied this problem, of which we mention a few below. Traditional adaptive control methods such as gain scheduling, model reference adaptive control (MRAC), and self tuning regulator (STR) designs have been applied to improve the system’s transient performance in [3]. A constrained predictive control algorithm has been presented in [4]. Intelligent control algorithms, including fuzzy control [5], neural control [6], [7] and genetic algorithms [8] have also been applied. A “static” and two “dynamic” sliding mode control (SMC) schemes for the coupled tanks system have been proposed by Almutairi and Zribi [9], and a second order sliding mode control (SOSMC) based on the super-twisting algorithm in [10]. SMC for the quadruple tank process [11] has also been developed by P. P. Biswas et al. [12], where a controller is designed based on the feedback linearization and SMC, that the authors claim outperforms traditional PI control. A highgain output feedback design for the quadruple tank process is described in [13], and an adaptive sliding mode design S. B. Prusty, U. C. Pati and K. K. Mahapatra are with the Department of Electronics and Communication Engineering, NIT, Rourkela, India,
[email protected]. S. Seshagiri is with the Department of Electrical and Computer Engineering, San Diego State University, San Diego, CA, USA,
[email protected].
978-1-4673-7993-9/16/$31.00 ©2016 IEEE
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II. DYNAMIC M ODEL A) SISO two tank system: The schematic diagram of a two-tank system is shown in Fig. 1. In the figure, q1 is the inlet flow rate into the tank 1, q12 is the flow rate from tank 1 to tank 2 through the valve and q2 is the outlet flow rate of tank 2 through a valve. The variables, h1 , and h2 denote the liquid levels of tank 1 and tank 2 respectively. The valve connecting the two tanks allows the water to flow from tank 1 to tank 2 and the valve connected at the output of tank 2 allows the water out to a reservoir. The control input to the plant is the inlet flow rate, q1 and the level in tank 2, h2 is the controlled output of the plant. A centrifugal pump is provided to supply water from the reservoir to the first tank.
Tank 3
Tank 4
h3
h4
Tank 1
Pump 1
Tank 2 Pump 2 h1
q1
h2
Resevoir
Fig. 2: Schematic diagram of the quadruple tank system Pump
Tank 1 Tank 2 h1
q12
mass balances and Bernoulli’s law, the dynamic model of the quadruple-tank system is as follows [11]: dh1 a3 √ a1 √ 2gh1 + A 2gh3 + γA1 k11 u1 = −A dt 1√ 1√ a4 a2 dh1 2gh2 + A 2gh4 + γA2 k22 u2 = −A dt 2 2 √ (4) (1−γ dh3 a3 2gh3 + A32 )k2 u2 = −A dt 3 √ a4 dh4 1 )k1 2gh4 + (1−γ = −A u1 dt A4 4
q2
h2
Reservoir
Fig. 1: Schematic diagram of the coupled tank system
where the Ai ’s are the cross sectional area of the tanks, ai ’s the cross-sectional areas of the outlet pipes, γi ’s the split fractions for the 2 valves, and g is the gravitational constant. The voltage applied to the pumps are ui and the corresponding flows ki ui . The parameters γ1 , γ2 ∈ (0, 1) are determined from how the valves are set prior to an experiment. As before, the diffeomorphic change of variables ξi = hi and (1 − γ2 )A2 (1 − γ1 )A1 h1 − h4 , η2 = h2 − h3 η1 = γ1 A4 γ2 A3
Using mass balances and Bernoulli’s law, the dynamic model of the two-tank system is as follows [9]: √ dh1 = qA1 √ − a2 (h1 − h2 ) dt √ (1) dh2 = a2 (h1 − h2 ) − a1 h2 dt √ √ where a1 = AA2 2g, a2 = AA12 2g, A is the cross sectional area of the tanks, A12 is the area of the coupling pipe, A2 is the area of the outlet pipe, and g is the gravitational constant. The flow rate, q1 cannot be negative because the pump can only pump water into the tank, i.e., q1 ≥ 0. The liquid flow rate q12 is assumed to be unidirectional from tank 1 to tank 2, so that h1 ≥ h2 . Defining the diffeomorphic change of variables x1 x2
= h2√ √ = a2 (h1 − h2 ) − a1 h2
transforms the system to normal form η˙ ξ˙i yi
(2)
We write
we rewrite the system (1) in standard normal form below x˙ 1 x˙ 2 y
= x2 ( a1 a2 ) a2 √x1 √ = 2 x2 +a1 x1 − = x1
a√ 1 x2 2 x1
− a22 +
(
a22 2A
)
[
a A = 11 0
u√ x2 +a1 x1
= ϕ(η, ξ) = bi (η, ξ) + aii ui = ξi ] [ γ1 k1 0 = A1 a22 0
0 γ2 k2 A2
(5)
] = ΓAˆ
where Aˆ is a nominal value for A; and note that A, Γ are diagonal matrices with entries bounded away from zero. The system has vector relative degree {1, 1}. The control design is discussed in the next section.
(3) B) M IM O quadruple tank system: The schematic diagram of the four tank system (FTS) is shown in Fig. 2. The variables h1 to h4 denote the liquid levels in the 4 tanks respectively. The aim is to control the levels h1 and h2 in the lower tanks with two centrifugal pumps. The process inputs are u1 and u2 . the input voltages to the pumps; note that each pump will influence the two outputs. As before, using
III. C ONTROL D ESIGN A) SISO two tank system: The control objective is to regulate the liquid level h2 of tank 2 to a desired set point level H. From (3), it is clear that the system is relative degree of 2, with no zero dynamics. We assume that the 147
measured outputs are the tank levels h1 and h2 , so that full-state feedback (that uses the state x) requires that the parameters a1 and a2 be known. In ideal SMC design, the sliding surface can be chosen as s = k 1 e1 + e2
the boundary layer. Accordingly, we replace the ideal SMC control (10) with the continuous sliding mode controller: ) ( k0 σ + k1 e1 + e2 def = −k sat(s/µ) (11) u = −k sat µ
(6)
where σ is the output of the “conditional integrator” ( ) s , σ(0) ∈ [−µ/k0 , µ/k0 ] σ˙ = −k0 σ + µ sat µ
def
where e1 = x1 − H, e2 = e˙ 1 = f (e) + g(e)u, and k1 > 0. This guarantees that when the motion is restricted to s = 0, the error e1 asymptotically converges to zero. Taking the derivative of (6) and using (3), we see that = ( k1 e2 +)f((e) + g(e)u, ) √ a1 a22 e1√+H − f (e) = ( 22 ) ( e2 +a1 e1 +H) a2 u √ g(e) = 2A e +a e +H
where k0 > 0. From (12) and (11), it is clear that inside the boundary layer |s| ≤ µ, σ˙ = k1 e1 + e2 , which implies that ei = 0 at equilibrium, i.e., (12) is the equation of an integrator that provides integral action “conditionally”, inside the boundary layer. As shown in [15], such a design provides asymptotic error regulation, while not degrading the transient performance, as is common in a conventional design that uses the integrator σ˙ = e1 . In the relative degree ρ = 2 case, the controller (11)- (12) is simply a specially tuned saturated PID controller with anti-windup (see [15, Section √ 6]). Our state-feedback design uses e = a (h1 − h2 ) − 2 2 √ a1 h2 , which assumes that the parameters a1 and a2 are known. If that is not the case, the control (11) can be extended to the output-feedback case by replacing e2 by its estimate eˆ2 , obtained using the high-gain observer (HGO) } eˆ˙ 1 = eˆ2 + α1 (e1 − eˆ1 )/ϵ, (13) eˆ˙ 2 = α2 (e1 − eˆ1 )/ϵ2
s˙
2
1
a1 e2 √ 2 e1 +H
− a22
(7)
1
Note that g(e) is physically bounded away from zero. It is standard to design the control u to have two components: a “nominal control” that cancels known terms in (7) and a “sliding component” that overbounds the uncertain terms, so that ss˙ < 0. Accordingly, define u = unom + usliding =
−k1 e2 − fˆ + v gˆ
(8)
where fˆ and gˆ are nominal values of f and g respectively. In order to design the switching term v, we rewrite the closedloop equation for s as
where ϵ > 0, and the positive constants α1 , α2 are chosen such that the roots of λ2 + α1 λ + α2 = 0 have negative real parts. To complete the design, we need to specify how k, µ and ϵ are chosen. As previously mentioned, the gain k can simply be chosen as the largest possible control magnitude, while µ and ϵ are chosen “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 the continuous SMC. Analytical results for stability and performance are given in [15]. B) M IM O quadruple tank system: The SISO design can be almost directly extended to the MIMO case, noting that A is diagonal, so that the equations are effectively decoupled. Remark 1: The SMC design in [15] is based on regulating the output errors to zero, and relying on an input-to-state stability (ISS) property to ensure that the zero dyanmics are stable. It is well-known [11] that the quadruple tank system is minimum-phase only if γ1 + γ2 > 1. Therefore, our design only applies to the minimum-phase case. To proceed with the design, let the desired values for the outputs be H1 , H2 , and define the error variables ei = hi − Hi , and the sliding surfaces
s˙ = ∆(e1 , e2 ) + γ(e1 , e2 )v and suppose sup ∆(·) γ(·) ≤ ρ(·), where the supremum is taken over any set of interest in the state-space. The switching term is then chosen as v = −β(e1 , e2 )sgn(s), β(·) = ρ(·) + ρ0 , ρ0 > 0
(9)
where sgn(·) is the standard signum function. It is easy to check that this guarantees that ss˙ ≤ −ρ0 |s|, from which it follows that the output error e1 asymptotically converges to zero. We note that one possible nominal control component design results from the choice ( aˆ1 aˆ2 ) ( aˆ2 √e1 +H ) 2 √ fˆ(e) = − 2√aˆe1 e+H −a ˆ22 1 ( 22 ) ( e2 +ˆa1 e1 +H) a ˆ2 u √ gˆ(e) = ˆ e +ˆ a e +H 2A 2
1
1
ˆ a with A, ˆ1 and a ˆ2 being nominal values of A, a1 and a2 respectively. However, in keeping with the “universal design” presented in [15], and to simplify the design, we choose unom = 0, and β(·) simply as the largest possible constant value the control can be. Thus the state-feedback ideal SMC design is simply s = k1 e1 + e2 , u = −k sgn(s)
(12)
(10)
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. In [15], we presented a design that introduces integral action “conditionally” inside
si = ki σi + ei
(14)
where ki > 0, and σi are the outputs of the conditional integrators ( ) si σ˙ i = −ki σi + µi sat (15) µi 148
def
Note that s˙ i = ki σ˙ i + bi (·) + aii ui = Fi (·) + aii ui . The control is then designed as = Aˆ−1 [−Fˆ (·) + v], where = −βi (·)sat(si /µi )
u vi
60
50
(16) 40 Control Signal, u
where, as before, the design of the sliding component gains βi (·) is based on making si s˙ i < 0. Again, as before, to simplify the design, we assume that Aˆ = I (so that Γ = A), Fˆ = 0, and simply choose β1 , β2 to be their maximum allowed values, i.e, the constraints on the allowable applied voltages to the 2 pumps. With this choice, the control (14)(16) then simplifies to ( ) ui = −βi sat ki σµii+ei , where ) ( (17) σ˙ i = −ki σi + µi sat ki σµii+ei
30
20
10
0
−10
0
50
100
150 Time (sec)
200
250
300
Fig. 4: Control signal for SMC
Analytical results for stability and performance recovery of ideal SMC are discussed in [15]. We present simulation results that validate our design in the next section.
with SMC with conditional integrator (CI) are shown in Fig. 5 and Fig. 6, and it is clear that the design achieves good regulation, and that the chattering is greatly reduced.
IV. S IMULATION R ESULTS A) SISO two tank system: The parameters of CTS are taken from [9]. The area of cross-section of tank 1 and tank 2 is same, i. e., A = 208.2 cm2 . The value of the parameter, A12 is 0.58 cm2 and the value of the parameter, A2 is 0.24 cm2 . The gravitational constant is 981 cm/sec2 . The simulations are carried out with the input constraint as 0 cm3 /sec ≤ u ≤ 50 cm3 /sec. The levels of the liquid have been measured through the level sensors. Then, the error is determined by the difference between the set point value and processed value of the liquid level. The nominal values used in the simulation are Aˆ = 249.84 cm2 , a ˆ1 = 0.04072 and a ˆ2 = 0.136. The controller parameters used in the simulation are chosen to be k1 = 0.08 and k = 50. The desired value of the output level of the system is taken as H = 6 cm. The simulation results of ideal SMC are shown in Fig. 3 and 4.
7
6
Output in cm
5
4
3
2
1
0
0
50
100
150 Time (sec)
200
250
300
7
Fig. 5: Response of liquid level in tank 2 using SMC with CI
6
Output in cm
5
Next, we demonstrate performance robustness of our design when the parameters of the system are not exactly known, and when disturbances enter into the system. The parameters A12 and A2 are increased by 25 % of their nominal values and also a step disturbance of amplitude -2 cm3 /sec is applied as an external disturbance to the system at time t = 200s. Remark 2: We cannot use the state-feedback design, since the sliding surface variable s requires x2 , which is computed from h1 and h2 through the unknown parameters a1 , a2 . Therefore, we replace s by its estimate sˆ, which in turn uses the estimate eˆ2 of e2 = x2 obtained using the HGO (13). The HGO parameters are chosen as α1 = 20, α2 = 100, and ϵ = 10−3 . The responses of output level of CTS using our method and the design in [9] are shown in Fig. 7. It can be observed
4
3
2
1
0
0
50
100
150 Time (sec)
200
250
300
Fig. 3: Response of liquid level in tank 2 using SMC It is clear that the output converges to the desired set point value H, but there is considerable control chattering, which can cause valve wear and tear. For comparison, the results 149
TABLE I: Comparison of performance indices 60
50
Control Signal, u
40
Method
Settling Time (sec)
Rise Time (sec)
MSE
SMC in [9]
113.1
52.1
4.8
SMC with CI
81.8
45.9
4.2
30
20
10
TABLE II: Comparison of error indices
0
−10
0
50
100
150 Time (sec)
200
250
300
Fig. 6: Control signal for SMC with CI
from the figure that SMC with CI controller has superior performance; it takes ∼82 secs to settle versus ∼113 secs in the design in [9]. The rise time of the output response in CI controller is ∼46 seconds, and that of the dynamics method developed in [9] is ∼52 seconds. There is no overshoot in both the cases. Classical measures such as settling time, rise
6
Output in cm
5
Reference SMC with CI Second Dynamics by Almutairi et al.
3
2
1
0
0
50
100
150 Time (sec)
200
250
IAE
ISE
ITAE
ITSE
SMC in [9]
114.5
254.8
2690
4188
SMC with CI
92.1
193.8
1807
2792
cross-section of tank 2 and tank 4 is same, i. e., A2 = A4 = 32 cm2 . The value of the parameters, a1 and a3 is 0.071 cm2 .The value of the parameters a2 and a4 is 0.057 cm2 . The gravitational constant is 981 cm/sec2 . The simulations are carried out with the input constraints as 0 cm3 /sec ≤ u1 ≤ 50 cm3 /sec and 0 cm3 /sec ≤ u2 ≤ 50 cm3 /sec. The levels of the liquid have been measured through the level sensors. The nominal values used in the simulation are Aˆ1 = 33.6 cm2 , Aˆ2 = 35.2 cm2 , Aˆ3 = 33.6 cm2 , Aˆ4 = 35.2 cm2 , a ˆ1 = 0.0568 cm2 , a ˆ2 = 0.0342 cm2 , a ˆ3 = 0.0568 cm2 and a ˆ4 = 0.0342 cm2 . The controller parameters used in the simulation are chosen to be k1 = 10, k2 = 25, β1 = 100 and β2 = 100. The desired value of the output levels of FTS are taken as H1 = 50 cm and H2 = 35 cm respectively. The decentralized PI controllers are applied to the FTS. The PI controllers are in the form ) ( 1 , m = 1, 2 (18) Gcm = Kpm 1 + Tim s
7
4
Type
The parameters of the PI controller are tuned manually based on simulations of the linear physical models of the FTS. It is easy to find the controller parameters that give good performance for the minimum phase case. The controller parameter values (Kp1 , Ti1 ) = (0.1, 55.6) and (Kp2 , Ti2 ) = (0.7, 40) are applied to FTS for comparing with SMC with CI controller. The response of output level h1 in tank 1 of FTS using different controllers is shown in Fig. 8. It is observed that both the controllers reach the set point, but the SMC with CI controller has less settling time as well as less overshoot as compared to the PI controller. The SMC with CI controller takes∼748 seconds to reach the desired set point, while the PI controller takes ∼905 seconds to settle at the desired set point. While comparing the percentage overshoot, the PI controller has overshoot of ∼13.8 % where as SMC with CI controller has zero overshoot. Simillarly, Fig. 9 shows the response of output level h2 in tank 2 of FTS. From the figure,
300
Fig. 7: Responses of output level of CTS time, and MSE for the two designs are tabulated in Table I. It is observed from the table that the SMC with CI controller shows superior performance in terms of settling time, rise time and MSE. The CI design results in a ∼28 % smaller settling time than the one in [9], and the rise time in [9] is ∼12 % more than that of our design. The error indices such as IAE, ISE, ITAE and ITSE for the design in [9] and our design are tabulated in Table II. From the table, it is clear that our design outperforms the other design wrt all the performance measures. B) M IM O f our tank system: The parameters of FTS are taken from [11]. The area of cross-section of tank 1 and tank 3 is same, i. e., A1 = A3 = 28 cm2 and the area of 150
ITAE and ITSE. Simulations demonstrate that the design recovers the performance robustness to system parameters and disturbances that ideal SMC does, but without the control chattering. Experimental results have also been obtained for the coupled-tank system [21] that corroborate the efficacy of the design; they are not presented here due to space constraints.
60
Output Level, h1 in cm
50
40
Set Point SMC with CI PI Controller
30
R EFERENCES [1] B. W. Bequette, Process Control: Modelling, Design and Simulation. Prentice Hall, 2003. [2] C. A. Smith and A. B. Corripio, Principles and Practice of Automatic Process Control. John Wiley and Sons, Inc., 1998. [3] E. Daniel et al., “Model reference adaptive control using simultaneous probing, estimation, and control,” IEEE Trans. Aut. Ctrl., vol. 55, no. 9, pp. 2014–2029, 2010. [4] N. K. Poulsen et al., “Constrained predictive control and its application to a coupled-tanks apparatus,” Int. Jnl. of Ctrl., vol. 74, no. 6, pp. 552– 564, 2001. [5] Z. Aydogmus, “Implementation of a fuzzy-based level control using SCADA,” Jnl. Expert Systems with Applications, vol. 36, pp. 6593– 6597, 2009. [6] T. Tani et al., “Neuro-fuzzy hybrid control system of tank level in petroleum plant,” IEEE Trans. Fuzzy Syst., vol. 4, no. 3, pp. 360–368, 1996. [7] S. Kamalasadan et al., “A neural network parallel adaptive controller for dynamic system control,” IEEE Trans. Instrum. Meas., vol. 56, no. 5, pp. 1786–1796, 2007. [8] K. A. Mohideen et al., “Real-coded genetic algorithm for system identification and tuning of a modified model reference adaptive controller for a hybrid tank system,” Jnl. Applied Mathematical Modeling, vol. 37, pp. 3829–3847, 2013. [9] N. Almutairi and M. Zribi, “Sliding mode control of coupled tanks,” Journal of Mechatronics, vol. 16, pp. 427–441, 2006. [10] R. Benayache et al., “Controller design using second order sliding mode algorithm with an application to a coupled-tank liquid-level system,” in 2009 IEEE Intl. Conf. Ctrl. & Aut., pp. 558–563, 2009. [11] K. H. Johansson, “The quadruple-tank process: A multivariable laboratory process with an adjustable zero,” IEEE Trans. Ctrl. Sys. Tech., vol. 8, no. 3, pp. 456–465, 2000. [12] P. P. Biswas et al., “Sliding mode control of quadruple tank process,” Journal of Mechatronics, vol. 19, pp. 548–561, 2009. [13] A. Gaaloul and F. M’Sahli, “High gain output feedback control of a quadruple tank process,” in 2008 IEEE MELECON Electrochemical Conf., pp. 23–28, 2008. [14] S. Larguech et al., “Improved sliding mode of a class of nonlinear systems: Application to quadruple tank system,” in 2013 Eur. Ctrl. Conf., pp. 3203–3208, 2013. [15] S. Seshagiri and H. Khalil, “Robust output feedback regulation of minimum-phase nonlinear systems using conditional integrators,” Automatica, vol. 41, no. 1, pp. 43–54, 2005. [16] S. Nazrullah and H. Khalil, “Robust stabilization of non-minimum phase nonlinear systems using extended high-gain observers,” IEEE Trans. Aut. Ctrl., vol. 56, no. 4, pp. 802–813, 2011. [17] E. Promtun and S. Seshagiri, “Sliding mode control of F-16 longitudinal dynamics,” in Proc. 2008 ACC, 2008. [18] S. Seshagiri, “Robust multivariable PI control: Applications to process control,” in 17th IFAC World Congress, (Seoul, S. Korea), July 2008. [19] H. Vo and S. Seshagiri, “Robust control of F-16 lateral dynamics,” in Proc. 2008 IECON, 2008. [20] S. Seshagiri, “Position control of permanent magnet stepper motors using conditional servocompensators,” IET Ctrl. Theory & Appl., vol. 3, no. 9, pp. 1196–1208, 2009. [21] S. B. Prusty, S. Seshagiri, U. C. Pati, and K. K. Mahapatra, “Sliding mode control of interacting two-tank system using conditional integrators,” Ctrl. Engg. Practice (submitted).
20
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0
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1000 1500 Time (sec)
2000
2500
Fig. 8: Response of liquid level in tank 1 we get settling time in case of SMC with CI controller is ∼1174 seconds and that of PI controller is ∼1249 seconds. The PI controller has an overshoot of ∼13.4 % while the SMC with CI controller has no overshoot. 40 35
Output Level, h2 in cm
30
Set Point SMC with CI PI Controller
25 20 15 10 5 0
0
500
1000 1500 Time (sec)
2000
2500
Fig. 9: Response of liquid level in tank 2 V. C ONCLUSION This paper presents the control of the liquid level in coupled tank system as well as four tank system using sliding mode control with conditional integrators. The integrator is designed in such a way that it provides integral action only conditionally, i.e. it provides integral action only inside the boundary layer. The controller is of PI/PID type with anti-windup, which is widely used in industrial practice. It robustly performs set-point regulation, with good transient performance, as quantified by classical measures such as rise time, settling time and error indices such as IAE, ISE,
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