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GPC Controller Design for an Intelligent Pneumatic Actuator
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Procedia Engineering 41 (2012) 657 – 663
GPC Controller design for an Intelligent Pneumatic Actuator Ahmad 'Athif Mohd Faudzi a,*, Nu’man Din Mustafaa, Khairuddin bin Osmanb, M. Asyraf Azmana, Koichi Suzumori c a b
Department of Mechatronics and Robotics Engineering, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Malaysia Department of Industrial Electronics, Faculty of Electrical and Electronics, Universiti Teknikal Malaysia Melaka, Hang Tuah J aya, 76100 Durian Tunggal, Melaka, Malaysia c Graduate School of Natural Science and Technology, Okayama University, Okayama, Japan
Abstract This paper proposed a Generalized Predictive Controller (GPC) to the Intelligent Pneumatic Actuator (IPA) in order to analyze the control approach and the performance of the actuator. First, an estimation model of the IPA is obtained by using a Reaction Curve M ethod and the controller is designed based on the model obtained. By using the exist ing function proposed by the previous researcher, the implementation and tuning process of the GPC become easier. The parameter needed for the function can be obtained from the model and from simple calculation as presented in this paper. The performance of the designed GPC controller is tested to the actuator position control and the results shows good accuracy and fast response. Lastly, Proportional-Integral-Derivative controller (PID) is used to validate the result.
© 2012 The Authors. Published by Elsevier Ltd. Select ion and/or peer-review under responsibility of the Centre o f Humanoid Robots and Bio-Sensor (HuRoBs), Faculty of Mechanical Engineering, Universiti Teknologi MARA. Keywords: Intelligent Pneumatic Actuator (IPA), GPC controller, PID controller.
1. Introduction Pneumatic actuator is a device that converts energy in the form of co mpressed air into motion. Lately, pneumat ic actuators have been largely used in numerous control applications in the industries. This is main ly because these actuators have advantages in their high power-to-weight ratio, relat ively low cost, easy to maintain, lighter, an d have simple structure compare to other actuator that available in the market [1]. Despite of all the advantages compared to electrical actuator used in robots and mach ines, it is d ifficult to control. This is due to the nonlinear factor involved such the nonlinearity o f the valve, compliance variation and generating force. GPC method was proposed by Clarke et al. [2] and it is widely used nowadays in various application in the industry. This is mainly due to the ability of the controller to control a p lant with various parameters, dead time and higher model order [3]. It is also recorded that GPC was successfully imp lemented to plants that have non-minimu m phase, unstable open loop, or model that are over parameterized or under parameterized by the estimation scheme [4]. Lotfi Chikh et al. in [5] used Generalized Predict ive force control fo r electropneumatic cylinder. Fro m the experiment al result, it shows that GPC was a good control approach and have high performance in term of capacity of tracking long duration static forces of high amp litudes. GPC controller also have been recorded for its quick response and accurate tracking based on research done by [6]. The capability of the GPC to control the pressure with system that used estimate parameter also have been recorded in [7]. In this paper, the aim is to design a controller for the Intelligent Pneumatic Actuator by using GPC. MATLAB – Simu link is used as the platform to develop the controller. It is hoped that GPC could eliminate the overshoot for open loop response and yield a good accuracy for pressure control. PID is used as reference because of its simp licity, easy to use and
* Corresponding author. Tel.: +607-5535291; fax: +607-5566272 E-mail address: [email protected]
1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.07.226
658
Ahmad ‘Athif Mohd Faudzi et al. / Procedia Engineering 41 (2012) 657 – 663
do not require the plant model to perform the controller [8, 9]. However, PID parameter is not adaptive and requires some optimal control method to increase its precision [10]. In this research, the parameter for PID is obtained by using MATLAB toolbox provided in the Simu lin k. The result for the GPC controller and PID are presented and discussed in the result and discussion section. The flow of the paper starts with introduction, Intelligent Pneumatic Actuator as plant, model estimation, GPC controller design, result and discussion and conclusion section. 2. Intelligent Pneumatic Actuator as Plant Intelligent Pneumatic Actuator (IPA) as shown in Fig. 1 was developed by A.A M. Faudzi et al. for research purposes [11, 12]. The application proposed was a seating apparatus called Pneu matic Actuator Seat ing System (PASS) where the system can imitate o ffice chair, stool and other chair shapes with stiffness and damping characteristics [13]. The pneumat ic actuator consists of five elements which are Programmable System on Ch ip (PSoC) as the controller, laser strip code with 0.169mm accuracy for the position feedback, optical encoder (A EDR – 8300) fo r read ing the laser strip code, pressure sensor (KOGANEI: PSU-EM-S) for reading the pressure inside the chamber and valves (KOGA NEI: EB10ES1 -PS-6W ) fo r controlling the injection of air into the chamber. The body part of the actuator is a linear double acting cylinder (KOGA NEI – HA Twinport Cy linders) with two air inlets and one exhaust outlet. The control algorith m is programmed direct ly into the microcontroller (PSo C) and it will control the output position based on the input fro m the sensors. The actuator has 200mm stroke and force up to 100N.
Laser strip code
PSoC microcontroller Optical encoder
Pressure sensor Valve
Fig. 1: Intelligent Pneumatic Actuator
3. Model Estimation Most equipment in the industries is very high order to model and difficult to control. The reason is that most industrial processes consists of many dynamic elements, usually first order, therefore the full model is of an order equal to the nu mber of elements [14]. According to [15] it is possible to estimate higher order model processes with first order process combined with dead t ime. Therefore, the p lant model is estimated by using Reaction Curve Method. Based on [14], the discrete transfer function for first-order type has the form
G( z 1 )
bz 1 d z 1 az 1
(1)
K (1 a)
(1)
where
a
e
T
W
d
Wd T
b
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Ahmad ‘Athif Mohd Faudzi et al. / Procedia Engineering 41 (2012) 657 – 663
Wd
W 1.5(t1 t2 )
(1)
1 1.5(t1 t2 ) 3
(2)
T is sampling time, t 1 is when the response reach 28.3% of the final value and t 2 is when the response reach 63.2% of the final value. Thus the model obtained is,
G( z 1 )
0.0491z 1 1 z 1 0.8261z 1
(3)
4. GPC Controller Design GPC algorith m is based on Controlled Auto-Regressive Integrated Moving Average (CARIMA) model and can be described after linearizat ion and considering operation around a particular set point of a SISO plant can be described as below:
A( z 1 ) y(t )
z d B( z 1 )u (t 1) C ( z 1 )
e(t ) '
(4)
where, u (t ) and y (t ) are the control input and output sequence of the plant, ' 1 z 1 , e(t ) is zero mean wh ite noise and d is dead time of the system. Meanwhile A, B and C are polynomial in the backward shift operator z 1 as following:
A( z 1 ) 1 a1z 1 a2 z 2 ... ana z na
(1)
B( z 1 )
b0 b1z 1 b2 z 2 ... bnb z nb
(1)
C ( z 1 ) 1 c1z 1 a2 z 2 ... cnc z nc
(5)
For simplicity, C ( z 1 ) is assumed to be 1, thus the following equation is obtained:
y(t 1)
(1 a) y(t ) ay(t 1) b'(t d ) H (t 1)
(6)
GPC algorithm consists of applying control sequence in order to minimize a multistage cost function as in Equation (7) ,[2].
J ( N1, N2 , N3 )
¦
N2 j N1
G ( j )[ yˆ (t j | t ) w(t j )]2 ¦ j 1 O ( j )['u(t j 1)]2 Nu
(7)
where u is the control input, N u is the control horizon, w is the reference value, yˆ is the plant predict ion on data up to t ime t, N 1 is the minimu m costing horizon with N1 control weighting [16].
d 1 , N 2 is the maximu m costing horizon with N 2
d N and λ is the
By using Simu link, the value for yˆ (t d j 1 | t ) and y(t d j 2 | t ) can be obtained easily, thus making the best expected value for yˆ (t d j ) according to [14] is given by:
yˆ (t d j | t )
(1 a) y(t d j 1 | t ) ayˆ (t d j 2) b'u(t j 1)
(8)
Equation (8) then is applied recursively j 1,2..., i and thus,
yˆ (t d i | t )
Gi ( z 1 ) yˆ (t d | t ) Di ( z 1 )'u(t i 1)
(9)
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Ahmad ‘Athif Mohd Faudzi et al. / Procedia Engineering 41 (2012) 657 – 663
Gi z 1 is of degree 1 and 'u(t ), 'u(t 1) 'u(t N 1) leads to
where
Di ( z 1 ) is of degree i 1 . Minimizing
Mu
Py Rw
J ( N1, N2 , Nu ) with respect to (10)
where
u [u(t ) 'u(t 1) 'u(t N 1)]T y [ yˆ (t d | t )
(1)
yˆ (t d 1 | t )]T
(1)
w [w(t d 1) w(t d 2) w(t d N )]T
(1)
M and R are matrices of dimension N x N, P o f d imension N x 2. When the future set point are unknown, w(t d i) is equal to current references, r (t ). Thus, the control increment is as Equation (11) , the equation for I y1 , I y 2 and I r1 according to the control scheme in Fig. 2 can be refer to [16].
'u(t )
I y1 yˆ (t d | t ) I y 2 yˆ (t d 1 | t ) I r1r (t )
Fig. 2: GPC Control Scheme
Fig. 3: GPC block diagram in Simulink
(11)
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Ahmad ‘Athif Mohd Faudzi et al. / Procedia Engineering 41 (2012) 657 – 663
5. Result and Discussion Choosing N1 5 , N 2 6 , N u 6 , O 0.95 and by using the plant model obtained in Equation (2), then the GPC algorith m is applied in the MATLAB for simulat ion by using Simu lin k as shown in Fig. 3. The simu lation is constructed based on the control scheme show in Fig. 2. The experimental results for the responses are shown in Fig. 4 and 5 below.
100
Position(mm)
100
9.085 9.09 9.095
0
Control Input(%)
GPC Input
99.5
50
0 500
5
10
15
20
25
30
Control Input 400 300
0
5
10
15 Time(s)
20
25
30
Fig. 4: Experimental result with step input
Position(mm)
101
100
100
0
99
9
9.5
10
-100 0
Control Input(%)
GPC Input
5
10
15
20
25
30
1000 Control Input 0 -1000
0
5
10
15 Time(s)
20
25
30
Fig. 5: Experimental result with square wave input
The result show a good performance with time response, TR =1.04s, settling time, TS =2s and dead time=0.1s for the for step input. The error fo r the step input as show in Fig. 4 is 0.15 and error for the square wave input as shown in Fig. 5 is 0.16. A lthough this controller used first order model as plant, the result showed that the controller manages to follo w the input given with accuracy up to 99.8%. In order to validate the controller, PID is used as reference and the comparison by using square wave input is shown Fig. 6.
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Position(mm)
101
100
100
0
99
9
9.5
GPC PID Input
10
-100 0
5
10
15 Time(s)
20
25
30
Fig. 6: Comparison between PID and GPC T able 1: Comparison between PID controller and GPC controller analysis results by using step input Analysis
PID controller
GPC controller
Percent Overshoot (%OS)
0%
0%
Dead T ime (TU)
0.1s
0.1s
Peak T ime (TP) = Settling Time (TS )
2.5s
2.0s
Rise T ime (TR)
1.2s
1.0s
Percent Steady State error (%ess)
0.001%
0.15%
In table 1, the results for both controllers show no percentage of overshoot (%OS) and this is mainly because the plant model used is first order. The results also show that both controllers have same dead time, Tv as the plant model used is similar. The GPC controller show better performance co mpared to PID with faster settling time, TS and rise time, TR . The percentage steady error of GPC controller is larger co mpared to PID controller but it is acceptable because still in the range of 2% of the final value. In Fig. 4 and Fig. 5 the GPC controller shows good response for step and square wave input. In order to imp lement other type of input such as sinus and random, the controller variable, N 1 , N 2 , N u and O need to be tuned. PID controller is a linear controller and does not taking into account the plant or model parameter. Thus the performance of PID controllers in nonlinear systems such as pneumatic actuator may vary and sometimes unstable. In the PID loop, the control signal is calculated fro m the erro r by cancelling out the current erro r directly by using gain (Proportional), the amount of time the error has continued uncorrected (Integral), and anticipate the future error fro m the rate of change of the error over time (Derivative). The error so metimes cannot be eliminated and as the result the error will grow bigger and the system will become unstable. Usually PID controller will give a fast and good performance, however in this case the plant used is pneumatic actuator and sometimes the nonlinearities and the constraint (i.e. the frict ion, the natural weight and the limitat ion of the valve) involved will make the plant unstable. GPC on the other hand, used past control and output signal to estimate the future control. In the GPC algorithm the plant parameter involved can be obtained fro m the estimated model as discussed in previous section. GPC algorithm has the capability to con trol plant with unstable open loop and constraint. 6. Conclusion In this paper the controller design using GPC and its performance have been recorded and analysed. The PID controller is used to validate the GPC controller performance. Both controllers have been successfully simulated in the MATLAB by using same model in Equation (3). The result is analysed and compared based on its percentage of overshoot (%OS), dead time (T ), peak time (T ), settling time (T ), rise time (T ) and percent steady state error (%ess ). A ll criteria except the percent steady state error show that the GPC controller has the capability to control the plant smoothly and with faster response compared to PID. In the future this research will be used as a comparison in the validation process for another controller and will be further improved in order to obtained better response. U
P
S
R
Acknowledgements The authors would like to thank Universit i Teknologi Malaysia (UTM) by UTM -NAS Grant No R.J130000.7723.4P008, Ministry of Higher Education (MOHE) Malaysia and Okayama University for their support.
Ahmad ‘Athif Mohd Faudzi et al. / Procedia Engineering 41 (2012) 657 – 663
References [1] A. Gentile, N. I. Giannoccaro and G. Reina, 2002. "Experimental tests on position control of a pneumatic actuator using on/off solenoid valves," in Industrial T echnology - IEEE ICIT '02. 2002 IEEE International Conference on, 2002, vol.1, p. 555 -559. [2] D.W. Clarke, C.M., and P.S. T uffs., 1987. Generalized predictive control, Part I: Basic algorithm and Part II: Extensions and interpretations., Automatica 23, p. 132 - 160. [3] Dajun, D., and L. Lixiong., 2007. “A Novel Implicit Generalized Predictive Control Algorithm” Control and Automation -ICCA 2007. IEEE International Conference on. 2007,vol.1,p. 2716-2770. [4] Zeybek, Z., and S. Çetinkaya., 2006. Generalized Delta Rule (GDR) algorithm with generalized predictive control (GPC) for optimum temperature tracking of batch polymerization. Chemical Engineering Science 61(20), p. 6691-6700. [5] Chikh, L.P., Philippe; Pierrot, Francois; Michelin, Micaël.,2010. “ A Generalized Predictive Force Controller for electropneumatic cylinders,” 8 th IFAC Symposium on Nonlinear Control System (2010) . [6] Qiang, S., and F. Liu., 2006. “The Direct Approach to Unified GPC Based on ARMAX/CARIMA/CARMA Model and Application for Pneum atic Actuator Control,” Innovative Computing, Information and Control 2006, ICICIC '06. First International Conference on 2006, Vol.1,p.336-339. [7] Chaewieang, P., K. Sirisantisamrit, and T. Thepmanee., 2008. “ Pressure control of pneumatic-pressure-load system using generalized predictive controller,” Mechatronics and Automation - ICMA 2008. IEEE International Conference on. 2008, p.788-791 [8] Kiam Heong, A., G. Chong, and L. Yun., 2005. PID control system analysis, design, and technology. Control System s Technology, IEEE T ransactionson 13(4), p. 559-576. [9] Silva, G.J., A. Datta., and S.P. Bhattacharyya., 2002. New results on the synthesis of PID controllers, Automatic Control, IEEE Transactions on 47(2), p. 241-252. [10 Jun, W., and H. Jian., 2009. “Fuzzy PID control of a wearable rehabilitation robotic hand driven by pneumatic muscles,” Micro-NanoMechatronics and Human Science-MHS 2009. International Symposium on. 2009,p.408-413. [11] A.A.M.Faudzi., K. Suzumori, and S. Wakimoto., 2008. “ Distributed Physical Human Machine Interaction Using Intelligent Pneumatic Cylinders,” Micro-Nano Mechatronics and Human Science- MHS 2008. International Symposium on. 2008,p.249-254. [12] A.A.M. Faudzi, K. Suzumori., and S. Wakimoto,, January 2009. Development of an Intelligent Pneumatic Cylinder for Distribut ed Physical HumanMachine Interaction Advanced Robotics, vol. 23, p. 203-225. [13] Ahmad 'Athif Mohd Faudzi, 2010. "Development of Intelligent Pneumatic Actuators and T heir Applications to Physical Human -Mechine Interaction System," Ph.D. thesis, T he Graduate School of Natural Science and Technology, Okayama Univ ersity, Japan. [14] Bordons, C., and E.F. Camacho.,1998. “ A Generalized Predictive Controller for a Wide Class of Industrial P rocesses,” Control Systems Technology, IEEE T ransactions on 6(3),p. 372-387. [15] P.B Deshpande., and R.H.A.,1981. Elements of Computer Process Control, Charlotte,USA. [16]Comacho, E.F., and Bordon, C., 1998. A generalized predictive controller for a wide class of industrial processes. IEEE Transactions on Robotics and Automation, 1998. 6(3),p.372-387.
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Procedia Engineering 41 (2012) 657 – 663
GPC Controller design for an Intelligent Pneumatic Actuator Ahmad 'Athif Mohd Faudzi a,*, Nu’man Din Mustafaa, Khairuddin bin Osmanb, M. Asyraf Azmana, Koichi Suzumori c a b
Department of Mechatronics and Robotics Engineering, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Malaysia Department of Industrial Electronics, Faculty of Electrical and Electronics, Universiti Teknikal Malaysia Melaka, Hang Tuah J aya, 76100 Durian Tunggal, Melaka, Malaysia c Graduate School of Natural Science and Technology, Okayama University, Okayama, Japan
Abstract This paper proposed a Generalized Predictive Controller (GPC) to the Intelligent Pneumatic Actuator (IPA) in order to analyze the control approach and the performance of the actuator. First, an estimation model of the IPA is obtained by using a Reaction Curve M ethod and the controller is designed based on the model obtained. By using the exist ing function proposed by the previous researcher, the implementation and tuning process of the GPC become easier. The parameter needed for the function can be obtained from the model and from simple calculation as presented in this paper. The performance of the designed GPC controller is tested to the actuator position control and the results shows good accuracy and fast response. Lastly, Proportional-Integral-Derivative controller (PID) is used to validate the result.
© 2012 The Authors. Published by Elsevier Ltd. Select ion and/or peer-review under responsibility of the Centre o f Humanoid Robots and Bio-Sensor (HuRoBs), Faculty of Mechanical Engineering, Universiti Teknologi MARA. Keywords: Intelligent Pneumatic Actuator (IPA), GPC controller, PID controller.
1. Introduction Pneumatic actuator is a device that converts energy in the form of co mpressed air into motion. Lately, pneumat ic actuators have been largely used in numerous control applications in the industries. This is main ly because these actuators have advantages in their high power-to-weight ratio, relat ively low cost, easy to maintain, lighter, an d have simple structure compare to other actuator that available in the market [1]. Despite of all the advantages compared to electrical actuator used in robots and mach ines, it is d ifficult to control. This is due to the nonlinear factor involved such the nonlinearity o f the valve, compliance variation and generating force. GPC method was proposed by Clarke et al. [2] and it is widely used nowadays in various application in the industry. This is mainly due to the ability of the controller to control a p lant with various parameters, dead time and higher model order [3]. It is also recorded that GPC was successfully imp lemented to plants that have non-minimu m phase, unstable open loop, or model that are over parameterized or under parameterized by the estimation scheme [4]. Lotfi Chikh et al. in [5] used Generalized Predict ive force control fo r electropneumatic cylinder. Fro m the experiment al result, it shows that GPC was a good control approach and have high performance in term of capacity of tracking long duration static forces of high amp litudes. GPC controller also have been recorded for its quick response and accurate tracking based on research done by [6]. The capability of the GPC to control the pressure with system that used estimate parameter also have been recorded in [7]. In this paper, the aim is to design a controller for the Intelligent Pneumatic Actuator by using GPC. MATLAB – Simu link is used as the platform to develop the controller. It is hoped that GPC could eliminate the overshoot for open loop response and yield a good accuracy for pressure control. PID is used as reference because of its simp licity, easy to use and
* Corresponding author. Tel.: +607-5535291; fax: +607-5566272 E-mail address: [email protected]
1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.07.226
658
Ahmad ‘Athif Mohd Faudzi et al. / Procedia Engineering 41 (2012) 657 – 663
do not require the plant model to perform the controller [8, 9]. However, PID parameter is not adaptive and requires some optimal control method to increase its precision [10]. In this research, the parameter for PID is obtained by using MATLAB toolbox provided in the Simu lin k. The result for the GPC controller and PID are presented and discussed in the result and discussion section. The flow of the paper starts with introduction, Intelligent Pneumatic Actuator as plant, model estimation, GPC controller design, result and discussion and conclusion section. 2. Intelligent Pneumatic Actuator as Plant Intelligent Pneumatic Actuator (IPA) as shown in Fig. 1 was developed by A.A M. Faudzi et al. for research purposes [11, 12]. The application proposed was a seating apparatus called Pneu matic Actuator Seat ing System (PASS) where the system can imitate o ffice chair, stool and other chair shapes with stiffness and damping characteristics [13]. The pneumat ic actuator consists of five elements which are Programmable System on Ch ip (PSoC) as the controller, laser strip code with 0.169mm accuracy for the position feedback, optical encoder (A EDR – 8300) fo r read ing the laser strip code, pressure sensor (KOGANEI: PSU-EM-S) for reading the pressure inside the chamber and valves (KOGA NEI: EB10ES1 -PS-6W ) fo r controlling the injection of air into the chamber. The body part of the actuator is a linear double acting cylinder (KOGA NEI – HA Twinport Cy linders) with two air inlets and one exhaust outlet. The control algorith m is programmed direct ly into the microcontroller (PSo C) and it will control the output position based on the input fro m the sensors. The actuator has 200mm stroke and force up to 100N.
Laser strip code
PSoC microcontroller Optical encoder
Pressure sensor Valve
Fig. 1: Intelligent Pneumatic Actuator
3. Model Estimation Most equipment in the industries is very high order to model and difficult to control. The reason is that most industrial processes consists of many dynamic elements, usually first order, therefore the full model is of an order equal to the nu mber of elements [14]. According to [15] it is possible to estimate higher order model processes with first order process combined with dead t ime. Therefore, the p lant model is estimated by using Reaction Curve Method. Based on [14], the discrete transfer function for first-order type has the form
G( z 1 )
bz 1 d z 1 az 1
(1)
K (1 a)
(1)
where
a
e
T
W
d
Wd T
b
659
Ahmad ‘Athif Mohd Faudzi et al. / Procedia Engineering 41 (2012) 657 – 663
Wd
W 1.5(t1 t2 )
(1)
1 1.5(t1 t2 ) 3
(2)
T is sampling time, t 1 is when the response reach 28.3% of the final value and t 2 is when the response reach 63.2% of the final value. Thus the model obtained is,
G( z 1 )
0.0491z 1 1 z 1 0.8261z 1
(3)
4. GPC Controller Design GPC algorith m is based on Controlled Auto-Regressive Integrated Moving Average (CARIMA) model and can be described after linearizat ion and considering operation around a particular set point of a SISO plant can be described as below:
A( z 1 ) y(t )
z d B( z 1 )u (t 1) C ( z 1 )
e(t ) '
(4)
where, u (t ) and y (t ) are the control input and output sequence of the plant, ' 1 z 1 , e(t ) is zero mean wh ite noise and d is dead time of the system. Meanwhile A, B and C are polynomial in the backward shift operator z 1 as following:
A( z 1 ) 1 a1z 1 a2 z 2 ... ana z na
(1)
B( z 1 )
b0 b1z 1 b2 z 2 ... bnb z nb
(1)
C ( z 1 ) 1 c1z 1 a2 z 2 ... cnc z nc
(5)
For simplicity, C ( z 1 ) is assumed to be 1, thus the following equation is obtained:
y(t 1)
(1 a) y(t ) ay(t 1) b'(t d ) H (t 1)
(6)
GPC algorithm consists of applying control sequence in order to minimize a multistage cost function as in Equation (7) ,[2].
J ( N1, N2 , N3 )
¦
N2 j N1
G ( j )[ yˆ (t j | t ) w(t j )]2 ¦ j 1 O ( j )['u(t j 1)]2 Nu
(7)
where u is the control input, N u is the control horizon, w is the reference value, yˆ is the plant predict ion on data up to t ime t, N 1 is the minimu m costing horizon with N1 control weighting [16].
d 1 , N 2 is the maximu m costing horizon with N 2
d N and λ is the
By using Simu link, the value for yˆ (t d j 1 | t ) and y(t d j 2 | t ) can be obtained easily, thus making the best expected value for yˆ (t d j ) according to [14] is given by:
yˆ (t d j | t )
(1 a) y(t d j 1 | t ) ayˆ (t d j 2) b'u(t j 1)
(8)
Equation (8) then is applied recursively j 1,2..., i and thus,
yˆ (t d i | t )
Gi ( z 1 ) yˆ (t d | t ) Di ( z 1 )'u(t i 1)
(9)
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Ahmad ‘Athif Mohd Faudzi et al. / Procedia Engineering 41 (2012) 657 – 663
Gi z 1 is of degree 1 and 'u(t ), 'u(t 1) 'u(t N 1) leads to
where
Di ( z 1 ) is of degree i 1 . Minimizing
Mu
Py Rw
J ( N1, N2 , Nu ) with respect to (10)
where
u [u(t ) 'u(t 1) 'u(t N 1)]T y [ yˆ (t d | t )
(1)
yˆ (t d 1 | t )]T
(1)
w [w(t d 1) w(t d 2) w(t d N )]T
(1)
M and R are matrices of dimension N x N, P o f d imension N x 2. When the future set point are unknown, w(t d i) is equal to current references, r (t ). Thus, the control increment is as Equation (11) , the equation for I y1 , I y 2 and I r1 according to the control scheme in Fig. 2 can be refer to [16].
'u(t )
I y1 yˆ (t d | t ) I y 2 yˆ (t d 1 | t ) I r1r (t )
Fig. 2: GPC Control Scheme
Fig. 3: GPC block diagram in Simulink
(11)
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5. Result and Discussion Choosing N1 5 , N 2 6 , N u 6 , O 0.95 and by using the plant model obtained in Equation (2), then the GPC algorith m is applied in the MATLAB for simulat ion by using Simu lin k as shown in Fig. 3. The simu lation is constructed based on the control scheme show in Fig. 2. The experimental results for the responses are shown in Fig. 4 and 5 below.
100
Position(mm)
100
9.085 9.09 9.095
0
Control Input(%)
GPC Input
99.5
50
0 500
5
10
15
20
25
30
Control Input 400 300
0
5
10
15 Time(s)
20
25
30
Fig. 4: Experimental result with step input
Position(mm)
101
100
100
0
99
9
9.5
10
-100 0
Control Input(%)
GPC Input
5
10
15
20
25
30
1000 Control Input 0 -1000
0
5
10
15 Time(s)
20
25
30
Fig. 5: Experimental result with square wave input
The result show a good performance with time response, TR =1.04s, settling time, TS =2s and dead time=0.1s for the for step input. The error fo r the step input as show in Fig. 4 is 0.15 and error for the square wave input as shown in Fig. 5 is 0.16. A lthough this controller used first order model as plant, the result showed that the controller manages to follo w the input given with accuracy up to 99.8%. In order to validate the controller, PID is used as reference and the comparison by using square wave input is shown Fig. 6.
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Position(mm)
101
100
100
0
99
9
9.5
GPC PID Input
10
-100 0
5
10
15 Time(s)
20
25
30
Fig. 6: Comparison between PID and GPC T able 1: Comparison between PID controller and GPC controller analysis results by using step input Analysis
PID controller
GPC controller
Percent Overshoot (%OS)
0%
0%
Dead T ime (TU)
0.1s
0.1s
Peak T ime (TP) = Settling Time (TS )
2.5s
2.0s
Rise T ime (TR)
1.2s
1.0s
Percent Steady State error (%ess)
0.001%
0.15%
In table 1, the results for both controllers show no percentage of overshoot (%OS) and this is mainly because the plant model used is first order. The results also show that both controllers have same dead time, Tv as the plant model used is similar. The GPC controller show better performance co mpared to PID with faster settling time, TS and rise time, TR . The percentage steady error of GPC controller is larger co mpared to PID controller but it is acceptable because still in the range of 2% of the final value. In Fig. 4 and Fig. 5 the GPC controller shows good response for step and square wave input. In order to imp lement other type of input such as sinus and random, the controller variable, N 1 , N 2 , N u and O need to be tuned. PID controller is a linear controller and does not taking into account the plant or model parameter. Thus the performance of PID controllers in nonlinear systems such as pneumatic actuator may vary and sometimes unstable. In the PID loop, the control signal is calculated fro m the erro r by cancelling out the current erro r directly by using gain (Proportional), the amount of time the error has continued uncorrected (Integral), and anticipate the future error fro m the rate of change of the error over time (Derivative). The error so metimes cannot be eliminated and as the result the error will grow bigger and the system will become unstable. Usually PID controller will give a fast and good performance, however in this case the plant used is pneumatic actuator and sometimes the nonlinearities and the constraint (i.e. the frict ion, the natural weight and the limitat ion of the valve) involved will make the plant unstable. GPC on the other hand, used past control and output signal to estimate the future control. In the GPC algorithm the plant parameter involved can be obtained fro m the estimated model as discussed in previous section. GPC algorithm has the capability to con trol plant with unstable open loop and constraint. 6. Conclusion In this paper the controller design using GPC and its performance have been recorded and analysed. The PID controller is used to validate the GPC controller performance. Both controllers have been successfully simulated in the MATLAB by using same model in Equation (3). The result is analysed and compared based on its percentage of overshoot (%OS), dead time (T ), peak time (T ), settling time (T ), rise time (T ) and percent steady state error (%ess ). A ll criteria except the percent steady state error show that the GPC controller has the capability to control the plant smoothly and with faster response compared to PID. In the future this research will be used as a comparison in the validation process for another controller and will be further improved in order to obtained better response. U
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Acknowledgements The authors would like to thank Universit i Teknologi Malaysia (UTM) by UTM -NAS Grant No R.J130000.7723.4P008, Ministry of Higher Education (MOHE) Malaysia and Okayama University for their support.
Ahmad ‘Athif Mohd Faudzi et al. / Procedia Engineering 41 (2012) 657 – 663
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