Active Disturbance Rejection Control
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HAN’s ACTIVE DISTURBANCE REJECTION CONTROL (ADRC) -AN ANALYSIS
ADRC INTRODUCTION PID CONTROL ADRC
◦ ◦ ◦ ◦
TRANSIENT TRAJECTORY GENERATION NOISE TOLERANT DIFFERENTIATION NON LINEAR FEEDBACK TOTAL DISTURBANCE ESTIMATION AND REJECTION
STABILITY ANALYSIS APPLICATIONS OF ADRC IMPLEMENTATION ISSUES CONCLUSION
ADRC
PASSIVE DISTURBANCE REJECTION CONTROL ◦ Deals with disturbance (internal and external) as one of the design issues. ◦
ACTIVE DISTURBANCE REJECTION CONTROL ◦ The disturbances which are mostly external are estimated by using an observer and thus can be canceled out thus making the control design disturbance free. ◦
ADRC
WHY ADRC? ◦ LIKE PID CONTROL, THE CONTROL LAW IS BASED ON ERROR AND NOT ON MODEL ◦ IT IS BASED ON STATE OBSERVER, AN EFFICIENT STRATEGY WHICH IS THE OUTCOME OF MODERN CONTROL APPROACH ◦ IT USES NON-LINEAR FEEDBACK THUS ENHANCING PERFORMANCE ◦ IT IS WELL SUITED TO DIGITAL COMPUTER TECHNOLOGY BASED APPLICATIONS AND IS AN OUTCOME OF EXPERIMENTAL DIGITAL SIMULATIONS
PID CONTROL DATES BACK TO 1920s IT IS DOMINANT IN INDUSTRIAL APPLICATIONS EVEN TODAY SIMPLICITY IS A MERIT NOT ABLE TO MEET DEMANDS OF MODERN INDUSTRY IN TERMS OF
◦ EFFICIENCY ◦ LACK AND COST OF SKILLED LABOUR ◦ DUE TO SIMPLICITY DOES NOT TAKE FULL ADVANTAGE OF MODERN DAY POWERFUL DIGITAL PROCESSORS
DISADVANTAGES OF PID CONTROLLERS
ERROR COMPUTATION IS A CUMBERSOME PROCESS FOR INPUT OTHER THAN STEP SIGNAL. NOISE GETS AMPLIFIED DUE TO DERIVATIVE CONTROL. THE CONTROL LAW IS IN THE FORM OF A LINEAR WEIGHTED SUM REQUIRING OVERSIMPLIFICATION AND THUS LOSS OF PERFORMANCE. INTEGRAL CONTROL MAY LEAD TO COMPLICATIONS OF SATURATION AND REDUCED STABILITY AMRGIN DUE TO INTRODUCTION OF PHASE LAG.
ADRC
TO OVERCOME THE DISADVANTAGES OF PID CONTROLLERS, ADRC IS PROPOSED WHICH CONSISTS OF FOUR STEPS ◦ A SIMPLE DIFFERENTIAL EQUATION IS USED AS A TRANSIENT TRAJECTORY GENERATOR ◦ THE DIFFERENTIATOR IS DESIGNED TO BE NOISE TOLERANT ◦ NON LINEAR CONTROL LAWS ARE USED FOR FEEDBACK ◦ TOTAL DISTURBANCE ESTIMATION AND REJECTION CONCEPT IS USED
HISTORY OF ADRC FIRST SYSTEMATICALLY INTRODUCED IN 2001 [8] FURTHER ELABORATED IN [9] PRACTICAL APPLICATION IN [11,12][13] EQUIVALENT INPUT DISTURBANCE [14] AND DISTURBANCE INPUT COUPLING [15] ARE SPECIAL CASES OF ADRC WHERE ONLY EXTERNAL DISTURBANCE IS CONSIDERED
TRANSIENT TRAJECTORY GENERATION FOR A DYNAMICAL SYSTEM, STEP INPUT MAY NOT BE APPROPRIATE FOR THE OUTPUT TO TRACK DUE TO ITS SUDDEN JUMP INPUT WITH TRANSIENT PROFILE REQUIRED
TRANSIENT TRAJECTORY GENERATION FOR EXAMPLE, THE DOUBLE INTEGRAL PLANT ẋ1 =x2
ẋ2 =u
IF |u|≤ r AND v IS THE DESIRED VALUE OF x1, THEN THE TIME OPTIMAL SOLUTION IS
u = −r sign(x1 − v + ((x2|x2|)/2r))
TRANSIENT TRAJECTORY GENERATION THUS THE DESIRED PROFILE CAN BE CHOSEN AS v̇1 =v2
v̇2 = −r sign(v1 − v + ((v2|v2|)/2r))
THE PARAMETER r CAN BE CHANGED TO SPEED UP OR SLOW THE PROFILE
NOISE TOLERANT DIFFERENTIATION
IN PID CONTROL, DIFFERENTIATION IS OBTAINED BY THE LAPLACE OPERATOR AS IN
NOISE TOLERANT DIFFERENTIATION
This can be rewritten as
The time domain solution is obtained by inverse Laplace Transform operation as
This can be approximated as
NOISE TOLERANT DIFFERENTIATION
Thus, if v(t) contains noise of the form n(t), y(t) will contain the noise term.
Since τ is small, this term will amount to amplification of n(t).
The amplification of noise in PID control is undesirable.
NOISE TOLERANT DIFFERENTIATION A different form of Laplace operator for differentiation is proposed by Han. The approximation
is used.
NOISE TOLERANT DIFFERENTIATION A particular second approximation of a differentiator is given by Or
Defining r=1/τ, and carrying out inverse Laplace Transform on above equation gives
Where y(t) tracks v(t), tracks , and r determines the speed.
NOISE TOLERANT DIFFERENTIATION
Consider that v(t) is the input signal to be differentiated. Then the previous equation can be expressed as
This gives the fastest tracking of v(t) and its derivative subject to an acceleration limit in the form of r. The above equation is denoted as “tracking differentiator”.
NON LINEAR FEEDBACK
The PID controller employs a linear feedback which is a combination of present, past and future samples of the tracking error, e.
However with this form, the tracking error approaches zero or converges to origin with infinite time.
NON LINEAR FEEDBACK
A non-linear feedback of the form
is proposed by Han which has the following advantages:-
NON LINEAR FEEDBACK
The tracking error can approach zero in finite time and much more quickly by the choice of α<1. Another advantage of this selection of α is that the steady state error reduces significantly without having to resort to integral feedback. Thus the disadvantages of integral control can be avoided. An extreme case is when α=0, which is equivalent to bang-bang control that ensures zero steady state error without requirement of integral control term in PID.
TOTAL DISTURBANCE ESTIMATION AND REJECTION
Consider the single-input-single-output (SISO) second order plant represented by the state equation
TOTAL DISTURBANCE ESTIMATION AND REJECTION To ensure that y tracks the input, the control, u, is chosen such that it overcomes the multivariable function f(x1, x2, w(t), t) in feedback control architecture. Hence the above function can be denoted as
F(t) is called the “total disturbance”.
TOTAL DISTURBANCE ESTIMATION AND REJECTION
F(t) can be treated as an additional state variable, x3. Thus denoting G(t)=Ḟ(t), the second order plant can be represented by a new set of state equations given by
TOTAL DISTURBANCE ESTIMATION AND REJECTION
The above plant is observable and hence an extended state observer (ESO) can be constructed by defining the tracking error to be of the form e=z1-y as follows:-
The gains β01 , β02 and β03 are selected such that the tracking error, e, approaches zero in finite time.
TOTAL DISTURBANCE ESTIMATION AND REJECTION
It can be shown that by choosing the control law as
The second order plant reduces to the following set of equations:-
This cascaded integral form can be easily controlled by making u0 as function of the tracking error and its derivative i.e., a PD controller. Thus the control problem is converted to a simple problem of estimation and rejection of the total disturbance.
FORMULATION OF ADRC
TO BE CONTINUED
APPLICATIONS OF ADRC
IMPLEMENTATION ISSUES
CONCLUSION
REFERENCES [1] J. Han, “Control theory: Model approach or control approach,” Syst. Sci. Math., vol. 9, no. 4, pp. 328–335, 1989, (in Chinese). [2] J. Han andW.Wang, “Nonlinear tracking-differentiator,” Syst. Sci. Math., vol. 14, no. 2, pp. 177–183, 1994, (in Chinese). [3] J. Han, “Nonlinear PID controller,” J. Autom., vol. 20, no. 4, pp. 487– 490, 1994, (in Chinese). [4] J. Han, “Extended state observer for a class of uncertain plants,” Control Decis., vol. 10, no. 1, pp. 85–88, 1995, (in Chinese). [5] J. Han, “Auto disturbances rejection controller and its applications,” Control Decis., vol. 13, no. 1, pp. 19–23, 1998, (in Chinese). [6] J. Han, “From PID to auto disturbances rejection control,” Control Eng., vol. 9, no. 3, pp. 13–18, 2002, (in Chinese). [7] J. Han, “Active disturbances rejection control technique,” Frontier Sci., vol. 1, no. 1, pp. 24–31, 2007, (in Chinese).
REFERENCES (CTD..) [8] Z. Gao, Y. Huang, and J. Han, “An alternative paradigm for control system design,” in Proc. 40th IEEE Conf. Decis. Control, 2001, vol. 5, pp. 4578–4585. [9] Z. Gao, “Active disturbance rejection control: A paradigm shift in feedback control system design,” in Proc. Amer. Control Conf., 2006, pp. 2399–2405. [10] B. Sun and Z. Gao, “A DSP-based active disturbance rejection control design for a 1-kW H-bridge DC–DC power converter,” IEEE Trans. Ind. Electron., vol. 52, no. 5, pp. 1271–1277, Oct. 2005. [11] Y. Su, B. Y. Duan, C. H. Zheng, Y. F. Zhang, G. D. Chen, and J. W. Mi, “Disturbance-rejection high-precision motion control of a Stewart platform,” IEEE Trans. Control Syst. Technol., vol. 12, no. 3, pp. 364–374, May 2004. [12] Y. Su, C. Zheng, and B. Duan, “Automatic disturbances rejection controller for precise motion control of permanent-magnet synchronous motors,” IEEE Trans. Ind. Electron., vol. 52, no. 3, pp. 814–823, Jun. 2005. [13] D. Sun, “Comments on active disturbance rejection control,” IEEE Trans. Ind. Electron., vol. 54, no. 6, pp. 3428–3429, Dec. 2007. [14] J.-H. She, F. Mingxing, Y. Ohyama, H. Hashimoto, and M. Wu, “Improving disturbance-rejection performance based on an equivalentinput-disturbance approach,” IEEE Trans. Ind. Electron., vol. 55, no. 1, pp. 380–389, Jan. 2008. [15] M. Valenzuela, J. M. Bentley, P. C. Aguilera, and R. D. Lorenz, “Improved coordinated response and disturbance rejection in the critical sections of paper machines,” IEEE Trans. Ind. Appl., vol. 43, no. 3, pp. 857–869, May/Jun. 2007. [16]Jingqing Han, ”From PID to Active Disturbance Rejection Control,” IEEE Transactions on Industrial Electronics, Vol. 56, No. 3, March 2009.
ADRC INTRODUCTION PID CONTROL ADRC
◦ ◦ ◦ ◦
TRANSIENT TRAJECTORY GENERATION NOISE TOLERANT DIFFERENTIATION NON LINEAR FEEDBACK TOTAL DISTURBANCE ESTIMATION AND REJECTION
STABILITY ANALYSIS APPLICATIONS OF ADRC IMPLEMENTATION ISSUES CONCLUSION
ADRC
PASSIVE DISTURBANCE REJECTION CONTROL ◦ Deals with disturbance (internal and external) as one of the design issues. ◦
ACTIVE DISTURBANCE REJECTION CONTROL ◦ The disturbances which are mostly external are estimated by using an observer and thus can be canceled out thus making the control design disturbance free. ◦
ADRC
WHY ADRC? ◦ LIKE PID CONTROL, THE CONTROL LAW IS BASED ON ERROR AND NOT ON MODEL ◦ IT IS BASED ON STATE OBSERVER, AN EFFICIENT STRATEGY WHICH IS THE OUTCOME OF MODERN CONTROL APPROACH ◦ IT USES NON-LINEAR FEEDBACK THUS ENHANCING PERFORMANCE ◦ IT IS WELL SUITED TO DIGITAL COMPUTER TECHNOLOGY BASED APPLICATIONS AND IS AN OUTCOME OF EXPERIMENTAL DIGITAL SIMULATIONS
PID CONTROL DATES BACK TO 1920s IT IS DOMINANT IN INDUSTRIAL APPLICATIONS EVEN TODAY SIMPLICITY IS A MERIT NOT ABLE TO MEET DEMANDS OF MODERN INDUSTRY IN TERMS OF
◦ EFFICIENCY ◦ LACK AND COST OF SKILLED LABOUR ◦ DUE TO SIMPLICITY DOES NOT TAKE FULL ADVANTAGE OF MODERN DAY POWERFUL DIGITAL PROCESSORS
DISADVANTAGES OF PID CONTROLLERS
ERROR COMPUTATION IS A CUMBERSOME PROCESS FOR INPUT OTHER THAN STEP SIGNAL. NOISE GETS AMPLIFIED DUE TO DERIVATIVE CONTROL. THE CONTROL LAW IS IN THE FORM OF A LINEAR WEIGHTED SUM REQUIRING OVERSIMPLIFICATION AND THUS LOSS OF PERFORMANCE. INTEGRAL CONTROL MAY LEAD TO COMPLICATIONS OF SATURATION AND REDUCED STABILITY AMRGIN DUE TO INTRODUCTION OF PHASE LAG.
ADRC
TO OVERCOME THE DISADVANTAGES OF PID CONTROLLERS, ADRC IS PROPOSED WHICH CONSISTS OF FOUR STEPS ◦ A SIMPLE DIFFERENTIAL EQUATION IS USED AS A TRANSIENT TRAJECTORY GENERATOR ◦ THE DIFFERENTIATOR IS DESIGNED TO BE NOISE TOLERANT ◦ NON LINEAR CONTROL LAWS ARE USED FOR FEEDBACK ◦ TOTAL DISTURBANCE ESTIMATION AND REJECTION CONCEPT IS USED
HISTORY OF ADRC FIRST SYSTEMATICALLY INTRODUCED IN 2001 [8] FURTHER ELABORATED IN [9] PRACTICAL APPLICATION IN [11,12][13] EQUIVALENT INPUT DISTURBANCE [14] AND DISTURBANCE INPUT COUPLING [15] ARE SPECIAL CASES OF ADRC WHERE ONLY EXTERNAL DISTURBANCE IS CONSIDERED
TRANSIENT TRAJECTORY GENERATION FOR A DYNAMICAL SYSTEM, STEP INPUT MAY NOT BE APPROPRIATE FOR THE OUTPUT TO TRACK DUE TO ITS SUDDEN JUMP INPUT WITH TRANSIENT PROFILE REQUIRED
TRANSIENT TRAJECTORY GENERATION FOR EXAMPLE, THE DOUBLE INTEGRAL PLANT ẋ1 =x2
ẋ2 =u
IF |u|≤ r AND v IS THE DESIRED VALUE OF x1, THEN THE TIME OPTIMAL SOLUTION IS
u = −r sign(x1 − v + ((x2|x2|)/2r))
TRANSIENT TRAJECTORY GENERATION THUS THE DESIRED PROFILE CAN BE CHOSEN AS v̇1 =v2
v̇2 = −r sign(v1 − v + ((v2|v2|)/2r))
THE PARAMETER r CAN BE CHANGED TO SPEED UP OR SLOW THE PROFILE
NOISE TOLERANT DIFFERENTIATION
IN PID CONTROL, DIFFERENTIATION IS OBTAINED BY THE LAPLACE OPERATOR AS IN
NOISE TOLERANT DIFFERENTIATION
This can be rewritten as
The time domain solution is obtained by inverse Laplace Transform operation as
This can be approximated as
NOISE TOLERANT DIFFERENTIATION
Thus, if v(t) contains noise of the form n(t), y(t) will contain the noise term.
Since τ is small, this term will amount to amplification of n(t).
The amplification of noise in PID control is undesirable.
NOISE TOLERANT DIFFERENTIATION A different form of Laplace operator for differentiation is proposed by Han. The approximation
is used.
NOISE TOLERANT DIFFERENTIATION A particular second approximation of a differentiator is given by Or
Defining r=1/τ, and carrying out inverse Laplace Transform on above equation gives
Where y(t) tracks v(t), tracks , and r determines the speed.
NOISE TOLERANT DIFFERENTIATION
Consider that v(t) is the input signal to be differentiated. Then the previous equation can be expressed as
This gives the fastest tracking of v(t) and its derivative subject to an acceleration limit in the form of r. The above equation is denoted as “tracking differentiator”.
NON LINEAR FEEDBACK
The PID controller employs a linear feedback which is a combination of present, past and future samples of the tracking error, e.
However with this form, the tracking error approaches zero or converges to origin with infinite time.
NON LINEAR FEEDBACK
A non-linear feedback of the form
is proposed by Han which has the following advantages:-
NON LINEAR FEEDBACK
The tracking error can approach zero in finite time and much more quickly by the choice of α<1. Another advantage of this selection of α is that the steady state error reduces significantly without having to resort to integral feedback. Thus the disadvantages of integral control can be avoided. An extreme case is when α=0, which is equivalent to bang-bang control that ensures zero steady state error without requirement of integral control term in PID.
TOTAL DISTURBANCE ESTIMATION AND REJECTION
Consider the single-input-single-output (SISO) second order plant represented by the state equation
TOTAL DISTURBANCE ESTIMATION AND REJECTION To ensure that y tracks the input, the control, u, is chosen such that it overcomes the multivariable function f(x1, x2, w(t), t) in feedback control architecture. Hence the above function can be denoted as
F(t) is called the “total disturbance”.
TOTAL DISTURBANCE ESTIMATION AND REJECTION
F(t) can be treated as an additional state variable, x3. Thus denoting G(t)=Ḟ(t), the second order plant can be represented by a new set of state equations given by
TOTAL DISTURBANCE ESTIMATION AND REJECTION
The above plant is observable and hence an extended state observer (ESO) can be constructed by defining the tracking error to be of the form e=z1-y as follows:-
The gains β01 , β02 and β03 are selected such that the tracking error, e, approaches zero in finite time.
TOTAL DISTURBANCE ESTIMATION AND REJECTION
It can be shown that by choosing the control law as
The second order plant reduces to the following set of equations:-
This cascaded integral form can be easily controlled by making u0 as function of the tracking error and its derivative i.e., a PD controller. Thus the control problem is converted to a simple problem of estimation and rejection of the total disturbance.
FORMULATION OF ADRC
TO BE CONTINUED
APPLICATIONS OF ADRC
IMPLEMENTATION ISSUES
CONCLUSION
REFERENCES [1] J. Han, “Control theory: Model approach or control approach,” Syst. Sci. Math., vol. 9, no. 4, pp. 328–335, 1989, (in Chinese). [2] J. Han andW.Wang, “Nonlinear tracking-differentiator,” Syst. Sci. Math., vol. 14, no. 2, pp. 177–183, 1994, (in Chinese). [3] J. Han, “Nonlinear PID controller,” J. Autom., vol. 20, no. 4, pp. 487– 490, 1994, (in Chinese). [4] J. Han, “Extended state observer for a class of uncertain plants,” Control Decis., vol. 10, no. 1, pp. 85–88, 1995, (in Chinese). [5] J. Han, “Auto disturbances rejection controller and its applications,” Control Decis., vol. 13, no. 1, pp. 19–23, 1998, (in Chinese). [6] J. Han, “From PID to auto disturbances rejection control,” Control Eng., vol. 9, no. 3, pp. 13–18, 2002, (in Chinese). [7] J. Han, “Active disturbances rejection control technique,” Frontier Sci., vol. 1, no. 1, pp. 24–31, 2007, (in Chinese).
REFERENCES (CTD..) [8] Z. Gao, Y. Huang, and J. Han, “An alternative paradigm for control system design,” in Proc. 40th IEEE Conf. Decis. Control, 2001, vol. 5, pp. 4578–4585. [9] Z. Gao, “Active disturbance rejection control: A paradigm shift in feedback control system design,” in Proc. Amer. Control Conf., 2006, pp. 2399–2405. [10] B. Sun and Z. Gao, “A DSP-based active disturbance rejection control design for a 1-kW H-bridge DC–DC power converter,” IEEE Trans. Ind. Electron., vol. 52, no. 5, pp. 1271–1277, Oct. 2005. [11] Y. Su, B. Y. Duan, C. H. Zheng, Y. F. Zhang, G. D. Chen, and J. W. Mi, “Disturbance-rejection high-precision motion control of a Stewart platform,” IEEE Trans. Control Syst. Technol., vol. 12, no. 3, pp. 364–374, May 2004. [12] Y. Su, C. Zheng, and B. Duan, “Automatic disturbances rejection controller for precise motion control of permanent-magnet synchronous motors,” IEEE Trans. Ind. Electron., vol. 52, no. 3, pp. 814–823, Jun. 2005. [13] D. Sun, “Comments on active disturbance rejection control,” IEEE Trans. Ind. Electron., vol. 54, no. 6, pp. 3428–3429, Dec. 2007. [14] J.-H. She, F. Mingxing, Y. Ohyama, H. Hashimoto, and M. Wu, “Improving disturbance-rejection performance based on an equivalentinput-disturbance approach,” IEEE Trans. Ind. Electron., vol. 55, no. 1, pp. 380–389, Jan. 2008. [15] M. Valenzuela, J. M. Bentley, P. C. Aguilera, and R. D. Lorenz, “Improved coordinated response and disturbance rejection in the critical sections of paper machines,” IEEE Trans. Ind. Appl., vol. 43, no. 3, pp. 857–869, May/Jun. 2007. [16]Jingqing Han, ”From PID to Active Disturbance Rejection Control,” IEEE Transactions on Industrial Electronics, Vol. 56, No. 3, March 2009.
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