Multi-loop PI Controller Design for Enhanced Disturbance Rejection in Multi-delay Process
Truong Nguyen Luan Vu and Moonyong Lee
Abstrac t—In this paper, a new design method is propod for multi-loop PI controllers in the multiple input, multiple output (MIMO) systems in two cas: t-point tracking and disturbance rejection. The generalized IMC-PID approach [1], which is extended from single input, single output (SISO) systems to MIMO systems, is considered to identify the tuning parameters of multi-loop PI controllers. However, there is not enough robustness in multi-delay systems which contain a lot of noi and disturbance. The propod design method can solve this problem by using the magnitude of nsitivity (Ms) theory. A simulation study is performed for the well-known process model and the respon performances compared favorably with some famous tuning methods.The results show that the propod method is superior to existing techniques for multi-delay process.
Keywords— Multi-loop PI controller, Multi-delay process, IMC-PID approach, Ms Criterion.
I.INTRODUCTION
T H E multi-loop PID/PI controller has been studied for many decades. In the 1980s, the famous tun
ing method for calculating multi-loop PID controller parameters was the Internal Model Control (IMC) [2], it was published by C.G. Economou and M. Morari. A typical method which related to the multi-loop IMC design method was propod by M.S. Basualdo and J. L. Marchetti [3], which considers to the interactions between the control loops. The biggest log modulus (BLT) tuning design method [4] was published by W. L. Luyben and it is still popular in process control today. In the 1990s, Loh et al. [5] studied the auto-tuning procedure for improving the clod-loop frequency respons in MIMO systems, and Jung et al. [6] prented the decentralized lambda tuning (DLT) design method with the same goal of improving stability and robustness. Recently, the generalized IMC-PID approach is designed for the multi-loop PID control systems by Lee et al. [1]. This approach is a variation of Lee et al. [7] which admitted to SISO systems. Many multi-loop tuning design methods exist for t-point tracking problems today. However, there are few methods available for disturbance rejection despite the fact that disturbance rejection is a more rious problem in industry. Therefore, we propod a new design method which proceeds from the generalized IMC-PID approach and Ms Criterion. The aim of this method is to design a multi-loop PI controller that enhances disturbance rejection as well as t-point tracking.
Manuscript received October 29, 2007: Revid version received April 29, 2008.
Truong Nguyen Luan Vu and Moonyong Lee are with the School of Display & Chemical Engineering, Yeungnam University, 214-1, Dae-dong, Gyeongsan , Gyeongbuk 712-749, Korea (corresponding author to provide In the multi-loop IMC control systems, the performance and
robustness of the clod-loop system largely depends on the
clod-loop time constant (λ). The optimal value for the
clod-loop time constant can be obtained by using Ms Criteria.
The propod method can be compensated the influence of
disturbance effectively by compensating the dominant poles in
the diagonal element of the process transfer function.
Fig.1 Block diagram for the multi-loop control system.
II.THE MULTI-LOOP PI CONTROLLER DESIGN
In the nxn multi-loop feedback control system shown in Fig.
1, the clod-loop respon to the t-point change is
()1
()()()()()()()()
c c
s s s s s s s s
−
==+
y H r I G G G G r
(1)
where H(s) is the clod-loop transfer function; G(s) is the
process transfer function which is open-loop stable; )(
~
s
c
G is
the multi-loop controller with diagonal elements only; y(s) and
r(s) are the controlled variable and the t-point, respectively.
Suppo that the desired clod-loop respon of the diagonal
elements in the multi-loop system is given by
]
,...,
,
[
)
(
~
2
1n
R
R
R
diag
s=
R (2)
kimiko
According to the design strategy of the IMC controller [1],
the desired clod-loop respon R i of the i th loop is prented
by
f
y()G()(1)
R()
r()(λ1)i
i ii
i n
i i
s s
s
xlneekos s
i
s
β
+
+
==
+
(3)
where G ii+ is the non-minimum part of G ii and chon to be the all pass form; i λis an adjustable constant for system performance and robustness; n i is chon for the IMC controller to be realizable. βi is designed to cancel the dominant poles in the diagonal process element.
()
111i i
ii+i n
i s = p G (s)(βs+)-λs+0= (4) Note that i λis analogous to the clod-loop time constant and thus determines the speed of the clod-loop respon. The multi-loop controller )(~
s c G
with integral term can be expresd in a Maclaurin ries as [)
(~~~1)(~3
2210s O s s s
s c c c c +++=G G G G ]
2
(5)
where can be considered as the integral, proportional, and derivative terms of the multi-loop PID controller, respectively.
01,c c c G G ,G As indicated from (5), the impact of proportional and
derivative terms (i.e ., 21~
,~c c G G ) dominates at high frequencies and thus they should be designed bad on the process characteristics at high frequencies. On the other hand, the
integral term 0~c G is dominating at low frequencies and thus
needs to be designed bad on the characteristics at low
frequencies.
In the multi-loop system, the characteristic of the clod-loop interaction is changed according to frequency range. Using this frequency-dependent properties of the clod-loop interactions, analytical design of the multi-loop PID controller can be largely simplified while it still takes the interaction effect fully into account as follows [1]:
At high frequencies, the magnitude of open loop gain becomes
()()c
j j ωωG G 1 and thus H (s) can be approximated to
)(~)()(~)())(~)(()(1s s s s s s s c c c G G G G G G I H ≈+=− 6)
It indicates that c0~G and c1~
G can be designed by considering
only the diagonal elements in G (s), which means the generalized IMC-PID method for the SISO system [7] can be applied to the design of the proportional and derivative terms in the multi-loop PID controller. Therefore, at high frequencies, the ideal multi-loop feedback controller to give the desired clod-loop respon )(~s R is given by
1
1))(~)((~)(~)(~−−−=s s s s c R I R G G (7) where ],...,,[)(~2211nn G G G diag s =G
Accordingly, the ideal multi-loop controller of the i th loop can be designed by
1(())(1)()(1)()(1i
6级真题
ii i ci n i ii i G s s G s s G s s βλβ−−++=)+−+ (8)
where G ii- is the minimum part of G ii . Since G ii+(0)=1, (8) can be rewritten in a Maclaurin ries
with an integral term as
))(02
)
0()0()0((1)(32'''
s s f s f f s s G i i i ci +++= (9) where f i (s) = G ci (s)s The standard PID control algorithm is given by 1
()(1)ci ci Di Ii G s K s s
you jump i jump是什么意思ττ=+莱昂纳多 oppo
+ (10) C omparing (9) with (10) gives the analytical tuning rules for the proportional gain of the multi-loop PI controller as follows:
(11) '(0)ci i K f =
At low frequencies, according to the design of the integral term 0~c G , the interaction effect between the control loops can not be neglected. Expansion of G (s) in a Maclaurin ries gives
)()(32210s O s s s +++=G G G G (12)
where 2/)0(";)0(';)0(210G G G G G G ===
By substituting (5) and (12) into (1), one can obtain H (s) as
)()~
()(2100s O s s c +−=−G G I H (13)
Furthermore, the desired clod-loop respon R ~
can also
be written in Maclaurin ries as
)()0('~
)0(~)(~2s O s s ++=R R R (14)
where I R =)0(~ becau 1)0(=+ii G
小学英语语法大全
By comparing the diagonal element of H (s) in (13) and
in (14) for the first-order s term, one can get the analytical tuning rule for the integral time constant of the multi-loop PID controller as follows (s)R
'1((0))((0))ii i i i ci Ii ii
G n K λβτ+−−+=−G (15)
Tuning formulae by (11) and (15) provide an important
advantage to solve the optimization problem for finding the PID parameter values: for a given process, all the PID parameters can be expresd by a single design parameter i λand thus the dimension of the arch space for
optimization is greatly reduced.
The lead term by (1)i s β+in (3) can cau an excessive overshoot in the t-point respon. The two degree of freedom structure can overcome this problem by designing a t-point filter q i as课外辅导学校
1
()(1fi i q s s β=+)
(16)
III. MS CRITERION FOR MIMO SYSTEMS
Ms tuning is the frequency-domain method which relates to the resonant peak Ms . Ms values are related to the resonant peak of the nsitivity function. The relative stability and robustness of a stable clod-loop system can be suggested by the magnitude of Ms . In 1996, Skogestad and Postlethwaite [8] employed Ms as a tool for measuring system robustness. In 1998, Astrom et al. [9] propod that the desirable values of Ms for SISO systems are in the range of 1.2 to 2. Ms tuning provides a limit for the clod-loop time constant for a model, and it allows the optimal controller parameters to be found. The nsitivity function in the multi-loop control system can be reprented by
-1c ()(+ (s )(s )) s =S I G G (17)
The nsitivity frequency respon can be found by tting s = j ω in term of ω and λas follows
[]-1
c (j ω,λ)+ (j ω,λ)(j ω,λ)=S I G G ⎥ (18)
The nsitivity function can be expresd by the matrix form as
{}(j ,)11121n 21
222n ij n1n2nn S S S S S S S S S S ωλ⎡⎤
⎢⎥⎢==⎢⎥⎢⎥⎣⎦S ""###" (19) The maximum nsitivity Ms is obtained as the maximum value of the nsitivity function over frequencies
{}{}
λ,ω 0
==max ij ij Ms S (j ω,λ)≥ Ms (20)
The peak magnitude of the nsitivity function can be expresd by the matrix form as
{}
11121n 21222n ij n1maxx
n2nn Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms ⎡⎤⎢⎥
⎢⎥==⎢⎥⎢⎥
⎣⎦""###"Ms (21) The propod Ms tuning method is aimed to improve the
performance and robustness of clod-loop frequency
respons in the multi-loop control system by finding an optimal λ. The multi-loop control system can also be made to meet the stability bounds and all the multi-loop PID parameters
can be expresd by a single design parameter i λ.This optimization problem in the frequency domain is
λ,ω 0
i
min ()
< ii ii low Ms Ms Ms ≥≥∑ (22)
where Ms low is the lower bound of the diagonal Ms and it also can be considered as optimizing value to minimize the integral absolute error (IAE). Fig. 2 shows the effects of Ms low on the overall
performance in the OR column [11]. It implies that at small values of Ms low , the IAE values are large. However, when Ms low increas to high values, the IAE values also increa. Fig.2 Effects of Ms low on the IAE: OR column. Our extensive simulation study shows that the desirable value of Ms low lies between 1.8 and 2. This range of Ms low can be ud for the trade-off between sluggish, overshoot,
oscillation, and minimizing IAE.
According to (22) it is easy to find the optimal value of λ which makes multi-loop control systems stable and robust not
only for t-point tracking but also for disturbance rejection.
IV. ROBUSTNESS STABILITY ANALYSIS
While the process model contains uncertainties in its parameters, the stability robustness of multi-loop control system become more importantly. Therefore, veral uncertainty models are considered to demonstrate the stability robustness of propod control system. For the multi-loop control system included a process output uncertainty as , stability robustness of the clod-loop system can be obtained by [12] )()]([s G s I o Δ+
{}1()1/[()()]()()o c c j I G j G j G j G j ωσωωω−Δ≤+ω(23)
where is stable and )(s o Δσis denoted the maximum
singular values of the clod-loop transfer function.
Fig.3 Stability regions of output uncertainties
cet4Figure 3 has shown the stability bounds for OR column with propod, BLT, and DLT controller, respectively. It is implied that the propod control system has the largest stability region of process output uncertainty at low frequencies. Therefore, the propod PI control system is more robustness stability than tho by BLT and DLT.
V. SIMULATION STUDY
In the following ca studies, we demonstrate our tuning rules with 2x2 and 3x3 systems from the open literature. The propod method is also compared with veral well-known tuning methods such as BLT and DLT tuning methods.
Example 1: Consider the Wardle and Wood (WW) column
which was studied by W. Luyben in 1986. This can be
reprented by
61880.1260.101601(481)(451)()0.0940.12381351s s s s e e s s s s e e s s −−−−2⎡⎤−⎢⎥
+++⎢
⎥=⎢⎥−⎢⎥
++⎣⎦
G (24)
By using (22), the optimum λi values can be obtained as 34.75 and 34.28 for loop 1 and 2, respectively. n i in (3) was chon as 2 for all loops according to the process model order. The value of 1.9 was chon for Mslow. Figure 4 shows the clod-loop frequency respon of the multi-loop PI control system for the WW column in the ca of step changes in disturbance. As shown in the Figure 4, the propod method provides more well-balanced and faster respons when compared to other existing methods. Besides, the propod method can give the minimum integral absolute error (IAE)
values which are shown in Table 1.
Fig.4 Clod-loop respon to quential step changes in
disturbance for WW column
Table 1
Tuning results by the propod PI method and various
methods: WW column
Propod BLT DLT K c
53.4, -20.9 27.4, -13.3 33.3,- 21.7 I τ
63.2, 38.8
41.4, 52.9
63.0, 39.0
Step changes in t-point
IAE 1 27.05,
36.41 31.96, 65.68 37.42, 29.93
IAE 2 18.9, 39.49 26.60, 87.92 29.73, 38.51
IAE t
121.85 212.16 135.59 Step changes in disturbance IAE 1 1.16, 0.56 1.49, 0.91 1.83, 0.67 IAE 2
0.24, 1.81
0.55, 3.67
0.35, 1.76
IAE t
3.76 3.76 6.62 6.62
4.61 4.61
IAE i : IAE for the step change in loop i . IAE t : sum of each IAE i .
Example 2: Consider the Ogunnaike and Ray (OR) column, a multi-product plant distillation column for paration of a binary ethanol-water mixture, was modeled experimentally in Ogunnaike et al. [11].The transfer function matrix of the OR column is given by
2.6
3.56.53129.29.40.660.60.00496.718.6419.0611.11 2.360.01() 3.251517.091
34.6846.20.89(11.611)8.15110.91(3.891)(18.81)s s s s s s s s
e e e s s s e e e s s s s e
e s e s s s s −−−−−−−−⎡⎤−−⎢⎥+++⎢⎥
−−⎢⎥=⎢⎥+++⎢⎥−+⎢⎥⎢+++⎣⎦marriage是什么意思
G s −+ (25)
The optimum λi values were found as 10.07, 8.78, and 2.3 for each loop, respectively. Step changes in t-point and disturbance were quentially made in the individual loops. The t-point filters can be found as
1,2,311
1
(){
,,}
(5.481)(3.431)(3.391)f q s s s s =+++
Figures 5 show the clod-loop respons for quential step changes in t-point and disturbance, respectively. Sequential step changes of magnitude 1, 1, and 10 were made to each loop.
Fig.5 Clod-loop respon to quential step changes in t-point for OR column
