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Decoupling Control Based on Active Disturbance Rejection Controller
Available online at www.sciencedirect.com
Energy Procedia 17 (2012) 214 – 220
2012 International Conference on Future Electrical Power and Energy Systems
Decoupling Control Based on Active Disturbance Rejection Controller Sun Lingfang,Liu Xuying School of Automation Engineering Northeast Dianli University Jilin,p.r.China
Abstract In order to reduce the complexity and calculated amount of those decoupling control methods based on the system model, active disturbance rejection controller (ADRC) was applied to decoupling of multivariable system. We designed the tracking-differentiator, the extended state observer and nonlinear state error feedback components. Model was established with Simulink and encapsulated as functional module. ADRC is independent with the controlled object, and take all uncertainties of the system as the disturbances to compensate. The simulation result shows that ADRC can achieve the decoupling control of multivariable system, and have well robustness. © by Elsevier Ltd. Selection and/or peer-review under responsibility of Hainan University. ©2012 2011Published Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer] Keywords:decoupling control; ADRC; Simulink;
1.
Introduction
Active disturbance rejection technology develops and enriches the essence of PID, i.e., the new practical technology eliminates error based on error, open up and use especial nonlinear-effects. Consequently, on those occasions that the conventional PID can be used, as long as which can be digitized, the control quality and control accuracy can be improved fundamentally by Active Disturbance Rejection Control (ADRC).Especially on those occasions that requires high-speed and high-precision control in the harsh environment, the ADRC technique shows its superiority even more. The ADRC technique[1] need the information of the controlled object are the object’s order, action scope of force, the number of input-output channels and interfaces, signal’s delay time, especially those features which have clear physical concept and is easy to get, such as the "time scale" that represent the system response speed ,the fair time scales can be charged with the same ADRC. Large number of digital simulation and field applications proved that if we generally descriptive the controlled object’s approximate model, and add some extreme disturbance to represent the controlled object, the computer
1876-6102 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Hainan University. doi:10.1016/j.egypro.2012.02.086
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
numerical simulation results can be directly applied to the actual object, because ADRC is completely independent of the specific mathematical model of the controlled object. 2.
The Principle of Adrc
ADRC is mainly composed with three core components: the tracking-Differentiator (TD),the Extended State Observer (ESO) and nonlinear state error feedback (NLSEF). TD TD is the role of arranging transition process, tracking the system input signal fast without overshoot, and generating a good differential signal. If we input a signal v(t) to TD, it will output two signals:x1,x2,which x1 track v(t), x x . 2
1
NLSEF NLSEF’s control purpose is to impose proper control power u over the system, so that the system’s output y tracks a pre-set value or give a set trajectory v (t). The "state error feedback," use the appropriate "nonlinear configurations" to achieve the nonlinear state error feedback control law. ESO Automatic estimation and compensation of disturbance are the most crucial part of the controller designing. Extended state observer [2] is used to estimate the system state, model and external disturbance. Extended state observer does not depend on the specific mathematical model of disturbance generated and does not need to directly measure its effect. State observer which base on the measured system input (control inputs) and system output (some state variables or state variables of the function) determine the internal state information of system for all devices. State observer structure is shown as Figure 1. U
object˄state variable X˅
Y
state observer Figure 1. state observer structure
For a general nonlinear system
x ( n ) f ( x, x , " , x ( n 1) , t ) bu (t ) ® ¯y x Its state space expression is
215
216
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
x1 x 2 ° °# ° ® x n 1 x n ° ° x n f ( x1 , x 2 , " x n , t ) bu °¯ y x1 The design of state observer has nothing to do with the object function, so the observer is completely generic on a range of objects. We can see from the references[3-6] that the linear system tracks the input signal, the linear state feedback parameters will get a large (r = 100), but with a nonlinear feedback tracks the same input signal, the required parameters r is only 20 with considerable tracking error , the difference between them is an order of magnitude, which shows the appropriate nonlinear feedback ‘s efficiency is higher than the linear feedback. Therefore, we can establish the following state observer by nonlinear feedback.
e1 ° z ° 1 ° z °° 2 ®" ° ° z ° n 1 ° °¯ z n 3.
z1 y z 2 E 01e1 z 3 E 02 e1
0.5
sign(e1 )
z n E 0 n 1 e1 E n e1
1 2 n 1
1 2 n 1
sign(e1 )
sign(e1 ) bu
Decoupling Control Based on Adrc
Suppose a multiple input-multiple output system, the amplification coefficient bij of the controlling amount is the function of the state and time bij ( x, x , t )
x1 ° x ° 2 °" ° ° xm ® ° y1 ° y2 ° °" °y ¯ m Suppose matrix B is reversible.
f1 ( x1 , x1 ," , x m ) b11u1 " b1m u m f 2 ( x1 , x1 ," , x m ) b21u1 " b2 m u m f m ( x1 , x1 ," , x m ) bm1u1 " bmm u m x1 x2 xm
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
B ( x, x, t )
§ b11 ( x, x , t ) " b1m ( x, x , t ) · ¨ ¸ # % # ¨ ¸ ¨ b ( x, x, t ) " b ( x, x , t ) ¸ mm © m1 ¹
The model part beyond the controlling amount f ( f 1 f 2 " f m )
[ f 1 f 2 ! f m ]T is ”dynamic
is “static coupling”. coupling”ˈU=B( We show the “virtual controlling amount” into equation (1), the system equations (1) becomes
x ® ¯y
f ( x, x , t ) U x
In this system, the relationship of I-channel’s input and output is
x ® ¯ yi
f i ( x1 , x1 , " , x m , t ) U i xi
f i ( x1 , x1 ,", x m , t ) is the “total disturbance” which act on the channel, U is the i y x i .This way, on the every channel, the relationship between virtual input of I channel, output is i In the equation (2),
controlling Ui and controlled output Yi is single input-single output, nowthat, we disolve the question of the controlled system’s dynamic coupling and static coupling. The decoupling process diagram is shown as figure 2.
Figure 2. ADRC decoupling control structure
In the control vector U and the output vector Y, we embed m ADRC, then we can basically achieve the decoupling control of multivariable system. The relationship between actual control and virtual control can use the following expression.
u
B 1 ( x, x , t )U
The dynamic coupling part of each component which is treated as the total disturbance of each channel is estimated and compensated. Therefore, we can not considerĀdynamic couplingāpart when we use ADRC decoupling control. A large number of simulation studies have shown that the decoupling control need the estimation accuracy of “static coupling” part is not high when we use ADRC decoupling, we only ensure the matrix reversibility, it will not have a significant impact on the controlling quality of the closed-loop. 4.
Decoupling Control of Adrc’S Matlab Simulation Take the following second-order system as an example:
217
218
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
x1 x12 x2 2 x1 x2 sign(sin(0.9t)) b11(t)u1 b12 (t)u2 ° °x2 x1 x2 x12 x2 2 cos(0.7t) b21(t)u1 b22 (t)u2 ® ° y1 x1 °y x ¯ 2 2
We can observe from the above equation that the system is a serious nonlinearity, strong coupling, time-varying, multivariable-input and multivariable-output system. So we adopt ADRC decoupling for he need of system’s controlling quality. For the demand of less estimated precision on the matrix B (x), when we guarantee its reversibility, we can suppose:
ªb b º ª 3 0.5cos(t) 1 0.2sin(0.8t) º B(t) « 11 12 » « » ¬b21 b22 ¼ ¬3 0.6sin(0.6t) 2 0.5cos(0.7t)¼ Based on the ADRC’s principle, use the simulink simulation[6] to establish simulation model(figure3) and the controlled object model(figure4).
Figure 3. ADRC overall structure simulation
Figure 4.
controlled object’s structure
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
In the ADRC control system’s simulation platform, the function of the ADRC’s tracking-differentiator (TD), the extended state observer(ESO) use in the form of S-Function. Suppose the channel one’s value of the system is unit step signal, channel two’s value is the stack of sin(t) and the square wave signal, the square wave signal’s frequency is 0.25Hz,amplitude is 0.5. After setting the parameters of each functional module, the structure of the simulation is as follows:
Figure 5. Channel 1’s input and output responses
Figure 7.
Channel 1’s disturbance and its estimation
Figure 9.
Channel1 and Channel2’s estimate error
Figure 6.
Channel 2’s input and output responses
Figure 8.
Channel2’s disturbance and its estimation
219
220
5.
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
Conclusion
Results based on the simulation analysis shows that ADRC can solve the control problem of MIMO system with serious non-linear, strong coupling and time-varying characteristic. MIMO system achieved decoupling completely, and the output can fast track the input without overshoot, and the system also can keep steady quickly under the condition of the sudden change of square wave amplitude. The ESO of the ADRC can estimate the state and disturbance of the system, the actual estimated value of disturbance and the estimate of the disturbance almost completely overlap, which is the basis of the system’s high precision. References [1]
Han Jingqing.Active Disturbance Rejection Control Technique—the technique for estimating and compensating the uncertainties.Beijing:National Defense Industry Press,2008:197-270.
[2]
Yi Huang,Jingqing Han.Analysis and design for the second order Nonlinear continuous exteded.state observer.Chinese Science Bulletin,2000,45(21):1938-1944
[3] [4]
Han Jingqing.Active Disturbance Rejection Controller and Applytion.Control and decision,1998,13(1) :19-23. Qiu Xiaolin. Simulink3.0 the dynamic modeling and simulation tools of system based on MATLAB. Xi'an jiaotong university press,2003:165-223.
[5] [6]
Yang Jin Ming. Auto disturbance rejection speed control of linear switched reluctance motor.IEEELAS㧘2005:2491-2497. HU Lin-jing; SUN Zheng-shun ,Build and Encapsulate Custom Block in SIMULINK[J], Acta Simulata Systematica Sinica, 2004,16(3)㧦488-491
Energy Procedia 17 (2012) 214 – 220
2012 International Conference on Future Electrical Power and Energy Systems
Decoupling Control Based on Active Disturbance Rejection Controller Sun Lingfang,Liu Xuying School of Automation Engineering Northeast Dianli University Jilin,p.r.China
Abstract In order to reduce the complexity and calculated amount of those decoupling control methods based on the system model, active disturbance rejection controller (ADRC) was applied to decoupling of multivariable system. We designed the tracking-differentiator, the extended state observer and nonlinear state error feedback components. Model was established with Simulink and encapsulated as functional module. ADRC is independent with the controlled object, and take all uncertainties of the system as the disturbances to compensate. The simulation result shows that ADRC can achieve the decoupling control of multivariable system, and have well robustness. © by Elsevier Ltd. Selection and/or peer-review under responsibility of Hainan University. ©2012 2011Published Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer] Keywords:decoupling control; ADRC; Simulink;
1.
Introduction
Active disturbance rejection technology develops and enriches the essence of PID, i.e., the new practical technology eliminates error based on error, open up and use especial nonlinear-effects. Consequently, on those occasions that the conventional PID can be used, as long as which can be digitized, the control quality and control accuracy can be improved fundamentally by Active Disturbance Rejection Control (ADRC).Especially on those occasions that requires high-speed and high-precision control in the harsh environment, the ADRC technique shows its superiority even more. The ADRC technique[1] need the information of the controlled object are the object’s order, action scope of force, the number of input-output channels and interfaces, signal’s delay time, especially those features which have clear physical concept and is easy to get, such as the "time scale" that represent the system response speed ,the fair time scales can be charged with the same ADRC. Large number of digital simulation and field applications proved that if we generally descriptive the controlled object’s approximate model, and add some extreme disturbance to represent the controlled object, the computer
1876-6102 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Hainan University. doi:10.1016/j.egypro.2012.02.086
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
numerical simulation results can be directly applied to the actual object, because ADRC is completely independent of the specific mathematical model of the controlled object. 2.
The Principle of Adrc
ADRC is mainly composed with three core components: the tracking-Differentiator (TD),the Extended State Observer (ESO) and nonlinear state error feedback (NLSEF). TD TD is the role of arranging transition process, tracking the system input signal fast without overshoot, and generating a good differential signal. If we input a signal v(t) to TD, it will output two signals:x1,x2,which x1 track v(t), x x . 2
1
NLSEF NLSEF’s control purpose is to impose proper control power u over the system, so that the system’s output y tracks a pre-set value or give a set trajectory v (t). The "state error feedback," use the appropriate "nonlinear configurations" to achieve the nonlinear state error feedback control law. ESO Automatic estimation and compensation of disturbance are the most crucial part of the controller designing. Extended state observer [2] is used to estimate the system state, model and external disturbance. Extended state observer does not depend on the specific mathematical model of disturbance generated and does not need to directly measure its effect. State observer which base on the measured system input (control inputs) and system output (some state variables or state variables of the function) determine the internal state information of system for all devices. State observer structure is shown as Figure 1. U
object˄state variable X˅
Y
state observer Figure 1. state observer structure
For a general nonlinear system
x ( n ) f ( x, x , " , x ( n 1) , t ) bu (t ) ® ¯y x Its state space expression is
215
216
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
x1 x 2 ° °# ° ® x n 1 x n ° ° x n f ( x1 , x 2 , " x n , t ) bu °¯ y x1 The design of state observer has nothing to do with the object function, so the observer is completely generic on a range of objects. We can see from the references[3-6] that the linear system tracks the input signal, the linear state feedback parameters will get a large (r = 100), but with a nonlinear feedback tracks the same input signal, the required parameters r is only 20 with considerable tracking error , the difference between them is an order of magnitude, which shows the appropriate nonlinear feedback ‘s efficiency is higher than the linear feedback. Therefore, we can establish the following state observer by nonlinear feedback.
e1 ° z ° 1 ° z °° 2 ®" ° ° z ° n 1 ° °¯ z n 3.
z1 y z 2 E 01e1 z 3 E 02 e1
0.5
sign(e1 )
z n E 0 n 1 e1 E n e1
1 2 n 1
1 2 n 1
sign(e1 )
sign(e1 ) bu
Decoupling Control Based on Adrc
Suppose a multiple input-multiple output system, the amplification coefficient bij of the controlling amount is the function of the state and time bij ( x, x , t )
x1 ° x ° 2 °" ° ° xm ® ° y1 ° y2 ° °" °y ¯ m Suppose matrix B is reversible.
f1 ( x1 , x1 ," , x m ) b11u1 " b1m u m f 2 ( x1 , x1 ," , x m ) b21u1 " b2 m u m f m ( x1 , x1 ," , x m ) bm1u1 " bmm u m x1 x2 xm
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
B ( x, x, t )
§ b11 ( x, x , t ) " b1m ( x, x , t ) · ¨ ¸ # % # ¨ ¸ ¨ b ( x, x, t ) " b ( x, x , t ) ¸ mm © m1 ¹
The model part beyond the controlling amount f ( f 1 f 2 " f m )
[ f 1 f 2 ! f m ]T is ”dynamic
is “static coupling”. coupling”ˈU=B( We show the “virtual controlling amount” into equation (1), the system equations (1) becomes
x ® ¯y
f ( x, x , t ) U x
In this system, the relationship of I-channel’s input and output is
x ® ¯ yi
f i ( x1 , x1 , " , x m , t ) U i xi
f i ( x1 , x1 ,", x m , t ) is the “total disturbance” which act on the channel, U is the i y x i .This way, on the every channel, the relationship between virtual input of I channel, output is i In the equation (2),
controlling Ui and controlled output Yi is single input-single output, nowthat, we disolve the question of the controlled system’s dynamic coupling and static coupling. The decoupling process diagram is shown as figure 2.
Figure 2. ADRC decoupling control structure
In the control vector U and the output vector Y, we embed m ADRC, then we can basically achieve the decoupling control of multivariable system. The relationship between actual control and virtual control can use the following expression.
u
B 1 ( x, x , t )U
The dynamic coupling part of each component which is treated as the total disturbance of each channel is estimated and compensated. Therefore, we can not considerĀdynamic couplingāpart when we use ADRC decoupling control. A large number of simulation studies have shown that the decoupling control need the estimation accuracy of “static coupling” part is not high when we use ADRC decoupling, we only ensure the matrix reversibility, it will not have a significant impact on the controlling quality of the closed-loop. 4.
Decoupling Control of Adrc’S Matlab Simulation Take the following second-order system as an example:
217
218
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
x1 x12 x2 2 x1 x2 sign(sin(0.9t)) b11(t)u1 b12 (t)u2 ° °x2 x1 x2 x12 x2 2 cos(0.7t) b21(t)u1 b22 (t)u2 ® ° y1 x1 °y x ¯ 2 2
We can observe from the above equation that the system is a serious nonlinearity, strong coupling, time-varying, multivariable-input and multivariable-output system. So we adopt ADRC decoupling for he need of system’s controlling quality. For the demand of less estimated precision on the matrix B (x), when we guarantee its reversibility, we can suppose:
ªb b º ª 3 0.5cos(t) 1 0.2sin(0.8t) º B(t) « 11 12 » « » ¬b21 b22 ¼ ¬3 0.6sin(0.6t) 2 0.5cos(0.7t)¼ Based on the ADRC’s principle, use the simulink simulation[6] to establish simulation model(figure3) and the controlled object model(figure4).
Figure 3. ADRC overall structure simulation
Figure 4.
controlled object’s structure
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
In the ADRC control system’s simulation platform, the function of the ADRC’s tracking-differentiator (TD), the extended state observer(ESO) use in the form of S-Function. Suppose the channel one’s value of the system is unit step signal, channel two’s value is the stack of sin(t) and the square wave signal, the square wave signal’s frequency is 0.25Hz,amplitude is 0.5. After setting the parameters of each functional module, the structure of the simulation is as follows:
Figure 5. Channel 1’s input and output responses
Figure 7.
Channel 1’s disturbance and its estimation
Figure 9.
Channel1 and Channel2’s estimate error
Figure 6.
Channel 2’s input and output responses
Figure 8.
Channel2’s disturbance and its estimation
219
220
5.
Sun Lingfang and Liu Xuying / Energy Procedia 17 (2012) 214 – 220
Conclusion
Results based on the simulation analysis shows that ADRC can solve the control problem of MIMO system with serious non-linear, strong coupling and time-varying characteristic. MIMO system achieved decoupling completely, and the output can fast track the input without overshoot, and the system also can keep steady quickly under the condition of the sudden change of square wave amplitude. The ESO of the ADRC can estimate the state and disturbance of the system, the actual estimated value of disturbance and the estimate of the disturbance almost completely overlap, which is the basis of the system’s high precision. References [1]
Han Jingqing.Active Disturbance Rejection Control Technique—the technique for estimating and compensating the uncertainties.Beijing:National Defense Industry Press,2008:197-270.
[2]
Yi Huang,Jingqing Han.Analysis and design for the second order Nonlinear continuous exteded.state observer.Chinese Science Bulletin,2000,45(21):1938-1944
[3] [4]
Han Jingqing.Active Disturbance Rejection Controller and Applytion.Control and decision,1998,13(1) :19-23. Qiu Xiaolin. Simulink3.0 the dynamic modeling and simulation tools of system based on MATLAB. Xi'an jiaotong university press,2003:165-223.
[5] [6]
Yang Jin Ming. Auto disturbance rejection speed control of linear switched reluctance motor.IEEELAS㧘2005:2491-2497. HU Lin-jing; SUN Zheng-shun ,Build and Encapsulate Custom Block in SIMULINK[J], Acta Simulata Systematica Sinica, 2004,16(3)㧦488-491