Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 1 (June 2016), 1-4

Undershoot Reduction in Discrete-Time ADRC of NMP Plant by Parameters OptimizationTong Wu

School of Automation and Electrical Engineering,
University of Science and Technology Beijing,#30 Xueyuan Rd,Haidian District,Beijing,100083,P. R. China
Weicun Zhang

School of Automation and Electrical Engineering,
University of Science and Technology Beijing, #30 Xueyuan Rd,Haidian District,Beijing,100083,P. R. China
E-mail: xiaobulu1992@163.com, weicunzhang@263.net
www.buaa.edu.cn

Abstract
Undershoot phenomena caused by unstable zeros of non-minimum-phase (NMP) plant is difficult to deal with inactive disturbance rejection control (ADRC). This paper proposes a new method to reduce undershoot by
optimizing the controller parameters of discrete-time ADRC system. Simulation results are given to verify theeffectiveness of the proposed scheme.
Keywords: ADRC, NMP System, Discrete Control, Parameters Optimization

1. Introduction
Active disturbance rejection control (ADRC), first
proposed by Han Jingqing, as a novel control design
method has been applied to solve various control
problems since 2000s1. The ADRC technique, including
Extended State Observer (ESO), mentioned in Han’s
book2 is a kind of non-linear technique which shows
great advantages in dynamic performance, robust
performance, and disturbance rejection performance.
But it confronts with much trouble in finding
appropriate tuning method to deal with ADRC
parameters for different plants. Gao Zhiqiang focuses on

the linearized ADRC and gives a bandwidthparameterization method to solve the linear ADRC
tuning problem. 3 The number of ADRC parameters tobe tuned can be decreased to two in this method whichlargely simplifies the task of ADRC design.
The discrete control technology as an emerging
technology has been widely applied in control system.
Various discrete techniques have been proposed to solve
control problems. 4, 5 In This paper, a discrete ADRC isdesigned to achieve the control task for NMP system
which having the undershoot problem due to theunstable zeros.6, 7 A modified method of using
appropriate non-linear function for parameters

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Tong Wu, Weicum Zhang

optimization to reduce undershoot will be discussed inthe following sections. For simplicity, the subject to be
discussed in this paper is the second order system.
2.

yields8Discrete Active Disturbance Rejection
Control

2.1. Discrete Extended State Observer
The linear extended state observer (LESO)3 can beconstructed asz = Az + Bu + L( y − yˆ )yˆ = Cz(1)

where0 1 0 0
A = 0 0 1  , B = b0  , 0 
0 0 0 
3 − 3β ( β − 1) 2 ( β + 5) H = [1 0 0] , L p = 2h( β − 1)3  − h2(7)

β = e −ω h(8)0where b is a constant.

 β1 
C = [1 0 0] , L =  β 2   β 3 (2)2.2. Discrete ADRC control law
The parameterized control law can be constructed as3, 8
u0 (k )= k p (r (k ) − z1 (k )) − kd z2 (k )Using ω0 -Parameterization method mentioned in Ref. 3,
the observer gain vector L can be get asu (k ) =(3)2=β1 3=
ω0 , β 2 3ω=ω0 30 , β3where ω0 is the bandwidth of the observer. And thevalue of b0 falls in the range from 20 to 100. The
extended state observer can be discretized as8, 9z (k + 1) = Φz (k ) + Γu (k ) + Lp ( y (k ) − yˆ (k ))yˆ(k ) = Hz (k )h2  h2 
 2 
2 h  , Γ = b×  h  ,
 
 

1 
0
 


1 h

Φ =0 1

0 0


u 0 ( k ) − z3 ( k )
b(10)

where
2=
ω0 2 ~ 5ω=
ω=2ωcc , kpc , kd(4)3. Optimization of ADRC Parameters

whereStep Response

0.7

f0.6(5)e0.5

ijkh0.3

h is the sample period. And the estimator gain vector
L p is determined by the equation

0.2

a0.1
0b-0.1-0.2

λ ( z )= zI − (Φ − L p H )= ( z − β )3

g0.4

Amplitude β1 h AhAt=
Φ e ,=
Γ ( ∫ e dt ) B, H
= C , L=pβ2 0 β3 (9)(6)

Combine equation (2) and (5), and solving (6) for L p

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dc02486101214

Time (sec)Figure 1 step response of NMP system(11)Undershoot Reduction in Discrete-TimeFor the controller design, we need a bigger proportional
gain at a, c, f, h and j, and a smaller one at b, e, g, i andk in Fig. 1. A bigger differential gain between a and b, cand d, f and g, h and i and j and k, and a smaller one
between b and c, e and f, g and h and i and j in Fig. 110.
Generally this is not realizable where controller
parameters are found being fixed. But we can choose
appropriate non-linear functions to optimize theproportional gain and differential gain of the controller
as a solution to this problem10.
We assume yd as the derivative of the output, yd 2 as
the second derivative of the output，and e(k ) as theerror between the reference and the output of the system.
Using non-linear function, controller parameters can beconstructed ase(k )= r − z1 (k )(12)z (k ) − z1 (k − 1)yd ( k ) = 1h(13)yd (k ) − yd (k − 1)h(14)2)+ e − yd(15)

dr−2 sign ( yd ) sign ( yd 2 ) e ( k )(16)

yd 2 ( k ) =K p = K pr × (1 +Kd =eydK(1 + e)where Kpr, Kdr is the original proportional gain andoriginal differential gain of the controller which can besettled using the method mentioned in (9) and (11). And
z1 (k ) is the state of the extended observer which is used
to estimate the output of the plant.

G( z) =y=(k ) 1.81 y (k − 1) − 0.8187 y (k − 2)
−0.004501u (k − 1) + 0.004546u (k − 2)z (k + 1) = Φz (k ) + Γu (k ) + Lp ( y (k ) − yˆ (k ))yˆ(k ) = Hz (k )(17)

Its corresponding z transfer function and difference
equation is shown as (18) and (19)

(19)

(20)

As is mentioned in Ref. 2, the extended state is fed backto cancel the model uncertainty. ω0 is selected as 100
rad/sec
andusing
bandwidth-parameterization,
parameters of discrete ESO are chosen asu (k ) =u 0 ( k ) − z3 ( k )b(21)1.25 × 10−5 1 0.005 1.25 ×10−5 

0.005  , Γ = b ×  0.005  ,Φ =0 1
0 001 
 1.2 H = [1 0 0] , L p =  86.8  2436.6 (22)where b = 120 .
Finally the controller can be designed as (23) to control
augmented system where r is the setpoint. The valueof ωc is 25 rad/sec .
u0 (k ) = K p (r − z1 (k )) − K d z2 (k )K p = ωc 2 × (1 +A simulation of NMP system is used to demonstrate thecontrol design procedure and its resulting performance.
The plant is modeled as(18)

The sample period h is taken as 0.005s. The discrete
extended state observer can be constructed as

4. Design Example−s + 2G (s) = 2s + 40 s + 400−0.004501z + 0.004546
z 2 − 1.81z + 0.8187

Kd =2)
e + e − ydyd2ω

(1 + ec−2 sign ( yd ) sign ( yd 2 ) e ( k ))(23)

The result for the simulation mentioned above is shownin figure 2. The dash line represents the step response ofregular discrete ADRC and the solid line represents thestep response of modified discrete ADRC. Intuitively,
the undershoot of modified discrete ADRC is lower thanthat of regular discrete ADRC.

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Tong Wu, Weicum Zhang

References
1.

1
0.8

2.

0.6
0.4
0.2

3.

0-0.2

modified discrete ADRC
regular discrete ADRC

-0.4
012

34

5

6

7894.
10

Figure 2 Step response of regular discrete ADRC andmodified discrete ADRC

5. Conclusion Remarks

5.

6.

In this paper, a method of ADRC parametersoptimization using nonlinear function has beendiscussed. It is shown that the modified discrete ADRC
has smaller undershoot than the regular one. Simulation
result has been shown to verify the effectiveness of the
proposed scheme.
Acknowledgements

7.
8.

9.

The author would like to thank the anonymous
reviewers for their constructive and insightfulcomments for further improving the quality of thiswork. This work was supported by the Major State
Basic Research Development Program (973 Program)(No. 2012CB821200).

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