1. ISA Transactions 40 (2001) 31±39
www.elsevier.com/locate/isatrans
Robust PID controller design for non-minimum
phase time delay systems
Ying J. Huang *, Yuan-Jay Wang
Institute of Electrical Engineering, Yuan Ze University, 135 Far-East Road, Chungli, Taiwan
Received 10 January 2000; received in revised form 6 July 2000; accepted 6 July 2000
Abstract
A robust PID controller for a non-minimum phase system subject to uncertain delay time is presented in this paper.
Utilizing the gain-phase margin tester method, a speci®cation-oriented parameter region in the parameter plane that
characterizes all admissible controller coecient sets can be obtained. The PID controller gains are then directly
selected from the parameter region. Henceforth, the designed controller can guarantee the system at least a pre-speci-
®ed safety margin to compensate for the instability induced by the time delay. A compromise between the robustness
and tracking performance of the system in the presence of time delay is achieved. Simulation results indicate that the
proposed method performs a good time response, and robustness is obtained e€ectively. # 2001 Elsevier Science Ltd.
All rights reserved.
Keywords: PID control; Non-minimum phase; Gain-phase margin tester method
1. Introduction processes. Subsequently, parameter plane methods
[4, 5] for the evaluation of the PID settings based
The PID controllers have been successfully on the given Gm and Pm speci®cations are pre-
applied to many industrial control systems. De sented. However, there are few systematic PID
Paor and O'Malley [1] derived PID controllers of tuning formulas for non-minimum phase systems,
the Ziegler±Nichols type for unstable processes especially with time delay.
with time delay, based on an optimal gain margin Gain margin and phase margin have always
(Gm) and an optimal phase margin (Pm). Later, played an important role concerning the robustness
Sha®ei and Shenton [2] presented a graphical tech- of systems. In this paper, the previous achievement
nique for calculating PID controller parameters. is extended to the non-minimum phase plant con-
According to the gain and phase margin speci®ca- taining an uncertain delay time with speci®cations
tions, simple rules were introduced by Ho and Xu in terms of gain and phase. Controllers designed to
[3] to tune the PID controller settings for unstable meet the gain phase margin speci®cations have been
demonstrated in the literature [4±6]. The gain-phase
* Corresponding author. Tel.: +886-3-463-8800, ext. 410; margin tester method [4] is adopted to test the sta-
fax: +886-3-463-3326. bility boundary in the parameter plane [7±9, 11] for
E-mail address: eeyjh@saturn.yzu.edu.tw (Y.J. Huang). any given gain or phase margin speci®cations.
0019-0578/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.
PII: S0019-0578(00)00036-7

2. 32 Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39
These margins serve as restrictions to scheduling phase systems and endows the system with robust
the controller. Through the de®nition of such safety margins in terms of gain and phase.
margins, not only the relative stability margin, but For a high-order non-minimum system which
also the absolute stability margin, can be guaran- contains a time delay element [10], its transfer
teed [2]. Henceforth, the speci®cation of the sys- function is shown as follows,
tem in terms of gain margin and phase margin is
interpreted into desired parameter area in a two-
…1 À 3:6s†exp…ÀTs†
dimensional parameter plane. The method intro- GP …s† ˆ ; …1†
duced in this paper is based on a search for such …5s ‡ 1†…s ‡ 1†…0:2s ‡ 1†…0:5s ‡ 1†
an aforementioned parameter area to achieve
compromise between good tracking performance where T is the delay time of the system. Using a
and system robustness with respect to external second-order approximation, the time domain and
disturbance. frequency domain speci®cations are approxi-
The advantage of the gain-phase margin tester mately converted into interval gain margins and
method is that various system performances phase margins [12]. Therefore, the control system
resulting from the tuning of the adjustable para- with a PID controller in a series connection with
meters can be realized completely. A speci®cation- the plant is expected to achieve the speci®cations
oriented parameter area, which characterizes all of 5 dB4Gm410 dB and 30 4Pm460 . Fig. 1
admissible stabilizing controller sets, can be shows the block diagram of the considered system.
obtained in the parameter plane. PID controller The transfer functions of the process and the con-
with coecients selected from the obtained para- troller are denoted as GP …s† and GC …s†, respec-
meter area stabilizes the non-minimum phase time tively. D…s† is the external disturbance.
delay systems with pre-speci®ed safety margins. An error-actuated PID controller has the gen-
Especially when the delay time is uncertain, this eral transfer function
method works e€ectively well. PID controller set-
tings could be tuned out o€-line in general. It can KI
avoid extensive or unnecessary on-line tuning and GC …s† ˆ KP ‡ ‡ KD s: …2†
s
makes the implementation of the controller easier.
After all, it is noted that this method can be applied The forward open-loop transfer function of the
to both stable and unstable systems of high order control system shown in Fig. 1 is
and where the controller design has considerable
¯exibility. N…s†
G0 …s† ˆ GC …s†GP …s†; ˆ : …3†
D…s†
2. Non-minimum phase time delay control system
By letting s ˆ j!, and Re‰G0 …j!†Š and Im‰G0 …j!†Š
Time delay occurs in the control system when be the real part and imaginary part of the G0 …j!†,
there is a delay between the commanded response respectively, one has
and the start of the output response [12]. The
delay causes a decreased phase margin which
implies a lower damping ratio and a more oscilla-
tory response for the closed-loop system. Further,
it decreases the gain margin, thus moving the sys-
tem closer to instability. In this section, a sys-
tematic algorithm is introduced for the
determination of the PID settings. The controller
is designed to compensate for the instability
induced by the time delay for the non-minimum Fig. 1. Block diagram of a typical PID control system.

3. Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39 33
G0 …j!† ˆ G0 …j!†ej ; …4† D…j!† À
1
G0 …j!†ej N…j!† ˆ 0: …7†
where
Let
q
G0 …j!† ˆ Re‰G0 …j!†Š2 ‡Im‰G0 …j!†Š2 ; …5†
A ˆ 1=G0 …j!†; …8†
È É ˆ ‡ 180: …9†
ˆ €G0 …j!† ˆ tanÀ1 Im‰G0 …j!†Š=Re‰G0 …j!†Š : …6†
When ˆ 0, A is the gain margin of the system,
Substituting (4) into (3), one obtains and when A ˆ 1, is the corresponding phase
margin. Now we de®ne the gain-phase margin
tester function as
F…j!† ˆ D…j!† ‡ AeÀj N…j!†: …10†
Eqs. (7)±(10) imply that the function F…j!†
Fig. 2. Block diagram of the control system with a gain-phase should always be equal to zero. This indicates that
margin tester. the gain margin and the phase margin of the PID
Fig. 3. R1 (ABCD) is the user-speci®ed parameter region. P1 (KP ˆ 0:4, KI ˆ 0:1, and KD ˆ 0:10) is the representative point with
delay time T ˆ 0:5 s.

4. 34 Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39
control system can be determined from the char- Xa …j!† ˆ …4:1!4 À 6:7!2 † ‡ j…0:5!5 À 9:3!3 ‡ !†;
acteristic equation.
By adding a so-called gain-phase margin tester …13†
AeÀj into the system as shown in Fig. 2, the
characteristic equation is Xb …j!† ˆ 3:6!2 ‡ j!; …14†
KI Xc …j!† ˆ 1 À j3:6!; …15†
1 ‡ AeÀj …KP ‡ ‡ KD s†
s
…1 À 3:6s†exp…ÀTs†
… † ˆ 0: …11† Xd …j!† ˆ À!2 ‡ j3:6!3 : …16†
…5s ‡ 1†…s ‡ 1†…0:2s ‡ 1†…0:5s ‡ 1†
From Eq. (12), letting 1 ˆ ‡ !T, one obtains
Noting that AeÀj ˆ Acos À jAsin, Eqs. (10) the following two stability equations,
and (11) give rise to
FR …j!† ˆ KP B1 ‡ KI C1 ‡ D1 ;
F…j!† ˆ Xa …j!† ‡ A…cos… ‡ !T† À jsin… ‡ !T†† ˆ Re…Xa † ‡ Acos1 …KP Re…Xb †
Á ‰KP Xb …j!† ‡ KI Xc …j!† ‡ KD Xd …j!†Š; …12† ‡ KI Re…Xc † ‡ KD Re…XD ††
‡ Acos1 …KP Im…Xb † ‡ KI Im…Xc †
where ‡ KD Im…XD ††; ˆ 0 …17†
Fig. 4. Bode diagrams with P1 selected (KP ˆ 0:40, KI ˆ 0:10, and KD ˆ 0:10).

5. Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39 35
FI …j!† ˆ KP B2 ‡ KI C2 ‡ D2 ; D2 ˆ Im…Xa † ‡ Acos1 KD Im…Xd †
ˆ Im…Xa † ‡ Acos1 …KP Im…Xb † ‡ KI Im…Xc †
‡ KD Im…XD †† À Asin1 …KP Re…Xb † À Asin1 KD Im…Xd †: …24†
‡ KI Re…Xc † ‡ KD Re…XD ††; ˆ 0; …18†
Note that Re…Xa †, Re…Xb †, Re…Xc †, and Re…Xd †
where are the real parts of Xa , Xb , Xc , and Xd , respec-
tively; and Im…Xa †, Im…Xb †, Im…Xc †, and Im…Xd †
B1 ˆ Acos1 Re…Xb † ‡ Asin1 Im…Xb †; …19† are the imaginary parts of Xa , Xb , Xc , and Xd ,
respectively.
C1 ˆ Acos1 Re…Xc † ‡ Asin1 Im…Xc †; …20† Let KD be a constant, and solving Eqs. (17) and
(18), one has
D1 ˆ Re…Xa † ‡ Acos1 KD Re…Xd †
C1 ÁD2 À C2 ÁD1
KP ˆ ; …25†
‡ Asin1 KD Im…Xd †; …21† B1 ÁC2 À B2 ÁC1
B2 ˆ Acos1 Im…Xb † À Asin1 Re…Xb †; …22†
D1 ÁB2 À D2 ÁB1
KI ˆ : …26†
C2 ˆ Acos1 Im…Xc † À Asin1 Re…Xc †; …23† B1 ÁC2 À B2 ÁC1
Fig. 5. Output response and load disturbance response of the controlled system (KP ˆ 0:40, KI ˆ 0:10, and KD ˆ 0:10).

6. 36 Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39
3. Parameter plane analysis locus in the plane is a boundary of the constant
gain margin. On the other hand, if A ˆ 1, and is
Let A ˆ 1 and ˆ 0, and set KD equal to a assumed equal to a constant value, then the locus
constant, then for various values of !, a locus in the plane is a boundary of constant phase mar-
representing the stability boundary of the system gin.
without the gain-phase margin tester can be plot- By varying one of the parameters, A; and !,
ted in the KP ±KI plane. The stability characteristics and ®xing the others, it suces to plot the con-
of two sides of the locus are completely di€erent. stant gain margin boundary and the constant
De®ne the Jacobian [6], J, of Eqs. (17) and (18) as phase margin boundary in the parameter plane.
Then exploiting the stability equations method
J ˆ B1 ÁC2 À B2 ÁC1 : …27† presented in Ref. [6], a speci®cation-oriented region
enclosed by the constant gain margin boundaries
and constant phase margin boundaries could be
By resorting to [11], it is concluded that if J 0, found. The region characterizes all feasible con-
then to the left of the stability boundary, facing troller parameter sets which guarantees the con-
the direction in which ! increases, is the stable trolled system robust margins, i.e. Gm and Pm of
parameter area. Similarly, to the right of the sta- the system. For every value of KD the parameter
bility boundary, facing the direction in which ! area can be found easily in the two-dimensional
increases, is the stable parameter region while parameter plane. The aforementioned area shows
J 0. Accordingly, the stability boundary isolates a useful relationship between the three parameters,
the parameter plane into stable and unstable KP ; KI and KD of the PID controller. The absolute
parameter regions, respectively. Further, if A is and relative stability margins can, in fact, be
assumed equal to a constant value and ˆ 0, the readily obtained. Trial and error evaluation is
Fig. 6. The user-speci®ed parameter regions, R2, R3, and R4 for system with di€erent delay time T ˆ 0:1, 1, and 2 s, respectively.

7. Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39 37
avoidable in such a case. Thus a lot of work can be speci®cation-oriented parameter area can also be
saved. obtained.
A representative point P1 with KP ˆ 0:40,
KI ˆ 0:10, and KD ˆ 0:10 is selected. The stability
4. Numerical results of the closed-loop system is proved referring to the
Bode diagram as shown in Fig. 4. Output response
The control result is inspected by the following and disturbance rejection response for step dis-
simulation. First, we assume the delay time of the turbance of the controlled system are demon-
system is ®xed at T ˆ 0:50 s. According to Eqs. strated in Fig. 5. It is seen that tracking error
(17)±(26), let KD ˆ 0:10, the constant phase mar- approaches zero and disturbance rejection ability
gin boundaries for ˆ 30 and 60 can be plotted is obvious.
as in Fig. 3. In a same way, for A ˆ 5 and 10 dB, Next, a non-minimum phase system subject to
the constant gain margin boundaries can also be uncertain time delay is inspected. By letting
plotted as in Fig. 3. The region ABCD shown in T ˆ 0:1, 1 and 2 s, and exploiting Eqs. (17)±(26),
Fig. 3 is the parameter area which constitutes of the speci®cation-oriented parameter region can be
all the possible parameter sets of the controller found, respectively. Consequently, as seen in Fig.
that guarantees the system at-least the pre- 6, one obtains three di€erent regions, R2, R3 and
speci®ed safety margins in terms of gain and R4 in the parameter plane. These regions are the
phase. For other values of KD , the corresponding speci®cation-oriented areas for di€erent delay
Fig.7. The gain margins and phase margins of the controlled system subject to the variation of the delay time, T, with the designed
PID controller (KP ˆ 0:41, KI ˆ 0:11, and KD ˆ 0:10).

8. 38 Y.J. Huang, Y.-J. Wang / ISA Transactions 40 (2001) 31±39
Fig. 8. Output responses for three di€erent delay time cases: T ˆ 0:1, 1, and 2 s. The representative point P2 (KP ˆ 0:41, KI ˆ 0:11,
and KD ˆ 0:10) is selected.
times. On the intersectional area of those three meter plane for the system with uncertain time
regions, we can freely choose an operation point. delay is introduced in this paper. The advantage of
For example, P2 (KP ˆ 0:41, KI ˆ 0:11 and this method is the guaranteed robustness with
KD ˆ 0:10) is selected. Here one already success- respect to plant variation and external disturbance.
fully obtains a robust PID controller for the non- Excessive on-line tuning can be signi®cantly alle-
minimum phase plant with uncertain delay time. viated. It promises the control system with good
Fig. 7 shows that the designed PID controller tracking and disturbance rejection behavior. One
maintains the time delay system with known var- can expect that this method of selecting PID con-
iation range of delay a robust safety margins. The troller settings can be applied to a wide range of
time responses in Fig. 8 demonstrate the robust- industrial applications.
ness of the designed controller in the case of
uncertain delay time. The designed robust PID
controller is seen to stabilize the system. References
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