Abstract

Actuator faults are inevitable in small reverse osmosis desalination plants. They may cause energy losses and reduce the quality of the freshwater, which may endanger human life. Model predictive control (MPC) is a model-based approach widely used to control process systems such as reverse osmosis, while considering a set of constraints. In this paper, three methods of predictive model controllers are considered for the control of a multi-input multi-output (MIMO) reverse osmosis desalination system in the presence of noise, model mismatch, and actuator fault. Formulation of enhanced constrained receding horizon predictive control via bounded data uncertainties (CRHPC-BDU) are extended for linear time-invariant MIMO systems. Permeate flow rate and conductivity of the water produced are controlled by a retentate valve and a bypass valve, respectively. The simulation results show the robustness of the suggested approach in the presence of both noise and uncertainties. CRHPC-BDU has a better performance subject to systems with model uncertainty and actuator fault up to a reasonable limit. By increasing the actuator fault up to 34%, the robustness of CRHPC-BDU is further highlighted in permeate conductivity, where the fluctuations of permeate conductivity dampen sooner than in the other two controllers.

INTRODUCTION

Utilizing sea water for drinking is a deep-routed wish for nations that suffer from fresh water shortage. The use of large water purification systems based on reverse osmosis (RO) desalination is growing for the purposes of providing drinking water for the food industry, the wine industry, car washes and other industries. Small RO desalination plants are used in operation rooms, the semiconductor industry and domestic applications.

RO desalination control systems have different tasks for each application. The most challenging part is when a multi-input multi-output (MIMO) system needs to be controlled. A system with more than one input and/or more than one output are known as MIMO. Alatiqi et al. (1989) proposed a multi loop control system which includes a pressure controller and two pH controllers. Dynamic models of RO plants were studied by Soltanieh & Gill (1981) and Al-Bastaki & Abbas (1999). The application of some advanced control systems for RO plants were studied by Assef et al. (1997); McFall et al. (2008b); and Hong Phuc et al. (2016). For RO plants many control designs have been studied, such as multi loop proportional-integral (PI) controller (Gambier et al. 2006), self-regulating proportional integral derivative (PID) based on genetic algorithms (Kim et al. 2008) and feed forward-feedback for disturbance rejection (McFall et al. 2008a). Model predictive control (MPC) is deployed for many RO plant applications. In Robertson et al. (1996) and Janghorban Esfahani et al. (2016) an MPC controller based on dynamic matrix control (DMC) was compared with a PID controller for RO plants. Some other MPC approaches are DMC design using system identification (Abbas 2006), MPC design for high capacity RO plant confronting membrane deformation (Bartman et al. 2009) and MPC design based on mathematical modelling for an RO plant (Ali et al. 2010). Other control methods are also used for RO plants where, for instance in Hong Phuc et al. (2016), a robust controller is designed based on two-degrees-of-freedom loop-shaping control methodology.

In addition to problems such as slow variation, model mismatch, measurement noise and disturbances, fault is a likely problem in RO systems. Faults decrease the water quality and increase the risk of hazard. The common faults in RO plants are actuator faults, membrane deformation and sensor faults. The most frequent are actuator faults, in the form of decreasing efficiency which stems from constant exposure to salty water. Fault tolerant control (FTC) systems are divided into active and passive FTCs. In active FTCs, the structure or parameters of the controller change with respect to the estimation of fault detection and isolation (FDI) filters. But passive FTCs are fixed robust controllers that do not use the information of FDI filters and can tolerate faults because they are robust. Active fault tolerant control for RO system was studied by McFall et al. (2007) and Gambier et al. (2010).

In this work, passive FTC is used to tolerate an actuator fault. It is essential to have a control system with a passive FTC, because a fixed controller has modest hardware and software requirements. Furthermore, in comparison to active FTC systems, passive FTC systems are less complex, and they can be designed to be more reliable based on classical reliability theory (Stoustrup & Blondel 2004). Passive FTCs have implemented in MPC frameworks for RO plants. In Lee (2011) three decades of MPC developments were explained, which supported our primary reason for selecting this control approach. Three MPC controllers, including generalized predictive control (GPC) (Clarke et al. 1987a, 1987b), constrained receding horizon predictive control (CRHPC) (Clarke & Scattolini 1991; Chow 1996), and enhanced robustness CRHPC via bounded data uncertainty (CRHPC-BDU) (Ramos et al. 2009), were considered. These controllers were designed for the small RO plant studied and identified by Gambier et al. (2010). GPC is a well-known controller of the MPC class that is simple, effective, and has a low computational cost, but in general, the stability of this controller is not guaranteed. CRHPC is similar to GPC except for output constraints. These constraints lead to guaranteed stability for exact systems. However, Manoso (1999) showed how GPC and CRHPC nominal stability may fail in systems with uncertainties and cause steady state convergence error accuracy. The CRHPC-BDU guarantees stability in the presence of model and measurement uncertainties. In this work, formulation of the CRHPC-BDU method for MIMO systems is was carried out and is presented in the Supplementary Material. The primary aim of this paper is to propose a novel robust model predictive control (RMPC) to be used in the FTC approach. The suggested approach employs the CRHPC-BDU controller, which is the robust and stable variant of GPC. The state-space model was utilized to design the CRHPC-BDU controller.

PLANT DESCRIPTION

Desalination systems based on RO are commonly used. Considering the system scale and the quality of inlet and outlet water, different units may be deployed for different systems. General RO system units include pretreatment, membrane assembly and post-treatment. The pretreatment unit includes flocculation, chlorination, pH value adjustment, and other treatments that are needed to reduce membrane fouling, inhibit further precipitation and growth of microorganisms, remove suspended particles and assure the safety of the process. A high pressure pump is needed for the feed water to overcome osmotic pressure and permeation of pure water through the membrane. A control valve is also needed to discharge brine water. The permeate water flows into the post-treatment unit to adjust the pH value again, add minerals, exchange ions, etc., as necessary. The whole process involves many control loops which vary from case to case. Two important variables of output are permeate flow rate and permeate conductivity, which are affected by pressure pump and retentate valve.

In this paper a small plant is considered that is used for drinking water purification in which the permeate flow rate is controlled by retentate control valve. The schematic of the case study is presented in Figure 1.

Figure 1

Schematic of the control loops process.

Figure 1

Schematic of the control loops process.

As can be seen in Figure 1, input variables are the brine and bypass flow, which are changed by the retentate and bypass valves, respectively. The input variables control the permeate flow rate and conductivity, which are the output of the system. In order to control permeate conductivity, Gambier et al. (2010) suggested that a bypass valve is installed to add a small amount of inlet water to outlet water.

A dynamic linear time invariant (LTI) state space model is identified for plants operating at 250 L/h for the outlet, 500 L/h for the inlet, 0.02 L/h for the bypass flow rates and 425 μS/cm for permeate conductivity. These rates occur when the valve opening for both valves is 50%. The discreet time LTI state space model equations are as follows: 
formula
(1)
 
formula
(2)
where model parameters for plant in the operation point with 0.015 seconds for the sample time were obtained as Table 1. Although the system is considered to be linear, some authors highlight the nonlinear behavior of RO plants due to membrane fouling. In practice, variation in some parameters, such as membrane fouling that cause nonlinearity, is small and can be eliminated by membrane washing once a week. Thus, it is promising that the behavior of the plant remains quite linear during a week when variations are modeled as small time varying parameter uncertainties. These uncertainties have a small range because they will be eliminated by membrane washing by the end of a week. Moreover, actuator fault is a common experience in this plant, and usually occurs in the retentate valve and affects the control. The presented model is shown below: 
formula
(3)
 
formula
(4)
where , , are model uncertainties, is actuator fault and is white noise. The control goal is to track reference signal in the presence of uncertainties, sensor noise and actuator fault with small steady state error values.
Table 1

Model parameters

 
 

CONTROL STRATEGIES

Simplicity and transparency of design, flexibility of controller structure and optimal performance are the advantages of the GPC method despite the lack of guaranteed stability. CRHPC provides guaranteed stability for nominal systems. The mathematical descriptions of the GPC and CRHPC approach are presented in the Supplementary Material.

CRHPC-BDU is presented in Ramos et al. (2009) for single-input single-output (SISO) systems. In this paper this method is expanded for MIMO systems. The CRHPC-BDU extension guarantees stability in the presence of bounded uncertainties. First, the augmented matrices for the MIMO system are defined. Consider the controlled auto-regressive integrated moving average (CARIMA) representation of a MIMO plant. 
formula
(5)
where is output, is input, is noise (in a case where is measurement noise) and . Consider following parameters: 
formula
(6)
 
formula
(7)
 
formula
(8)
 
formula
(9)
 
formula
(10)
 
formula
(11)
 
formula
(12)
 
formula
(13)
where is the k-th step response coefficient of the transfer function linking the j-th input to i-th output, N is prediction horizon, is control horizon, is the number of output constraints of i-th output, is the reference signal of i-th output at and is the free response of i-th output at .
The solution of the optimization problem with the following cost function and constraints is used for CRHPC-BDU: 
formula
(14)
 
formula
(15)
where , and the operators and define deviation and 2-norm respectively. The input variations are calculated using the following equation: 
formula
(16)
In the above equation, is the solution particular to the constraints. It is calculated by solving the following nonlinear equations concurrently using a numerical method: 
formula
(17)
 
formula
(18)
and is calculated by the following equation: 
formula
(19)
where is the null space of . Also and are calculated by solving the following nonlinear equations concurrently using a numerical method: 
formula
(20)
 
formula
(21)
 
formula
(22)

SIMULATION RESULTS

In this section, the aforementioned three methods, GPC, CRHPC and CRHPC-BDU, are deployed for the RO plant with the purpose of testing and comparing MPC controllers for the RO plant for optimality and robustness. Parameters of the controllers are shown in Table 2 and further descriptions are included in the Supplementary Material. These parameters were adjusted equally to give a fair comparison between control methods and were chosen in such a way that CRHPC stabilizing conditions were satisfied and control signals were not exceeded. For this plant , , and guarantee CRHPC stability, as stated in the Supplementary Material.

Table 2

Controller parameters

DescriptionEXP. 1EXP. 2EXP. 3EXP. 4EXP. 5EXP. 6EXP. 7
Horizon parameters  Prediction horizon 27 27 100 100 100 100 100 
 Control horizon 25 25 80 80 80 80 80 
 Output condition horizon 12; i = 1,2 12; i = 1,2 12; i = 1,2 12; i = 1,2 12; i = 1,2 12; i = 1,2 12; i = 1,2 
Cost function weightings  Output weighting matrix ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 
 Input weighting matrix in CRHPC ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 
 Input weighting parameter in GPC 
 Input weighting parameter in CRHPC-BDU 
CRHPC-BDU
uncertainty bounds 
 Maximum deviation of in CRHPC-BDU 
 Maximum deviation of in CRHPC-BDU 
 Maximum deviation of in CRHPC-BDU 0.1 0.1 0.1 0.1 0.1 0.1 0.1 
 Maximum deviation of in CRHPC-BDU 0.1 0.1 0.1 0.1 0.1 0.1 0.1 
Reference signals  Flow rate reference signal 350 L/h 350 L/h 350 L/h 350 L/h 350 L/h 350 L/h 350 L/h 
 Permeate conductivity reference signal 420 μS/cm 420 μS/cm 420 μS/cm 420 μS/cm 420 μS/cm 420 μS/cm 420 μS/cm 
Simulation plant parameters  Noise power 0.05 0.05 0.05 0.05 0.05 0.05 
 Actuator fault 30% 34% 
 Matched uncertainty in A matrix 0.03 0.04 0.03 0.03 
 Matched uncertainty in B matrix 0.02 0.03 0.02 0.02 
 Matched uncertainty in C matrix 0.06 0.07 0.06 0.06 
DescriptionEXP. 1EXP. 2EXP. 3EXP. 4EXP. 5EXP. 6EXP. 7
Horizon parameters  Prediction horizon 27 27 100 100 100 100 100 
 Control horizon 25 25 80 80 80 80 80 
 Output condition horizon 12; i = 1,2 12; i = 1,2 12; i = 1,2 12; i = 1,2 12; i = 1,2 12; i = 1,2 12; i = 1,2 
Cost function weightings  Output weighting matrix ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 
 Input weighting matrix in CRHPC ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 ; i = 1,2 
 Input weighting parameter in GPC 
 Input weighting parameter in CRHPC-BDU 
CRHPC-BDU
uncertainty bounds 
 Maximum deviation of in CRHPC-BDU 
 Maximum deviation of in CRHPC-BDU 
 Maximum deviation of in CRHPC-BDU 0.1 0.1 0.1 0.1 0.1 0.1 0.1 
 Maximum deviation of in CRHPC-BDU 0.1 0.1 0.1 0.1 0.1 0.1 0.1 
Reference signals  Flow rate reference signal 350 L/h 350 L/h 350 L/h 350 L/h 350 L/h 350 L/h 350 L/h 
 Permeate conductivity reference signal 420 μS/cm 420 μS/cm 420 μS/cm 420 μS/cm 420 μS/cm 420 μS/cm 420 μS/cm 
Simulation plant parameters  Noise power 0.05 0.05 0.05 0.05 0.05 0.05 
 Actuator fault 30% 34% 
 Matched uncertainty in A matrix 0.03 0.04 0.03 0.03 
 Matched uncertainty in B matrix 0.02 0.03 0.02 0.02 
 Matched uncertainty in C matrix 0.06 0.07 0.06 0.06 

The plant was simulated under seven scenarios, as shown in Table 2. The experiments procedure was designed with the aim of studying the plant behavior under various situations, including noise, model mismatch and actuator fault.

Experiment 1: In the first experiment, noise, fault and uncertainty were not included. Small N and , which satisfy the stabilizing conditions of CRHPC, provided a promising performance. It should be noted that large N and impose more computational burden. The results shown in Figure 2 indicate that CRHPC provided slightly faster convergence compared to GPC and CRHPC-BDU. CRHPC had the best performance because of its precise mathematical structure, and CRHPC-BDU had the worst performance because it is designed to be more robust than optimum.

Figure 2

Input and outputs of the nominal model by three controllers with small computational cost for 15 seconds.

Figure 2

Input and outputs of the nominal model by three controllers with small computational cost for 15 seconds.

Experiment 2: In the second experiment, model and plant were equivalent except for output measurement noise. Figure 3 shows that CRHPC was very sensitive to measurement noise and had an unacceptable performance despite the guarantee of stability. CRHPC-BDU was more robust and had an acceptable performance. In this case, GPC was recommended, because it had the least computational burden and was robust to noise.

Figure 3

Input and outputs of plant subject to measurement noise by three controllers with small computational cost for 15 seconds.

Figure 3

Input and outputs of plant subject to measurement noise by three controllers with small computational cost for 15 seconds.

Experiment 3: In the third experiment, and were increased, which subsequently increased the computational burden. The results in Figure 4 show that bigger horizons improved the controllers performance and made the behavior of CRHPC similar to GPC. CRHPC-BDU settling time decreased significantly in this case.

Figure 4

Input and outputs of plant subject to measurement noise by controllers with increased computational cost for 15 seconds.

Figure 4

Input and outputs of plant subject to measurement noise by controllers with increased computational cost for 15 seconds.

Experiment 4: In the fourth experiment, the model was considered to be uncertain and no fault occurred. The stability and steady state performance of the system could have been be affected by uncertainty and noise. Simulation results in Figure 5 shown that CRHPC-BDU was more robust, with lower fluctuation compared to the others, despite its steady state error.

Figure 5

Input and outputs of plant subject to measurement noise and small uncertainties by three controllers for 15 seconds.

Figure 5

Input and outputs of plant subject to measurement noise and small uncertainties by three controllers for 15 seconds.

Experiment 5: In this experiment the amount of uncertainty increased and outputs in Figure 6 show that CRHPC and GPC become unstable while CRHPC-BDU successfully controlled the system.

Figure 6

Outputs of plant subject to measurement noise and large uncertainties by three controllers for 15 seconds.

Figure 6

Outputs of plant subject to measurement noise and large uncertainties by three controllers for 15 seconds.

The five experiments above show that control objectives are reachable for both certain and uncertain conditions. However, for large amounts of uncertainty the performance of controller was not acceptable. In order to study the performance of the design in the presence of described actuator fault, two more experiments were performed for 30 and 34% of valve decay.

Experiment 6: In the sixth experiment, the actuator fault was considered to be 30% at t = 9 seconds. In Figure 7, the effect of the fault is clear, particularly in permeate conductivity. This case shows that controllers had a robust performance for this amount of fault and the quality of responses in CRHPC-BDU was more promising despite the fact that the existence of the fault caused steady state error in responses.

Figure 7

Input and outputs of plant subject to measurement noise, uncertainties and 30% fault by three controllers for 15 seconds.

Figure 7

Input and outputs of plant subject to measurement noise, uncertainties and 30% fault by three controllers for 15 seconds.

Experiment 7: By increasing the actuator fault up to 34% at t = 9 seconds, the robustness of CRHPC-BDU was further highlighted in permeate conductivity, as shown in Figure 8, where the fluctuations of permeate conductivity dampened sooner in comparison to the other two controlling approaches discussed earlier.

Figure 8

Permeate conductivity in the context of measurement noise, uncertainties and 34% fault by three controllers for 16 seconds.

Figure 8

Permeate conductivity in the context of measurement noise, uncertainties and 34% fault by three controllers for 16 seconds.

All simulations were performed by a PC with one quad-core Intel Core i7 processor and 16 GB RAM. Computational time values for , and for each sample of input signals were 0.0081 seconds for GPC, 0.0162 seconds for CRHPC and 0.0812–0.0302 seconds for CRHPC-BDU. Although the time for CRHPC BDU is greater than for the other two methods because of numerical optimization loops, it is acceptable in comparison to the plant sample time (0.15 seconds). Further experiments showed that for faults larger than 34% all controllers were not able to dampen the fluctuations.

CONCLUSION

In this paper, the comparative study between the three controllers, CRHPC, GPC, and CRHPC-BDU, were studied to show the performance of the proposed approach. These controllers were analyzed by their closed-loop stability, nominal performance, measurement noise, model mismatch, and capability of being used as passive FTCs for small actuator faults. Moreover, an expanded version of CRHPC-BDU for MIMO systems was proposed. The simulation was performed to show that the proposed method can tolerate the fault because of its robustness. The actuator fault was increased in the simulation model to measure the performance of the experiment.

The results revealed that GPC had the least computational cost with poor performance, and CRHPC-BDU provided robustness for controlling the system, though with more computational cost. CRHPC provided nominal stability, while its performance was not better than the other two methods. These controllers were used as passive FTCs for the RO plant with uncertainty in the model. CRHPC-BDU gave better results than the other two controllers, and the actuator fault was tolerated up to 34%.

NOTES

The authors declare no competing financial interest.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this paper is available online at https://dx.doi.org/10.2166/ws.2020.043.

REFERENCES

REFERENCES
Abbas
A.
2006
Model predictive control of a reverse osmosis desalination unit
.
Desalination
194
(
1–3
),
268
280
.
https://doi.org/10.1016/J.DESAL.2005.10.033
.
Alatiqi
I. M.
Ghabris
A. H.
Ebrahim
S.
1989
System identification and control of reverse osmosis desalination
.
Desalination
75
(
January
),
119
140
.
https://doi.org/10.1016/0011-9164(89)85009-X
.
Al-Bastaki Nader
M.
Abbas
A.
1999
Modeling an industrial reverse osmosis unit
.
Desalination
126
(
1–3
),
33
39
.
https://doi.org/10.1016/S0011-9164(99)00152-6
.
Al-haj Ali
M.
Ajbar
A.
Ali
E.
Alhumaizi
K.
2010
Robust model-based control of a tubular reverse-osmosis desalination unit
.
Desalination
255
(
1–3
),
129
136
.
https://doi.org/10.1016/J.DESAL.2010.01.003
.
Assef
J. Z.
Watters
J. C.
Deshpande
P. B.
Alatiqi
I. M.
1997
Advanced control of a reverse osmosis desalination unit
.
Journal of Process Control
7
(
4
),
283
289
.
https://doi.org/10.1016/S0959-1524(97)00004-8
.
Bartman
A. R.
McFall
C. W.
Christofides
P. D.
Cohen
Y.
2009
Model Predictive Control of Feed Flow Reversal in a Reverse Osmosis Desalination Process
. In:
2009 American Control Conference
.
IEEE
, pp.
4860
4867
. .
Chow
C.-M.
1996
An Approach to Multivariable Predictive Control with Stability Guarantees
. In:
UKACC International Conference on Control. Control ‘96, 1996
.
IEE.
, pp.
1350
1355
.
Clarke
D. W.
Mohtadi
C.
Tuffs
P. S.
1987a
Generalized predictive control – part I. The basic algorithm
.
Automatica
23
(
2
),
137
148
.
https://doi.org/10.1016/0005-1098(87)90087-2
.
Clarke
D. W.
Mohtadi
C.
Tuffs
P. S.
1987b
Generalized predictive control – part II extensions and interpretations
.
Automatica
23
(
2
),
149
160
.
https://doi.org/10.1016/0005-1098(87)90088-4
.
Clarke
D. W.
Scattolini
R.
1991
Constrained receding-horizon predictive control
.
IEE Proceedings D Control Theory and Applications
138
(
4
),
347
.
https://doi.org/10.1049/ip-d.1991.0047
.
Gambier
A.
Miksch
T.
Badreddin
E.
2010
Fault-Tolerant Control of a Small Reverse Osmosis Desalination Plant with Feed Water Bypass
. In:
Proceedings of the 2010 American Control Conference
.
IEEE
, pp.
3611
3616
. .
Gambier
A.
Wellenreuther
A.
Badreddin
E.
2006
Optimal Control of a Reverse Osmosis Desalination Plant Using Multi-Objective Optimization
. In:
2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control
.
IEEE
, pp.
1368
1373
. .
Janghorban Esfahani
I.
Ifaei
P.
Rshidi
J.
Yoo
C.
2016
Control performance evaluation of reverse osmosis desalination system based on model predictive control and PID controllers
.
Desalination and Water Treatment
57
(
55
),
26692
26699
.
https://doi.org/10.1080/19443994.2016.1191776
.
Kim
J.-S.
Kim
J.-H.
Park
J.-M.
Park
S.-M.
Choe
W.-Y.
Heo
H.
2008
‘Auto Tuning PID Controller Based on Improved Genetic Algorithm for Reverse Osmosis Plant,’ November
. .
Lee
J. H.
2011
Model predictive control: review of the three decades of development
.
International Journal of Control, Automation and Systems
9
(
3
),
415
424
.
https://doi.org/10.1007/s12555-011-0300-6
.
Manoso
C.
1999
Robust Stability Analysis of GPC and CRHPC Using the Theory of Extreme Point Results
. In:
Proceedings of the 1999 American Control Conference
. .
McFall
C. W.
Bartman
A.
Christofides
P. D.
Cohen
Y.
2008a
Control of a Reverse Osmosis Desalination Process at High Recovery
. In:
2008 American Control Conference
.
IEEE
, pp.
2241
2247
. .
McFall
C. W.
Bartman
A.
Christofides
P. D.
Cohen
Y.
2008b
Control and monitoring of a high recovery reverse osmosis desalination process
.
Industrial & Engineering Chemistry Research
47
(
17
),
6698
6710
.
https://doi.org/10.1021/ie071559b
.
McFall
C. W.
Christofides
P. D.
Cohen
Y.
Davis
J. F.
2007
Fault-tolerant control of a reverse osmosis desalination process
.
IFAC Proceedings Volumes
40
(
5
),
161
166
.
https://doi.org/10.3182/20070606-3-MX-2915.00145
.
Phuc
H.
Duc
B.
You
S.-S.
Lim
T.-W.
Kim
H.-S.
2016
Robust water quality controller for a reverse osmosis desalination system
.
Water Science and Technology: Water Supply
16
(
2
),
324
332
. .
Ramos
C.
Martínez
M.
Sanchis
J.
Salcedo
J. V.
2009
Robust constrained receding-horizon predictive control via bounded data uncertainties
.
Mathematics and Computers in Simulation
79
(
5
),
1452
1471
.
https://doi.org/10.1016/j.matcom.2008.06.002
.
Robertson
M. W.
Watters
J. C.
Desphande
P. B.
Assef
J. Z.
Alatiqi
I. M.
1996
Model based control for reverse osmosis desalination processes
.
Desalination
104
(
1–2
),
59
68
.
https://doi.org/10.1016/0011-9164(96)00026-4
.
Soltanieh
M.
Gill
W. N.
1981
Review of reverse osmosis membranes and transport models
.
Chemical Engineering Communications
12
(
4–6
),
279
363
.
https://doi.org/10.1080/00986448108910843
.
Stoustrup
J.
Blondel
V. D.
2004
Fault tolerant control: a simultaneous stabilization result
.
IEEE Transactions on Automatic Control
49
(
2
),
305
310
.
https://doi.org/10.1109/TAC.2003.822999
.

Supplementary data