PI CONTROLLER DESIGN FOR TIME
DELAY SYSTEMS
M. Bakošová, K. Vaneková, J. Závacká
Department of Information Engineering and Process Control, Institute
of Information Engineering, Automation and Mathematics, Faculty of
Chemical and Food Technology, Slovak University of Technology in
Bratislava, Radlinského 9, 812 37 Bratislava, Slovak Republic
fax: +421 2 52496469 and e-mail: [email protected],
[email protected], [email protected]
Abstract: The paper presents an approach to PI controller design for systems with varying time
delay, which can be interpreted as systems with interval parametric uncertainty. The transfer
function of the controlled system is modified by approximation of the time delay term by its
Pade expansion or Taylor expansion in the numerator. PI controllers, that are able to assure
robust stability of the feedback closed loop with the modified controlled system are found
by the graphical method. Then the pole-placement method is used to specify those controller
parameters, which assure certain quality of the control performance. Designed PI controllers are
implemented using PLC SIMATIC S7-300 and tested by experiments on a laboratory electronic
equipment with varying time delay.
Keywords: PI controller, pole-placement method, robust control, time-delay system
1. INTRODUCTION
Presence of a time delay in an input-output relation
is a common property of many technological processes.
Plants with time-delays can often not be controlled using
controllers designed without consideration of a time delay.
These controllers can destabilize the closed-loop system.
There are several approaches for time delay treatment.
One of them is based on using time-delay compensators
based on Smith predictor, see e.g. Majhi and Atherton
(1998). The other approach is based on an approximation
of a time delay. After the approximation, a modified
transfer function of the controlled process is obtained
and this transfer function does not contain the term
representing the time delay. Then, classical approaches to
controller design can be used, as e.g. polynomial approach
(Dostál et al. (2000)), robust control approach (Zhong
(2006)) and others.
PI controllers belong to the most frequently used controllers in industry. A method for PI controller design
for systems with varying time delay and parametric uncertainty is presented in this paper. The method combines several known approaches. The term representing
the time delay in the transfer function of the controlled
system is approximated by Taylor expansion in numerator
or Padé expansion (Dostál et al. (2000)) at first. The
modified transfer function with interval polynomials in the
numerator and the denominator is obtained. Then, the
graphical method is used, which designes robust PI controllers for systems with interval parametric uncertainty.
The method is based on obtaining robust stability regions
for parameters of PI controllers (Tan and Kaya (2003),
Závacká et al. (2007)). In the last step, the PI controllers
from robust stability regions are selected using the poleplacement method (Mikleš and Fikar (2007))
2. CONTROLLED SYSTEM
Consider a controlled time-delay system with parametric
uncertainty described by the transfer function
[K − , K + ] −[D− ,D+ ]s
G(s) = − +
e
(1)
[T , T ]s + 1
The reason for the controlled system description (1) consists in the fact that many aperodic processes can be
identified in this form.
One of possibilities for treating the system (1) is to transform the transfer function into the form of rational function only, i.e. the time-delay term is approximated, see
e.g. Dostál et al. (2000). Various approximations of the
time-delay term can be used, e.g. the first order or the
higher order Taylor expansions in the denominator or in
the numerator, the first order or the higher order Padé expansions, etc. Then, the mutual dependence between polynomial coefficients in the numerator and the denominator
of the modified transfer function can be ignored and the
polynomials in the transfer function can be overbounded
by interval polynomials. In other words, the time-delay
system (1) can be approximated by the system without
the time delay described by the transfer function
Pm − + i
[bi , bi ]s
N (s, b)
m≤n
(2)
G(s, b, a) =
= Pni=0 −
+ i
D(s, a)
i=0 [ai , ai ]s
where N (s, b), D(s, a) are polynomials in s with coefficients changing their values between lower and upper
+ − +
bounds b−
i , b i , ai , ai .
Fk (ω) = −ω 2 Nko (−ω 2 )
Gk (ω) = Nke (−ω 2 )
Hk (ω) = Nke (−ω 2 )
Ik (ω) = Nko (−ω 2 )
Fig. 1. Control system
2
(9)
2
Jl (ω) = ω Dlo (−ω )
Kl (ω) = −Dle (−ω 2 )
3. ROBUST PI CONTROLLER DESIGN
we can write (7) and (8) as
Consider the closed-loop feedback control system depicted
in Fig. 1, where G(s, b, a) is the controlled system (2), C(s)
is the controller, w is the reference, u is the control input
and y is the controlled output.
In compliance with the sixteen plant theorem (Barmish
(1994)), a first order controller
C(s) = kp +
ki
s
(3)
robustly stabilizes an interval plant (2) if and only if it
stabilizes its 16 Kharitonov plants, which are defined as
Gkl (s) =
Nk (s)
Dl (s)
(4)
where k ∈ {1, 2, 3, 4}, l ∈ {1, 2, 3, 4} and N1 (s) to N4 (s)
and D1 (s) to D4 (s) are the Kharitonov polynomials for
the numerator and the denominator of the interval system
(2) (Barmish (1994)).
The method for finding robust stabilizing PI controllers is
based on plotting the stability boundary of the closed loop
in the plane of controller parameters for frequency ω ≥ 0
(D-partition method (Neymark (1978)).
Substituting s = jω into (4) and decomposing the numerator and the denominator polynomials of (4) into their even
and odd parts leads to
Gkl (jω) =
Nke (−ω 2 ) + jωNko (−ω 2 )
Dle (−ω 2 ) + jωDlo (−ω 2 )
(5)
where even parts of polynomials are represented by terms
with even powers of ω and odd parts are defined analogically.
kp Fk (ω) + ki Gk (ω) = Jl (ω)
kp Hk (ω) + ki Ik (ω) = Kl (ω)
(10)
From these equations, parameters of the PI controller are
expressed in the form
Jl (ω)Ik (ω) − Kl (ω)Gk (ω)
Fk (ω)Ik (ω) − Gk (ω)Hk (ω)
Kl (ω)Fk (ω) − Jl (ω)Hk (ω)
ki =
Fk (ω)Ik (ω) − Gk (ω)Hk (ω)
kp =
(11)
(12)
For the choice ω ≥ 0, we obtain sets of kp and ki from
(11) and (12). After plotting the dependence of ki on kp ,
the stability boundary locus is obtained, which splits the
(kp , ki )-plane into several regions. Then a pair (kp , ki ) is
chosen in each region and the stability of the closed loop
with the chosen PI controller is tested, e.g. by calculation
of closed-loop poles. The stable regions are those, where
pairs (kp , ki ) give closed-loop poles with negative real
parts. The approach enables to find regions of PI contollers
that stabilize the plant Gkl (s).
To avoid potential problems with proper choice of frequency ω ≥ 0, the frequency axis can be divided into
several intervals (Söylemez et al. (2003)) by the real values
of ω which fulfill
Im[G(jω)] = 0
(13)
and such intervals are then sufficient for testing.
To obtain PI controllers, which robustly stabilize the
system (2), the over described approach must be applied
onto all Kharitonov plants. The robust stability region
is obtained as the intersection of stability regions of all
Kharitonov plants (Tan and Kaya (2003), Závacká et al.
(2007)).
The closed loop characteristic equation for (5) and (3) is
∆(jω) = [−kp ω 2 Nko (−ω 2 ) + ki Nke (−ω 2 )−
−ω 2 Dlo (−ω 2 )] + jω[kp Nke (−ω 2 )+
+ki Nko (−ω 2 ) + Dle (−ω 2 )] = 0
3.1 Robust PI controller synthesis for the system of the
first order
(6)
Then, equating the real and imaginary parts of (6) to zero,
we obtain
kp (−ω 2 )Nko (−ω 2 ) + ki Nke (−ω 2 ) = ω 2 Dlo (−ω 2 ) (7)
kp Nke (−ω 2 ) + ki Nko (−ω 2 ) = −Dle (−ω 2 )
After denoting
(8)
Suppose that the controlled system is described by the
transfer function
+
− +
[b−
1 , b1 ]s + [b0 , b0 ]
(14)
G(s, b, a) = −
− +
[a1 , a+
1 ]s + [a0 , a0 ]
and it is necessary to find robust PI controller (3). The
total number of Kharitonov plants (4) for the 1st order
system (14) is 4 and k ∈ {1, 2} and l ∈ {1, 2}. The closed
loop characteristic equation (6) for e.g. the Kharitonov
plant G11 is
−
− 2
2
∆(jω) = (−kp b−
1 ω + ki b0 − a1 ω )+
−
−
+ jω(kp b0 + ki b−
1 + a0 ) = 0
(15)
Equating the real and imaginary parts of (15) to zero
gives following expressions for calculating of kp , ki in the
dependence on the frequency ω
−
− 2
2
b−
1 ω kp − b0 ki = −a1 ω
(16)
+
(17)
After the calculation of sets of kp and ki for chosen
intervals of ω, the plot representing the stability boundary
locus can be driven and the stability region of kp and ki
can be found. The robust stability region is obtained as the
intersection of stability regions obtained for all Kharitonov
plants.
b−
0 kp
b−
1 ki
= −a−
0
3.2 Robust PI controller synthesis for the system of the
second order
Suppose that the controlled system is described by the
transfer function
[b− , b+ ]s + [b− , b+ ]
G(s) = − + 12 1 − +0 0 − +
(18)
[a2 , a2 ]s + [a1 , a1 ]s + [a0 , a0 ]
and it is necessary to find robust PI controller (3). The
total number of Kharitonov plants (4) for the 2nd order
system (18) is 6 and k ∈ {1, 2} and l ∈ {1, 2, 3}. The closed
loop characteristic equation (6) for e.g. the Kharitonov
plant G11 is
closed-loop pole. Specifying suitable values of parameters
ξ, ω0 , α in (22)-(24) leads to the controller which can assure the desired quality of the control response. To obtain
unique solution, the system of equations for calculation
of controller parameters has to be the system with zero
degree of freedom. If higher order characteristic equation
is considered, then any of parameters ξ, ω0 or α can be
added to unknown variables.
4.1 Pole-placement method for the system of the first order
Suppose that for the controlled system (25)
b1 s + b0
(25)
G(s) =
a1 s + a0
it is necessary to find PI controller (3). The closed loop
characteristic equation can be written as (23).
The closed loop characteristic equation for the controlled
system (25) and the PI controller (3) has also the form
a0 + b0 kp + b1 ki
b0 ki
s2 +
s+
=0
(26)
a1 + b1 kp
a1 + b1 kp
After the choice of ξ and ω0 and comparison of coefficients
in (23) and (26), parameters of the PI controller can be
computed from (27) and (28)
(b0 − 2ξω0 b1 )kp + b1 ki = 2ξω0 a1 − a0
∆(jω) =
+
− 2
+ ki b−
0 − a1 ω )+
−
−
2
jω(kp b0 + ki b1 − a−
2ω
−b1 ω02 kp
2
(−kp b−
1ω
+
a−
0)
(20)
+
−
(21)
After the calculation of sets of kp and ki for chosen
intervals of ω, the plot representing the stability boundary
locus can be driven and the stability region of kp and
ki can be found. The robust stability region is obtained
as the intersection of stability regions obtained for all 16
Kharitonov plants.
b−
0 kp
b−
1 ki
= a−
2ω
a−
0
The pole-placement control design belongs to the class
of well-known analytical methods where transfer function
of the controlled process is known (Mikleš and Fikar
(2007)). In this method, only the closed-loop denominator
that assures stability is specified. The advantage of this
approach is its usability for a broad range of systems. If
the controller is of PID structure then the characteristic
equation can be one of the following
(s +
(28)
4.2 Pole-placement method for the system of the second
order
Suppose that for the controlled system (29)
b1 s + b0
(29)
G(s) =
2
a2 s + a1 s + a0
it is necessary to find PI controller (3). The closed loop
characteristic equation can be written as (24).
The closed loop characteristic equation for the controlled
system (29) and PI controller (3) has also the form
a1 + b1 kp 2 a0 + b1 ki + b0 kp
b0 ki
s3 +
s +
s+
= 0 (30)
a2
a2
a2
After comparison of coefficients in (24) and (30), parameters of the PI controller can be computed from (31)-(33)
4. POLE-PLACEMENT METHOD
s + ω0 = 0
(22)
s + 2ξω0 s + ω02 = 0
α)(s2 + 2ξω0 s + ω02 ) = 0
(23)
2
(27)
= 0 (19)
Equating the real and imaginary parts of (19) to zero
gives following expressions for calculating of kp , ki in the
dependence on the frequency ω
−
− 2
2
b−
1 ω kp − b0 ki = −a1 ω
+
b0 ki = a1 ω02
(24)
or the combination of (23) and (24), where ξ is relative
damping, ω0 natural undamped frequency, and α is a
b1 kp = a2 α + 2ξω0 a2 − a1
b0 kp + b1 ki = 2ξω0 a2 α +
ω02 a2
− a0
b0 ki = ω02 a2 α
(31)
(32)
(33)
It is clear that unique solution kp ki is obtained if only two
of parameters ξ, ω0 or α are chosen.
5. CONTROLLED LABORATORY PROCESS
Controlled laboratory process (Fig. 2) is an electronic
model of a linear system with a varying time delay
(Babirád (2006)). The process was identified from step
responses measured in various working areas by Strejc
0.12
0.1
ki
0.08
0.06
0.04
0.02
0
−1
−0.5
0
kp
0.5
1
Fig. 2. Controlled laboratory process
method (Mikleš and Fikar (2007)) in the form of a transfer
function of the first order with time delay
−
+
K
G(s) = − +
e−[D ,D ]s
(34)
[T , T ]s + 1
where K is the gain of the process, which is constant, and
T − , T + , D− , D+ are the minimum and maximum values of
the time constant and the time delay of the process with
following values
K = 0.97
T
−
(35)
= 16s = 0.27min,
D = 16s = 0.27min,
−
+
= 19s = 0.31min
(36)
D = 39s = 0.65min
(37)
T
+
Using the first order Taylor expansion in numerator, the
modified transfer function of the controlled system is
−K[D− , D+ ]s + K
G(s) =
(38)
[T − , T + ]s + 1
and after overbounding polynomials in (38) with interval
polynomials, the transfer function of the first order is
obtained in the following form
G(s) =
[−37.83, −15.52]s + 0.97
[16, 19]s + 1
(39)
Similarly, using the first order Padé expansion, the modified transfer function of the controlled system is
G(s) =
−K [D
[T − , T + ] [D
− ,D + ]
s2 +
,D+ ]
s+K
2
−
+
−
([T , T + ] + [D 2,D ] )s
−
+1
(40)
and after overbounding polynomials in (40) with interval
polynomials, the transfer function of the second order is
obtained in the following form
[−18.91, −7.76]s + 0.97
G(s) =
.
(41)
[128, 370.5]s2 + [24, 38.5]s + 1
2
5.1 PI controller design for the controlled process
Methods described in Section 3, Section 4, and in Tan
and Kaya (2003), Závacká et al. (2007) are used for PI
controller design.
Consider the controlled system (39). The robust stability
region of PI controller parameters is found at first. The
first Kharitonov plant is obtained for the maximum time
delay and has the form
Fig. 3. Choices of PI controllers for the 1st order system
−37.83s + 0.97
(42)
16s + 1
The expressions for calculation of kp and ki in the dependence on ω are
G11 (s) =
605.28ω 2 − 0.97
(43)
1431.1ω 2 + 0.94
53.35ω 2
ki =
(44)
1431.1ω 2 + 0.94
After a suitable choice of interval for ω, the stability
boundary locus as the dependence of ki on kp is plotted
and the stability region is found by the choice of testing
points in two obtained regions. Analogical procedure must
be repeated for all Kharitonov plants that can be created
for (39). The stability boundaries for Kharitonov plants
are depicted in Fig. 3 by blue lines. The stability regions
lie in the intersection of regions below the boundaries.
kp =
In the second step the pole-placement method is used for
the reasonable choice of the controller from the robust
stability region. The quality of the control response can
be prescribed by the choice of the relative damping ξ and
the natural undamped frequency ω0 in (23). The more
important parameter from our point of view is the relative
damping ξ and PI controllers were designed for ξ = 1 for
the aperiodic and ξ = 0.5 for the periodic control response.
The natural undamped frequncy ω0 was chosen in the same
interval as the frequency ω. Expressions for obtaining PI
controller parameters in dependence on prescribed ξ and
ω0 are
31.04ξω0 + 605.28ω02 − 0.97
(45)
73.39ξω0 + 1431.1ω02 + 0.94
53.35ω02
(46)
ki =
73.39ξω0 + 1431.1ω02 + 0.94
After calculation of kp and ki for chosen ξ and ω0 , the
curves representing the loci of PI controllers were plotted
(Fig. 3). The curve for ξ = 0.5 is represented by magenta
line and for ξ = 1 by green line. Then two PI controllers
where choosen as it is shown in Tab. 1 and Fig. 3 (red
stars).
kp =
Consider now the controlled system (41). The robust
stability region of PI controllers, that stabilize (41), is
shown in Fig. 4, and it is found as the intersection of
stability regions of all Kharitonov plants of (41). Then,
the pole-placement method is used and the loci of PI
Table 1. Controller parameters for the 1st
order system
C1
C2
ξ
0.5
1
kp
0.370
0.371
16
14
ki
0.0224
0.0232
12
w,y [%]
10
0.25
8
6
0.2
4
w
y − Dmin
y − Dmax
y − Dnom
0.15
k
i
2
0
0
0.1
2
4
time[min]
6
8
10
0.05
Fig. 5. Control responses for controller C1
0
−1
0
1
2
kp
3
4
5
Fig. 4. Choices of PI controllers for the 2nd order system
Table 2. Controller parameters for the 2nd
order system
C3
C4
ξ
0.5
1
kp
0.543
0.235
ki
0.0234
0.0111
controllers are found so, that the relative damping ξ and
the natural undamped frequency ω0 in (24) are chosen.
PI controllers are designed for ξ = 1 for the aperiodic
control response and ξ = 0.5 for the periodic control
response. The natural undamped frequency ω0 is chosen
in the same interval as ω. The real pole α is added to
unknown variables, and so the set of equations (31)-(33)
has unique solution. The calculated real pole α has to be
checked during computation, because it has to lie in the
left half of the complex plane. After calculation of kp and
ki for chosen ξ and ω0 , the curves representing the loci of
PI controllers are plotted (Fig. 4). The curve for ξ = 0.5
is represented by magenta line and for ξ = 1 by green line
Then two PI controllers are chosen as it is shown in Tab. 2
and Fig. 4 (red stars).
It is necessary to note that designed PI controllers robustly
stabilize only interval processes (39) and (41). Similarly,
the control responses with prescribed properties will be
obtained only for the process used in the pole-placement
design.
The possibility to use designed PI controllers for control
of time-delay processses was verified by experiments.
6. CONTROL OF A LABORATORY PROCESS
6.1 SIMATIC S7-300
Industrial control system SIMATIC S7-300 (Kožka and
Kvasnica (2001)), (Siemens (1996)) was used for implementation of designed PI controllers.
SIMATIC includes programming (STEP7) and visualization (WinCC) software, which are used for programming
of programmable logic controller (PLC). The organization
block OB35 includes a function block of a PID controller
(FB41).
6.2 Experimental results
The laboratory process (Fig. 2) was controlled by the
designed PI controllers (Tab. 1) and (Tab. 2).
Twelve various control experiments are presented. Control
responses of the process obtained using PI controllers C1
- C4 for three different values of time delay are shown
in Figs. 5, 6, 7 and 8. Here, w is the setpoint, y
the controlled output and Dmin , Dmax , Dnom are the
minimum, maximum and nominal time delays of the
controlled system, respectively.
More ”aggressive” PI controllers are controllers C1 , C2 ,
design of which was based on the Taylor aproximation
of the time delay. The controller C1 (ξ = 0.5) lead to
periodic control response for maximum and nominal time
delay, but the response for the minimum time delay was
aperiodic. The controller C2 (ξ = 1) was able to assure the
aperiodic control response for minimum and nominal time
delays. The control response for maximum time delay was
periodic.
Less ”aggressive” PI controllers are C3 , C4 , design of
which was based on Padé approximation of the time
delay. The controller C4 (ξ = 1) was able to assure
the aperiodic control response for all time delays. The
controller C3 (ξ = 0.5) lead to periodic control response
for nominal and maximum time delay, but the response
for the minimum time delay was aperiodic. The reason
is, that the pole-placement method was applied onto the
model with maximum time delay.
According to the presented results, it can be stated that all
four tested PI controllers are able to control the laboratory
time-delay process with acceptable behavior.
7. CONCLUSION
A laboratory electronic equipment was identified as a system with time delay and parametric uncertainty. For this
process, PI controllers were designed via combination of
two methods, the graphic method for robust PI controller
design for interval plants and the pole-placement method.
ACKNOWLEDGEMENTS
The authors are pleased to acknowledge the financial
support of the Scientific Grant Agency of the Slovak
Republic under grants No. 1/0537/10 and 1/0071/09 and
the Slovak Research and Development Agency under the
contract No. APVV-0029-07.
12
10
w,y [%]
8
6
REFERENCES
4
w
y − Dmin
y − Dmax
y − Dnom
2
0
0
1
2
3
4
time[min]
5
6
7
Fig. 6. Control responses for controller C2
12
w,y [%]
10
8
6
4
w
y − Dmin
y − Dmax
y − Dnom
2
0
0
1
2
3
time[min]
4
5
6
Fig. 7. Control responses for controller C3
12
10
w,y [%]
8
6
4
w
y − Dmin
y − Dmax
y − Dnom
2
0
0
1
2
3
4
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5
6
7
8
Fig. 8. Control responses for controller C4
The pole-placement method was used for the choice of
PI controllers from the robust stability region. Designed
PI controllers were implemented using the industrial control system SIMATIC and tested by experiments on the
laboratory equipment. Although designed PI controller
do not assure robust stabilizability of the original timedelay process, obtained experimental show that proposed
approach leads to the design of PI controllers that are
suitable for control of real processes with varying timedelay and parametric uncertainty.
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