Indirect Adaptive Control of Droplet Dispensing in Digital Microfluidic
Indirect Adaptive Control of Droplet Dispensing in Digital Microfluidic Systems Arash Edalatnoor, Afshin Izadian, Senior Member, IEEE, Masoud Vaezi Energy Systems and Power Electronics Laboratory Purdue School of Engineering and Technology, IUPUI [email protected]
Abstract— Digital microfluidics systems require advanced controllers to operate accurately since their parameters are subjected to change in environment and over time. Due to imperfect manufacturing processes, their fabricated system parameters may become different from the destined values. Hence, estimation based controllers are required to identify the system parameters. The electrowetting on dielectric can be precisely controlled to dispense desirable and repeatable droplets. However, the system parameters and their variation over time makes the control system challenging. This paper describes the application of an indirect adaptive trajectory controller for digital Pico-Droplet dispensing system. Forgetting factor recursive least square estimator is used to estimate the system parameters including capacitance and resistance of the occupying droplet between electrodes. Indirect adaptive technique is used to measure and control the droplet volume on the dispensing electrodes. Simulations of the estimator, tracking performance of dispensed droplet volume and the controller’s control effort are provided to demonstrate an accurate and high performance control approach.
I.
INTRODUCTION
Digital microfluidic systems are increasingly used in LabOn-Chip devices and commercial electronics. Since these devices are made in small scale, they can advance the speed of processes and highly reduce the cost of experiments [1]-[3]. In many applications such as drug delivery or quantitative analysis, small scale handing of liquids is the key in safe and accurate operation of the device. Less expensive and highly functional material have been introduced and used in digital microfluidics. However, stability and reliability of device operation depends on the effectiveness of the active controllers [4]-[6]. Using real time parameter estimation, the slowly time varying parameters and device behavioral variations that affect the droplet dispensing can be identified and proper control actions can be taken [4], [7]. Since the size of droplets is highly dependent on the volume of liquid reservoir, it is important to control the droplet volume through various techniques. The parameters of pico-droplet systems are also subjected to change in environment over time, with viscosity of liquid, air gap, and operating frequency of the device. For accurate, rapid, and repeatable droplet dispensing different control techniques including model reference adaptive control, model predictive and proportional integral controllers have been used [8]-[11]. A droplet volume control technique is based on the footprint of the droplet formation using optical techniques [6].
978-1-4799-0223-1/13/$31.00 ©2013 IEEE
Open loop and closed loop controllers can be used to control and estimate the system behavior. Fixed gain proportional-integral (PI) controllers have been considered one of the most common ways to control basic systems. However, the PI controller performance highly depends on the accuracy of the plant parameters. Therefore, they cannot perform accurately in highly sensitive pico-droplet dispensing systems as their parameters and hence the controller gains vary over time [12], [13]. Robust methods can also be used to increase the ability of a system to adapt itself to parameter uncertainty or disturbance. Robust methods can identify the system variations and update the controller’s gains. In general, adaptive controllers can be extremely helpful to control such devices since they can be adapted to the controlled system parameters [14]-[16]. The use of model reference adaptive control has been reported earlier [17]. However, often models cannot include all system details. In addition, the system response may have some delays associated with viscous liquid dispensing, making the system response non-minimum phase. Therefore, for perfect tracking performance, adaptive controllers need a parameter estimation unit that can recursively identify the plant parameters and an adaptation unit to constantly adjust the controller gains [14]. Forgetting factor least square method can be used to estimate and identify the system parameters. By implementing the FFRLS method in indirect adaptive control scheme, perfect droplet dispensing performance can be obtained [21]. In this paper, forgetting factor RLS and indirect adaptive controller have been used to measure and control a premodeled digital microfluidic system. This method has led to improve the functionality and accuracy of the system. Simulations of tracking performance and controller effort of the system have been demonstrated to analyze the effectiveness of the indirect adaptive control system in digital microfluidics. II.
DIGITAL MICROFLUIDIC SYSTEM MODELLING
The model for digital microfluidic system has been made using the electrical equivalent circuit of the system [17]. The voltage which will be applied across the plates leads to the electromechanical actuation force, electro-wetting-ondielectric (EWOD) and dielectrophoresis (DEP) [10], [11].
3578
where u and y are system input and output respectively, Z(s) and R(s) are the numerator and denominator of the transfer function. These polynomials for an nth order system with relative degree n-m can be described as
Sn
R(s)
an 1s n
1
... a1s a0
(4)
Z ( s ) bm s m ... b1 b0
(5)
Parametric model of the pico-droplet system can be obtained from (1)as
Fig. 1. Schematic diagram of a digital microfluidic system.
These forces can cause pico-droplet to move across the plates and electrodes. The output of the system will be voltage on the dielectric layer between air and the liquid. Applying voltage to each electrode generates forces on the droplet and attracts it to move in the same direction as electrodes. The droplet will fill the gap between plates and change the output voltage. The transfer function of the system is defined as the voltage drop across the droplet Vout to the input voltage Vin.
A B s C C D s C
y u
(6)
where A , B , D , are system parameters which are described as C C C
P1, P2, P3:
y
P1s P2
u
s P3
(7)
The system (7) can be described in a suitable parametric representation as *T
z
(8)
*
where is the matrix of unknowns and matrix of measurement signals as follows
is the normalized
Fig. 2. Equivalent circuit of the system
Using equivalent circuit above the transfer function can be expressed as:
Vout Gm ( s ) Vin
A
Gm ( s )
C
(1
As B Cs D B
D
A
C ) D
s
C
s
P1 , P2 , P3
(9)
1 u, u, y s 1 s 1 s 1
(1)
1
Considering (3), the Laplace representation of the system can be obtained as
(2)
where A, B, C, and D are system parameters which will be constitute based on a capacitive-resistive circuit. To control the droplet dispensing, the parameters should be accurately known. Perfect estimation of these parameters will update the system model and the controller gains [17]. III.
A B D , , C C C
*
Dividing
sy
sp1u
p2 u
both
sides
by
p3 y
(10)
normalizing
signal
) n , where n is the degree of system results in
(s) (s
the normalized system representation. This will guarantee the boundedness of the measurement and output signals as follows:
PARAMETRIC MODEL PRESENTTION
The droplet dispensing system (2) can be represented as a first-order zero-relative degree system. A parametric model of the system is as follows: y
Z ( s) u R( s )
(3)
3579
s s 1
y
p1 , p2 , p3
s s 1
u,
1 s 1
u,
1 y s 1
T
(11)
IV.
PARAMETER ESTIMATION USING FORGETTING FACTOR RECURSIVE LEAST SQUARE
yp
Recursive least square is used to identify the parameters of digital microfluidic system. This technique can be written as
Pe ,
( 0)
0
,
(12)
r The controller gains for ( K , from (17) as follows:
where 0 is the initial value of the parameters, and e is the estimation error calculated by
z
e
T
.
(13)
m The covariance matrix P is updated recursively by using the forgetting factor as 2 s
T
P
P
P
m
P , P ( 0)
2 s
P0
, (15) with a positive 0 as normalizing coefficient. The performance of the FFRLS will depend on the choice of and as they affect the system delay and estimation speed [18]. m
V.
K
1
,
(17)
1
and
2
) can be obtained
m1 p1
(18)
Solving closed loop transfer functions for the plant and model, two values will be considered for K. The other controller coefficients will be: m3 p3 bm m3 bm p3 (19) 1 p3 1
(14)
where 0 is the forgetting factor, and m s2 is the normalizing signal, defined as 2 s
ym r
bm m3 bm p3 m3 p3 p2 ( p3 1)
2
and
1
p32
(20)
0.
T
1
INDIRECT ADAPTIVE CONTROLLER
The control will use the estimator parameters to control the droplet volume. Tracking performance of the system highly depends on the quality of parameter estimation. Using perfect estimation values, the controller gains can be obtained.
VI.
CONTROLLER AND ESTIMATOR SIMULATION RESULTS
To guarantee the stability and parameter estimation convergence, the must be persistently exciting [18]. The ms
control performance highly depends on forgetting and normalizing factors and . There is an acceptable range for the value by which the increment of forgetting factor, the tracking and estimation time will be advanced. In these simulations is chosen between 1 and 30. The parameter estimation profile for each system parameter is shown in Fig. 4-6. There are two different configurations for parameter estimations and control performance with 10 , 1 , and the other configuration is 10 , 30 . P1 3 2.5
Value
2
Fig. 3. Block Diagram of Indirect Adpative Controller with Estimator
The closed-loop transfer function from the plant output yp to the reference signal r without the estimator is:
1 0.5
yp r
1.5
K( ps p2 ) 1 (1
2
1
p1 )s
( p3 bm
1
1
p2
2
p2 )s bm p3
2
p2
1
p3
(16)
In order to get the zero error tracking performance, the plant and model should generate similar outputs. Therefore, the controller should generate appropriate signal to drive the plant as model dictates. Therefore, the plant and model output to input signals should match:
0
Estimated Value Actual Value 0
0.5
Fig. 4. Plant parameter identification
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1 Time (ms)
=10 and
1.5
2 4
x 10
=1
P2
P2
10
10
8
8 Value
12
Value
12
6
4
4 2 0
6
2
Estimated Value Actual Value 0
0.2
0.4
0.6
0.8
Fig. 5. Plant parameter identification
1 1.2 Time (ms)
=10 and
1.4
1.6
0
1.8
2
Estimated Value Actual Value 0
200
400
600
800
4
x 10
1000 1200 Time (ms)
Fig. 8. Plant parameter identification
=10 and
1400
1600
1800
2000
=30
=1 P3 5
P3 5
4
4 3 Value
Value
3
2
2 1
1
0
Estimated Value Actual Value
0 0
0.2
0.4
0.6
0.8
1 1.2 Time (ms)
Fig. 6. Plant parameter identification
=10 and
1.4
1.6
-1
1.8
2
Estiamted Value Actual Value 0
200
400
600
1400
1600
1800
2000
Fig. 9. Plant parameter identification
=10 and
=30
=1
In Fig. 10-13 tracking performance and the error of the system are shown. It is seen that at larger forgetting factor values, the convergence rate of parameters and system tracking increases.
P1 3
600
2.5
500
Tracking Performance Model Output Plant Output
400
Value
2 Value
1000 1200 Time (ms)
4
x 10
In Fig. 7-9 it is seen with higher values for forgetting factor system parameter estimation will be improved and the estimator quickly converges to actual values.
1.5 1
300 200 100
0.5 0
800
0
Estimated Value Actual Value
-100
0
200
400
600
800
1000 1200 Time (ms)
Fig. 7. Plant parameter identification
=10 and
1400
=30
1600
1800
2000
0
0.2
0.4
0.6
0.8
1 1.2 Time (ms)
Fig. 10. Tracking Performance of Adaptive Controller
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1.4
1.6
1.8
2 4
x 10
=10 and
=1
Plant parameter estimation and trajectory tracking has been improved by using higher forgetting factor values considering its limit to keep system stable. Since digital microfluidics systems require controllers to operate accurately, indirect adaptive controllers can improve the droplet dispensing by compensating the system parameter variations. By updating indirect adaptive controller and using forgetting factor RLS estimations, system parameters can be accurately obtained. This application is suitable for uncertain parameter systems.
600
400
Error
200
0
In this study, forgetting factor RLS identified system parameters with different tracking performance. FFRLS method can improve the controller effort by varying forgetting factor value. Fig. 14 compares the controller effort for =1 and =30 with same value for =10.
-200
-400
0
0.2
0.4
0.6
Fig. 11. Error of the system for
0.8
1 1.2 Time (ms)
=10 and
1.4
1.6
1.8
2 4
x 10
=1 Control Effort 300
Comparing paramter estimations for two different forgetting factors it can be obtained that for value of 30 plant paramters can be estimated less than 2 seconds and for value of 1 it takes about 20 seconds to estimate these values. Since the estimation is more precise, the tracking error is reduced improving the control performance.
Alpha =10 Beta =1 250
200
Value
150
Tracking Performance 600 Model Output Plant Output
500
Alpha =10 Beta=30
100
50
0
Value
400 -50
300
0.4
0.6
0.8
1 Time (ms)
1.2
1.4
1.6
1.8
2 4
x 10
Higher values for forgetting factor improve the system tracking performance and control effort. Comparing first 0.75 second of Fig. 14, it is seen that higher values for enhance the system control effort. Since the error of the controller highly depends on control effort, higher forgetting factor value can advance the system error.
100
0
0.2
0.4
0.6
0.8
1 1.2 Time (ms)
Fig. 12. Tracking Performance of Adaptive Controller
1.4
1.6
1.8
2 4
x 10
=10 and
=30
VII. CONCLUSION Forgetting factor recursive least square estimates the parameters of systems with different orders and relative degrees. Having system parameters known perfectly picodroplet system can be controlled using indirect adaptive method. The controller can be adapted to any system parameter changes to improve the droplet dispensing system performance through better parameter eestimation and tracking.
600 500 400 300 Error
0.2
Fig. 14. Control effort for different forgetting factors.
200
0
0
200 100
REFERENCES
0
[1]
-100 -200
0
0.2
0.4
0.6
Fig. 13. Error of the system for
0.8
1 1.2 Time (ms)
=10 and
1.4
1.6
1.8
2
[2]
4
x 10
=30
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A. Izadian, L. Hornak, P. Famouri, ”Adaptive Control of MEMS, Devices” in Proc. Int. Conf. on Intelligent Systems and Control, ISC, 2006, August 14-16, 2006. D. Chatterjee, H. Shepherd, and R. L. Garrell,” Electromechanical model for actuating liquids in a two-plate droplet microfluidic device,” Lab on a Chip, vol.9, 2009, pp. 1219-1229.
[3]
Mais J. Jebrail,a Michael S. Bartschb and Kamlesh D. Patel, “Digital microfluidics: a versatile tool for applications in chemistry, biology and medicine”, published online 10 May 2012. [4] 1--J. W. Judy, “Microelectromechanical Systems (MEMS) - Their Design, Fabrication, and Broad Range of Application”, Journal of Smart Materials, vol. 10, no. 6, December 2001, pp. 1115-1134J. [5] 2--S. B. Brown, C. Muhlstein, C. Chui, C. Abnet, “Testing of MEMS material properties and stability,” IEEE Readiness Technology Conference, Aug. 1998 pp. 161 – 162 [6] J. Gong and C.-J. Kim,”All-electric droplet generation on-chip with realtime feedback control for EWOD digital microfluidics,” Lab-on Chip, vol. 8, 2008, pp. 898-906. [7] A. Izadian, P. Famouri, “Reliability Enhancement of Micro Comb Resonators under Fault Conditions,” IEEE Transaction on Control System Technology, vol. 16, no. 4, July 2008, pp 726-734. [8] A. Izadian, P. Khayyer, and P. Famouri, “Fault Diagnosis of Time Varying Parameter Systems with Application in MEMS LCRs,” IEEE Transaction on Industrial Electronics, Invited Paper, vol. 56, no. 4, April 2009, pp. 973-978. [9] A. Izadian, P. Famouri, “Intelligent Fault Diagnosis of MEMS Micro Comb Resonators Using Multiple-Model Adaptive Control Technique,” IEEE Transaction on Control System Technology, (In Press, 2010) [10] A. Izadian, “Automatic Control and Fault Diagnosis of MEMS Lateral Comb Resonators,” PhD. Dissertation, West Virginia University, Lane Department of Computer Science and Electrical Engineering, May 2008.
[11] T. B. Jones, J. D. Fowler, Y. S. Chang, C. J. Kim, ”Frequency Based Relationship of Electrowetting and Dielectrophoretic Liquid Microactuation,” Langmuir 2003, vol. 19, pp. 7646-7651 [12] L. Wang, J. M. Dawson, L. A. Hornak, P. Famouri, R. Ghaffarian , Real Time Transitional control of a MEMS comb resonator. IEEE transaction on Aerospace and electronics. Vol. 40, No. 2, April 2004. [13] L. Wang, J. M. Dawson, J. Chen, P. Famouri, and L. A. Hornak, stroke Length control of a MEMS device. ISIE2000, Cholula, Puebla, Mexico. [14] A. Izadian and P. Famouri,” Reliability Enhancement of MEMS Lateral Comb Resonators under Fault Conditions,” IEEE Transaction on Control System Technology, vol. 16, no. 4, July 2008, pp. 726-734. [15] R. P. Leland, “Adaptive Control of a MEMS Gyroscope Using Lyapunov Methods,” IEEE Trans. on Control Systems Technology, vol. 14, no. 2, pp. 278 – 283, March, 2006. [16] A. Izadian, L. A. Hornak, P. Famouri, ”Trajectory Control of Lateral Comb Resonators under Fault Conditions,” in Proc. 2007 American Cont. Conf., July 2007. [17] A.Izadian, ” Pico-droplet dispensing control in digital microfluidic systems” 49th IEEE conference December 15-17 2010. [18] P. Ioannou, B. Fidan, Adaptive Control Tutorial, Siam, 2006.
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Abstract— Digital microfluidics systems require advanced controllers to operate accurately since their parameters are subjected to change in environment and over time. Due to imperfect manufacturing processes, their fabricated system parameters may become different from the destined values. Hence, estimation based controllers are required to identify the system parameters. The electrowetting on dielectric can be precisely controlled to dispense desirable and repeatable droplets. However, the system parameters and their variation over time makes the control system challenging. This paper describes the application of an indirect adaptive trajectory controller for digital Pico-Droplet dispensing system. Forgetting factor recursive least square estimator is used to estimate the system parameters including capacitance and resistance of the occupying droplet between electrodes. Indirect adaptive technique is used to measure and control the droplet volume on the dispensing electrodes. Simulations of the estimator, tracking performance of dispensed droplet volume and the controller’s control effort are provided to demonstrate an accurate and high performance control approach.
I.
INTRODUCTION
Digital microfluidic systems are increasingly used in LabOn-Chip devices and commercial electronics. Since these devices are made in small scale, they can advance the speed of processes and highly reduce the cost of experiments [1]-[3]. In many applications such as drug delivery or quantitative analysis, small scale handing of liquids is the key in safe and accurate operation of the device. Less expensive and highly functional material have been introduced and used in digital microfluidics. However, stability and reliability of device operation depends on the effectiveness of the active controllers [4]-[6]. Using real time parameter estimation, the slowly time varying parameters and device behavioral variations that affect the droplet dispensing can be identified and proper control actions can be taken [4], [7]. Since the size of droplets is highly dependent on the volume of liquid reservoir, it is important to control the droplet volume through various techniques. The parameters of pico-droplet systems are also subjected to change in environment over time, with viscosity of liquid, air gap, and operating frequency of the device. For accurate, rapid, and repeatable droplet dispensing different control techniques including model reference adaptive control, model predictive and proportional integral controllers have been used [8]-[11]. A droplet volume control technique is based on the footprint of the droplet formation using optical techniques [6].
978-1-4799-0223-1/13/$31.00 ©2013 IEEE
Open loop and closed loop controllers can be used to control and estimate the system behavior. Fixed gain proportional-integral (PI) controllers have been considered one of the most common ways to control basic systems. However, the PI controller performance highly depends on the accuracy of the plant parameters. Therefore, they cannot perform accurately in highly sensitive pico-droplet dispensing systems as their parameters and hence the controller gains vary over time [12], [13]. Robust methods can also be used to increase the ability of a system to adapt itself to parameter uncertainty or disturbance. Robust methods can identify the system variations and update the controller’s gains. In general, adaptive controllers can be extremely helpful to control such devices since they can be adapted to the controlled system parameters [14]-[16]. The use of model reference adaptive control has been reported earlier [17]. However, often models cannot include all system details. In addition, the system response may have some delays associated with viscous liquid dispensing, making the system response non-minimum phase. Therefore, for perfect tracking performance, adaptive controllers need a parameter estimation unit that can recursively identify the plant parameters and an adaptation unit to constantly adjust the controller gains [14]. Forgetting factor least square method can be used to estimate and identify the system parameters. By implementing the FFRLS method in indirect adaptive control scheme, perfect droplet dispensing performance can be obtained [21]. In this paper, forgetting factor RLS and indirect adaptive controller have been used to measure and control a premodeled digital microfluidic system. This method has led to improve the functionality and accuracy of the system. Simulations of tracking performance and controller effort of the system have been demonstrated to analyze the effectiveness of the indirect adaptive control system in digital microfluidics. II.
DIGITAL MICROFLUIDIC SYSTEM MODELLING
The model for digital microfluidic system has been made using the electrical equivalent circuit of the system [17]. The voltage which will be applied across the plates leads to the electromechanical actuation force, electro-wetting-ondielectric (EWOD) and dielectrophoresis (DEP) [10], [11].
3578
where u and y are system input and output respectively, Z(s) and R(s) are the numerator and denominator of the transfer function. These polynomials for an nth order system with relative degree n-m can be described as
Sn
R(s)
an 1s n
1
... a1s a0
(4)
Z ( s ) bm s m ... b1 b0
(5)
Parametric model of the pico-droplet system can be obtained from (1)as
Fig. 1. Schematic diagram of a digital microfluidic system.
These forces can cause pico-droplet to move across the plates and electrodes. The output of the system will be voltage on the dielectric layer between air and the liquid. Applying voltage to each electrode generates forces on the droplet and attracts it to move in the same direction as electrodes. The droplet will fill the gap between plates and change the output voltage. The transfer function of the system is defined as the voltage drop across the droplet Vout to the input voltage Vin.
A B s C C D s C
y u
(6)
where A , B , D , are system parameters which are described as C C C
P1, P2, P3:
y
P1s P2
u
s P3
(7)
The system (7) can be described in a suitable parametric representation as *T
z
(8)
*
where is the matrix of unknowns and matrix of measurement signals as follows
is the normalized
Fig. 2. Equivalent circuit of the system
Using equivalent circuit above the transfer function can be expressed as:
Vout Gm ( s ) Vin
A
Gm ( s )
C
(1
As B Cs D B
D
A
C ) D
s
C
s
P1 , P2 , P3
(9)
1 u, u, y s 1 s 1 s 1
(1)
1
Considering (3), the Laplace representation of the system can be obtained as
(2)
where A, B, C, and D are system parameters which will be constitute based on a capacitive-resistive circuit. To control the droplet dispensing, the parameters should be accurately known. Perfect estimation of these parameters will update the system model and the controller gains [17]. III.
A B D , , C C C
*
Dividing
sy
sp1u
p2 u
both
sides
by
p3 y
(10)
normalizing
signal
) n , where n is the degree of system results in
(s) (s
the normalized system representation. This will guarantee the boundedness of the measurement and output signals as follows:
PARAMETRIC MODEL PRESENTTION
The droplet dispensing system (2) can be represented as a first-order zero-relative degree system. A parametric model of the system is as follows: y
Z ( s) u R( s )
(3)
3579
s s 1
y
p1 , p2 , p3
s s 1
u,
1 s 1
u,
1 y s 1
T
(11)
IV.
PARAMETER ESTIMATION USING FORGETTING FACTOR RECURSIVE LEAST SQUARE
yp
Recursive least square is used to identify the parameters of digital microfluidic system. This technique can be written as
Pe ,
( 0)
0
,
(12)
r The controller gains for ( K , from (17) as follows:
where 0 is the initial value of the parameters, and e is the estimation error calculated by
z
e
T
.
(13)
m The covariance matrix P is updated recursively by using the forgetting factor as 2 s
T
P
P
P
m
P , P ( 0)
2 s
P0
, (15) with a positive 0 as normalizing coefficient. The performance of the FFRLS will depend on the choice of and as they affect the system delay and estimation speed [18]. m
V.
K
1
,
(17)
1
and
2
) can be obtained
m1 p1
(18)
Solving closed loop transfer functions for the plant and model, two values will be considered for K. The other controller coefficients will be: m3 p3 bm m3 bm p3 (19) 1 p3 1
(14)
where 0 is the forgetting factor, and m s2 is the normalizing signal, defined as 2 s
ym r
bm m3 bm p3 m3 p3 p2 ( p3 1)
2
and
1
p32
(20)
0.
T
1
INDIRECT ADAPTIVE CONTROLLER
The control will use the estimator parameters to control the droplet volume. Tracking performance of the system highly depends on the quality of parameter estimation. Using perfect estimation values, the controller gains can be obtained.
VI.
CONTROLLER AND ESTIMATOR SIMULATION RESULTS
To guarantee the stability and parameter estimation convergence, the must be persistently exciting [18]. The ms
control performance highly depends on forgetting and normalizing factors and . There is an acceptable range for the value by which the increment of forgetting factor, the tracking and estimation time will be advanced. In these simulations is chosen between 1 and 30. The parameter estimation profile for each system parameter is shown in Fig. 4-6. There are two different configurations for parameter estimations and control performance with 10 , 1 , and the other configuration is 10 , 30 . P1 3 2.5
Value
2
Fig. 3. Block Diagram of Indirect Adpative Controller with Estimator
The closed-loop transfer function from the plant output yp to the reference signal r without the estimator is:
1 0.5
yp r
1.5
K( ps p2 ) 1 (1
2
1
p1 )s
( p3 bm
1
1
p2
2
p2 )s bm p3
2
p2
1
p3
(16)
In order to get the zero error tracking performance, the plant and model should generate similar outputs. Therefore, the controller should generate appropriate signal to drive the plant as model dictates. Therefore, the plant and model output to input signals should match:
0
Estimated Value Actual Value 0
0.5
Fig. 4. Plant parameter identification
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1 Time (ms)
=10 and
1.5
2 4
x 10
=1
P2
P2
10
10
8
8 Value
12
Value
12
6
4
4 2 0
6
2
Estimated Value Actual Value 0
0.2
0.4
0.6
0.8
Fig. 5. Plant parameter identification
1 1.2 Time (ms)
=10 and
1.4
1.6
0
1.8
2
Estimated Value Actual Value 0
200
400
600
800
4
x 10
1000 1200 Time (ms)
Fig. 8. Plant parameter identification
=10 and
1400
1600
1800
2000
=30
=1 P3 5
P3 5
4
4 3 Value
Value
3
2
2 1
1
0
Estimated Value Actual Value
0 0
0.2
0.4
0.6
0.8
1 1.2 Time (ms)
Fig. 6. Plant parameter identification
=10 and
1.4
1.6
-1
1.8
2
Estiamted Value Actual Value 0
200
400
600
1400
1600
1800
2000
Fig. 9. Plant parameter identification
=10 and
=30
=1
In Fig. 10-13 tracking performance and the error of the system are shown. It is seen that at larger forgetting factor values, the convergence rate of parameters and system tracking increases.
P1 3
600
2.5
500
Tracking Performance Model Output Plant Output
400
Value
2 Value
1000 1200 Time (ms)
4
x 10
In Fig. 7-9 it is seen with higher values for forgetting factor system parameter estimation will be improved and the estimator quickly converges to actual values.
1.5 1
300 200 100
0.5 0
800
0
Estimated Value Actual Value
-100
0
200
400
600
800
1000 1200 Time (ms)
Fig. 7. Plant parameter identification
=10 and
1400
=30
1600
1800
2000
0
0.2
0.4
0.6
0.8
1 1.2 Time (ms)
Fig. 10. Tracking Performance of Adaptive Controller
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1.4
1.6
1.8
2 4
x 10
=10 and
=1
Plant parameter estimation and trajectory tracking has been improved by using higher forgetting factor values considering its limit to keep system stable. Since digital microfluidics systems require controllers to operate accurately, indirect adaptive controllers can improve the droplet dispensing by compensating the system parameter variations. By updating indirect adaptive controller and using forgetting factor RLS estimations, system parameters can be accurately obtained. This application is suitable for uncertain parameter systems.
600
400
Error
200
0
In this study, forgetting factor RLS identified system parameters with different tracking performance. FFRLS method can improve the controller effort by varying forgetting factor value. Fig. 14 compares the controller effort for =1 and =30 with same value for =10.
-200
-400
0
0.2
0.4
0.6
Fig. 11. Error of the system for
0.8
1 1.2 Time (ms)
=10 and
1.4
1.6
1.8
2 4
x 10
=1 Control Effort 300
Comparing paramter estimations for two different forgetting factors it can be obtained that for value of 30 plant paramters can be estimated less than 2 seconds and for value of 1 it takes about 20 seconds to estimate these values. Since the estimation is more precise, the tracking error is reduced improving the control performance.
Alpha =10 Beta =1 250
200
Value
150
Tracking Performance 600 Model Output Plant Output
500
Alpha =10 Beta=30
100
50
0
Value
400 -50
300
0.4
0.6
0.8
1 Time (ms)
1.2
1.4
1.6
1.8
2 4
x 10
Higher values for forgetting factor improve the system tracking performance and control effort. Comparing first 0.75 second of Fig. 14, it is seen that higher values for enhance the system control effort. Since the error of the controller highly depends on control effort, higher forgetting factor value can advance the system error.
100
0
0.2
0.4
0.6
0.8
1 1.2 Time (ms)
Fig. 12. Tracking Performance of Adaptive Controller
1.4
1.6
1.8
2 4
x 10
=10 and
=30
VII. CONCLUSION Forgetting factor recursive least square estimates the parameters of systems with different orders and relative degrees. Having system parameters known perfectly picodroplet system can be controlled using indirect adaptive method. The controller can be adapted to any system parameter changes to improve the droplet dispensing system performance through better parameter eestimation and tracking.
600 500 400 300 Error
0.2
Fig. 14. Control effort for different forgetting factors.
200
0
0
200 100
REFERENCES
0
[1]
-100 -200
0
0.2
0.4
0.6
Fig. 13. Error of the system for
0.8
1 1.2 Time (ms)
=10 and
1.4
1.6
1.8
2
[2]
4
x 10
=30
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Mais J. Jebrail,a Michael S. Bartschb and Kamlesh D. Patel, “Digital microfluidics: a versatile tool for applications in chemistry, biology and medicine”, published online 10 May 2012. [4] 1--J. W. Judy, “Microelectromechanical Systems (MEMS) - Their Design, Fabrication, and Broad Range of Application”, Journal of Smart Materials, vol. 10, no. 6, December 2001, pp. 1115-1134J. [5] 2--S. B. Brown, C. Muhlstein, C. Chui, C. Abnet, “Testing of MEMS material properties and stability,” IEEE Readiness Technology Conference, Aug. 1998 pp. 161 – 162 [6] J. Gong and C.-J. Kim,”All-electric droplet generation on-chip with realtime feedback control for EWOD digital microfluidics,” Lab-on Chip, vol. 8, 2008, pp. 898-906. [7] A. Izadian, P. Famouri, “Reliability Enhancement of Micro Comb Resonators under Fault Conditions,” IEEE Transaction on Control System Technology, vol. 16, no. 4, July 2008, pp 726-734. [8] A. Izadian, P. Khayyer, and P. Famouri, “Fault Diagnosis of Time Varying Parameter Systems with Application in MEMS LCRs,” IEEE Transaction on Industrial Electronics, Invited Paper, vol. 56, no. 4, April 2009, pp. 973-978. [9] A. Izadian, P. Famouri, “Intelligent Fault Diagnosis of MEMS Micro Comb Resonators Using Multiple-Model Adaptive Control Technique,” IEEE Transaction on Control System Technology, (In Press, 2010) [10] A. Izadian, “Automatic Control and Fault Diagnosis of MEMS Lateral Comb Resonators,” PhD. Dissertation, West Virginia University, Lane Department of Computer Science and Electrical Engineering, May 2008.
[11] T. B. Jones, J. D. Fowler, Y. S. Chang, C. J. Kim, ”Frequency Based Relationship of Electrowetting and Dielectrophoretic Liquid Microactuation,” Langmuir 2003, vol. 19, pp. 7646-7651 [12] L. Wang, J. M. Dawson, L. A. Hornak, P. Famouri, R. Ghaffarian , Real Time Transitional control of a MEMS comb resonator. IEEE transaction on Aerospace and electronics. Vol. 40, No. 2, April 2004. [13] L. Wang, J. M. Dawson, J. Chen, P. Famouri, and L. A. Hornak, stroke Length control of a MEMS device. ISIE2000, Cholula, Puebla, Mexico. [14] A. Izadian and P. Famouri,” Reliability Enhancement of MEMS Lateral Comb Resonators under Fault Conditions,” IEEE Transaction on Control System Technology, vol. 16, no. 4, July 2008, pp. 726-734. [15] R. P. Leland, “Adaptive Control of a MEMS Gyroscope Using Lyapunov Methods,” IEEE Trans. on Control Systems Technology, vol. 14, no. 2, pp. 278 – 283, March, 2006. [16] A. Izadian, L. A. Hornak, P. Famouri, ”Trajectory Control of Lateral Comb Resonators under Fault Conditions,” in Proc. 2007 American Cont. Conf., July 2007. [17] A.Izadian, ” Pico-droplet dispensing control in digital microfluidic systems” 49th IEEE conference December 15-17 2010. [18] P. Ioannou, B. Fidan, Adaptive Control Tutorial, Siam, 2006.
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