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Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC
Journal Pre-proof Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC Binwen Zhang, Wen Tan, Jian Li
PII: DOI: Reference:
S0019-0578(19)30482-3 https://doi.org/10.1016/j.isatra.2019.11.002 ISATRA 3389
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ISA Transactions
Received date : 17 June 2019 Revised date : 1 November 2019 Accepted date : 2 November 2019 Please cite this article as: B. Zhang, W. Tan and J. Li, Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC. ISA Transactions (2019), doi: https://doi.org/10.1016/j.isatra.2019.11.002. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Journal Pre-proof *Title page showing Author Details
Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC Binwen Zhang, Wen Tan, Jian Li
pro of
School of Control & Computer Engineering. North China Electric Power University. Beijing 102206, China
Abstract
A control strategy combining generalized active disturbance rejection control (GADRC) with Smith predictor (SP) is investigated for time-delayed systems. It takes all available model information into consideration, including the plant dynamic and the time delay, thus inherits the advantages of both SP-ADRC and GADRC. Since SP-GADRC can be transformed into a two-degree-of-freedom (TDF) feedback control structure, it is simply interpreted and tuned through the equivalent TDF internal model control (TDF-IMC) by using
re-
the delay-free part of the system as the nominal model for the IMC design and selecting the bandwidths of controller and extended state observer (ESO) for SP-GADRC as the inverse of the time constants of the setpoint filter and the disturbance-rejection filter in IMC, respectively. The analysis and tuning is of tutorial value for practitioners and engineers, and the effectiveness is validated by a few comparative simulations.
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Keywords: Tuning, smith predictor, active disturbance rejection control (ADRC), time-delay, internal model control (IMC)
Email addresses: [email protected] (Binwen Zhang), [email protected] (Wen Tan), [email protected] (Jian Li)
Preprint submitted to ISA Transactions
November 1, 2019
Journal Pre-proof *Highlights (for review)
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Smith predictor (SP) based generalized active disturbance rejection control (SPGADRC) is proposed for time-delayed systems. The structure of SP-GADRC is transformed into a two-degree-of-freedom internal model control (TDF-IMC), where the nominal model for the IMC design is delay-free. SP-GADRC is equivalent to IMC if the bandwidths of SP-GADRC are selected as the inverse of the time constants of the set-point filter and the disturbance-rejection filter in IMC. The tuning method of SP-GADRC based on IMC is tested for three examples, and simulation results show that they can achieve good performance.
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Journal Pre-proof *Blinded Manuscript - without Author Details Click here to view linked References
pro of
Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC
Abstract
A control strategy combining generalized active disturbance rejection control (GADRC) with Smith predictor (SP) is investigated for time-delayed systems. It takes all available model information into consideration, including the plant dynamic and the time delay, thus inherits the advantages of both SP-ADRC and GADRC. Since SP-GADRC can be transformed into a two-degree-of-freedom (TDF) feedback control structure, it is simply interpreted and tuned through the equivalent TDF internal model control (TDF-IMC) by using the delay-free part of the system as the nominal model for the IMC design and selecting the bandwidths of controller and extended
re-
state observer (ESO) for SP-GADRC as the inverse of the time constants of the setpoint filter and the disturbance-rejection filter in IMC, respectively. The analysis and tuning is of tutorial value for practitioners and engineers, and the effectiveness is validated by a few comparative simulations.
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Keywords: Tuning, smith predictor, active disturbance rejection control (ADRC), time-delay, internal model control (IMC) 1. Introduction
the closed-loop bandwidth. There has been a growing interest in active disturbance re-
Industrial processes such as chemical processes and electric jection control (ADRC), which is a new control technique first power systems, exhibit time delay characteristics, due to energy proposed by Han [10] to provide an alternative practical control
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exchange, mass transportation, signal processing and communi- method to PID and has become an attractive control algorithm. It cation, etc. [1]. The main limitation of systems with time de- is a very useful observer-based disturbance attenuation technique, lay is that time delay will introduce additional phase lag, reduce although its idea is simple. In ADRC, not only the external disturthe phase margin, making the closed-loop system bandwidth and bances but also the unmodeled dynamics and internal uncertainperformance very limited [2]. Therefore, the control of time- ties are viewed as a ‘generalized disturbance’ then treated as an delayed processes is very challenging and has attracted exten- extended state, so they can be estimated via an nonlinear extended sive attentions, such as PID control [3] and PID cascaded with state observer (NESO), and compensate it in the nonlinear error
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lead compensator [4]. In addition, a series of dead-time com- feedback control law (NEFCL). However, the original ADRC is pensators (DTC) are proposed in [5]. Internal model control based on nonlinear functions, thus has a complex structure and (IMC) [6] and modified IMC [7] have been designed for stable needs excessive tuning procedure in engineering applications. To time-delayed processes and integrating, unstable processes with overcome these difficulties, Gao [11] proposed the linear ADRC time delay, respectively. Besides, a variety of Smith predictor (LADRC) by using linear functions in ADRC, and introduced the (SP) based methods[8, 9] have been developed to deal with time- idea of bandwidth, which converts the complex parameter tuning delayed systems. The main advantage of SP based methods is that procedure into regulating the two bandwidths: ωo of ESO and ωc the delay can be removed from the closed-loop, and thus improve of feedback controller. Since then, many theoretical analysis and Preprint submitted to ISA Transactions
November 1, 2019
Journal Pre-proof
choose the model of the plant in IMC as the delay-free model, the
engineering applications [12, 13, 14, 15, 16] are reported.
With the development of ADRC, researchers have been con- tuning bandwidths of SP-GADRC are just the inverse of the two sidering extending ADRC applications to time-delayed systems. filters’ time constant of IMC. This is undoubtedly a good news
pro of
The simplest and straightforward idea is to neglect the time de- for engineers who are already familiar with the IMC control. lay or approximate it using a rational transfer function, and the
We arrange the structure of this paper as follows. The de-
error caused by the approximation can be included in the ”gen- tailed introductions of GADRC controller and TDF-IMC will be eralized disturbance”, but the effect is not ideal [17] and the pa- reviewed in section 2. In Section 3, the structure and tuning relarameter b need to retuned. The fundamental factor that ADRC tionship between SP-GADRC and TDF-IMC will be investigated. has limitations is that due to the time delay, the two signals (u Section 4 will be the case studies to illustrate the effectiveness of and y) entering the ESO are out of synchronization, resulting the proposed tuning procedure. The conclusions will be given in the inaccurate estimation of the ”generalized disturbance” and the last Section.
re-
reducing the control performance. Therefore, several modifications have been proposed to cancel the effect of the time delay in
ADRC design and improve the stability. In [18], Smith predictor 2. Preliminary is adopted to advance the output signal to achieve synchroniza-
tion, which is denoted as SP based ADRC (SP-ADRC); In [19], a
2.1. Generalized ADRC(GADRC)
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delay-designed ADRC (DD-ADRC) structure is used by adding an approximated delay to the control input signal to achieve syn-
chronization; In [20], the ESO is modified with an extended state
predictor observer (ESPO), resulting in a PO-ADRC structure.
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These three modified ADRC methods have been rigorously investigated and compared in [21].
Although ADRC has applied successfully in many systems, it
A nth-order system with uncertainties is represented as:
is noticed only the order of the plant has been used and most part of the model information has been treated as unknown. However, the control performance can be improved if we consider the known plant information in the ESO design, leading to the gener-
x˙ 1 = x2 x˙ 2 = x3 .. . x˙ n = −an−1 xn − · · · − a1 x2 − a0 x1
(1)
+bm u(m) + · · · + b1 u˙ + b0 u
+f x1 , · · · , xn , u, · · · , u(m) , d y = x1
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alized ADRC (GADRC) or model-based ADRC [22, 23, 24, 25]. where x = [x1 · · · xn ]T is the state vector, u is the control sig-
In this paper, we will investigate the SP based GADRC (SP- nal, y is the measured output, d is the external disturbance, and GADRC), which adds the available model dynamics and time- f x1 , · · · , xn , u, · · · , u(m) , d is the ‘generalized disturbance’,
delay to the controller design. We will first transform the SP- including both unknown internal dynamics and external disturGADRC to the two-degree-of-freedom (TDF) feedback control bance, and usually m ≤ n. The system also has the following structure as TDF-IMC, then the relationship between them is re- transfer function representation vealed on the perspective of state space. Further, it is shown G (s) =
SP-GADRC is absolutely equivalent to the TDF-IMC, when we 2
Y (s) bm sm + · · · + b1 s + b0 = n U (s) s + an−1 sn−1 + · · · + a1 s + a0
(2)
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and the corresponding controllable state-space realization B with A
=
B
=
C
=
h
h
u(t) =
0
1
0
···
0
0 .. .
0 .. .
1 .. .
··· .. .
0 .. .
0
0
0
···
1
−a0
−a1
−a2
−an−1
0
0
···
1
b1 b0
··· iT
···
0
b0 bm b0
1×n
01×(n−1−m)
i
(1):
(3)
where AL = Ae − Lo Ce , rˆ(t) = [ 0 0
Ee =
01×n 0n×1 1
0
0 ]T is a
Ko =
h
k1
···
kn
1
i
/b0 =
h
¯o K
1
i
/b0
(10)
Note that Lo and Ko are parameters to be determined in the
re-
the tuning of the observer bandwidth ωo and the controller band-
0(n−1)×1 1
0 h , Ce = C
thus can release some burden of ESO and greatly improve the
B , , Be = 0 i
Remark 1: GADRC can incorporate more plant information,
control performance [23]. In addition, there are only two tuning
0
width ωc [11].
(4)
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A
···
filtered signal of the setpoint r(t), and controller gain vector is.
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e
Ae =
(9)
GADRC. For simplicity, the tuning process is transformed into
z˙ (t) = A z (t) + B u (t) + E f˙ (t) e e e y (t) = C z (t)
zˆ˙ (t) = A zˆ (t) + B u (t) + L y (t) L e o u (t) = K (ˆ r (t) − zˆ (t)) o
1×n
ed state plant is defined as:
(8)
Finally, we can obtain the following GADRC controller for
Regard differentiable f as an extended state, then the extend-
where
k1 (r − z1 ) + · · · + kn (r(n−1) − zn ) − zn+1 b0
pro of
−1
C(sI − A)
,z =
h
x
f
parameters, thus it is convenient for engineering applications. 2.2. TDF-IMC
(5)
iT
and the estimation of state vector z can be realized through the following form of ESO
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zˆ˙ (t) = A zˆ (t) + B u (t) + L (y (t) − yˆ (t)) e e o yˆ (t) = C zˆ (t)
(6)
e
where
Lo =
h
β1
β2
is the gain vector of the ESO.
···
βn
βn+1
iT
Figure 1: TDF-IMC structure
IMC is an effective process control design method. Figure 1
(7)
illustrates the structure of TDF-IMC control, where, P is the real controlled plant and P0 is the model of the plant, Q and Qd are the setpoint-tracking ontroller and the disturbance-rejection con-
The control input to attenuate disturbance can be designed as troller, respectively. In TDF-IMC, the main purpose is to design 3
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Q and Qd according to the plant model P0 , and the design process
and then we can get
for a general plant is as follows [26]. Q (s) =
1) Decompose P0 into two parts PM and PA . PM is the min-
C1 (s) Fr (s) , 1 + Po (s) C2 (s)
Qd (s) =
C2 (s) C1 (s) Fr (s)
(16)
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imum phase part (PM is invertible), while PA is the allpass part (which means PA is non-minimum phase part and its magnitude is unity), i.e., (11)
P0 (s) = PM (s) PA (s)
3. TDF-IMC interpretation of SP-GADRC 2) Design the setpoint-tracking controller Q as Q (s) = PM −1 (s)
1 r (λs + 1) s
(12)
where rs is the relative degree of PM , and λ is a tuning param-
re-
For the output time-delayed plant, we modify the GADRC
eter of setpoint-tracking filter (STF) with the desired setpoint re- in(9) as follows: sponse is 1/(λs + 1)
rs
and generally, a larger λ will produce a
slower tracking response and slower disturbance rejection perfor-
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mance. 3) Design the disturbance-rejection controller Qd as αP sP + · · · + α1 s + 1 Qd (s) = r (λd s + 1) d
zˆ˙ (t) = A zˆ (t) + B u (t) + L y (t − τ ) L e o u (t) = K (ˆ r (t) − zˆ (t))
(17)
o
It is obvious that, due to the output delay, the signals entering
(13) the ESO (u and y) are not synchronized, which will affect the estimation ability of the ESO and reduce the control performance.
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where λd is the parameter of disturbance-rejection filter (DRF), Therefore, direct application of GADRC is not a sensible choice and usually a smaller λd will result a faster disturbance rejection
for delayed systems. Meanwhile, a modification, which is donat-
performance with a larger control effort, and rd is the order of ed as SP-ADRC, has been proposed for the traditional ADRC to DRF 1/(λd s + 1)
rd
. At the same time, assume that Qd needs improve the control performance for time-delayed systems. Since
to cancel the poles (p1 , · · · pP ) of P0 , and αP , · · · α1 can be ob- it considers combining GADRC with SP to propose a new control
tained by solving (14)
In the structure of SP-GADRC, as shown in Figure 2, the
(14)
predictive value of the system output yp which is obtained by the
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(1 − P0 QQd ) |s=p1 ,···pP = 0
strategy, it is denoted as SP-GADRC.
It is easy to obtain the TDF-IMC controller, which can be SP, goes into the ESO rather than the actual output y(t − τ ), thus represented as
Q (s) u(s) = r (s) 1 − Po (s) Q (s) Qd (s) Q (s) Qd (s) − y (s) 1 − Po (s) Q (s) Qd (s) =: C1 (s) Fr (s) r (s) − C2 (s) y (s)
achieving the synchronization of input signals and overcoming the limitation of GADRC for the delayed systems. yp (s) = y (s) + u (s) G (s) − u(s)G(s)e−τ s
(15) where G(s) is the model of P as shown in (2). 4
(18)
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The setpoint transfer function is −1
y (s) = C(sI − A + BKo )
BFr (s) r (s)
pro of
=Tr (s) Fr (s) r (s)
(23)
ideally, we expect the y(s) to track the setpoint without offset, i.e., Tr (s) Fr (s) =1. If there is no filtering requirement for the setpoint, for the sake of simplicity, it is usually selected rˆ (s) = Fr (0) r (s), and then we have Fr (0) = −
Figure 2: The structure of SP-GADRC
Now the model-based ESO of SP-GADRC can be designed
¯o C A − BK
−1
B
(24)
As shown in (21), SP-GADRC can be interpreted through the
re-
based on (6):
1
TDF structure. In other words, according to the (16), we can
zˆ˙ (t) = A zˆ (t) + B u (t) + L y (t) L e o p u (t) = K (ˆ r (t) − zˆ (t))
(19)
u (s) =
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o
rewrite (21) as follows:
By taking Laplace transform of (19) with setting initial conditions to zero, we will get
sˆ z (s) = AL zˆ (s) + Be u (s) + Lo yp (s) u (s) = K (ˆ r (s) − zˆ (s))
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o
QSP QSP QdSP r (s) − y (s) (25) 1 − Po QSP QdSP 1 − Po QSP QdSP
with the extended model defined in (4), the time-delayed system model can be express as
(20)
P0 (s) = G(s)e−τ s = Ce (sI − Ae )
−1
Be e−τ s
(26)
Substituting zˆ (s) in (20), the following equation can be ob- and then, solving the QSP and QdSP with tained:
u (s) = C1SP (s) rˆ (s) − C2SP (s)y (s)
= C1SP (s) Fr (s) r (s) − C2SP (s)y (s)
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where
−1 ˆe C (s) = 1 − K sI − A Be 1SP o −1 C ˆ L0 2SP (s) = Ko sI − Ae rˆ (s) = Fr (s) r (s) ˆ Ae = Ae − Be Ko − Lo Ce
. C2SP −1 = Ko (sI − AL ) Lo Fr C1SP Fr
(27)
C1SP Fr −1 = 1 − Ko (sI − AK ) Be Fr 1 + P0 C2SP
(28)
QdSP = (21)
QSP =
where AK = Ae − Be Ko .
(22)
4. Parameter tuning of SP-GADRC via TDF-IMC In general, the SP-GADRC parameters Ko and Lo can be ob-
and Fr (s) is the step filter, which is used to make sure the steady-
tained based on the bandwidth idea using the controller band-
state output error for the setpoint tracking is zero.
width ωc the and observer bandwidth ωo [11]. And in this section 5
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the relationship between the IMC parameters and the parameters Then rewrite (30) as follows of the SP-GADRC will be first analyzed. Section 4.1 shows that Q (s) =
Q(s) can be related to the parameters of SP-GADRC via (38) and
D(s)Fr N (s)(s+1/λ)n−m
N (s)(s+1/λ)n−m −D(s) N (s)(s+1/λ)n−m
pro of
Section 4.2 shows that Qd (s) can be related to the parameters of
= 1−
Fr (31)
SP-GADRC via (50). Thus by tuning the parameter for Q(s) and where
D (s) = sn + an−1 sn−1 + · · · + a1 s + a0 N (s) = (sm + · · · + b1 /bm s + b0 /bm ) F = 1/λn−m b
Qd (s), we can obtain the parameters of SP-GADRC via (53). First, the TDF-IMC structure for the SP-GADRC is considered. If we know the accurate plant model, i.e. P (s) =
r
G (s) e−τ s , we can get
(32)
m
Let AI = Ae , BI = Be and
P¯ (s) = P (s) + G(s)(1 − e−τ s ) = G(s)
(29)
KI =
re-
From (29), we can see that SP-GADRC transforms the time-
h
kI1
kI2
···
kIn
1
i
/b0
delayed system into a plant without delay through the SP, there- we get fore, in the equivalent IMC structure, the plant P0 is selected with
|sI − (AI − BI KI )|
delay-free one as shown in (2), and the Q and Qd are obtained
where,
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the TDF-IMC structure for SP-GADRC.
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based on it as described in section 2.3. Figure 3 detailed describe
(33)
= s sn + (an−1 + kIn ) sn−1 + · · · + (a0 + kI1 )
n−i kIi = Cn−m
b−n+m+i−1 bm
i = 1, · · · n
1 n−i λ
n+1−i + Cn−m
1 n+1−i λ
− ai−1 , (34)
Then let n−m
|sI − (AI − BI KI )| = s (s + z1 ) · · · (s + zm ) (s + 1/λ) = sN (s) (s + 1/λ)
n−m
(35) where, z1 , · · · , zm are zeros of plant.
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Finally, we can obtain
Figure 3: The TDF-IMC structure for SP-GADRC
N (s) (s + 1/λ)
n−m
− D (s) = kn sn−1 + · · · + k2 s + k1 (36)
4.1. State space realization of Q(s)
where,
According to the procedure of the IMC design, we can design
n−i kIi = Cn−m
Q(s) as Q (s) =
G(s) (λs + 1)
n−m
i = 1, · · · n
(30) 6
b−n+m+i−1 bm
1 n−i λ
n+1−i + Cn−m
1 n+1−i λ
− ai−1 , (37)
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In summary, (30) can be expressed as
where
4.2. State space realization of Qd (s)
ϕ (s) = sn−m + µn−m−1 sn−m−1 + · · · + µ1 s + µ0 (38) 1 1 1 1 + Cn+1 µn−m−1 = Cn−m λ λd − an−1 1 1 1 2 1 1 2 + µn−m−2 = Cn−m C n+1 λd Cn−m λ − an−1 µn−m−1 λ .. . n−m 1 n−m n−m 1 n−m 1 1 µ0 = Cn−m + − an−1 µ1 C n+1 λd Cn−m λ λ (42)
pro of
sn +···+k2 s2 +k1 s Q (s) = 1 − kn|sI−(A Fr I −BI KI )| −1 = Fr 1 − KI (sI − AI + BI KI ) BI
According to (41), we can obtain
re-
αn sn + · · · + α1 s + 1 = n−m n−m n+1 s+ s + λ1 λ λd
Following the procedure of IMC design, a disturbance rejection controller is obtained as n
αn s + · · · + α1 s + 1 (λd s + 1)
n+1
n−m
λ
(39)
Qd (s) =
For the SP-GADRC, the (n + 1)th-order of ESO is needed for controlling the nth-order plant, so the same order of Qd (s) is selected to match it.
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n+1 s + λ1d n−m 1 n+1 − λ1 (αn sn + · · · + α1 s + 1) λd 1 n−m λ
= s (s + p1 ) · · · (s + pn ) ϕ (s) n
n−1
=s s + an−1 s
s+
1 n−m λ
s+
1 λd
n+1 n+1
(s + 1/λd )
− sD (s) ϕ (s) (44)
In order to better demonstrate the certification process, n − 1, and let CI iT , βIn βIn+1
Ce , LI
=
=
(45)
according to the Ackerman formula, the ESO’s gain vector LI can be calculated as:
αn sn +···+α1 s+1 n−m n+1 |s=p1 ,···pn =0 (λs+1) n+1 (λd s+1) n+1 1 1 n−m 1 1 n−m s+ − (αn sn +···+α1 s+1) s+ (λ) ( λ) λd λd = n+1 1 n−m 1 s+ λ (s+ λ )
s+
(43)
n+1
1 − P QQd
which means
− sD (s) ϕ (s)
ψ (s) = |sI − (AI − LI CI )| = (s + 1/λd )
(14), and it can be expressed as
d
we take m = h βI1 βI2 · · ·
As mentioned before, αn , · · · , α1 can be obtained by solving
=1−
n+1
and then (39) can be expressed as
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Qd (s) =
1 λd
CI
CI AI LI = ψ (AI ) .. . CI AnI
(40)
−1
0 0 .. . 1
(46)
With (37) and (46), we can obtain
(41)
KI (sI − (AI − LI CI ))
+ · · · + a0 ϕ (s)
7
−1
LI =
dn sn + · · · d1 s + d0 (s + 1/λd )
n+1
(47)
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where n+1
1 1 1 n+1 1 + Cn+1 − a + − a − a n−1 λ n−1 n−2 λd λ λd n+1 3 2 1 1 1 1 1 n+1 3 2 + + − a − a dn−1 = λb1m − a Cn+1 λd Cn+1 λd n−2 λ n−1 n−3 λd λ λd dn =
1 λbm
1 λd
2 Cn+1
1 λd
2
pro of
.. .
n+1 n+2−i n+1−i n+2−i n+1−i 1 1 1 1 1 1 n+1 d = + − a + − a − a Cn+1 Cn+1 i i−1 λ n−1 n−2 λbm λd λd λd λ λd .. . n+1 n+1 n n+1 1 1 1 1 1 1 n+1 n d = + − a + − a Cn+1 λd Cn+1 λd 1 0 λ n−1 λbm λd λ λd n+1 n+1 n+1 1 1 d0 = 1 Cn+1 λ1d λbm λd λ
It is easy to verify that
(48)
re-
then we have
n+1
dn sn + · · · d1 s + d0 =Fr (αn sn + · · · α1 s + 1)/(λd )
(49)
QdSP = Qd ,
QSP = Q
(54)
which means that under the assumption of (53), SP-GADRC is
In other words, the state space realization of Qd can be expressed equivalent to IMC. The two bandwidths in SP-ADRC are the in
Qd (s) = KI (sI − AI + LI CI ) 4.3. Tuning of SP-GADRC via IMC
verse of the two tuning parameters in IMC. Therefore, it is a good
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as
−1
LI
.
(50) news for engineers who are already familiar with IMC.
Fr
on Ko , while Qd is related to both Ko and Lo . Therefore, it is
As suggested in [18], two sets of gains can be obtained using
more reasonable to tune controller gain Ko and then Lo , that is,
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bandwidth idea in SP-GADRC, i.e., βi 0 s of Lo are chosen such
for IMC, we should tune λ and then λd . The following steps are
that the observer bandwidth is ωo , and ki 0 s of Ko are chosen such
suggested for tuning SP-GADRC parameters via IMC:
that the controller bandwidth is ωc . For ESO, all observer poles
1) Determine the plant model P0 (s), e.g.
are placed at −ωo , we can get
n+1
P0 (s) = G (s) e−τ s
(51)
|sI − (Ae − Lo Ce )| =(s + ωo )
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and for the final state feedback control system, we place the m
3) Determine the desired tracking performance of the closed. n−m loop system 1 (λs + 1) by choosing λ.
4) Determine the disturbance rejection filter constant λd to
n−m
|sI − (Ae − Be Ko )| =s (s + z1 ) · · · (s + zm ) (s + ωc )
(52) have the desired disturbance-rejection performance while ensuring the closed-loop stability.
Let 1 1 ωc = , ωo = λ λd
(55)
2) Design Q(s) and Qd (s) according to G(s).
poles to the zeros of the plant, and the remaining (n − m) poles are placed at -ωc ,
It can be observed form (27)and (28) that Q only depends
(53)
5) Let ωc = 1/λ, ωo = 1/λd . 8
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5. Simulation QdSP (s) =
34.71s + 225 (s + 15)
(62)
2
The responses of P1 (s) are shown in Figure 4 and Figure 5,
To further demonstrate the tuning processes of SP-GADRC,
pro of
three examples are provided. For a fair comparison, a robustness and a unit step setpoint change and a unit step load disturbance indicator ε [27] is adopted. That is to say, by tuning the param- is added at t=0s and t=30s separately. Because TDF-IMC and eters to make sure that controlled systems have the same robust- SP-GADRC are equivalent, their responses are completely coness (or adaptability to parameter change), and then compare the incident. Moreover, when the model information is known and accurate, the response of SP-ADRC is very close to that of SP-
performance.
(56) GADRC. However, when there is a 6% mismatch in the delay
ε = sup (kSk∞ + kT k∞ ) ω
time τ , we can observe from Figure 5 that response of SP-ADRC
where, 1 kSk∞ = max 1+L(jω) , ω
L(jω) kT k∞ = max 1+L(jω) ω
re-
is oscillatory, while SP-GADRC remains smooth. (57)
1.2
1
and L(jω) is the frequency response of loop transfer function
0.8
y(t)
of ε.
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L(s). Generally, the robustness decreases by increasing the value
Case 1. A very simple first-order plant but with a very large
0.6 0.4
time delay is studied in [18], and parameters of SP-ADRC are tuned as ωc =12, ωo =36, b0 =10 with ε=2.91.
urn a
0.1 P1 (s) = e−5s 0.1s + 1
SP-GADRC SP-ADRC TDF-IMC
0.2 0 0
time constants in TDF-IMC are chosen as
60
0.8
y(t)
Jo
50
1
0.6 0.4
(60)
SP-GADRC,+6% SP-ADRC,+6% SP-GADRC,-6% SP-ADRC,-6%
0.2 0
and the controllers in the equivalent TDF-IMC are 1.7s + 17 s + 1.7
40
(59)
first-order SP-GADRC are tuned as
QSP (s) =
30 t/s
1.2
with the robustness indicator ε=2.91. Then the parameters of
ωc =1/0.59, ωo =1/0.0667
20
Figure 4: Responses of P1 (s)
For P1 (s), to obtain the same robustness as SP-ADRC, the
λ= 0.59, λd = 0.0667
10
(58)
0
(61)
10
20
30 t/s
40
50
60
Figure 5: Responses of P1 (s) with model uncertainties
9
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Case 2. A third-order time-delayed plant was controlled by output responses and the controller responses are shown in FigSP-ADRC with ωc =0.3, ωo =1.2, b0 =0.5, and the robustness indi- ure 6 and Figure 7. We can see that responses of SP-GADRC and cator ε=2.7 in [18].
TDF-IMC are indistinguishable with the same disturbance rejec-
(s + 1)
pro of
P2 (s) =
tion capability, and SP-GADRC outperforms SP-ADRC on the
e−5s
(63) response speed and smoothness of the controller output. Then we
3
selected Pm (s) =
e−6s 2s+1
as the G(s) in SP to further show the dis-
According to the tuning procedure of IMC, the parameters are turbance rejection characteristics, which are given in Figure 8 and tuning as: Figure 9. It is obvious that both controllers can achieve satisfac(64) tory control performance, but the response speed of SP-GADRC
λ= 1.25, λd = 0.3448
is still faster.
with the ε=2.7, so the parameters of SP-GADRC are (65)
1
re-
ωc =1/1.25, ωo =1/0.3448
0.5
and the controllers in the equivalent TDF-IMC are
0
SP-GADRC SP-ADRC TDF-IMC
-0.5
3
77.49s + 223.3s +216.7s + 70.73 s4 + 11.6s3 + 50.46s2 + 97.56s + 70.73
1
-2 -2.5 -3
(67)
-3.5 0
-0.5 -1 -1.5
-2.5 -3 0
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-2
100
200 t/s
100
200 t/s
300
400
Figure 7: Controllers output responses for P2 (s)
1 SP-GADRC, P m
0.5
SP-ADRC, Pm
0 -0.5
y(t)
0
-1
-1.5
SP-GADRC SP-ADRC TDF-IMC
0.5
y(t)
2
urn a
QdSP (s) =
(66)
y(t)
0.512s3 + 1.536s2 + 1.536s + 0.512 s3 + 2.4s2 + 1.92s + 0.512
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QSP (s) =
-1 -1.5
300
400
-2 -2.5
Figure 6: Responses of P2 (s)
-3 0
100
200
300
t/s
For P2 (s) a step reference is added at t=200s, and step-type disturbances are added at t=100s and t=300s, respectively. The
Figure 8: Responses of P2 (s) with Pm
10
400
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change of the setpoint, while the output of SP-GADRC maintains
1 SP-GADRC, P m
0.5
smooth.
SP-ADRC, Pm
0
1.2 1
pro of
y(t)
-0.5 -1
0.8
-1.5
0.6
y(t)
-2 -2.5
0.4 0.2
-3 -3.5
0
0
100
200 t/s
300
400
SP-GADRC DOB configuration strategy TDF-IMC
-0.2 -0.4
Figure 9: Controllers output responses for P2 (s) with Pm
re-
0
Case 3. [28] applied a DOB configuration strategy with ε=2.446 to a second-order non-minimum phase system.
20
40 t/s
60
80
Figure 10: Responses of P3 (s)
30
(68)
20
Using the proposed method, the parameters of SP-GADRC is
ωo = 0.6
urn a
ωc = 0.43,
(69)
with ε=2.4426. The controllers in the equivalent TDF-IMC are QSP (s) =
QdSP
2.58s2 + 2.15s + 0.43 s2 + 1.43s + 0.43
1.257s2 + 1.061s + 0.216 (s) = 3 s + 1.8s2 + 1.08s + 0.216
15
y(t)
tuned as:
SP-GADRC DOB configuration strategy TDF-IMC
25
lP
−s + 1 P3 (s) = e−s (2s + 1) (3s + 1)
10 5 0 -5
(70)
0
20
40 t/s
60
80
Figure 11: Controllers output responses for P3 (s)
(71) 6. Conclusion
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so the time constant for the setpoint filter in TDF-IMC is λ=1/0.43 and that for the disturbance rejection filter is λd =1/0.6.
We combine model-assisted ESO with Smith predictor to gen-
As shown in Figure 10 and Figure 11, a unit step-type ref- erate SP-GADRC for time-delayed systems. By transforming SPerence is loaded at t=10s, then a unit step-type disturbance at GADRC to the TDF-IMC control structure, in which the nominal t=40s. It can be observed that the undershoot of SP-GADRC is model is chosen as the delay-free one, the bandwidth parameters much smaller than the DOB configuration strategy, and the out- can be tuned through the two time constants of STF and DRF in put response curve is smoother. At the same time, the controller the equivalent IMC. Therefore, it provides us with a new idea of output of DOB configuration strategy is more sensitive with the tuning parameters of SP-GADRC: the controller bandwidth ωc is 11
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[8] M. R. Matausek, A. D. Micic, A modified smith predic-
width ωo is the inverse of the time constant λd of the DRF. The
tor for controlling a process with an integrator and long
proposed tuning method is tested in three examples, showing its
dead-time, IEEE Transactions on Automatic Control 41 (8)
effectiveness in the control design.
(1996) 1199–1203.
pro of
the inverse of the time constant λ of the STF and the ESO band-
[9] D. G. Padhan, S. Majhi, Modified smith predictor based cas-
Acknowledgement
cade control of unstable time delay processes, ISA Transac-
This work was supported by the National Natural Science Foundation of China under grant 61573138 and the Fundamental Research Funds for the Central Universities under grant
tions 51 (1) (2012) 95–104.
[10] J. Han, From PID to active disturbance rejection control, IEEE Transactions on Industrial Electronics 56 (3) (2009)
2019QN050.
900–906.
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*Conflict of Interest Statement
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pro of
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled ‘Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC’.
PII: DOI: Reference:
S0019-0578(19)30482-3 https://doi.org/10.1016/j.isatra.2019.11.002 ISATRA 3389
To appear in:
ISA Transactions
Received date : 17 June 2019 Revised date : 1 November 2019 Accepted date : 2 November 2019 Please cite this article as: B. Zhang, W. Tan and J. Li, Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC. ISA Transactions (2019), doi: https://doi.org/10.1016/j.isatra.2019.11.002. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Journal Pre-proof *Title page showing Author Details
Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC Binwen Zhang, Wen Tan, Jian Li
pro of
School of Control & Computer Engineering. North China Electric Power University. Beijing 102206, China
Abstract
A control strategy combining generalized active disturbance rejection control (GADRC) with Smith predictor (SP) is investigated for time-delayed systems. It takes all available model information into consideration, including the plant dynamic and the time delay, thus inherits the advantages of both SP-ADRC and GADRC. Since SP-GADRC can be transformed into a two-degree-of-freedom (TDF) feedback control structure, it is simply interpreted and tuned through the equivalent TDF internal model control (TDF-IMC) by using
re-
the delay-free part of the system as the nominal model for the IMC design and selecting the bandwidths of controller and extended state observer (ESO) for SP-GADRC as the inverse of the time constants of the setpoint filter and the disturbance-rejection filter in IMC, respectively. The analysis and tuning is of tutorial value for practitioners and engineers, and the effectiveness is validated by a few comparative simulations.
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Keywords: Tuning, smith predictor, active disturbance rejection control (ADRC), time-delay, internal model control (IMC)
Email addresses: [email protected] (Binwen Zhang), [email protected] (Wen Tan), [email protected] (Jian Li)
Preprint submitted to ISA Transactions
November 1, 2019
Journal Pre-proof *Highlights (for review)
pro of
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Smith predictor (SP) based generalized active disturbance rejection control (SPGADRC) is proposed for time-delayed systems. The structure of SP-GADRC is transformed into a two-degree-of-freedom internal model control (TDF-IMC), where the nominal model for the IMC design is delay-free. SP-GADRC is equivalent to IMC if the bandwidths of SP-GADRC are selected as the inverse of the time constants of the set-point filter and the disturbance-rejection filter in IMC. The tuning method of SP-GADRC based on IMC is tested for three examples, and simulation results show that they can achieve good performance.
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Journal Pre-proof *Blinded Manuscript - without Author Details Click here to view linked References
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Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC
Abstract
A control strategy combining generalized active disturbance rejection control (GADRC) with Smith predictor (SP) is investigated for time-delayed systems. It takes all available model information into consideration, including the plant dynamic and the time delay, thus inherits the advantages of both SP-ADRC and GADRC. Since SP-GADRC can be transformed into a two-degree-of-freedom (TDF) feedback control structure, it is simply interpreted and tuned through the equivalent TDF internal model control (TDF-IMC) by using the delay-free part of the system as the nominal model for the IMC design and selecting the bandwidths of controller and extended
re-
state observer (ESO) for SP-GADRC as the inverse of the time constants of the setpoint filter and the disturbance-rejection filter in IMC, respectively. The analysis and tuning is of tutorial value for practitioners and engineers, and the effectiveness is validated by a few comparative simulations.
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Keywords: Tuning, smith predictor, active disturbance rejection control (ADRC), time-delay, internal model control (IMC) 1. Introduction
the closed-loop bandwidth. There has been a growing interest in active disturbance re-
Industrial processes such as chemical processes and electric jection control (ADRC), which is a new control technique first power systems, exhibit time delay characteristics, due to energy proposed by Han [10] to provide an alternative practical control
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exchange, mass transportation, signal processing and communi- method to PID and has become an attractive control algorithm. It cation, etc. [1]. The main limitation of systems with time de- is a very useful observer-based disturbance attenuation technique, lay is that time delay will introduce additional phase lag, reduce although its idea is simple. In ADRC, not only the external disturthe phase margin, making the closed-loop system bandwidth and bances but also the unmodeled dynamics and internal uncertainperformance very limited [2]. Therefore, the control of time- ties are viewed as a ‘generalized disturbance’ then treated as an delayed processes is very challenging and has attracted exten- extended state, so they can be estimated via an nonlinear extended sive attentions, such as PID control [3] and PID cascaded with state observer (NESO), and compensate it in the nonlinear error
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lead compensator [4]. In addition, a series of dead-time com- feedback control law (NEFCL). However, the original ADRC is pensators (DTC) are proposed in [5]. Internal model control based on nonlinear functions, thus has a complex structure and (IMC) [6] and modified IMC [7] have been designed for stable needs excessive tuning procedure in engineering applications. To time-delayed processes and integrating, unstable processes with overcome these difficulties, Gao [11] proposed the linear ADRC time delay, respectively. Besides, a variety of Smith predictor (LADRC) by using linear functions in ADRC, and introduced the (SP) based methods[8, 9] have been developed to deal with time- idea of bandwidth, which converts the complex parameter tuning delayed systems. The main advantage of SP based methods is that procedure into regulating the two bandwidths: ωo of ESO and ωc the delay can be removed from the closed-loop, and thus improve of feedback controller. Since then, many theoretical analysis and Preprint submitted to ISA Transactions
November 1, 2019
Journal Pre-proof
choose the model of the plant in IMC as the delay-free model, the
engineering applications [12, 13, 14, 15, 16] are reported.
With the development of ADRC, researchers have been con- tuning bandwidths of SP-GADRC are just the inverse of the two sidering extending ADRC applications to time-delayed systems. filters’ time constant of IMC. This is undoubtedly a good news
pro of
The simplest and straightforward idea is to neglect the time de- for engineers who are already familiar with the IMC control. lay or approximate it using a rational transfer function, and the
We arrange the structure of this paper as follows. The de-
error caused by the approximation can be included in the ”gen- tailed introductions of GADRC controller and TDF-IMC will be eralized disturbance”, but the effect is not ideal [17] and the pa- reviewed in section 2. In Section 3, the structure and tuning relarameter b need to retuned. The fundamental factor that ADRC tionship between SP-GADRC and TDF-IMC will be investigated. has limitations is that due to the time delay, the two signals (u Section 4 will be the case studies to illustrate the effectiveness of and y) entering the ESO are out of synchronization, resulting the proposed tuning procedure. The conclusions will be given in the inaccurate estimation of the ”generalized disturbance” and the last Section.
re-
reducing the control performance. Therefore, several modifications have been proposed to cancel the effect of the time delay in
ADRC design and improve the stability. In [18], Smith predictor 2. Preliminary is adopted to advance the output signal to achieve synchroniza-
tion, which is denoted as SP based ADRC (SP-ADRC); In [19], a
2.1. Generalized ADRC(GADRC)
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delay-designed ADRC (DD-ADRC) structure is used by adding an approximated delay to the control input signal to achieve syn-
chronization; In [20], the ESO is modified with an extended state
predictor observer (ESPO), resulting in a PO-ADRC structure.
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These three modified ADRC methods have been rigorously investigated and compared in [21].
Although ADRC has applied successfully in many systems, it
A nth-order system with uncertainties is represented as:
is noticed only the order of the plant has been used and most part of the model information has been treated as unknown. However, the control performance can be improved if we consider the known plant information in the ESO design, leading to the gener-
x˙ 1 = x2 x˙ 2 = x3 .. . x˙ n = −an−1 xn − · · · − a1 x2 − a0 x1
(1)
+bm u(m) + · · · + b1 u˙ + b0 u
+f x1 , · · · , xn , u, · · · , u(m) , d y = x1
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alized ADRC (GADRC) or model-based ADRC [22, 23, 24, 25]. where x = [x1 · · · xn ]T is the state vector, u is the control sig-
In this paper, we will investigate the SP based GADRC (SP- nal, y is the measured output, d is the external disturbance, and GADRC), which adds the available model dynamics and time- f x1 , · · · , xn , u, · · · , u(m) , d is the ‘generalized disturbance’,
delay to the controller design. We will first transform the SP- including both unknown internal dynamics and external disturGADRC to the two-degree-of-freedom (TDF) feedback control bance, and usually m ≤ n. The system also has the following structure as TDF-IMC, then the relationship between them is re- transfer function representation vealed on the perspective of state space. Further, it is shown G (s) =
SP-GADRC is absolutely equivalent to the TDF-IMC, when we 2
Y (s) bm sm + · · · + b1 s + b0 = n U (s) s + an−1 sn−1 + · · · + a1 s + a0
(2)
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and the corresponding controllable state-space realization B with A
=
B
=
C
=
h
h
u(t) =
0
1
0
···
0
0 .. .
0 .. .
1 .. .
··· .. .
0 .. .
0
0
0
···
1
−a0
−a1
−a2
−an−1
0
0
···
1
b1 b0
··· iT
···
0
b0 bm b0
1×n
01×(n−1−m)
i
(1):
(3)
where AL = Ae − Lo Ce , rˆ(t) = [ 0 0
Ee =
01×n 0n×1 1
0
0 ]T is a
Ko =
h
k1
···
kn
1
i
/b0 =
h
¯o K
1
i
/b0
(10)
Note that Lo and Ko are parameters to be determined in the
re-
the tuning of the observer bandwidth ωo and the controller band-
0(n−1)×1 1
0 h , Ce = C
thus can release some burden of ESO and greatly improve the
B , , Be = 0 i
Remark 1: GADRC can incorporate more plant information,
control performance [23]. In addition, there are only two tuning
0
width ωc [11].
(4)
urn a
A
···
filtered signal of the setpoint r(t), and controller gain vector is.
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e
Ae =
(9)
GADRC. For simplicity, the tuning process is transformed into
z˙ (t) = A z (t) + B u (t) + E f˙ (t) e e e y (t) = C z (t)
zˆ˙ (t) = A zˆ (t) + B u (t) + L y (t) L e o u (t) = K (ˆ r (t) − zˆ (t)) o
1×n
ed state plant is defined as:
(8)
Finally, we can obtain the following GADRC controller for
Regard differentiable f as an extended state, then the extend-
where
k1 (r − z1 ) + · · · + kn (r(n−1) − zn ) − zn+1 b0
pro of
−1
C(sI − A)
,z =
h
x
f
parameters, thus it is convenient for engineering applications. 2.2. TDF-IMC
(5)
iT
and the estimation of state vector z can be realized through the following form of ESO
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zˆ˙ (t) = A zˆ (t) + B u (t) + L (y (t) − yˆ (t)) e e o yˆ (t) = C zˆ (t)
(6)
e
where
Lo =
h
β1
β2
is the gain vector of the ESO.
···
βn
βn+1
iT
Figure 1: TDF-IMC structure
IMC is an effective process control design method. Figure 1
(7)
illustrates the structure of TDF-IMC control, where, P is the real controlled plant and P0 is the model of the plant, Q and Qd are the setpoint-tracking ontroller and the disturbance-rejection con-
The control input to attenuate disturbance can be designed as troller, respectively. In TDF-IMC, the main purpose is to design 3
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Q and Qd according to the plant model P0 , and the design process
and then we can get
for a general plant is as follows [26]. Q (s) =
1) Decompose P0 into two parts PM and PA . PM is the min-
C1 (s) Fr (s) , 1 + Po (s) C2 (s)
Qd (s) =
C2 (s) C1 (s) Fr (s)
(16)
pro of
imum phase part (PM is invertible), while PA is the allpass part (which means PA is non-minimum phase part and its magnitude is unity), i.e., (11)
P0 (s) = PM (s) PA (s)
3. TDF-IMC interpretation of SP-GADRC 2) Design the setpoint-tracking controller Q as Q (s) = PM −1 (s)
1 r (λs + 1) s
(12)
where rs is the relative degree of PM , and λ is a tuning param-
re-
For the output time-delayed plant, we modify the GADRC
eter of setpoint-tracking filter (STF) with the desired setpoint re- in(9) as follows: sponse is 1/(λs + 1)
rs
and generally, a larger λ will produce a
slower tracking response and slower disturbance rejection perfor-
lP
mance. 3) Design the disturbance-rejection controller Qd as αP sP + · · · + α1 s + 1 Qd (s) = r (λd s + 1) d
zˆ˙ (t) = A zˆ (t) + B u (t) + L y (t − τ ) L e o u (t) = K (ˆ r (t) − zˆ (t))
(17)
o
It is obvious that, due to the output delay, the signals entering
(13) the ESO (u and y) are not synchronized, which will affect the estimation ability of the ESO and reduce the control performance.
urn a
where λd is the parameter of disturbance-rejection filter (DRF), Therefore, direct application of GADRC is not a sensible choice and usually a smaller λd will result a faster disturbance rejection
for delayed systems. Meanwhile, a modification, which is donat-
performance with a larger control effort, and rd is the order of ed as SP-ADRC, has been proposed for the traditional ADRC to DRF 1/(λd s + 1)
rd
. At the same time, assume that Qd needs improve the control performance for time-delayed systems. Since
to cancel the poles (p1 , · · · pP ) of P0 , and αP , · · · α1 can be ob- it considers combining GADRC with SP to propose a new control
tained by solving (14)
In the structure of SP-GADRC, as shown in Figure 2, the
(14)
predictive value of the system output yp which is obtained by the
Jo
(1 − P0 QQd ) |s=p1 ,···pP = 0
strategy, it is denoted as SP-GADRC.
It is easy to obtain the TDF-IMC controller, which can be SP, goes into the ESO rather than the actual output y(t − τ ), thus represented as
Q (s) u(s) = r (s) 1 − Po (s) Q (s) Qd (s) Q (s) Qd (s) − y (s) 1 − Po (s) Q (s) Qd (s) =: C1 (s) Fr (s) r (s) − C2 (s) y (s)
achieving the synchronization of input signals and overcoming the limitation of GADRC for the delayed systems. yp (s) = y (s) + u (s) G (s) − u(s)G(s)e−τ s
(15) where G(s) is the model of P as shown in (2). 4
(18)
Journal Pre-proof
The setpoint transfer function is −1
y (s) = C(sI − A + BKo )
BFr (s) r (s)
pro of
=Tr (s) Fr (s) r (s)
(23)
ideally, we expect the y(s) to track the setpoint without offset, i.e., Tr (s) Fr (s) =1. If there is no filtering requirement for the setpoint, for the sake of simplicity, it is usually selected rˆ (s) = Fr (0) r (s), and then we have Fr (0) = −
Figure 2: The structure of SP-GADRC
Now the model-based ESO of SP-GADRC can be designed
¯o C A − BK
−1
B
(24)
As shown in (21), SP-GADRC can be interpreted through the
re-
based on (6):
1
TDF structure. In other words, according to the (16), we can
zˆ˙ (t) = A zˆ (t) + B u (t) + L y (t) L e o p u (t) = K (ˆ r (t) − zˆ (t))
(19)
u (s) =
lP
o
rewrite (21) as follows:
By taking Laplace transform of (19) with setting initial conditions to zero, we will get
sˆ z (s) = AL zˆ (s) + Be u (s) + Lo yp (s) u (s) = K (ˆ r (s) − zˆ (s))
urn a
o
QSP QSP QdSP r (s) − y (s) (25) 1 − Po QSP QdSP 1 − Po QSP QdSP
with the extended model defined in (4), the time-delayed system model can be express as
(20)
P0 (s) = G(s)e−τ s = Ce (sI − Ae )
−1
Be e−τ s
(26)
Substituting zˆ (s) in (20), the following equation can be ob- and then, solving the QSP and QdSP with tained:
u (s) = C1SP (s) rˆ (s) − C2SP (s)y (s)
= C1SP (s) Fr (s) r (s) − C2SP (s)y (s)
Jo
where
−1 ˆe C (s) = 1 − K sI − A Be 1SP o −1 C ˆ L0 2SP (s) = Ko sI − Ae rˆ (s) = Fr (s) r (s) ˆ Ae = Ae − Be Ko − Lo Ce
. C2SP −1 = Ko (sI − AL ) Lo Fr C1SP Fr
(27)
C1SP Fr −1 = 1 − Ko (sI − AK ) Be Fr 1 + P0 C2SP
(28)
QdSP = (21)
QSP =
where AK = Ae − Be Ko .
(22)
4. Parameter tuning of SP-GADRC via TDF-IMC In general, the SP-GADRC parameters Ko and Lo can be ob-
and Fr (s) is the step filter, which is used to make sure the steady-
tained based on the bandwidth idea using the controller band-
state output error for the setpoint tracking is zero.
width ωc the and observer bandwidth ωo [11]. And in this section 5
Journal Pre-proof
the relationship between the IMC parameters and the parameters Then rewrite (30) as follows of the SP-GADRC will be first analyzed. Section 4.1 shows that Q (s) =
Q(s) can be related to the parameters of SP-GADRC via (38) and
D(s)Fr N (s)(s+1/λ)n−m
N (s)(s+1/λ)n−m −D(s) N (s)(s+1/λ)n−m
pro of
Section 4.2 shows that Qd (s) can be related to the parameters of
= 1−
Fr (31)
SP-GADRC via (50). Thus by tuning the parameter for Q(s) and where
D (s) = sn + an−1 sn−1 + · · · + a1 s + a0 N (s) = (sm + · · · + b1 /bm s + b0 /bm ) F = 1/λn−m b
Qd (s), we can obtain the parameters of SP-GADRC via (53). First, the TDF-IMC structure for the SP-GADRC is considered. If we know the accurate plant model, i.e. P (s) =
r
G (s) e−τ s , we can get
(32)
m
Let AI = Ae , BI = Be and
P¯ (s) = P (s) + G(s)(1 − e−τ s ) = G(s)
(29)
KI =
re-
From (29), we can see that SP-GADRC transforms the time-
h
kI1
kI2
···
kIn
1
i
/b0
delayed system into a plant without delay through the SP, there- we get fore, in the equivalent IMC structure, the plant P0 is selected with
|sI − (AI − BI KI )|
delay-free one as shown in (2), and the Q and Qd are obtained
where,
urn a
the TDF-IMC structure for SP-GADRC.
lP
based on it as described in section 2.3. Figure 3 detailed describe
(33)
= s sn + (an−1 + kIn ) sn−1 + · · · + (a0 + kI1 )
n−i kIi = Cn−m
b−n+m+i−1 bm
i = 1, · · · n
1 n−i λ
n+1−i + Cn−m
1 n+1−i λ
− ai−1 , (34)
Then let n−m
|sI − (AI − BI KI )| = s (s + z1 ) · · · (s + zm ) (s + 1/λ) = sN (s) (s + 1/λ)
n−m
(35) where, z1 , · · · , zm are zeros of plant.
Jo
Finally, we can obtain
Figure 3: The TDF-IMC structure for SP-GADRC
N (s) (s + 1/λ)
n−m
− D (s) = kn sn−1 + · · · + k2 s + k1 (36)
4.1. State space realization of Q(s)
where,
According to the procedure of the IMC design, we can design
n−i kIi = Cn−m
Q(s) as Q (s) =
G(s) (λs + 1)
n−m
i = 1, · · · n
(30) 6
b−n+m+i−1 bm
1 n−i λ
n+1−i + Cn−m
1 n+1−i λ
− ai−1 , (37)
Journal Pre-proof
In summary, (30) can be expressed as
where
4.2. State space realization of Qd (s)
ϕ (s) = sn−m + µn−m−1 sn−m−1 + · · · + µ1 s + µ0 (38) 1 1 1 1 + Cn+1 µn−m−1 = Cn−m λ λd − an−1 1 1 1 2 1 1 2 + µn−m−2 = Cn−m C n+1 λd Cn−m λ − an−1 µn−m−1 λ .. . n−m 1 n−m n−m 1 n−m 1 1 µ0 = Cn−m + − an−1 µ1 C n+1 λd Cn−m λ λ (42)
pro of
sn +···+k2 s2 +k1 s Q (s) = 1 − kn|sI−(A Fr I −BI KI )| −1 = Fr 1 − KI (sI − AI + BI KI ) BI
According to (41), we can obtain
re-
αn sn + · · · + α1 s + 1 = n−m n−m n+1 s+ s + λ1 λ λd
Following the procedure of IMC design, a disturbance rejection controller is obtained as n
αn s + · · · + α1 s + 1 (λd s + 1)
n+1
n−m
λ
(39)
Qd (s) =
For the SP-GADRC, the (n + 1)th-order of ESO is needed for controlling the nth-order plant, so the same order of Qd (s) is selected to match it.
urn a
Jo
n+1 s + λ1d n−m 1 n+1 − λ1 (αn sn + · · · + α1 s + 1) λd 1 n−m λ
= s (s + p1 ) · · · (s + pn ) ϕ (s) n
n−1
=s s + an−1 s
s+
1 n−m λ
s+
1 λd
n+1 n+1
(s + 1/λd )
− sD (s) ϕ (s) (44)
In order to better demonstrate the certification process, n − 1, and let CI iT , βIn βIn+1
Ce , LI
=
=
(45)
according to the Ackerman formula, the ESO’s gain vector LI can be calculated as:
αn sn +···+α1 s+1 n−m n+1 |s=p1 ,···pn =0 (λs+1) n+1 (λd s+1) n+1 1 1 n−m 1 1 n−m s+ − (αn sn +···+α1 s+1) s+ (λ) ( λ) λd λd = n+1 1 n−m 1 s+ λ (s+ λ )
s+
(43)
n+1
1 − P QQd
which means
− sD (s) ϕ (s)
ψ (s) = |sI − (AI − LI CI )| = (s + 1/λd )
(14), and it can be expressed as
d
we take m = h βI1 βI2 · · ·
As mentioned before, αn , · · · , α1 can be obtained by solving
=1−
n+1
and then (39) can be expressed as
lP
Qd (s) =
1 λd
CI
CI AI LI = ψ (AI ) .. . CI AnI
(40)
−1
0 0 .. . 1
(46)
With (37) and (46), we can obtain
(41)
KI (sI − (AI − LI CI ))
+ · · · + a0 ϕ (s)
7
−1
LI =
dn sn + · · · d1 s + d0 (s + 1/λd )
n+1
(47)
Journal Pre-proof
where n+1
1 1 1 n+1 1 + Cn+1 − a + − a − a n−1 λ n−1 n−2 λd λ λd n+1 3 2 1 1 1 1 1 n+1 3 2 + + − a − a dn−1 = λb1m − a Cn+1 λd Cn+1 λd n−2 λ n−1 n−3 λd λ λd dn =
1 λbm
1 λd
2 Cn+1
1 λd
2
pro of
.. .
n+1 n+2−i n+1−i n+2−i n+1−i 1 1 1 1 1 1 n+1 d = + − a + − a − a Cn+1 Cn+1 i i−1 λ n−1 n−2 λbm λd λd λd λ λd .. . n+1 n+1 n n+1 1 1 1 1 1 1 n+1 n d = + − a + − a Cn+1 λd Cn+1 λd 1 0 λ n−1 λbm λd λ λd n+1 n+1 n+1 1 1 d0 = 1 Cn+1 λ1d λbm λd λ
It is easy to verify that
(48)
re-
then we have
n+1
dn sn + · · · d1 s + d0 =Fr (αn sn + · · · α1 s + 1)/(λd )
(49)
QdSP = Qd ,
QSP = Q
(54)
which means that under the assumption of (53), SP-GADRC is
In other words, the state space realization of Qd can be expressed equivalent to IMC. The two bandwidths in SP-ADRC are the in
Qd (s) = KI (sI − AI + LI CI ) 4.3. Tuning of SP-GADRC via IMC
verse of the two tuning parameters in IMC. Therefore, it is a good
lP
as
−1
LI
.
(50) news for engineers who are already familiar with IMC.
Fr
on Ko , while Qd is related to both Ko and Lo . Therefore, it is
As suggested in [18], two sets of gains can be obtained using
more reasonable to tune controller gain Ko and then Lo , that is,
urn a
bandwidth idea in SP-GADRC, i.e., βi 0 s of Lo are chosen such
for IMC, we should tune λ and then λd . The following steps are
that the observer bandwidth is ωo , and ki 0 s of Ko are chosen such
suggested for tuning SP-GADRC parameters via IMC:
that the controller bandwidth is ωc . For ESO, all observer poles
1) Determine the plant model P0 (s), e.g.
are placed at −ωo , we can get
n+1
P0 (s) = G (s) e−τ s
(51)
|sI − (Ae − Lo Ce )| =(s + ωo )
Jo
and for the final state feedback control system, we place the m
3) Determine the desired tracking performance of the closed. n−m loop system 1 (λs + 1) by choosing λ.
4) Determine the disturbance rejection filter constant λd to
n−m
|sI − (Ae − Be Ko )| =s (s + z1 ) · · · (s + zm ) (s + ωc )
(52) have the desired disturbance-rejection performance while ensuring the closed-loop stability.
Let 1 1 ωc = , ωo = λ λd
(55)
2) Design Q(s) and Qd (s) according to G(s).
poles to the zeros of the plant, and the remaining (n − m) poles are placed at -ωc ,
It can be observed form (27)and (28) that Q only depends
(53)
5) Let ωc = 1/λ, ωo = 1/λd . 8
Journal Pre-proof
5. Simulation QdSP (s) =
34.71s + 225 (s + 15)
(62)
2
The responses of P1 (s) are shown in Figure 4 and Figure 5,
To further demonstrate the tuning processes of SP-GADRC,
pro of
three examples are provided. For a fair comparison, a robustness and a unit step setpoint change and a unit step load disturbance indicator ε [27] is adopted. That is to say, by tuning the param- is added at t=0s and t=30s separately. Because TDF-IMC and eters to make sure that controlled systems have the same robust- SP-GADRC are equivalent, their responses are completely coness (or adaptability to parameter change), and then compare the incident. Moreover, when the model information is known and accurate, the response of SP-ADRC is very close to that of SP-
performance.
(56) GADRC. However, when there is a 6% mismatch in the delay
ε = sup (kSk∞ + kT k∞ ) ω
time τ , we can observe from Figure 5 that response of SP-ADRC
where, 1 kSk∞ = max 1+L(jω) , ω
L(jω) kT k∞ = max 1+L(jω) ω
re-
is oscillatory, while SP-GADRC remains smooth. (57)
1.2
1
and L(jω) is the frequency response of loop transfer function
0.8
y(t)
of ε.
lP
L(s). Generally, the robustness decreases by increasing the value
Case 1. A very simple first-order plant but with a very large
0.6 0.4
time delay is studied in [18], and parameters of SP-ADRC are tuned as ωc =12, ωo =36, b0 =10 with ε=2.91.
urn a
0.1 P1 (s) = e−5s 0.1s + 1
SP-GADRC SP-ADRC TDF-IMC
0.2 0 0
time constants in TDF-IMC are chosen as
60
0.8
y(t)
Jo
50
1
0.6 0.4
(60)
SP-GADRC,+6% SP-ADRC,+6% SP-GADRC,-6% SP-ADRC,-6%
0.2 0
and the controllers in the equivalent TDF-IMC are 1.7s + 17 s + 1.7
40
(59)
first-order SP-GADRC are tuned as
QSP (s) =
30 t/s
1.2
with the robustness indicator ε=2.91. Then the parameters of
ωc =1/0.59, ωo =1/0.0667
20
Figure 4: Responses of P1 (s)
For P1 (s), to obtain the same robustness as SP-ADRC, the
λ= 0.59, λd = 0.0667
10
(58)
0
(61)
10
20
30 t/s
40
50
60
Figure 5: Responses of P1 (s) with model uncertainties
9
Journal Pre-proof
Case 2. A third-order time-delayed plant was controlled by output responses and the controller responses are shown in FigSP-ADRC with ωc =0.3, ωo =1.2, b0 =0.5, and the robustness indi- ure 6 and Figure 7. We can see that responses of SP-GADRC and cator ε=2.7 in [18].
TDF-IMC are indistinguishable with the same disturbance rejec-
(s + 1)
pro of
P2 (s) =
tion capability, and SP-GADRC outperforms SP-ADRC on the
e−5s
(63) response speed and smoothness of the controller output. Then we
3
selected Pm (s) =
e−6s 2s+1
as the G(s) in SP to further show the dis-
According to the tuning procedure of IMC, the parameters are turbance rejection characteristics, which are given in Figure 8 and tuning as: Figure 9. It is obvious that both controllers can achieve satisfac(64) tory control performance, but the response speed of SP-GADRC
λ= 1.25, λd = 0.3448
is still faster.
with the ε=2.7, so the parameters of SP-GADRC are (65)
1
re-
ωc =1/1.25, ωo =1/0.3448
0.5
and the controllers in the equivalent TDF-IMC are
0
SP-GADRC SP-ADRC TDF-IMC
-0.5
3
77.49s + 223.3s +216.7s + 70.73 s4 + 11.6s3 + 50.46s2 + 97.56s + 70.73
1
-2 -2.5 -3
(67)
-3.5 0
-0.5 -1 -1.5
-2.5 -3 0
Jo
-2
100
200 t/s
100
200 t/s
300
400
Figure 7: Controllers output responses for P2 (s)
1 SP-GADRC, P m
0.5
SP-ADRC, Pm
0 -0.5
y(t)
0
-1
-1.5
SP-GADRC SP-ADRC TDF-IMC
0.5
y(t)
2
urn a
QdSP (s) =
(66)
y(t)
0.512s3 + 1.536s2 + 1.536s + 0.512 s3 + 2.4s2 + 1.92s + 0.512
lP
QSP (s) =
-1 -1.5
300
400
-2 -2.5
Figure 6: Responses of P2 (s)
-3 0
100
200
300
t/s
For P2 (s) a step reference is added at t=200s, and step-type disturbances are added at t=100s and t=300s, respectively. The
Figure 8: Responses of P2 (s) with Pm
10
400
Journal Pre-proof
change of the setpoint, while the output of SP-GADRC maintains
1 SP-GADRC, P m
0.5
smooth.
SP-ADRC, Pm
0
1.2 1
pro of
y(t)
-0.5 -1
0.8
-1.5
0.6
y(t)
-2 -2.5
0.4 0.2
-3 -3.5
0
0
100
200 t/s
300
400
SP-GADRC DOB configuration strategy TDF-IMC
-0.2 -0.4
Figure 9: Controllers output responses for P2 (s) with Pm
re-
0
Case 3. [28] applied a DOB configuration strategy with ε=2.446 to a second-order non-minimum phase system.
20
40 t/s
60
80
Figure 10: Responses of P3 (s)
30
(68)
20
Using the proposed method, the parameters of SP-GADRC is
ωo = 0.6
urn a
ωc = 0.43,
(69)
with ε=2.4426. The controllers in the equivalent TDF-IMC are QSP (s) =
QdSP
2.58s2 + 2.15s + 0.43 s2 + 1.43s + 0.43
1.257s2 + 1.061s + 0.216 (s) = 3 s + 1.8s2 + 1.08s + 0.216
15
y(t)
tuned as:
SP-GADRC DOB configuration strategy TDF-IMC
25
lP
−s + 1 P3 (s) = e−s (2s + 1) (3s + 1)
10 5 0 -5
(70)
0
20
40 t/s
60
80
Figure 11: Controllers output responses for P3 (s)
(71) 6. Conclusion
Jo
so the time constant for the setpoint filter in TDF-IMC is λ=1/0.43 and that for the disturbance rejection filter is λd =1/0.6.
We combine model-assisted ESO with Smith predictor to gen-
As shown in Figure 10 and Figure 11, a unit step-type ref- erate SP-GADRC for time-delayed systems. By transforming SPerence is loaded at t=10s, then a unit step-type disturbance at GADRC to the TDF-IMC control structure, in which the nominal t=40s. It can be observed that the undershoot of SP-GADRC is model is chosen as the delay-free one, the bandwidth parameters much smaller than the DOB configuration strategy, and the out- can be tuned through the two time constants of STF and DRF in put response curve is smoother. At the same time, the controller the equivalent IMC. Therefore, it provides us with a new idea of output of DOB configuration strategy is more sensitive with the tuning parameters of SP-GADRC: the controller bandwidth ωc is 11
Journal Pre-proof
[8] M. R. Matausek, A. D. Micic, A modified smith predic-
width ωo is the inverse of the time constant λd of the DRF. The
tor for controlling a process with an integrator and long
proposed tuning method is tested in three examples, showing its
dead-time, IEEE Transactions on Automatic Control 41 (8)
effectiveness in the control design.
(1996) 1199–1203.
pro of
the inverse of the time constant λ of the STF and the ESO band-
[9] D. G. Padhan, S. Majhi, Modified smith predictor based cas-
Acknowledgement
cade control of unstable time delay processes, ISA Transac-
This work was supported by the National Natural Science Foundation of China under grant 61573138 and the Fundamental Research Funds for the Central Universities under grant
tions 51 (1) (2012) 95–104.
[10] J. Han, From PID to active disturbance rejection control, IEEE Transactions on Industrial Electronics 56 (3) (2009)
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*Conflict of Interest Statement
Journal Pre-proof Conflict of interest statement
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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled ‘Tuning of Smith predictor based generalized ADRC for time-delayed processes via IMC’.