Design of PI controllers for achieving time and frequency domain specifications simultaneously

ISA Transactions® Volume 45, Number 4, October 2006, pages 529–543

Design of PI controllers for achieving time and frequency domain
specifications simultaneously
Serdar Ethem Hamamci* Nusret Tan
Department of Electrical and Electronics Engineering, Engineering Faculty, Inonu University, 44280 Malatya, Turkey
͑Received 19 September 2005; accepted 20 March 2006͒

Abstract
This paper deals with the design of PI controllers which achieve the desired frequency and time domain specifications
simultaneously. A systematic method, which is effective and simple to apply, is proposed. The required values of the
frequency domain performance measures namely the gain and phase margins and the time domain performance
measures such as settling time and overshoot are defined prior to the design. Then, to meet these desired performance
values, a method which presents a graphical relation between the required performance values and the parameters of the
PI controller is given. Thus, a set of PI controllers which attain desired performances can be found using the graphical
relations. Illustrative examples are given to demonstrate the benefits of the method presented. © 2006 ISA—The
Instrumentation, Systems, and Automation Society.

Keywords: Frequency and time domain performances—map; PI control; Control system design; Gain margin; Phase margin; Overshoot;
Settling time

1. Introduction gives a high overshoot and a long settling time.
Therefore, other methods such as the refined
Ziegler-Nichols method, pole-zero cancellation
PID controllers are used extensively in the pro-
method, and ISTE performance criteria have been
cess industries because of their robust perfor-
proposed to improve the performance of control
mance and simplicity. Indeed, more than 90% of
systems which especially have a time delay. It is
the control loops are PID and most loops are in
considered in these methods directly either fre-
fact PI since derivative action is not used very
quency domain specifications such as gain margin
often ͓1͔. Several methods ͑see Refs. ͓2–6͔ and
͑GM͒ and phase margin ͑PM͒ or time domain
references therein͒ for determining the parameters
specifications like maximum overshoot ͑MO͒ and
of these controllers have been developed during
settling time ͑ts͒. However, one cannot specify all
past 60 years. Although most of these methods
provide acceptable performances for some transfer of these specifications before the design while us-
functions of the systems, there is not a general ing these methods.
method for tuning the parameters of these control- The basic approach for the design of the control-
lers. For example, the Ziegler-Nichols method ler in the modern control theory is first to obtain
which is still widely used in industries for tuning the entire set of stabilizing controllers as in the
Youla parameterization, and then to search over
the set to get a controller that meets the given
*Corresponding author. E-mail address: performance criteria. Several methods have been
shamamci@inonu.edu.tr, ntan@inonu.edu.tr recently reported on the computation of all the sta-

0019-0578/2006/$ - see front matter © 2006 ISA—The Instrumentation, Systems, and Automation Society.

530 S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543

bilizing P, PI, and PID controllers ͓7–10͔. How-
ever, beyond stabilization it is important to design
controllers that guarantee specified performance
measures such as gain and phase margins, settling
time, and overshoot. Although there are some clas-
sical formulas that exist which link frequency and
time domain performances such as ␰ Ϸ PM/ 100,
these approximate formulas do not work for time
delay systems or high order control systems. It
seems that the coefficient diagram method ͑CDM͒ Fig. 1. A standard block diagram of the CDM control
combined with the methods for the computation of system.
stabilizing controllers can be a good candidate for
designing controllers which attain the time and ts in the same plane, ͑k p , ki͒ plane, is obtained.
frequency domain specifications simultaneously. This graphical representation is called the fre-
CDM is a polynomial approach which was devel- quency and time domain performances (FTDP)
oped and introduced for a good transient response map. Thus, one can easily obtain a set of PI con-
of the control systems by Manabe ͓11͔. The most trollers which attains the desired frequency and
important properties of this method are the adap- time domain performances using the FTDP map.
tation of the polynomial representation for both The paper is organized as follows: The CDM is
the plant and controller, the use of the two-degree- revisited in Section 2. The proposed controller de-
of-freedom ͑2DOF͒ control system structure, the sign method is explained in Section 3. Simulation
nonexistence ͑or very small͒ of the overshoot in examples are given in Section 4. Section 5 in-
the step response of the closed-loop system, the cludes some concluding remarks.
determination of the desired settling time and the
maximum overshoot at the start and to continue 2. Coefficient diagram method
the design accordingly, and good robustness of the
control system with respect to the parameter The standard block diagram of the CDM control
changes. system is shown in Fig. 1, where r is the reference
In this paper, a new method is proposed for the input, y is the output, d is the disturbance, and u is
computation of a set of PI controllers which give the control signal. N p͑s͒ and D p͑s͒ are the nu-
the prescribed frequency and time domain perfor- merator and denominator polynomials of the
mance criteria such as gain margin, phase margin, transfer function of the plant and have not any
overshoot, and settling time simultaneously. In this common factors. A͑s͒ is the forward denominator
method, using the stability boundary locus ap- polynomial while B͑s͒ and F͑s͒ are the feedback
proach of Refs. ͓10,12͔, all the stabilizing values numerator and the reference numerator polynomi-
of the parameters of the PI controller are first ob- als of the controller transfer function. Since the
tained. This stability region is called the global transfer function of the CDM controller has two
stability region. In addition to this, all the stabiliz- numerators, the control system resembles to a
ing PI controllers within the global stability region 2DOF system structure. A͑s͒ and B͑s͒ are de-
which provide desired frequency domain perfor- signed as to satisfy the desired transient behavior
mance measures are identified. These subsets of and defined as
the global stability region are called the local sta- p
bility region. Then, the settling time and overshoot A ͑ s ͒ = ͚ l is i ,
which are very important time domain perfor- i=0
mance measures are chosen for time domain
q
specifications. Using the CDM, some explicit for-
mulations are obtained between the PI controller B ͑ s ͒ = ͚ k is i ͑1͒
i=0
parameters and the time domain specifications.
For this, a method is given for removing errors in the polynomial forms while prefilter F͑s͒ is de-
due to approximation used for the time delay term. termined as the zero order polynomial and used to
Finally, a graphical relation on GM, PM, MO, and provide the steady-state gain. Better performance

S. E. Hamamci and N. Tan / ISA Transactions 45, (2006) 529–543 531

Fig. 2. ͑a͒ The effect of the equivalent time constant on the rate of the closed-loop time response. ͑b͒ The effect of the stability
index, ␥1 on the time response shape.

can be expected when using a 2DOF structure, the target characteristic polynomial of the closed-
because it can focus on both tracking the desired loop system. From Eq. ͑4͒, the characteristic poly-
reference signal and disturbance rejection. Un- nomial in Eq. ͑3͒ can be expressed in terms of ␶
stable pole-zero cancellations and the use of more and ␥i as

ͫͭ ͚ ͩ ͟ ͪ ͮ ͬ
numbers of integrators are also avoided in imple-
mentations with this structure. The output of the n i−1
1
CDM control system in Fig. 1 is defined as Ptarget͑s͒ = a0 ͑␶s͒i + ␶s + 1 .
i=2 j=1 ␥i−j
j
N p͑ s ͒ F ͑ s ͒ A ͑ s ͒ N p͑ s ͒
y= r+ d, ͑2͒ ͑5͒
P͑s͒ P͑s͒
where P͑s͒ is the characteristic polynomial of the The equivalent time constant specifies the time re-
closed-loop system. This polynomial is a Hurwitz sponse speed ͑settling time, especially͒. The sta-
polynomial with real positive coefficients and de- bility indices affect the stability and the shape of
scribed by the time response ͑overshoot, especially͒. For ex-
n ample, consider a characteristic polynomial whose
P ͑ s ͒ = A ͑ s ͒ D p͑ s ͒ + B ͑ s ͒ N p͑ s ͒ = ͚ a is i, ai Ͼ 0. degree is chosen as n = 2. According to this poly-
i=0 nomial, choosing ␥1 = 3 as constants and changing
͑3͒ the equivalent time constant in the interval of ͑1–
4͒, the unit step responses of the control system
In the literature, it is reported that there are some are shown in Fig. 2͑a͒. It is seen in this figure that
certain relations between characteristic polynomial ␶ is very effective on the settling time of the unit-
coefficients and important time domain measures step response. If the equivalent time constant is
such as settling time and overshoot ͓13–15͔. increased, the settling time is also increased. On
Manabe integrated these studies with the basic the contrary, if ␶ is reduced, the system time re-
principles of his method and determined two im- sponse can be accelerated as desired. When the
portant design parameters. These parameters are choice ␶ = 1 is considered and ␥1 is changed in the
equivalent time constant ͑␶͒ and stability indices interval of ͑0.5–5͒, the change of the time re-
͑␥i͒ which are defined as sponse shapes are shown in Fig. 2͑b͒. This figure
␶ = a1/a0, ␥i = a2/͑ai+1ai−1͒ , indicates that the stability index ␥1 is much effec-
i
tive on the response shape and stability. If ␥1 is
͑4͒
made bigger, the stability of the control system is
i = 1 ϳ ͑n − 1͒, ␥0 = ␥n = ϱ.
increased and the overshoot is zero. But ␥1 is of a
It is important that these parameters are specified the small size, the stability decreases and over-
before the design and then used for determining shoot is nonzero in this case.


lim C͑s͒ = ϱ and lim C f ͑s͒/C͑s͒ = 0. ͑6͒
s→0 s→0

If C͑s͒ includes an integrator then these condi-
tions can be satisfied. Thus, C͑s͒ can be chosen as
k i k ps + k i
C͑s͒ = k p + = ͑7͒
s s
Fig. 3. A 2DOF control system structure.
in the type of the conventional PI element and
C f ͑s͒ is an appropriate element satisfying Eq. ͑6͒.
For a special case, it is recommended that stan- The proposed design approach consists of a num-
dard Manabe values for the time response without ber of distinct steps, which can be summarized as
overshoot and with the smallest settling time are follows.
used for the CDM design. Stability indices are
chosen as ␥i = ͕2.5, 2 , 2 , . . . , 2͖ for i = 1 ϳ ͑n − 1͒n ͑I͒ Frequency response performance
Ͼ 3 in this form. For a third order system ␥i ͑I a͒ Computation of global stability region
= ͕2.7, 2͖ and ␥1 = 3 in a second order system,
Consider a 2DOF control system shown in Fig.
overshoot is zero. When the standard Manabe val-
3 where
ues are chosen, the settling time is about
͑2.5– 3͒␶. The selection of the standard values can N ͑ s ͒ −␪s
be relaxed according to the various performance G a͑ s ͒ = G ͑ s ͒ e −␪s = e ͑8͒
D͑s͒
requirements.
and a PI controller of the form of Eq. ͑7͒. The
problem is to compute the global stability region
3. Controller design which includes all the parameters of the PI con-
troller of Eq. ͑7͒ which stabilize the given system.
The proposed method which is used to design a The closed-loop characteristic polynomial P͑s͒ of
PI controller satisfying the required time and fre- the system, i.e., the numerator of 1 + C͑s͒Ga͑s͒,
quency domain specifications consists of two can be written as
steps. In the first step, a method is proposed to P͑s͒ = sD͑s͒ + ͑k ps + ki͒N͑s͒e−␪s
compute the global and the local stability regions
using the stability boundary locus approach = ansn + an−1sn−1 + ¯ + a1s + a0 , ͑9͒
͓10,12͔. In the second step, the CDM method is
where all or some of the coefficients ai, i
used to design PI controllers for which the step
= 0 , 1 , 2 , . . . , n are the function of k p, ki, and e−␪s
responses have a required overshoot and an ac-
depending on the order of N͑s͒, and D͑s͒ polyno-
ceptable settling time. As a result of combining
mials. In the parameter space approach, there are
these two steps, the FTDP map is obtained. The
three possibilities for a root of a stable polynomial
FTDP map, which is a graphical tool, shows the
to cross over the imaginary axes ͑to become un-
relation between the stabilizing parameters of the
stable͒:
PI controllers and the chosen frequency and time
domain performance criteria on the same ͑k p , ki͒ ͑a͒ Real Root Boundary: A real root crosses
plane. Thus, one can choose a PI controller pro- over the imaginary axis at s = 0. Thus, the
viding all of the desired GM, PM, MO, and ts real root boundary can be obtained from
specification values together. substituting s = 0 in P͑s͒ of Eq. ͑9͒ which
A general schema of the 2DOF control system is gives a0 = 0.
shown in Fig. 3. This representation is the rational ͑b͒ Infinite Root Boundary: A real root crosses
equivalence of the control system given in Fig. 1. over the imaginary axis at s = ϱ. Thus, the
Here, C͑s͒ = B͑s͒ / A͑s͒ is the main controller and infinite root boundary can be characterized
C f ͑s͒ = F͑s͒ / B͑s͒ is the set point filter ͓16͔. It can by taking an = 0 from Eq. ͑9͒.
be shown that the steady-state error to the unit step ͑c͒ Complex Root Boundary: Eq. ͑9͒ becomes
change and the unit step disturbance become zero unstable at s = j ␻ when its roots cross the
robustly if imaginary axis which means that the real


and the imaginary parts of Eq. ͑9͒ become k p͓␻Ne cos͑␻␪͒ + ␻2No sin͑␻␪͔͒
zero simultaneously. Thus, the complex
root boundary can be obtained as follows: + ki͓␻No cos͑␻␪͒ − Ne sin͑␻␪͔͒ = − ␻De .
͑12b͒
Decomposing the numerator and the denomina-
tor polynomials of G͑s͒ in Eq. ͑8͒ into their even Let
and odd parts, and substituting s = j ␻, gives Q͑␻͒ = ␻Ne sin͑␻␪͒ − ␻2No cos͑␻␪͒ ,
N e͑ − ␻ 2͒ + j ␻ N o͑ − ␻ 2͒ R͑␻͒ = Ne cos͑␻␪͒ + ␻No sin͑␻␪͒ , ͑13a͒
G͑ j ␻͒ = . ͑10͒
D e͑ − ␻ 2͒ + j ␻ D o͑ − ␻ 2͒
X ͑ ␻ ͒ = ␻ 2D o ,
For simplicity ͑−␻2͒ will be dropped in the fol-
S͑␻͒ = ␻Ne cos͑␻␪͒ + ␻2No sin͑␻␪͒ ,
lowing equations. Thus, the closed-loop character-
istic polynomial of Eq. ͑9͒ can be written as ͑13b͒
U͑␻͒ = ␻No cos͑␻␪͒ − Ne sin͑␻␪͒, Y ͑␻͒ =
P͑ j␻͒ = ͓͑kiNe − k p␻2No͒cos͑␻␪͒ − ␻De .
+ ␻͑kiNo + k pNe͒sin͑␻␪͒ − ␻ Do͔ 2
Then, Eqs. ͑12a͒ and ͑12b͒ can be written as
+ j͓␻͑kiNo + k pNe͒cos͑␻␪͒ − ͑kiNe k pQ ͑ ␻ ͒ + k iR ͑ ␻ ͒ = X ͑ ␻ ͒ ,

− ␻2k pNo͒sin͑␻␪͒ + ␻De͔ = R P + jI P = 0. k pS ͑ ␻ ͒ + k iU ͑ ␻ ͒ = Y ͑ ␻ ͒ . ͑14͒
͑11͒ From these equations
X͑␻͒U͑␻͒ − Y ͑␻͒R͑␻͒
Then, equating the real and imaginary parts of kp = ,
P͑j␻͒ to zero, two equation are obtained as Q͑␻͒U͑␻͒ − R͑␻͒S͑␻͒

Y ͑␻͒Q͑␻͒ − X͑␻͒S͑␻͒
k p͓− ␻2No cos͑␻␪͒ + ␻Ne sin͑␻␪͔͒ ki = . ͑15͒
Q͑␻͒U͑␻͒ − R͑␻͒S͑␻͒
+ ki͓Ne cos͑␻␪͒ + ␻No sin͑␻␪͔͒ = ␻ Do , 2
Substituting Eqs. ͑13a͒ and ͑13b͒ into Eq. ͑15͒, the
͑12a͒ PI controller parameters are obtained as

͑␻2NoDo + NeDe͒cos͑␻␪͒ + ␻͑NoDe − NeDo͒sin͑␻␪͒
kp = , ͑16͒
− ͑ N 2 + ␻ 2N 2͒
e o

␻2͑NoDe − NeDo͒cos͑␻␪͒ − ␻͑NeDe + ␻2NoDo͒sin͑␻␪͒
ki = . ͑17͒
− ͑ N 2 + ␻ 2N 2͒
e o

The stability boundary locus, l͑k p , ki , ␻͒ can be the stability boundary locus has been obtained
constructed in the ͑k p , ki͒ plane using Eqs. ͑16͒ then it is necessary to test whether stabilizing con-
and ͑17͒. If at any particular frequency value the trollers exist or not since the stability boundary
denominator of Eqs. ͑16͒ and ͑17͒ Ne + ␻2N2 = 0,o
locus, l͑k p , ki , ␻͒, the real root and inﬁnite root
then this value of frequency must not be used. In boundary lines ͑if they are exist͒ may divide the
this case, a discontinuous stability boundary locus parameter plane ͓͑k p , ki͒ plane͔ into stable and un-
will be obtained and this will not be a problem for stable regions. It can be found that the line ki = 0 is
the computation of stabilizing controllers. Once the real root boundary line obtained from substi-


tuting ␻ = 0 into Eq. ͑9͒ and equating it to zero ␻ ͑ N oD e − N eD o͒
since a real root of P͑s͒ of Eq. ͑9͒ can crossover tan͑␻␪͒ = = f ͑␻͒ . ͑19͒
N eD e + ␻ 2N oD o
the imaginary axis at s = 0. Generally, the order of
D͑s͒ is greater than order of N͑s͒ for proper trans- Thus, ␻c is the solution of Eq. ͑19͒ in the interval
fer function. Therefore, there will be no infinite ͑0 , ␲͒. By plotting tan͑␻␪͒ and f͑␻͒ vs ␻, it can
root boundary line. be seen that ␻c is the smallest value of ␻ at which
It can be seen that the stability boundary locus is plots of tan͑␻␪͒ and f͑␻͒ intersect with each
dependent on the frequency ␻ which varies from 0
other. If there is more than one real value of ␻
to ϱ. However, one can consider the frequency
which satisfies Eq. ͑19͒ then the frequency axis
below the critical frequency, ␻c, or the ultimate
can be divided into a finite number of intervals
frequency since the controller operates in this fre-
and by testing each interval the stability region can
quency range. Thus, the critical frequency can be
be computed.
used to obtain the stability boundary locus over a
possible smaller range of frequency such as ␻ ͑I b͒ Computation of local stability regions
෈ ͓0 , ␻c͔. Since the phase of G p͑s͒ at s = j␻c is
equal to −180°, one can write Phase and gain margins are two important fre-

ͩ ͪ ͩ ͪ
quency domain performance measures which are
␻No ␻Do widely used in the classical control theory for the
tan−1 − tan−1 − ␻␪ = − ␲
Ne De controller design. Consider Fig. 3 with a gain-
phase margin tester ͓17͔, Gc͑s͒ = Ae−j␾, which is
͑18͒
connected in the feed forward path. From Eqs.
or ͑16͒ and ͑17͒,

͑␻2NoDo + NeDe͒cos͑h͒ + ␻͑NoDe − NeDo͒sin͑h͒
kp = , ͑20͒
− A ͑ N 2 + ␻ 2N 2͒
e o

␻2͑NoDe − NeDo͒cos͑h͒ − ␻͑NeDe + ␻2NoDo͒sin͑h͒
ki = , ͑21͒
− A ͑ N 2 + ␻ 2N 2͒
e o

where h = ␻␪ + ␾. Here, the gain-phase margin the industry can be mostly described by a FOPDT
tester is not an actual part of the system which is model such as
included in the system in order to obtain the sta-
bility regions for prespecified values of the gain N m͑ s ͒ −␪s K −␪s
G m͑ s ͒ = e = e , ͑22͒
and phase margins. To obtain the stability bound- D m͑ s ͒ Ts + 1
ary locus for a given value of gain margin A, one
needs to set ␾ = 0 in Eqs. ͑20͒ and ͑21͒. On the where K is the gain, T is the time constant, and ␪
other hand, setting A = 1 in Eqs. ͑20͒ and ͑21͒, one is the dead time. The experimental identification
can obtain the stability boundary locus for a given of these models using many techniques is well de-
phase margin ␾. Thus, the local stability regions scribed in Ref. ͓4͔. It is necessary to use the
for specified gain and phase margins can be iden- FOPDT model for the proposed method, since the
tified within the global stability region. CDM requires high order controllers for high or-
͑II͒ Time response performance der models. The term e−␪s which also represents
The CDM technique is used to obtain time domain the time delay in Eq. ͑22͒ is approximated as
performances. However, in this case, it is neces- e−␪s Ϸ 1 − ␪s using the first-order Taylor numerator
sary to use the first-order plus dead time ͑FOPDT͒ approximation. The Taylor numerator approxima-
model of the plant. The processes encountered in tion is used in order to not affect the denominator


Fig. 4. Time delay approximation for corrected and uncorrected cases: ͑a͒ selection of k p and ki parameters and ͑b͒ the step
responses for selected k p and ki values.

of Eq. ͑22͒. If the Taylor denominator approxima- by Eq. ͑24͒ are replaced in Eq. ͑3͒. Hence, a poly-
tion or Pade approximation is used, theirs denomi- nomial depending on the parameters k p and ki is
nators lead to a higher order of the approximate obtained. For the FOPDT model transfer function,
transfer function of the plant and consequently to the characteristic polynomial of the control system
more complex resulting controllers. The results is determined as
obtained in Section 4 show that the first-order Tay-
lor numerator approximation is acceptable and P͑s͒ = ͑T − K␪k p͒s2 + ͑1 − ␪Kki + Kk p͒s + Kki .
gives good results. Thus, for the model transfer
function of Eq. ͑22͒, plant polynomials are ͑25͒
N p͑s͒ = − K␪s + K, D p͑s͒ = Ts + 1. ͑23͒
Using Eq. ͑4͒, ␥1 and ␶ can be obtained as
For a selection of good controllers, the degrees of
the controller polynomials in Eq. ͑1͒ get impor- ␥1 = ͑1 + Kk p − K␪ki͒2/Kki͑T − K␪k p͒ ,
tance. The most important fact that affects the de-
grees is the existence of a disturbing signal and its ͑26a͒
type. It is advised that the minimum degree poly-
nomials are chosen depending on the type of the
disturbance. In this paper, the controller polynomi- ␶ = ͑1 + Kk p − K␪ki͒/Kki . ͑26b͒
als are chosen for the step disturbance signal. In
this case, the controller polynomials have the For Eqs. ͑26a͒ and ͑26b͒, ki can be found for ␥1
forms and ␶ separately as

A͑s͒ = l1s, B ͑ s ͒ = k 1s + k 0 . ͑24͒
Table 1
Thus, the configuration in Fig. 3 is transformed to ␥1 and k values for some overshoot values.
a PI-based control system. From Eqs. ͑7͒ and ͑24͒, MO
it can be seen that k1 = k p, k0 = ki and l1 = 1. ͑%͒ ␥1 values k values ͑ts Х k␶͒
͑II a͒ Computation of the PI controller parameters
for the time domain performance. 20 0.8 11.2
In the design, a feedback controller is chosen by 10 1.4 6
the pole-placement technique and then, a set point 5 1.9 5.2
filter is determined so as to match the steady-state 0a 3a 3a
0 5 3.9
gain of the closed-loop system. According to this,
a
the controller polynomials which are determined Standard Manabe values.


Fig. 5. For K = 1, T = 1, and ␪ = 2 in Eq. ͑22͒: ͑a͒ ki-k p curves of ␥1 and ␶, ͑b͒ the step responses for ␶ = 4 and different values
of ␥1 selected from Table 1.

ki = ͓K͑2␪ + 2K␪k p + ␥1T − ␥1K␪k p͒ may not be exactly obtained since an approxima-
tion for time delay has been used. Therefore, a
− ͱ⌬͔/͑2K2␪2͒ for ␥1 , ͑27a͒ correction process should be implemented. To do
this, a second order Taylor numerator approxima-
where ⌬ = K2͓͑2␪ + 2K␪k p + ␥1T − ␥1K␪k p͒2
tion, e−␪s Ϸ 1 − ␪s + 0.5␪2s2 is used for the same
− 4␪2͑1 + Kk p͒2͔ and
controller of Eq. ͑24͒. Repeating the same math-
ki = ͑1 + Kk p͒/͑␶K + K␪͒ for ␶ . ͑27b͒ ematical derivations for ␶ and ␥1, it can be seen
that Eq. ͑26b͒ remains same but Eq. ͑26a͒ is
From the global stability region obtained in step changed to the form
I a, an interval for k p can be found as ͑k pmin
-k pmax͒. Using this interval and Eqs. ͑27a͒ and ␥1 = ͑1 + Kk p − K␪ki͒2/Kki͑T − K␪k p + 0.5K␪2ki͒ .
͑27b͒, ki-k p curves can be plotted. The values of k p ͑28͒
and ki parameters at the intersection of two curves
provide desired ␶ vs ␥1 values. At this point one Using the higher order approximations for correc-
should be careful that desired ␶-␥1 speciﬁcations tion does not effect the relations for ␶ and ␥1. ki

Fig. 6. For K = 1, T = 1, and ␪ = 2 in Eq. ͑22͒: ͑a͒ ki-k p curves of ␥1 and ␶, ͑b͒ the step responses for ␥1 = 3 and different values
of ␶.


uncorrected cases are shown in Fig. 4͑a͒. From
this figure, it can be seen that ͑ki, k p͒ values which
give the desired specifications are different. The
step responses for PI controllers obtained from
corrected and uncorrected ki-k p curves are shown
in Fig. 4͑b͒ which indicate that the step response
for the uncorrected case have a 3.2% overshoot
and settling time of 18.2 s, however, the step re-
sponse for the corrected case has no overshoot and
a settling time of 11.9 s. These values of the set-
tling time have been computed using a tolerance
band of 2% for the steady-state value ͓18͔. Thus,
the error due to the time delay approximation can
be removed by this way.
The reference numerator polynomial F͑s͒ which is
Fig. 7. The global stability region including all of the sta-
defined as the prefilter element is chosen to be
bilizing PI parameters for the given plant in Eq. ͑32͒.
F͑s͒ = ͉ P͑s͒/Nm͑s͉͒s=0 = ki . ͑30͒
values for this case can be obtained as
This way, the value of the error that may occur in
ki = ͓K͑2␪ + 2K␪k p + ␥1T − ␥1K␪k p͒ the steady-state response of the closed-loop sys-
tem is reduced to zero. Finally, the set point filter
− ͱ⌬͔/͓K2␪2͑2 − ␥1͔͒ for ␥1 , ͑29a͒ in Fig. 3 is obtained by
where ⌬ = K2͓͑2␪ + 2K␪k p + ␥1T − ␥1K␪k p͒2 F͑s͒ ki
+ 2␪ ͑2 − ␥1͒͑1 + Kk p͒2͔ and
2 C f ͑s͒ = = . ͑31͒
B ͑ s ͒ k ps + k i
ki = ͑1 + Kk p͒/͑␶K + K␪͒ for ␶ . ͑29b͒
Note that the parameters of C f ͑s͒ depend on the PI
Thus, the PI controller parameters which nearly parameters directly. Therefore, the designer does
give the desired ␶ and ␥1 values can be obtained. not need the extra calculation for the set point fil-
For example, let us choose K = 1, T = 1, and ␪ = 2 ter.
in Eq. ͑22͒ and set ␥1 = 3 and ␶ = 4 for a step re- ͑II b͒ Choice of the key parameter values for the
sponse which has no overshoot and approximately desired time domain properties.
12 s settling time. ki-k p curves for corrected and One of the most important properties of the pro-

Fig. 8. Local stability regions: ͑a͒ for some GM values and ͑b͒ for some PM values.


Fig. 9. ͑a͒ ki-k p curves for different ␥1 values in Table 1. ͑b͒ The step responses of these ␥1 values.

posed method is that the settling time and over- different values of ␶ are shown in Fig. 6 which
shoot information of the control system are deter- clearly shows that there is no overshoot and the
mined at the beginning before starting to design. It settling time values determined in Table 1 are also
is generally aimed that the time response of the met.
control system must have no overshoot and de- ͑III͒ Building of the FTDP map
sired settling time. Table 1 shows the relation be- The FTDP map is built by plotting the frequency
tween ts and ␶ for the desired settling time for the domain stability regions and the time domain per-
various overshoot properties. These values of the formance curves in the same ͑k p, ki͒ plane. The
general relation are obtained by normalizing the frequency domain stability regions include the
characteristic polynomial of Eq. ͑5͒ for n = 2 and global and local stability regions which can be ob-
␶ = 1. For a transfer function with K = 1, T = 1, and tained using Eqs. ͑16͒, ͑17͒, ͑20͒, and ͑21͒, and the
␪ = 2 in Eq. ͑22͒, the step responses for ␶ = 4 and time domain performance curves which corre-
different values of ␥1 selected from Table 1 are spond to the various overshoot and settling time
shown in Fig. 5 where it can be seen that the maxi- values can be plotted using Eqs. ͑29a͒ and ͑29b͒.
mum overshoot values given in Table 1 are exactly Thus, the designer can choose any points, ͑k p, ki͒
met. Similarly, the step responses for ␥1 = 3 and values, from the FTDP map which satisfy the de-

Fig. 10. ͑a͒ Four different PI controllers in the GMϾ 2 region and on the ␥1 = 3 curve. ͑b͒ The unit step responses without
overshoot for the selected PI controllers.


Table 2
The time domain and frequency domain specifications cor-
respond to selected four points.

1 2 3 4

kp 3.0941 2.5002 2.0601 1.7201
ki 1.1573 0.8216 0.6142 0.4768
MO ͑%͒ 0 0 0 0
ts ͑s͒ 6.2 8.8 10.8 12.7
GM 2.3 2.95 3.6 4.4
PM ͑°͒ 38.8 47.7 54.6 60

͑2͒ finding ki = g͑k p , ␶͒ curves which have dif-
Fig. 11. A local stability region with 2.3Ͻ GMϽ 5 and ferent values of settling time using Eq.
30° Ͻ PMϽ 60° ͑filled in gray color͒ and ␥1 = 3 curve ex- ͑29b͒.
pressed no overshoot property.

• Step 3. Plotting the frequency domain perfor-
sired frequency and time domain properties. After
mance regions together with the time domain
building the FTDP map and choosing the control-
performance curves, thus, obtaining the FTDP
ler parameters, the control system shown in Fig. 3
map and finding the desired PI controllers from
is simulated using the actual process and the de-
this map.
signed PI controller. However, for some transfer
functions there are not any ͑k p, ki͒ values which
give the desired performances because of the hard
properties of the plant. For such situations, it is
4. Simulation examples
necessary to relax the desired frequency and time
domain specifications. 4.1. Example 1
In view of the earlier developments, the FTDP-
A first order process with a time delay is chosen
map design procedure linking the frequency and
as
time domain performances is summarized as fol-
lows for the ease of reference: 1
G a͑ s ͒ = e−s , ͑32͒
5s + 1
• Step 1. Computation of the frequency response
performance regions in the ͑k p, ki͒-plane for which has a pole near the imaginary axis. The aim
Eq. ͑8͒: is to investigate the PI controller parameters which
have a satisfying frequency and time domain
͑1͒ construction of the global stability region specifications such as gain margin, phase margin,
using Eqs. ͑16͒ and ͑17͒, the stability overshoot, and settling time. For this, it will be
boundary locus l͑k p , ki , ␻͒ and the real root made use of the FTDP map explained in Section 3.
boundary line ki = 0; The global stability region which is shown in
͑2͒ obtaining the local stability regions corre- Fig. 7 is computed using the procedure given in
spond to desired GM and PM values using Section 3I a. The global stability region includes
Eqs. ͑20͒ and ͑21͒. all PI controllers which stabilize the given system.
However, one needs to choose suitable k p and ki
values within this region to obtain the required
• Step 2. Computation of the time performance
frequency and time domain performances. The lo-
curves in the global stability region using
cal stability regions for the specified gain and
FOPTD model of actual plant given in Eq. ͑8͒:
phase margins can be identified within the global
͑1͒ obtaining ki = f͑k p , ␥1͒ curves which have stability region using the procedure given in Sec-
different values of overshoot using Eq. tion 3I b as shown in Figs. 8͑a͒ and 8͑b͒. If the
͑29a͒; local regions for the GM and PM values are com-


Fig. 12. ͑a͒ Global and desired local stability regions. ͑b͒ Time domain curves related to the overshoot and settling time.

bined, the intersection local regions can be ob- in Fig. 10͑a͒ give unit step responses without an
tained satisfying both the GM and PM together. overshoot for ␥1 = 3 as shown in Fig. 10͑b͒.
Now, it is important that how the PI parameters Eq. ͑29b͒ can be used to obtain the desired set-
which provide the desired frequency domain per- tling time. The four intersection points in Fig.
formances in the local region will exhibit time do- 10͑a͒ express four different settling time values
main performances. Using Eq. ͑29a͒ to get the for ␥1 = 3. The intersection points of ␥1 and ␶
overshoot information of the control system for curves give important information: ␥1 represents
different ␥1 values in Table 1, Fig. 9͑a͒ is ob- the overshoot, and the settling time obtained from
tained. According to this ␥1 values, the step re- ts = k␶. Thus, the values of k p and ki correspond to
sponses are shown in Fig. 9͑b͒ for k p = 1 and cor- the intersection points that give the values of the
responding ki values which are computed from the overshoot and the settling time. The step responses
␥1 curves. From the step responses, one can see for these points are shown in Fig. 10͑b͒. It can be
that the values of the resultant overshoots are seen that the settling time increases when moving
equal to the results given in Table 1. For example, from point 1 to point 4 as shown in Fig. 10͑b͒. It
for ␥1 = 1.4, the expected value of the overshoot has been observed that the values of the settling
from Table 1 is about 10% and this value is ex- time and overshoot given in Table 1 are met.
actly obtained as shown in Fig. 9͑b͒. Similarly, for The local region with properties of 2.3Ͻ GM
␥1 = 3 a response without an overshoot is expected Ͻ 5 and 30° Ͻ PMϽ 60° is shown in Fig. 11.
and it can be seen that which is also met.
It is vital to point out that all the values of k p
and ki obtained over any ␥1 curve do not neces-
sarily give a response having its overshoot prop-
erty given in Table 1. For satisfactory perfor-
mance, the gain margin should be greater than 2,
and the phase margin should be between 30° and
60° ͓18͔. However, the simulation results showed
that especially the gain margin is very effective on
the overshoot property. The part of any ␥1 curves
within the stability region for GMϾ 2 gives k p
and ki values for which the step responses have the
overshoot values given in Table 1. Note that GM
= 2 is a boundary value and causes a time response
with a little overshoot. For example, the PI con-
trollers which correspond to points 1, 2, 3, and 4 Fig. 13. FTDP map for example 2.


Fig. 14. Set point and disturbance responses for ͑a͒ points 1, ͑b͒ 2 and ͑c͒ 3.

Four points are selected from this ﬁgure within the neously. However, for some transfer functions
shaded region. The frequency and time domain there is not any ͑k p , ki͒ values which give desired
properties for each point are given in Table 2. performances. For example, there is not any
Thus, it can be seen that all PI controllers for ͑k p , ki͒ value for the example studied earlier which
which the gain margin is between 2.3 and 4.4, the simultaneously provide the gain margin to be 5
phase margin is between 38.3° and 60°, the set- and a response without the overshoot, because this
tling time is between 6.2 and 12.7 s, and with no property is out of the desired region. For such situ-
overshoot is between the points of 1–4 on ␥1 = 3 ations, it is necessary to relax the desired fre-
curve within the local region. quency and time domain speciﬁcations.
From the simulation studies using the FTDP-
4.2. Example 2
map method, it has been observed that there are
many ͑k p , ki͒ values which satisfy the proper time Consider a higher order process with a large
and frequency domain performances simulta- time delay as

1
G a͑ s ͒ = e−4s . ͑33͒
͑s + 1͒͑0.5s + 1͒͑0.25s + 1͒͑0.125s + 1͒

The aim is to design PI controllers which make the local stability regions are obtained as shown in
gain margin of the control system greater than 2.3 Fig. 12͑a͒. For time domain performances it is
and the phase margin between 30° and 60°. Using necessary to use the the FOPDT model which is
Eqs. ͑16͒, ͑17͒, ͑20͒, and ͑21͒ the global and the


Table 3 ͑iii͒ The FTDP map gives a visual chance to
The time domain and frequency domain specifications cor- see which parts of the global stability re-
respond to the selected three points. gion is important for the required design
1 2 3 specifications.
͑iv͒ A set of PI controllers which provide pre-
kp 0.2463 0.1295 0.1034 defined performance specifications can be
ki 0.1317 0.1194 0.1055 computed from FTDP-map.
MO ͑%͒ 5 10 5
ts ͑s͒ 25.2 29.3 31 The given simulation examples clearly show
GM 2.6 2.8 3.1 that the results presented are useful for the design
PM ͑°͒ 57 58.5 60 of PI controllers.
It is known that the mathematical techniques for
obtaining the exact correlation between the step
given in Ref. ͓19͔ as Gm͑s͒ = e−4.462s / ͑1.521s + 1͒. transient response and frequency response are
␥1 and ␶ curves which are directly related to the available but they are very laborious and of little
time domain performances as explained earlier can practical value ͓18͔. However, the method pre-
be seen from Fig. 12͑b͒. Combining Figs. 12͑a͒ sented in this paper has a potential to show a
and 12͑b͒, the FTDP map shown in Fig. 13 is ob- graphical relation between the frequency and time
tained. From this figure, it is clear that it is not domain performances of high order or time delay
possible to obtain a step response without an over- systems. Therefore, the results obtained can be ap-
shoot since the ␥1 = 3 curve does not pass through plied for many real applications. The extension of
the shaded region. Looking at Fig. 13, it is seen the method to the PID controller will be very im-
that the shaded region is approximately bounded portant in the control system design.
by ␥1 = 1.9 and ␥1 = 0.8 which give a 5% and 20%
overshoot as stated in Table 1, respectively. There-
fore, all PI controllers in the shaded region give References
the desired time and frequency domain perfor-
mances. Three points are selected within this re- ͓1͔ Åström, K. J. and Hägglund, T., The future of PID
control. Control Eng. Pract. 9, 1163–1175 ͑2001͒.
gion which are indicated by 1, 2, and 3 as shown ͓2͔ Ziegler, J. G. and Nichols, N. B., Optimum settings for
in Fig. 13. Figs. 14͑a͒–14͑c͒ show the time re- automatic controllers. Trans. ASME 64, 759–768
sponses of the selected three points according to ͑1942͒.
the set point changes applied at t = 0 s and the dis- ͓3͔ ÅströmK. J., Hägglund, T., Hang, C. C., and Ho, W.
K., Automatic tuning and adaptation for PID
turbance applied at t = 70 s. The time and fre-
controllers—A survey. Control Eng. Pract. 1, 699–
quency domain specifications of these points are 714 ͑1993͒.
shown in Table 3. ͓4͔ Åström, K. J. and Hägglund, T., PID Controllers:
Theory, Design, and Tuning. Instrument Society of
5. Conclusions America, Research Triangle Park, NC, 1995.
͓5͔ Zhuang, M. and Atherton, D. P., Automatic tuning of
optimum PID controllers. IEE Proc.-D: Control
A new method which is simple to use has been Theory Appl. 140, 216–224 ͑1993͒.
presented to design PI controllers giving the de- ͓6͔ Ho, W. K., Hang, C. C., and Cao, L. S., Tuning of PID
sired frequency and time domain performances. controllers based on gain and phase margins specifica-
One of the most important features of the pro- tions. Automatica 31, 497–502 ͑1995͒.
posed method is that the desired frequency and ͓7͔ Datta, A., Ho, M. T., and Bhattacharyya, S. P., Struc-
ture and Synthesis of PID Controllers. Springer, New
time domain specifications can be specified before York, 2000.
the design. It has been shown that: ͓8͔ Ackermann, J. and Kaesbauer, D., Design of robust
PID controllers. Proceedings of the 2001 European
͑i͒ The global and the local stability regions Control Conference, 2001, pp. 522–527.
can be computed using the stability ͓9͔ Söylemez, M. T., Munro, N., and Baki, H., Fast cal-
boundary locus approach. culation of stabilizing PID controllers. Automatica 39,
͑ii͒ The values of the parameters of PI con- 121–126 ͑2003͒.
͓10͔ Tan, N., Kaya, I., and Atherton, D. P., Computation of
troller which achieve desired values of stabilizing PI and PID controllers. Proceedings of the
settling time and overshoot can be esti- CCA2003 IEEE Conf. on the Contr. Appl., 2003.
mated using the CDM. ͓11͔ Manabe, S., Coefficient Diagram Method. Proceed-


ings of the 14th IFAC Symposium on Automatic Con- Serdar Ethem Hamamci re-
trol in Aerospace, Seoul, 1998. ceived B.Sc. and M.Sc. degrees
from Erciyes University, Kayseri,
͓12͔ Tan, N., Computation of stabilizing PI and PID con- Turkey, in 1992 and Firat Univer-
trollers for processes with time delay. ISA Trans. 44, sity, Elazig, Turkey, in 1997, re-
213–223 ͑2005͒. spectively. He obtained a Ph.D.
͓13͔ Naslin, P., Essentials of Optimal Control. Boston degree from Firat University, in
2002. He is currently working as
Technical, Cambridge, 1969. a research assistant in the depart-
͓14͔ Lipatov, A. and Sokolov, N., Some sufﬁcient condi- ment of electrical and electronics
tions for stability and instability of continuous linear engineering at Inonu University,
stationary systems. Autom. Remote Control ͑Engl. Malatya, Turkey. His primary area
Transl.͒ 39, 1285–1291 ͑1979͒. of research is polynomial control
methods ͑especially the coefﬁ-
͓15͔ Kim, Y. C., Keel, L. H., and Bhattacharyya, S. P., cient diagram method͒ and their applications.
Transient response control via characteristic ratio as-
signment. IEEE Trans. Autom. Control 48͑12͒, 2238–
2244 ͑2003͒. Nusret Tan was born in Malatya,
͓16͔ Chen, C. T., Analog and Digital Control System De- Turkey, in 1971. He received a
B.Sc. degree in electrical and
sign: Transfer Function, State-Space and Algebraic
electronics engineering from Hac-
Methods. Saunders College, New York, 1992. ettepe University, Ankara, Turkey,
͓17͔ Argoun, M. B. and Bayoumi, M. M., Robust gain and in 1994. He received a Ph.D. de-
phase margins for interval uncertain systems. Proceed- gree in control engineering from
ings of the 1993 Canadian Conference on Electrical University of Sussex, Brighton,
UK, in 2000. He is currently
and Computer Engineering 1993, pp. 73–78. working as an associate professor
͓18͔ Ogata, K., Modern Control Engineering. Prentice- in the department of electrical and
Hall, Englewood Cliffs, NJ, 1970. electronics engineering at Inonu
͓19͔ Mann, G. K. I., Hu, B.-G., and Gosine, R. G., Time- University, Malatya, Turkey. His
primary research interest lies in
domain based design and analysis of new PID tuning the area of systems and control.
rules. IEE Proc.: Control Theory Appl. 148͑3͒, 251–
261 ͑2001͒.

Design of PI controllers for achieving time and frequency domain specifications simultaneously

Recomendados

Recomendados

Más contenido relacionado

La actualidad más candente

La actualidad más candente (20)

Destacado

Destacado (8)

Similar a Design of PI controllers for achieving time and frequency domain specifications simultaneously

Similar a Design of PI controllers for achieving time and frequency domain specifications simultaneously (20)

Más de ISA Interchange

Más de ISA Interchange (20)

Último

Último (20)

Design of PI controllers for achieving time and frequency domain specifications simultaneously