ISA Transactions 50 (2011) 268–276

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ISA Transactions
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An analytical method for PID controller tuning with specified gain and phase
margins for integral plus time delay processes
Wuhua Hu a , Gaoxi Xiao a,∗ , Xiumin Li b
a
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore
b
Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong

article info abstract

Article history: In this paper, an analytical method is proposed for proportional–integral/proportional–derivative/
Received 3 May 2010 proportional–integral–derivative (PI/PD/PID) controller tuning with specified gain and phase margins
Received in revised form (GPMs) for integral plus time delay (IPTD) processes. Explicit formulas are also obtained for estimating
3 December 2010
the GPMs resulting from given PI/PD/PID controllers. The proposed method indicates a general form of the
Accepted 3 January 2011
Available online 1 February 2011
PID parameters and unifies a large number of existing rules as PI/PD/PID controller tuning with various
GPM specifications. The GPMs realized by existing PID tuning rules are computed and documented as a
Keywords:
reference for control engineers to tune the PID controllers.
IPTD processes © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
PI/PD/PID controller tuning
Gain margin
Phase margin
Crossover frequencies

1. Introduction exploration of a general solution for the PI/PD/PID controller tuning

on an IPTD process in this paper.
Proportional–integral–derivative (PID) control has been widely Tuning PI/PD/PID controllers based on gain and phase margin
applied in industry—more than 90% of the applied controllers (GPM) specifications has been extensively studied in the literature
are PID controllers [1–4]. In the absence of the derivative [1,14,20–23]. However, general analytic solutions of the controller
action, proportional–integral (PI) control is also broadly deployed, parameters are not available, because of the nonlinearity and solv-
since in many cases the derivative action cannot significantly ability of such problems. The existing solutions are limited by as-
enhance the performance or may not be appropriate for the suming certain constraints on GPM or by approximations that are
noisy environment [1–4]. Another special form of PID control valid only for certain regions of process parameters [1,14,20–22].
without the integral action, proportional–derivative (PD) control This paper is devoted to solving the PI/PD/PID parameters for an
is also applied [1–4]. Unlike the previous two cases, however, IPTD process with a specified GPM. Different from the existing re-
PD control cannot achieve zero steady-state error subject to load sults, analytic solutions are obtained for the whole domain of the
disturbances, which limits its applications [1–4]. process parameters. The derived PI/PD/PID tuning formulas unify a
large number of existing rules as PI/PD/PID controller tuning with
Due to the prevailing applications of PI/PD/PID control, research
various GPM specifications. As reverse solutions, explicit expres-
on tuning PI/PD/PID controllers has been of much interest in
sions of GPMs for given PI/PD/PID settings on an IPTD process are
the past decades [1–6]. As a particular case, tuning PI/PD/PID
also obtained. These GPM formulas estimate GPMs with high accu-
controllers for integral plus time delay (IPTD) processes has
racy and are applied to estimate the GPM attained by each relevant
attracted a lot of attention, dating back to 1940s and lasting even
PI/PD/PID tuning rule collected in [1].
today [1,7–18]. Lots of results have been accumulated. There are
The rest of the paper is organized as follows. In Section 2, an-
more than fifty PI/PD/PID tuning rules for IPTD processes according
alytic expressions of the PI/PD/PID parameters with a specified
to a survey made by O’Dwyer [1]. The actual number is even much GPM and the reverse solution of GPM for a given PI/PD/PID setting
higher [7–9,18,19]. Close observations reveal that many of these are derived. During the derivations, numerical evaluations are em-
rules are sharing a common form. Such observations motivate our ployed to validate any approximations involved. In Section 3, the
derived PI/PD/PID formulas are applied to unify the existing rules
as PI/PD/PID controller tuning with different GPM specifications,
∗ Corresponding author. Tel.: +65 6790 4552; fax: +65 6793 3318. and the derived GPM formulas are applied to estimate the GPMs at-
E-mail address: egxxiao@ntu.edu.sg (G. Xiao). tained by the existing rules. Finally, Section 4 concludes the paper.

0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.isatra.2011.01.001
 W. Hu et al. / ISA Transactions 50 (2011) 268–276 269

where (α, β) is solved from

 α
 φm = arctan α − arctan β,
β



(8)

A = β 1+α
2 2
.

 m

α2 1 + β 2
Fig. 1. Control system loop.
The solution (α, β) is a constant pair corresponding to a specified
2. Derivation of the PI/PD/PID tuning formulas and the GPM GPM which can easily be solved using a numerical solver, e.g., the
formulas solver ‘fsolve’ in Matlab. The solution is unique, if there is any,
since α > tan φm and β > 0 which ensure positive crossover
The ideal unity-feedback control system is considered, as frequencies and PI parameters. The initial guess of (α, β) for the
shown in Fig. 1, where Gc (s) denotes a PI/PD/PID controller and numerical solver can be any pair of large enough positive numbers,
Gp (s) denotes an IPTD process. Specifically, the transfer functions
e.g., (2 tan φm , 2 tan φm ), (5, 5) (as used in the later numeric tests),
are
etc.
Gp (s) = Kp e−τ s /s, τ > 0, (1) Therefore (7) gives explicit expressions of the PI parameters
where Kp is the process gain and τ the time delay, and (Kc , Ti ) in terms of the process parameters (Kp , τ ). For conve-
   nience, (7) is called PI tuning formula. Note that the crossover fre-
1 quencies ωp and ωg are also explicitly given in (7).

 Kc 1 + , PI controller;
sTi As an inverse problem, we compute the GPM resulting from a



Gc (s) = Kc (1 + Td s), PD controller; (2) given PI controller for an IPTD process. Still based on (3)–(6), the

 
1
 expression of GPM, namely GPM-PI formula, is obtained as follows:
+ T s , PID controller,

K 1 +

c d
sTi ωg = α/Ti ,


 p = β/T
ω i,

where Kc , Ti and Td are the proportional, integral and derivative



parameters, respectively. With this closed-loop system, explicit β 2 1 + α2 (9)
PI/PD/PID parameters are solved for achieving a given GPM. While Am = 2 ,
α 1 + β2


the way of properly specifying a GPM depends on specific design


φm = arctan α − ωg τ ,

requirements and demands extra studies, it is assumed that a GPM
has been specified by the designer throughout the paper. Some where
brief discussions on this are given in the next section.  
Although PI and PD controller tunings are special cases of   
γ2 4
PID controller tuning, their tuning formulas and corresponding α=  1+ 1+ 2 , with γ := Kp Kc Ti , (10)
GPM formulas are derived independently, adopting different 2 γ
approximations for accuracy and simplicity.
(the negative α is omitted) and β is solved from
2.1. PI tuning formula and GPM-PI formula arctan β = θ β, with θ := τ /Ti . (11)

Suppose the GPM of the closed-loop system is specified as Solution (9) also gives expressions of the gain and phase crossover
(Am , φm ), where Am denotes the gain margin and φm denotes the frequencies. As indicated by the above equations, the phase margin
phase margin. Given a PI controller in (2), the PI parameters (Kc , Ti ) φm is explicitly expressed; however, deriving the gain margin Am
are to be solved. According to the definition of GPM, we have requires first solving (11) for β . Although a numerical solution can
be used, for ease of application an approximate analytic solution is
arg[G(jωp )] = −π + arctan(ωp Ti ) − ωp τ = −π , (3)
proposed. According to Appendix A.1, such a solution is

Kc Kp 1 + ωp2 Ti2
 
π 16λB θ
 
1
= |G(jωp )| = , (4) β= , if 0 < θ ≤ θB ,

 1+ 1−
Am ωp2 Ti 4θ π2



  (12)
1 + ωg2 Ti2

Kc Kp

 1 120
1 = |G(jωg )| = , β = − 95, if θB < θ < 1,

(5)
 −5 +
ωg2 Ti 2θ θ
φm = arg[G(jωg )] + π = arctan(ωg Ti ) − ωg τ , (6) where λB = 0.917 and θB = 0.582. The constraint 0 < θ <
1 is imposed to ensure a positive solution for β . With β given
where ωp and ωg are the phase and the gain crossover frequencies, in (12), both Am and ωp in (9) are then explicitly expressed. The
respectively. Due to the nonlinearity of the equations, the four
above solution of (α, β) meanwhile justifies the uniqueness of the
variables ωg , ωp , Kc and Ti are normally analytically unsolvable,
solution to (8).
preventing derivation of a general PI tuning formula [1]. By
To evaluate the accuracy of (12) as the solution of (11), numeric
introducing two intermediate variables, however, all these four
tests are carried out. Without loss of generality, let Kp = 1. For
variables can be solved. Specifically, let α := ωg Ti and β := ωp Ti .
different (τ , Am , φm ), the PI parameters are first calculated by the
From (3)–(6), the following solution is obtained:
PI tuning formula. With these PI parameters, the realized GPMs are
1 then estimated by the GPM-PI formula, using β ’s estimated by (12).

ωg = τ (arctan α − φm ),

 The estimated GPMs are compared with the originally specified
ω = arctan β = β ω ,

GPMs correspondingly, so that the accuracy of the approximations


p g
τ α (7) is tested. In the computation, the parameters are chosen randomly
αω g as τ ∈ (0, 1] (which loses no generality since the PI tuning
,

Kc = √


Kp 1 + α 2 formula and GPM-PI formula both apply regardless of the process



Ti = α/ωg , parameters), Am ∈ (1, 12] and φm ∈ (10°, 70°]. The numerical

270 W. Hu et al. / ISA Transactions 50 (2011) 268–276

Fig. 2. GPMs estimated by GPM-PI formula versus true GPMs specified randomly a priori (50 tests), where the dots denote the estimated points and the circles denote the
true points.

results are shown in Fig. 2, where the relative estimation error The solution (α ′ , β ′ ) is unique since α ′ > 0 and β ′ > 0 which
(R.e.e.) is defined as R.e.e. := (the estimated value — the true make sure positive crossover frequencies and PD parameters. The
value)/the true value. Since α and φm are exactly derived by the initial guess of (α ′ , β ′ ) for the numerical solver can be any pair of
GPM-PI formula, their estimation errors are omitted in the figure, large enough positive numbers, e.g., (5, 5), (10, 10), etc. Therefore,
as it remains the same for later discussions on PD and PID controls. (17) gives the PD tuning formula.
The results indicate that the estimation errors of Am ’s are normally Inversely, given an IPTD process in (1) and a PD controller in
within 2% and thus validate (9) adopting the approximate solution (2), the resultant GPM and crossover frequencies of the closed-loop
of β by (12). system are derived from (13)–(16) as

ωg = α ′ /Td ,


ωp = β / Td ,
2.2. PD tuning formula and GPM-PD formula ′




Given a specified GPM (Am , φm ), an IPTD process in (1) and a PD
β ′ 1 + α ′2 (19)
Am = ′ ,
α 1 + β ′2

controller in (2), the PD parameters (Kc , Td ) are to be solved. The



φm = arctan α − ωg τ + π /2,
′

definition of GPM leads to

arg[G(jωp )] = −π /2 + arctan ωp Td − ωp τ = −π , (13) where


1 α′ = γ ′2 /(1 − γ ′2 ), with γ ′ := Kp Kc Td , (20)

= |G(jωp )| = Kc Kp 1 + ωp2 Td2 /ωp , (14)
Am and β is solved from
′

1 = |G(jωg )| = Kc Kp 1+ω 2 2
g Td /ωg , (15) arctan β ′ = θ ′ β ′ − π /2, with θ ′ := τ /Td . (21)

φm = arg[G(jωg )] + π = π /2 + arctan ωg Td − ωg τ , (16) Since deriving the gain margin requires solving β from (21), an
′

approximate analytic solution is proposed for it. Divide the domain

where the variables are defined the same as those in Section 2.1. of β ′ into two: 0 < β ′ ≤ 1 (β ′ being small) and β ′ > 1 (β ′ being
By introducing two new variables α ′ := ωg Td and β ′ := ωp Td in a large). In the former domain, use the approximation
similar way to that for the PI case, the parameters are solved from
(13)–(16) such that arctan β ′ ≈ λβ ′ , with λ := π /4, (22)
 1 π  and in the latter domain use the approximation
ωg = arctan α ′ + − φm ,
τ π λ π λ

2

arctan β ′ = ≈ − ′.

π  β′ − arctan (23)

ω = 1 arctan β ′ +
 
= ′ ωg , β′ β

p 2 2
τ ω 2 α (17)

K = √
g
, Solve (22) and (23), respectively, and express the applicable
c
domains in terms of θ ′ , an approximate solution of (21) is derived

Kp 1 + α ′2




as

Td = α ′ /ωg ,   
π θ ′λ

where the constant pair (α , β ) is solved from the equations
′ ′ 4
β = ′ 1 + 1 − 2 , if 0 < θ ′ ≤ θB′ ,

 ′

2θ π

π α′  π (24)

φ = arctan α ′
+ − arctan β ′
+ , π
β ′ = , if θ ′ > θB′ , where θB′ := π /2 + λ.

m

β′
 
2 2

2(θ ′ − λ)

(18)

A = β 1 + α .
′ ′2
Therefore, (19) gives the GPM-PD formula, where the intermediate

 m

α ′ 1 + β ′2 variables α ′ and β ′ are expressed in (20) and (24), respectively.
 W. Hu et al. / ISA Transactions 50 (2011) 268–276 271

Fig. 3. GPMs estimated by GPM-PD formula versus true GPMs specified randomly a priori (50 tests), where the dots denote the estimated points and the circles denote the
true points.

Meanwhile the solution of (α ′ , β ′ ) justifies the uniqueness of the α

 
1

ωg = τ arctan 1 − kα 2 + H (1 − kα )π − φm ,
 2
solution to (18) for a given GPM.



To evaluate the accuracy of (24) as a solution of (21), β β
  
ωp = 1 arctan

( β )π = ωg ,
 2
+ H 1 − k

numeric computations are carried out to test it. The IPTD process
τ 1 − kβ 2 α (30)
parameters and the GPMs are specified in a similar way to those αωg
K = ,

c

for the PI case (refer to Section 2.1). Analogously, the results are
Kp (1 − kα 2 )2 + α 2

 


obtained and shown in Fig. 3, which demonstrate the accuracy of
Ti = α/ωg ,


the GPM-PD formula adopting β ′ estimated by (24).

Td = kTi
where (α, β) is solved from the following equations
2.3. PID tuning formula and GPM-PID formula α
φm = arctan 1 − kα 2 + H (1 − kα )π
 2


 − α arctan β
  
Given a specified GPM (Am , φm ), an IPTD process in (1) and a PID

( β 2
)π ,

+ H 1 − k
controller in (2), the PID parameters (Kc , Ti , Td ) can be solved. The β 1 − kβ 2 (31)
 
definition of GPM leads to 

 β 2 (1 − kα 2 )2 + α 2
.

Am = 2

−π = arg[G(jωp )] α (1 − kβ 2 )2 + β 2
ω p Ti The solution (α, β) is unique for ensuring positive crossover
= −π + arctan + H (1 − ωp2 Ti Td )π − ωp τ , (25)
1 − ωp2 Ti Td frequencies and PID parameters subject to a given k. This is justified
 by an explicit solution of (α, β) in terms of the PID parameters as
1 Kc Kp (1 − ωp2 Ti Td )2 + ωp2 Ti2 presented later. The initial guess of (α, β) for a numerical solver
= |G(jωp )| = , (26) to solve (31) can be any pair of large enough positive numbers,
Am ωp2 Ti e.g., (5, 5), (10, 10), etc.
 Eq. (30) is the PID tuning formula. Note that when solving
Kc Kp (1 − ωg2 Ti Td )2 + ωg2 Ti2 (31), depending on the value of k, four different cases need to be
1 = |G(jωg )| = , (27) considered: (1) 1 − kα 2 > 0, 1 − kβ 2 > 0; (2) 1 − kα 2 > 0,
ωg2 Ti
1 − kβ 2 < 0; (3) 1 − kα 2 < 0, 1 − kβ 2 > 0; and (4) 1 − kα 2 < 0,
φm = arg[G(jωg )] + π 1 − kβ 2 < 0. If none of these cases gives a solution, we may take
ωg Ti (31) as having no solution for (α, β) and the GPM should be re-
= arctan + H (1 − ωg2 Ti Td )π − ωg τ , (28) specified to other values; or an alternative solution can be obtained
1 − ωg2 Ti Td
such that the attained GPM is in certain sense (e.g., the least square
sense) closest to the specified one.
where the function H (•) is defined as
Inversely, given an IPTD process in (1) and a PID controller in
(2), the resultant GPM and crossover frequencies of the closed-loop
0, if t ≥ 0,

H (t ) := (29) system are derived from (25)–(28) as
1, if t < 0.
ω = α/T ,
g i
Since there are five unknowns (ωg , ωp , Kc , Ti , Td ), but only ωp = β/Ti,





four equations, one additional condition is required for a unique β 2 (1 − kα 2 )2 + α 2

solution. In the literature, normally it assumes that Td = kTi and Am = , (32)
 α 2 (1 − kβ 2 )2 + β 2
k ∈ (0, 0.5] [1,2]. By defining α and β the same as those in α



φm = arctan + H (1 − kα 2 )π − ωg τ ,


Section 2.1, the parameters are solved from (25)–(28) such that
1 − kα 2
272 W. Hu et al. / ISA Transactions 50 (2011) 268–276

Fig. 4. GPMs estimated by GPM-PID formula versus true GPMs specified randomly a priori (50 tests), where the dots denote the estimated points and the circles denote the
true points.

where α and β are the respective solutions of the two equations: with xB := 1.5, x′B := 1.0 and λ(t ) := (arctan t )/t; and

(γ 2 k2 − 1)α 4 + γ 2 (1 − 2k)α 2 + γ 2 = 0, and (33) √ √

  
3 3
U := R + D + R − D, if D ≥ 0;

β
+ H (1 − kβ 2 )π = θ β, (38)

U := 2 R − D cos(ϕ/√ 3),
6
arctan (34) 2
1 − kβ 2
with ϕ := arctan( −D/R) + H (R)π , if D < 0,

where γ and θ are defined in (10) and (11). Eqs. (33)–(34) can be
solved numerically. Alternatively, their approximate solutions can with
be obtained as below.
For (33), noticing the common conditions that k ≤ 0.5 and D := Q 3 + R2 , Q := (3a1 − a22 )/9,
γ k < 1 as adopted by a large number of existing rules [1], its R := (9a2 a1 − 27a0 − 2a32 )/54, (39)
unique solution (the negative solution is omitted) is obtained as a0 := π /(kθ ), a1 := (λ′B − θ )/(kθ ), a2 := −π /θ .
To summarize, (32) gives the GPM-PID formula, with the
   

 γ2 4 intermediate variables α and β being expressed by (35) and
α= 1 − 2k + 1 − 4k + 2 . (35)
2(1 − γ 2 k2 ) γ (36), respectively. By the way, the solution of (α, β) justifies the
uniqueness of the solution to (31) for a given GPM.
When k = 0, this solution reduces to (10), namely the solution for
the case of PI control. Remark 1. (a) Since the boundary conditions in (36) are implicit,
For (34), according to Appendix A.2, an approximate solution is the candidate solutions are calculated in turn until a valid one is
obtained as obtained. (b) Refer to the end of Appendix A.2 for a less accurate
    yet simpler approximate solution of (34). 
  2
 1 1 3 1 3 12
 
   − 2+ + 2 − 3 , if β < βB ; Numerical computations are carried out to evaluate the accu-
θ θ kθ

2 k k




 racy of (36) as the solution of (34). The IPTD process parameters
and the GPMs are specified in a similar way to those for the PI case

  
π 16λB (θ − λB k)

 
(see Section 2.1). Numerical results are obtained for different val-

,

 1+ 1−
β = 4(θ − λB k) π2

ues of k as shown in Figs. 4 and 5. Since the estimation errors are
√ (36)
 if βB ≤ β < 1/ k;
 normally within 5%, the results validate the calculation of Am in the
GPM-PID formula based on the β approximated by (36).

   
π 16λ′B (θ − λ′B k)


,

1+ 1−

4(θ − λ′B k) π2


3. Application to unifying the existing tuning rules

√


 if 1/ k < β ≤ β ′ ;


 B
−a2 /3 + U , if β > βB′ ,

Rules of tuning PI/PD/PID controllers for an IPTD process have
been accumulated in the past decades. These rules are based
where
 on various requirements and specifications on performance and
λB := λ(1/xB ), βB := ( 1 + 4kx2B − 1)/(2kxB ), robustness of the closed-loop system and were derived with
 (37) various methods [1]. However, most of them can be unified by
λ′B := λ(1/x′B ), βB′ := ( 1 + 4kx′B2 + 1)/(2kx′B ), the tuning formulas presented above. From the PI, PD, PID tuning
 W. Hu et al. / ISA Transactions 50 (2011) 268–276 273

Fig. 5. Relative estimation errors of the results in Fig. 4.

formulas respectively in (7), (17), and (30), we see that the PID enhanced disturbance rejection) [18] about (2.9, 42.5°) for an IPTD
parameters have a common form of process, when the recommended settings are adopted for both
methods.
k1
Kc = , Ti = k2 τ , Td = k3 τ , (40) Finally, we apply the GPM-PI/PD/PID formulas derived in the
Kp τ last section to estimate the GPMs realized by relevant PI/PD/PID
where the parameters k1 , k2 , k3 are specifically tuning rules as collected in [1]. The GPM-PI/PD/PID formulas
indicate that any PI/PD/PID controllers with the same (k1 , k2 , k3 ) in
α(arctan α − φm )
PI controller: k1 = √ , (40) result in the same GPM, regardless of the process parameters.
1 + α2 This enables numeric computation of the exact GPM realized by
α each rule in the form of (40). To compare, GPM attained by each
k2 = , k3 = 0;
arctan α − φm rule is computed by using both the GPM-PI/PD/PID formula and
arctan α ′ + π /2 − φm the numeric approach. The results are documented in the link [24],
PD controller: k1 = √ , k2 = ∞, which take more than four pages to present and hence are omitted
1 + α ′2 (41) here. The results show that various GPMs are achieved by the
α′ existing tuning rules. Note that the larger the gain margin or the
k3 = ;
arctan α ′ + π /2 − φm smaller the phase margin is, the more aggressive yet less robust
αϕα α the closed-loop performance will be. The summary of such GPMs
PID controller: k1 =  , k2 = , thus provides a rich reference for control engineers to tune PID
(1 − kα 2 )2 + α 2 ϕα
controllers. Meanwhile the results verify that the GPM-PI/PD/PID
k3 = kk2 . formulas are accurate for GPM estimations.
Here ϕα := arctan 1−αkα 2 + H (1 − kα 2 )π − φm , and α , α ′ and α
for the PI, PD and PID controllers are determined from (8), (18) and 4. Conclusion
(31), respectively.
The common form of PI/PD/PID parameters in (40) indicates
For an IPTD process, PI/PD/PID tuning formulas with specified
that different rules employing different values of (k1 , k2 , k3 ) are
GPM were obtained and so were GPM-PI/PD/PID formulas for
realizing different GPMs which consequently lead to various
estimating GPM resulting from a given PI/PD/PID controller. The
closed-loop performances. This gives a unified interpretation
tuning formulas indicate a common form of the PID parameters and
to the vast variety of PI/PD/PID tuning rules accumulated in
the literature [1]. From this viewpoint, PI/PD/PID control design unify a large number of tuning rules as PI/PD/PID controller tuning
on an IPTD process is essentially choosing a proper GPM with various GPM specifications. The GPM formulas accurately
or parameter set (k1 , k2 , k3 ). The GPM or parameter set can estimate the GPM realized by each relevant PI/PD/PID tuning rule
be selected via performance optimization subject to design as collected in [1] and the results are summarized in the link [24].
constraints. Depending on the specific performance index and The results show that a variety of GPMs are attained by the existing
design constraints, the solution may differ from case to case and rules. Since the rules were developed based on various criteria
particular studies are required. A summary of various designs can and methods, the summary of their resulting GPMs provides a rich
be found in [1]. In particular, the well-known SIMC rule [12] uses reference for control engineers to tune PID controllers, helping to
a GPM of about (3.0, 46.9°) and the improved SIMC rule (with select a rule or GPM for a specific design.
274 W. Hu et al. / ISA Transactions 50 (2011) 268–276

Acknowledgements

The authors deeply appreciate A. O’Dwyer for sharing the

editable electronic tables presented in his book [1] on pages 76–82
and 259–261, which contributed a lot to the supplemental material
documented in the link [24]. The authors also thank S. Skogestad
for the insightful comments and helpful discussions on the draft
of this paper. And comments from anonymous reviewers are
acknowledged for having helped improve the quality of this paper.

Appendix. Approximate analytic solutions of β’s for (11) and

(34)

To solve (11) and (34) for approximate solutions, first consider

approximating the following equation.
x = tan y, y ∈ (−π /2, π /2). (42)
Divide the domain of y into two parts: Fig. 6. The maximal absolute values of the relative errors of the approximate
solutions, as functions of the boundary point xb .
D1 := (− arctan xb , arctan xb ), and
(43)
D2 := (−π /2, − arctan xb ] ∪ [arctan xb , π /2),
may be adopted. Consider the third-order Taylor expansion case
where xb ≥ 1 is a boundary value. Since (42) has odd solutions,
where (44) and (45) are replaced respectively by
it is sufficient to consider solving it in the domain consisting of
D1r := [0, arctan xb ) and D2r := [arctan xb , π /2).
In D1r , approximate (42) by the Taylor expansion of tan y to the x = tan y ≈ y + y3 /3, and (50)
fifth order, giving e1 (y) = (y + y /3)/ tan y − 1.
3
(51)
x = tan y ≈ y + y /3 + 2y /15,
3 5
(44) Keep (46) unchanged. By deducting similarly as above, the approx-
of which the relative approximation error is imation boundaries are obtained as xB ≈ 1.500 and λB ≈ 0.882,
at which the maximum absolute values of the relative errors by the
e1 (y) := (y + y3 /3 + 2y5 /15)/ tan y − 1. (45) two different approximations equal each other at 13.38%.
In D2r ,
first convert (42) into the arctangent form and then
approximate it by
A.1. An approximate solution of (11)
y = arctan x = π /2 − arctan z ≈ π /2 − λb z , (46)
where z := x−1 and λb := λ(1/xb ) and λ(•) is a function defined In particular, let x := β > 0 and y := θ β > 0 in (42). From
as (44) and (46), an approximate solution of (42) can be obtained as
follows:
λ(t ) := (arctan t )/t , t ∈ (0, +∞). (47)
 
The corresponding relative approximation error is 
1 120

β = − 95, 0 < β < βB ,

−5 +


e2 (z ) := tan(π /2 − λb z )/z −1 − 1 = z / tan(λb z ) − 1. (48) 2θ   θ  (52)
Note that (i) to be consistent with e1 (y), the tangents of both sides π 16λB θ
β = 4θ 1 + 1 − π 2 , β ≥ βB ,



of (46) are taken to calculate e2 (z ); and (ii) the Taylor expansion is

not used in D2r since it is hard to attain high accuracy; and (iii) in
D2r it has z ∈ (0, x− 1 where λB = 0.917 and βB = 1.848. Alternatively, by specifying
b ].
From (45) and (48), it can be easily proved that e1 (y) < 0, the conditions of θ , the solution (52) can be re-expressed as
de1 (y)/dy ≤ 0, e2 (z ) > 0 and de2 (z )/dz ≤ 0. Thus the maximum  
absolute values of e1 (y) and e2 (y) are, respectively, π 16λB θ
 
β= , if 0 < θ ≤ θB ,

 1+ 1−
4θ π2

‖e1 (y)‖∞ = −e1 (arctan xb ), and,
 

(49) (53)
‖e2 (z )‖∞ = lim e2 (z ) = 1/λ − 1. 
 
z →0  1 120
β = − 95, if θB < θ < 1,

 −5 +
Here ‖e1 (y)‖∞ and ‖e2 (z )‖∞ are both functions of xb , as shown 2θ θ
in Fig. 6, where the intersection point is numerically obtained as
xB := xb ≈ 1.848. At this point, the maximum absolute values of where
the relative errors by the two different approximations equal each
π2 1 π λB
  
other at 9.10%, and λB := λb = λ(1/xB ) ≈ 0.917. θB := min , − ≈ 0.582. (54)
For y being an explicit function of x, e.g., y = 2x, by taking xB and 16λB βB 2 βB
λB as the boundary parameters for the above two approximations,
an approximate solution of (42) can be obtained by solving either Note that for (52), as the boundaries of the applying regions of
(44) or (46) for x. θ do not coincide, for simplicity θB is taken as the one calculated
In addition, notice that in some cases where y is an explicit from the second equation of (52). The validity of the approximate
function of x, (44) may prevent an analytic solution of x. As a solution of (42) by (53) is demonstrated by the exemplary results
compromised solution, a lower-order Taylor expansion of tan y shown in Fig. 2.
 W. Hu et al. / ISA Transactions 50 (2011) 268–276 275

A.2. An approximate solution of (34) where

√ √
  
3 3
To solve (34), two different
√ cases are considered separately as U := R + D + R − D, if D ≥ 0;

follows (The point β = 1/ k is undefined in the equations and is (67)

U := 2 R2 − D cos(ϕ/√3),
6

therefore omitted.):

with ϕ := arctan( −D/R) + H (R)π , if D < 0.


β Here H (•) is the function defined in (29), and D and R keep the
arctan = θ β, if 1 − kβ 2 > 0; (55)
1 − kβ 2 same as those in (65).
With (57), (59) and (66), the approximate solution of (34) is thus
β
arctan = θ β − π , if 1 − kβ 2 < 0. (56) obtained as follows
1 − kβ 2   
2

Let x := β/(1 − kβ ) and y := θ β in (42). From (46) and (50)
2

1 1 3 1 3 12
 
+ 2 − 3 , if β < βB ;
 
  − 2+


an approximate solution of (55) is derived as 

 2 k θ k θ kθ


    

    
π 16λB (θ − λB k)
 2  
1 1 3 1 3 12
  
,
 
β=  − 2 + + 2 − 3 , 1+ 1−
  
β = 4(θ − λB k) π2
 
θ θ kθ

2 k k (68)



 if βB ≤ β < /;

  1
if 0 < β < βB ,  (57)
   
π 16λ′B (θ − λ′B k)


 
π 16λB (θ − λB k) ,
  
  1+ 1−
β= , 4(θ − λ′B k) π2
 
1+ 1−
 
4(θ − λB k) π2
 
√

 

√  if 1/ k < β ≤ βB′ ;

 

if βB ≤ β < 1/ k,
 
−a2 /3 + U , if β > βB′ ,
where where the intermediate variables, λB and βB , λ′B and βB′ , a2 and U,
 are defined in (58), (60), and {(62), (65), (67)}, respectively. Since
λB := λ(1/xB ), βB := ( 1 + 4kx2B − 1)/(2kxB ), (58) it is hard to give the piecewise conditions of (68) in terms of θ as
that in (53), the candidate solutions are calculated in a top-down
with xB := 1.5 and λ(•) being defined in (47).
sequence until a feasible β is obtained; if no feasible solution is
To solve (56), the approximation skills used in (22)–(23)
achieved, (34) will be taken as having no solution, or a numerical
are adopted. Specifically, by applying the skill used in (23), an
solution to it has to be tried.
approximate solution of (56) is obtained as Additionally, another simpler yet less accurate approximate
solution for (34) can be derived. The main idea√is as follows. For
  
π 16λ′B (θ − λ′B k)
β= 1+ 1− , the case of (55) and the case of (56) with 1/ k < β ≤ βB′
4(θ − λ′B k) π2 (Here βB′ is of a different value from that in (68).), the approximate
√ solutions remain the same as those in (57) and (59), respectively;
if 1/ k < β ≤ βB′ , (59) and for the case of (56) with β > βB′ , first (56) is approximated
where by replacing ‘‘1 − kβ 2 ’’ with −kβ 2 (requiring that kβB′2 ≫ 1—here
 kβB′2 = 10 is used, by selecting a proper boundary point x′B ). Then
λ′B := λ(1/x′B ), βB′ := ( 1 + 4kx′B2 − 1)/(2kx′B ), (60) by applying the same skill as that in (22), a less accurate yet simpler
approximate solution of (34) can be obtained. Specifically, it is as
with x′B := 1.0 and λ(•) being defined in (47). And for the case follows:
where β > βB′ , by applying the skill used in (22) the following 
equation of β is obtained:
  
 2

1 1 3 1 3 12
 
+ 2 − 3 , if β < βB ;
 

   − 2+
β 3 + a2 β 2 + a1 β + a0 = 0,  2 k θ θ kθ

(61) 
 k



where   
π 16λB (θ − λB k)
 

,

a2 := −π /θ , a1 := (λ′B − θ )/(kθ ), a0 := π /(kθ). 1+ 1−

(62)
 4(θ − λB k) π2



√

Eq. (61) is a standard cubic equation with real coefficients, and its β= if βB ≤ β < 1/ k; (69)
feasible solution (being real and positive) is obtained as

π 16λ′B (θ − λ′B k)


,

1+ 1−

β = −a2 /3 + S + T ,

(63) 4(θ − λ′B k) π2




 √
1/ k < β ≤ βB′ ;

where  if 


  
π 4λ′′B θ

√ √
  

3 3
D, D, , if β > βB′ ,

S := R+ T := R− (64)
 1+ 1−
2θ kπ 2

with
where λB := λ(1/xB ), λ′B := λ(1/x′B ), λ′′B := λ(x′B ), βB :=
D := Q + R ,
3 2
Q := (3a1 − a22 )/9, √

(65) ( 1 + 4kx2B − 1)/(2kxB ) and βB′ := 10/k, with xB := 1.5,
R := (9a2 a1 − 27a0 − 2a32 )/54.
x′B := βB′ /(kβB′2 − 1) and λ(•) being defined in (47). As expected, the
Since D in (64) may be negative, leading to complex numbers in the √
estimated β may not be accurate when β > 1/ k, but it is found
calculations which should be avoided in applications, the solution
to be able to achieve the final goal of estimating the gain margin
(63) is expressed in an alternative way such that
Am with satisfactory accuracy. The relative estimation errors are
β = −a2 /3 + U , (66) mostly within 7%. Exemplary results are shown in Fig. 7.
276 W. Hu et al. / ISA Transactions 50 (2011) 268–276

Fig. 7. Typical relative estimation errors of β and Am , with β being estimated by (69).

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