Active Disturbance Rejection
Control Analysis from A
P
Practical
i lP
Point
i off Vi
View
Mingwei Sun, Qi Xu, and Zengqiang Chen
Nankai University
2013 July 25
Outline




Relationship
R l ti hi with
ith robust
b t control
t l
Dynamic
y
performance
p
investigation
g
Sensitivity of the bandwidth of ESO
A tuning
i software
f
ffor fli
flight
h controll
Guideline: Most theoretical investigations were based on the
p
y plant.
p
In this manner,, several
first-order-plus-time-delay
valuable results could be obtained.
Relationship with robust control (1)
ωo2  uPD θ 
δ p = u PD +
−


s + 2ωo  s
Kc 
u PD
ke (θ r − θ ) − k d θ
=
= K e (θ r − θ ) − K d θ
Kc
Equivalent LADRC flight control
How to tune the control parameters in a systematic way?

β2 
δ p = K e 1 +
 (θ r − θ ) − K d
s ( s + β1 ) 


β2   1 β2 
θ
1 +
θ −
s ( s + β1 ) 
K c s + β1

Its generalized form
K = [ K e , K d , K c−1 ]
Relationship with robust control (2)
We
θr
+
−
e
Wu
C
Wθ
P
δp
A + B1d + B2δ p
 x = Ax

 yr = Cr x + Dr , d d + Dr ,u δ p

 ys = Cs x + Ds d
Tys , d − Dr , d
∞
< γh
θ
Robust Static Output Feedback
( A + B2 KCs )T P + P ( A + B2 KCs ) P ( B1 + B2 KDs ) (Cr + Dr ,u KCs )T 


T
T
(
B
+
B
KD
)
P
−
γ
I
(
D
KD
)
1
2
s
h
r ,u
s

<0


−γ h I
Cr + Dr ,u KCs
Dr ,u KDs


The parameter tuning problem can be reformulated as a BMI
optimization problem.
problem
The hardness of the SOF problem implies the hardness of
analysis
l i off LADRC
LADRC.
Return
Background


How to obtain a monotone and nonnonovershoot nondecreasing
g step
p response
p
for a
wide range of industrial processes in
practice? This problem has practical value
for many chemical and petrochemical
plants.
l t
The first
first--order lag
g was considered here.
(1)
Reduced--order ESO based LADRC
Reduced
Theorem 1 For the closed-loop system generated by rLADRC,
its step-response is monotone nondecreasing without
overshoot iff one of the following conditions holds:
Performance robustness (1)
multiplicative perturbation
Theorem 2 The dynamic
y
performance
p
robustness can be
maintained iff one of the following conditions holds:
Performance robustness (2)
Theorem 3 With identical control parameters, we have:
Second--order ESO based LADRC
Second
the closed-loop system has a monotone
nondecreasing step response.
response
Return
Observation

Extended
d d state observer
b
((ESO)
SO) iis the
h k
key
part of ADRC. A fast convergent ESO is
helpful to estimate the unun-modeled
dynamics and external disturbances with
high accuracy. However in practice, over
fast observer may lead to oscillation or
instability. This seems to be a bottleneck.
How to investigate?
g
Reason
The coupling
Th
li effects
ff
b
between estimation
i
i and
d controll
should be carefully investigated for such extended observer
based control methods.
methods
g of adaptive
p
control: the p
parameter
Rethinking
estimation and control design interconnection leads to a
complicated nonlinear dynamics. Rohrs example.
LADRC is a linear scheme, how instability emerges?
Th relationship
The
l i hi b
between h
hen and
d egg.
Time--delay sensitivity
Time



Communication delay.
Many existing lags with relatively small time
constants are thrown away.
All these
h
reductions
d i
iinevitably
i bl result
l iin
comprehensive input timetime-delay uncertainty
which is one of the most severe factors
ccausing
us g instability.
s b y.
Approximate timetime-delay sensitivity
analysis based on PM (1)
G p (s) =
Plant
kp
τ s +1
e − Ls
ke (r − y ) − kv y − z2
u=
b
Controller
PI tuning
g based on
stability margin
Lmax = φm / ωg = φm ( Am2 −1) L / (φm + π ( Am −1) / 2)
Approximate timetime-delay sensitivity
analysis based on PM (2)
∂Lmax
τ ke (ke − 2ωo )
2
2
0=
−
−
−
2
3
∂ωo
( k e + ωo )
k p ( k e + ωo )
k p ( k e + ωo ) 4
ωo,opt = (τ ke − 1 − 2k p ke ± τ 2 ke2 + 1 − 2τ ke − 6τ k p ke2 ) /(2k p )
The time-delay tolerance will be reduced with the increase of
the bandwidth of ESO
This is the coupling effects between control and observer.
Accurate analysis (1)
E li i method:
Explicit
h d the
h original
i i l ADRC
Implicit method: the Extended Observer based control where the control term is
also included in the extended state.
Accurate analysis (2)
Critical oscillatory frequency
Accurate analysis (3)
Accurate analysis (4)
Upper limit estimation for the bandwidth of ESO
Lower limit estimation for the bandwidth of ESO
Accurate analysis (5)
Explicit method
Implicit method
IMC analysis (1)
IMC analysis (2)
Effects of b
Effects of bandwidth
Tendency
Return
T i software
Tuning
ft
based
b d on stability
t bilit margin
i ttester
t
20
15
A=1/3
kd
10
feasible part
stability boundary
5
PM=45deg
0
A=3.0
-5
-10
0
100
200
300
kp
400
500
600
The End
Thank you for your attendance!
Criticisms are all welcome.
Email: sun
[email protected]
mingwei@aliyun com