1. ISA Transactions 39 (2000) 419±431
www.elsevier.com/locate/isatrans
Robust PID tuning strategy for uncertain plants based on the
Kharitonov theorem
Ying J. Huang *, Yuan-Jay Wang
Institute of Electrical Engineering, Yuan Ze University, 135 Far-East RD., Chungli, 320 Taiwan
Abstract
In this paper, the Kharitonov theorem for interval plants is exploited for the purpose of synthesizing a stabilizing
controller. The aim here is to develop a controller to simultaneously stabilize the four Kharitonov-de®ned vortex
polynomials. Di€erent from the prevailing works, the controller is designed systematically and graphically through the
search of a non-conservative Kharitonov region in the controller coecient parameter plane. The region characterizes
all stabilizing PID controllers that stabilize an uncertain plant. Thus the relationship between the Kharitonov region
and the stabilizing controller parameters is manifest. Extensively, to further guarantee the system with certain robust
safety margins, a virtual gain phase margin tester compensator is added. Stability analysis is carried out. The control
system is proved to maintain robustness at least to the pre-speci®ed margins. The synthesized controller with coe-
cients selected from the obtained non-conservative Kharitonov region can stabilize the concerned uncertain plants and
ful®ll system speci®cations in terms of gain margins and phase margins. # 2000 Elsevier Science Ltd. All rights
reserved.
Keywords: Kharitonov theorem; Parameter plane; PID controller; Interval plants
1. Introduction theorem [3] and box theorem [4] then suggested
that the set of transfer functions generated by
In the past years, many important results in the changing its perturbed coecients in the pre-
area of robust control for uncertain plants have scribed ranges corresponds to a box in the para-
been based on the Kharitonov's celebrated theorem meter space and is referred to ``interval plants''.
[1±6]. The theorem investigates the stability char- Further, the Kharitonov theorem is generalized to
acteristics of the interval systems via four vortex obtain a ``Kharitonov region'' [5,6] in the complex
polynomials with real coecients varying in a plane for the robust stability of linear uncertain
bounded range. Kharitonov extended his results systems. In order to guarantee the uncertain sys-
into interval polynomials with complex coecients tems a stronger stability characteristic, a virtual
later. Based on the Kharitonov theorem, the edge compensator was introduced such that the closed
loop systems can maintain a suitable gain margin
(Gm) and phase margin (Pm) [7]. Recently, atten-
* Corresponding author. Tel.: +886-3-4638800 ext. 410; fax: tion has been given to the formulation of P, PI
+886-3-4633326. and PID controllers to stabilize an interval plant
E-mail address: eeyjh@saturn.yzu.edu.tw (Y.J. Huang). family [8].
0019-0578/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.
PII: S0019-0578(00)00026-4

2. 420 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431
Despite the various results concerning the synthesizing ®xed and low order controllers, that
robust stability for uncertain plants, existing simultaneously stabilize a given interval plant
methods in the area of parametric robust control family in the parametric robust control area, is
are of analysis type which are passively concerned bene®cial. To achieve this goal, a systematic and
only if a family of Kharitonov polynomials are graphical robust controller design procedure is
Hurwitz or not. It turns out that a technique for established based on the Kharitonov-de®ned vortex
Fig. 1. Feedback control system with a gain phase margin tester.
Fig. 2. Stability boundary for GI …s†.

3. Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 421
polynomials. A speci®c Kharitonov region can be also ful®lls several speci®cations simultaneously.
obtained in the parameter plane using the para- For every one of the four vortex polynomials, it
meters of the controller as the axes. Therefore, the suces to plot the constant gain margin and con-
stability characteristic for the uncertain system in stant phase margin boundaries in the parameter
question, with respect to the adjustment of the plane. The overlapped region of speci®cation-
controller parameters, is obvious. The aforemen- satis®ed area for each Kharitonov polynomial is
tioned region constitutes the whole admissible the useful parameter area for the selection of con-
stabilizing PID controllers. troller parameters, where the whole uncertain
Further, for the purpose of endowing the system, plant can be stabilized.
a robust performance in the case of parameter The advantage of the developed method is that
variation, it is of interest to insert speci®ed gain the procedure to design a robust stabilizing con-
margins and phase margins into the characteristic troller is systematic and straightforward. A trial
equations, such that the resulted system will and error process is unnecessary. All exploitable
maintain at least the pre-speci®ed robust margins. controller parameters can be obtained from the
The proposed method in this paper not only pro- non-conservative Kharitonov region. Controllers
vides a necessary and sucient condition for the selected from the region guarantee the uncertain
Hurwitz stability of an interval polynomial set but systems a pre-speci®ed safety margin. Moreover,
Fig. 3. Stability boundary for G1 …s† with di€erent value of KD .

4. 422 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431
the intrinsic property of system stability corre- ``interval plants'' [4]. The controller C(s) is
sponding to the drift of the polynomial coecients designed to simultaneously stabilize the system
can be analyzed manifestly in the demonstrated and track the command signal. The gain phase
parameter planes. margin tester AeÀj is applied in the forward path,
where A and are subject to gain margin and
phase margin speci®cations, respectively. Let
2. Mathematical description
F…s† ˆ 1 ‡ AeÀj C…s†G…s† …P†
Consider the feedback control system as shown
in Fig. 1. The general expression of the plant G…s† is and
a0 ‡ a1 s ‡ a2 s2 ‡ Á Á Á ‡ am sm Fi …s† ˆ 1 ‡ AeÀj C…s†Gi …s†Y i ˆ 1Y Á Á Á Y 4Y …Q†
G…s† ˆ Y …I†
b0 ‡ b1 s ‡ b2 s 2 ‡ Á Á Á bn s n
where Gi …s† are the edge interval plants. If the
where am Tˆ 0, bn Tˆ 0, ai P ‰aÀ Y a‡ Š, bj P ‰bÀ Y b‡ Š,
i i j j controller C…s† can be designed with imposed spe-
and n5m. A set of transfer functions can be gen- ci®cations on gain margins and phase margins,
erated by changing the perturbed coecients in then it is of no doubt that the compensated system
the prescribed ranges, which correspond to boxes will maintain robust performance to some degree.
in the parameter plane. This is referred to as To show this, we present the following theorem.
Fig. 4. A speci®c Kharitonov region (hatched area) in the KP ÀKI plane.

5. Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 423
Theorem 1. Consider the control system as shown in p1 …s† ˆ dÀ ‡ dÀ s ‡ d‡ s2 ‡ d‡ s3 ‡ dÀ s4 ‡ Á Á Á Y
0 1 2 3 4 …S†
Fig. 1, with the de®nitions (1) and (2), the entire
family G…s† is stabilizable by a controller C…s† if and
only if each interval plant Gi …s† is stabilizable by p2 …s† ˆ dÀ ‡ d‡ s ‡ d‡ s2 ‡ dÀ s3 ‡ dÀ s4 ‡ Á Á Á Y
0 1 2 3 4 …T†
that same controller C…s†.
p3 …s† ˆ d‡ ‡ dÀ s ‡ dÀ s2 ‡ d‡ s3 ‡ d‡ s4 ‡ Á Á Á Y
0 1 2 3 4 …U†
Proof. Consider the characteristic polynomial of
the closed-loop system shown in Fig. 1, i.e. the
numerator of F…s†, p4 …s† ˆ d‡ ‡ d‡ s ‡ dÀ s2 ‡ dÀ s3 ‡ d‡ s4 ‡ Á Á Á Y …V†
0 1 2 3 4
ˆ
n
p…s† ˆ di s i Y …R†
iˆ0 which are associated with Gi …s† in (3), respectively.
It follows that the entire family is Hurwitz stable
where di are the characteristic coecients, and if and only if the four vortex polynomials (4)±(7)
dÀ 4di 4d‡ . According to the Kharitonov theo-
i i are all Hurwitz stable. Hence one can determine
rem [1,4], every interval polynomial in the family all possible stabilizing controllers Ci …s† for each
is Hurwitz if and only if the following four Khar- vortex polynomial, and then take their intersection
„ „ „
itonov polynomials are Hurwitz, C…s† ˆ C1 …s† C2 …s† C3 …s† C4 …s† to obtain the
Fig. 5. Output responses for the PID controlled interval system.

6. 424 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431
non-conservative robust parametric controller for The problem of characterizing all stabilizing PID
the uncertain plant G…s†. controllers for the entire family G…s† is to deter-
mine KP , KI and KD such that all the closed loop
In the following section, the controller sets, which characteristic polynomials are Hurwitz.
stabilize the uncertain plant, are demonstrated to lie Consider the same uncertain system in [9],
within a speci®c Kharitonov region in the parameter
plane. A strict Kharitonov region, which guarantees 5X2…s ‡ 2†
the system a robust margin, can be obtained. G…s† ˆ X …IH†
s… s3 ‡ b2 s2 ‡ b1 s ‡ b0 †
3. PID controller design for uncertain systems Assume that the coecients b0 Y b1 and b2 of the
denominator lie within the following bounds, where
For the uncertain plant (1), a PID controller 9X54b0 411X5Y 124b1 415, and 3X54b2 44X8. The
C…s† is developed with the exploitation of a gain desired speci®cations for the control system are sup-
phase margin tester in the forward loop. Let the posed to be 5db4Gm410dbY and 30 4Pm460 . By
controller C…s† be inserting a gain phase margin tester AeÀj in the
KD s2 ‡ KP s ‡ KI forward loop, it is found that the characteristic
C…s† ˆ X …W† polynomial is
s
Fig. 6. A speci®cation-oriented parameter area for G1 …s† with pre-speci®ed robust margins in gain and phase.

7. Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 425
p…s† ˆ s5 ‡ b2 s4 ‡ b1 s3 ‡ b0 s2 p3 …s† ˆ s5 ‡ 3X5s4 ‡ 15s3 ‡ 11X5s2
 Â
‡ 5X2KD s3 ‡ …5X2KP ‡ 10X4KD †s2 ‡ 5X2KD s3 ‡ …5X2KP ‡ 10X4KD †s2
à Ã
‡ …5X2KI ‡ 10X4KP †s ‡ 10X4KI ‰A…™os À j sin †ŠX ‡ …5X2KI ‡ 10X4KP †s ‡ 10X4KI ‰A…™os À j sin †ŠY
…II† …IR†
From (11), in four Kharitonov polynomials result p4 …s† ˆ s5 ‡ 4X8s4 ‡ 15s3 ‡ 9X5s2
Â
p1 …s† ˆ s5 ‡ 4X8s4 ‡ 12s3 ‡ 9X5s2 ‡ 5X2KD s3 ‡ …5X2KP ‡ 10X4KD †s2
 Ã
‡ 5X2KD s3 ‡ …5X2KP ‡ 10X4KD †s2 ‡ …5X2KI ‡ 10X4KP †s ‡ 10X4KI ‰A…™os À j sin †ŠX
à …IS†
‡ …5X2KI ‡ 10X4KP †s ‡ 10X4KI ‰A…™os À j sin †ŠY
…IP†
The aim here is to ®nd that all possible sets of
p2 …s† ˆ s5 ‡ 3X5s4 ‡ 12s3 ‡ 11X5s2 KP , KI , and KD , which make the characteristic
 polynomials (12)±(15) to be Hurwitz stable. Take
‡ 5X2KD s3 ‡ …5X2KP ‡ 10X4KD †s2
à p1 …s† for instance, substituting s ˆ j3 and equating
‡ …5X2KI ‡ 10X4KP †s ‡ 10X4KI ‰A…™os À j sin †ŠY the real part and imaginary part of (12) to zero,
…IQ† respectively, one obtains
Fig. 7. A non-conservative Kharitonov region in the parameter plane with pre-speci®ed gain margin and phase margin speci®cations.

8. 426 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431
C1 D2 À C2 D1 D2 ˆ sm…q1 …3Y KD ††Y …PQ†
KP ˆ Y …IT†
B1 C2 À B2 C1
and
D1 B2 À D2 B1
KI ˆ Y …IU†
B1 C2 À B2 C1 q1 …3Y KD † ˆ 4X834 À 5X2KD Asin33
À …11X5 ‡ 10X4KD A™os†32
where  Ã
j 35 À …12 ‡ 5X2KD A™os†33 ‡ 10X4KD Asin32 Y
B1 ˆ ‚e…q2 …3††A™os ‡ sm…q2 …3††AsinY …IV† …PR†
C1 ˆ ‚e…q3 …3††A™os ‡ sm…q3 …3††AsinY …IW† q2 …3† ˆ À5X232 ‡ j10X43Y …PS†
D1 ˆ ‚e…q1 …3Y KD ††Y …PH† q3 …3† ˆ 10X4 ‡ j5X23X …PT†
B2 ˆ sm…q2 …3††A™os À ‚e…q2 …3††AsinY …PI†
Referring to (16) and (17), JÁB1 C2 À B2 C1 is
C2 ˆ sm…q3 …3††A™os À ‚e…q3 …3††AsinY …PP† the Jacobian. Let KD ˆ 0X01Y A ˆ 1, and ˆ 0, a
Fig. 8. Bode diagrams for the PID controlled uncertain plants G1 …s† and G2 …s†.

9. Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 427
stability boundary is plotted as shown in Fig. 2. pi …s†, i ˆ 2, 3, 4 are portrayed in Fig. 4, and Ji
The stability characteristics of the considered (i ˆ 1, 2, 3, 4) are the corresponding Jacobian of
polynomial are totally di€erent to the left and each vortex polynomial. We can, therefore, obtain
right of the boundary. Resorting to [10,11], it is a speci®c Kharitonov region in the plotted para-
concluded that if J b 0, then to the left of the sta- meter plane by ®nding the overlapped region, the
bility boundary facing the direction in which 3 hatched area, as shown in Fig. 4. The overlapped
increases is the stable area. Similarly, to the right area of the boundaries constitutes the entire fea-
of the stability boundary facing the direction in sible controller sets that can stabilize each parti-
which 3 increases is the stable region while J ` 0. cular selected edge polynomial. Within it, the KP ,
Then a graphical stability region in the controller KI , and KD of the PID controller can be arbitrarily
coecient parameter space as seen in Fig. 2 can be selected. To show that the obtained controller
obtained. For various KD , a family of stability parameter area can eciently compensate the
boundaries is portrayed in Fig. 3. It is realized that uncertain plant, a representative point P1 (KP ˆ 2,
with a larger KD , the stability region for the con- KI ˆ 1, KD ˆ 0X01) is picked for demonstration.
sidered plant would be larger. The resulting step responses are shown in Fig. 5. It
For example, choose KD ˆ 0X01. Following the is obvious that the designed controller stabilizes
same procedure, the other stability boundaries for all the four interval plants.
Fig. 9. Bode diagrams for the PID controlled uncertain plants G3 …s† and G4 …s†.

10. 428 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431
Table 1 end, the constant gain margin boundaries and
Gain margins and phase margins for the family of uncertain constant phase margin boundaries are plotted in
polynomials with designed PID controller (KP ˆ 1X72, the KP ÀKI plane. It isolates speci®ed gain margin
KI ˆ 0X05, KD ˆ 0X01)
and phase margin regions in the parameter plane
b2 b1 b0 Gain margin (db) Phase margin (deg) to allow ¯exible choice of the controller para-
3.5 12 9.5 5.5293 42.1600
meters. For the edge polynomial p1 …s†, the hatched
11.5 5.1832 50.6060 area as shown in Fig. 6 is the suitable area for
selecting the stabilizing controller coecients.
15 9.5 9.0530 44.9010
11.5 8.7540 54.0440 Concurrently, system speci®cations can be
achieved.
4.8 12 9.5 6.0765 30.5700 A non-conservative Kharitonov region is
11.5 6.0978 38.0700 obtained, as seen in Fig. 7, by searching for the
15 9.5 10.0000 37.9260 intersectional region of the speci®cation-limited
11.5 9.9410 51.7380 areas from all edge polynomials pi …s†, i ˆ 1, 2, 3, 4.
This region constitutes all the possible stabilizing
PID controller sets for system with robust safety
margins. In other words, the KP , KI , and KD can
Successively, we provide the uncertain system be readily chosen from the non-conservative
with a robust performance through specifying the Kharitonov region. A representative point P2
gain margins and phase margins. Toward this (KP ˆ 1X72, KI ˆ 0X05, KD ˆ 0X01) is selected.
Fig. 10. The variation of gain margins subject to parameteric change of the control system with the selected PID controller
(KP ˆ 1X72, KI ˆ 0X05, KD ˆ 0X01).

11. Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431 429
Bode diagrams are plotted in Figs. 8 and 9. It is conservative Kharitonov region in the coecient
veri®ed that the designed controller can really sta- plane.
bilize the interval system with robust margins. Output responses are shown in Fig. 12. For a
Gain margins and phase margins for the con- standard second system, the speci®cations in terms
sidered uncertain plant with designed PID con- of gain margin and phase margin are de®ned
troller (KP ˆ 1X72, KI ˆ 0X05, KD ˆ 0X01) are exactly corresponding to time domain perfor-
tabulated in Table. 1. The resulting gain margins mance requirement. However, for the considered
and phase margins subject to the uncertain coe- system, to inherit the same design idea, it is of
cients b0 , b1 , and b2 with the selected controller are much interest to de®ne interval gain margin and
illustrated in Fig. 10 and Fig. 11, respectively. phase margin speci®cations such that the time
Obviously, the speci®cations 5db4Gm410dbY and domain performance can also be guaranteed to
30 4Pm460 are satis®ed. Note that for any some degree. From Fig. 12, it is seen that the rise
other KD , the whole exploitable PID coecients time is less than 2 s and the maximum overshoot is
can be obtained, similarly, by searching the non- less than 50%. Henceforth, the linkage of the time
Fig. 11. The variation of phase margins subject to parameteric change of the control system with the selected PID controller
(KP ˆ 1X72, KI ˆ 0X05, KD ˆ 0X01).

12. 430 Y.J. Huang, Y.-J. Wang / ISA Transactions 39 (2000) 419±431
Fig. 12. Output responses for the four interval plants with KP ˆ 1X72, KI ˆ 0X05, KD ˆ 0X01.
performance and the gain and phase margin spe- The relationship between the varying coecients
ci®cations can also be evinced. of the uncertain characteristic polynomial and its
corresponding stable area in the plotted parameter
plane is studied. Simulation results reveal the
4. Conclusions remarkable performance and the e€ectiveness of
the developed method. In future work, the devel-
The so-called interval plants are often used for oped method will be extended to uncertain systems
dealing with uncertain dynamical systems. A with adjustable parameters and time delay, and
robustness stability analysis is useful to provide systems with inherent nonlinearities.
some ®nite checking of the stability of the closed
loop system so that one can design the controller to
stabilize the whole uncertain plant. However, much
of the present success in the control of interval
systems is restricted to the analysis issue. In this References
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