transactions of the institute of measurement and control-2012-rahimian-487-98

Article

Application of stability region centroidsin robust PI stabilization of a class ofsecond-order systems

Mohammad Amin Rahimian and Mohammad Saleh Tavazoei

AbstractIn this paper we offer a tuning method for the design of stabilizing PI controllers that utilizes the stability region centroid in the controller parameter

space. To this end, analytical formulas are derived to describe the stability boundaries of a class of relative-degree-one linear time invariant second-

order systems, the stability region of which has a closed convex shape. The so-called centroid stable point is then calculated analytically and the resultant

set of algebraic formulas are utilized to tune the controller parameters. The freedom to choose the surface density function in the calculation of

centroid stable point provides the designer with the possibility to incorporate optimal or robustness requirements in the controller design process.

The proposed method uses the stability regions in the controller parameter space to ensure closed-loop stability, and, while offering robust stability

properties, it does not rely on predetermined information with regard to the nature or range of parameter variations and coefficient uncertainty

bounds. Being situated away from the boundaries of the stability region in the controller parameter space, controllers designed based on the centroid

method are both robust and non-fragile.

KeywordsNumerator dynamics, optimal tuning, PI controller, robust tuning, second-order system, stability region, stabilizing controllers, surface density

Introduction

Favoured for their simple structure, eectiveness and robust-

ness, PID controllers remain of paramount importance in thecontrol of industrial processes (Astrom and Hagglund, 1995,2005). Moreover, good man-machine interfaces for the speci-cation of controller structure and parameters, as well as

ecient tuning tools, are necessary requirements for wide-spread adoption of any industrial controller; and methodsthat can expedite, ameliorate and simplify the tuning process

are highly sought after (Astrom and Hagglund, 2001).Among various design criteria to be satised by a control-

ler, closed-loop stability is the most basic, and the stability of

nal closed-loop systems for both fractional-order and inte-ger-order PID controllers have been the subject of many pre-vious studies (Nusret Tan et al., 2006; Hamamci, 2007; Fang

et al., 2009). The set of controller parameters yielding a stableclosed-loop system is known as the stability region, and a lotcan be learned about PID control through analysis of stabilityregions. The problem of nding all stabilizing PID controllers

for a given plant, which leads to the stability domains in thecontrollers parameter space, has traditionally been dealt withusing Nyquist plot, stability boundary locus, characteristic

equation and frequency-based methods. Applications of theHermiteBiehler theorem (Caponetto et al., 2010), as well aselaborate polynomial calculations (Hohenbichler and

Ackermann, 2003a; Soylemez and Baki, 2003;Hohenbichler, 2009), have been the driving force behindsome of the recent results on this topic. The author in

Hamamci (2008) has introduced a method for investigating

the stabilization of fractional-order PI controllers that is basedon plotting the global stability region in the (kp, ki) plane. A

similar method is adopted by the authors in Hamamci andKoksal (2010) to derive the stability regions for fractional-order PD controllers in the (kp, kd) plane. The authors inRahimian and Tavazoei (2010a,b) have provided an extension

of the method used in Hamamci (2008) to plot the stabilityregions for the integer-order approximations of PIl and PDm

controllers. The same methods form the basis of the design

scheme used in Rahimian et al. (2010) for the stabilization oftwo-mass drive systems with elastic coupling.

The main theme of this paper is the calculation of the cen-

troid point for the stable region in the controller parameterspace. This so-called centroid stable point serves as the designchoice, according to which controller parameters are set. The

centroid stable point has the advantage that it lies at the farth-est possible distance from the boundaries of the stability regionand will therefore ensure the robust stability of the resultingclosed-loop control system. Accordingly, while oering robust

stability properties, the proposed method does not rely on anyinformation pertaining to the nature or range of parametervariations. Furthermore, the freedom to choose the surface

Electrical Engineering Department, Sharif University of Technology, Iran

Corresponding author:

Mohammad Saleh Tavazoei, Electrical Engineering Department, Sharif

University of Technology, Tehran, Iran

Email: [email protected]

Transactions of the Institute of

Measurement and Control

34(4) 487498

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density in the calculation of centroid stable point provides amore exible tuning strategy, and therefore an easier way toachieve control requirements such as optimality or robustness.

The remainder of this paper is organized as follows. Themethod of Hohenbichler and Ackermann (2003b), Hamamci(2008) and Hamamci and Koksal (2010) is used in the nextsection to derive an analytical description for the boundaries

of the stability region in a closed-loop system comprised of aPI controller and a strictly proper second-order plant transferfunction. Next, in the section titled Centroid stable point

and the proposed tuning formulas, a set of conditions areset forth under which the stability region described in its pre-ceding section has a closed convex shape, for which the cen-

troid exists and is meaningful stability-wise. Following thespecication of constraints on the plant parameters, a con-stant value for the surface density function is assumed and the

centroid of the stability region is analytically calculated. Thesection titled Choice of surface density for robust and optimaltuning elaborates on the eects of the choice of surface densityon the calculation of the centroid stable point, and illustrates

how various surface density functions can be harnessed to pro-vide the designed control system with robustness and optimalityproperties. To this end, a specic example, where surface den-

sity function is adjusted to procure optimal disturbance rejec-tion properties, is discussed analytically, and algebraic formulasfor the corresponding centroid stable point are proered. Using

constant and tailored surface density functions, two choices ofcentroid stable points are computed for an example system insections Centroid stable point and the proposed tuning for-mulas and Choice of surface density for robust and optimal

tuning respectively, and the corresponding closed-loop sys-tems are simulated and compared in the section titledSimulation results and discussion. A second example in the

latter section compares the performance of the proposed tuningrules with classical ZieglerNichols rules. Lastly, some conclud-ing remarks are provided in the nal section concerning the

scope and possible extensions of the proposed approach.

Stability regions for the PI controller

Consider the basic closed-loop control system depicted inFigure 1, where y is the output and r is the reference input.

The inputoutput relation for the closed-loop system of

Figure 1 in the Laplace domain is given by

YsRs

CsGs1 CsGs , 1

with G(s) and C(s) indicating the plant and compensator

transfer functions, respectively; the characteristic equation

for the closed-loop system will be derived from setting thedenominator of (1) to zero, and is as follows:

1 CsGs 0: 2

The numerator of (2) is called the characteristic polynomialand is denoted by P(s). This paper focuses on the problem of

controlling a general second-order plant given by

Gs as bs2 cs d , 3

with a PI controller given by

Cs kp kis: 4

Numerous real-world processes can be shown to observe thetransfer function in (3), and second-order processes withnumerator dynamics are the focus of Seborg et al. (2004:

Section 6.1). There, it is explained that second-order systemswith right-half-plane (RHP) zeros exhibit the inverse-response phenomenon, where the direction of an initial

response to a step input contradicts that of its nal steadystate. The authors in Seborg et al. (2004) then describe twopractical scenarios, involving the liquid levels in a distillation

column and the temperature of a chemical reactor withexothermic reactions. In both cases, competing dynamiceects that operate on two dierent time scales lead to theinverse-response behaviour. Accordingly, second-order sys-

tems with numerator dynamics that exhibit inverse-responseor overshoot behaviour can occur whenever two physicaleects act on the process output variable in dierent ways

and with dierent time scales.Another classical example of a second-order process with

numerator dynamics is a mass-spring-dashpot (MSD) system

in series, where a mass, m, is attached to a spring with con-stant k, which is attached to a dashpot with damping coe-cient h, which is attached to a wall. Accordingly, the transfer

function from the force F, acting on the mass m, to its speed n,is given by

nsFs

hs kmhs2 mks kh : 5

Moreover, several of the plants that are investigated in the

literature are formulated in form of (3). These include thetransfer function from motor torque to load torque in atwo-mass drive system with elastic coupling in Rahimian

et al. (2010), and the transfer function from steeringangle to tilt angle for the unrideable bicycle in Klein(1989). Other examples from robotics include the free loadlinear actuator model for the biomimetic robot in Robinson

et al. (1999) and the transfer function of the end eectormotion/command position for the single-link manipulatormodel in Xu and Paul (1988). The transfer function

from the elevator deection (de) to the pitch rate (q) inight control systems (Oosterom and Babuska, 2006;El-Mahallawy et al., 2011) also follows the same dynamics

as described by (3).

r C(s) G(s) y+

Figure 1 A basic closed-loop control system, using unity negative

feedback.

488 Transactions of the Institute of Measurement and Control 34(4)


In addition to the aforementioned examples, in which realworld plants are modelled by the same type of transfer func-tions as (3), there are several cases where approximation,

order-reduction or identication methods lead to a formula-tion of the process dynamics that is in accordance with (3).The transfer function for every First-Order Plus Dead-Time(FOPDT) system, in which the time delay is replaced by a

rst-order Pade approximation, takes the form of a second-order system with a non-minimum phase zero in the numera-tor, as does the transfer function for every Second-Order Plus

Dead-Time (SOPDT) system, in which the exponential term isapproximated by the rst two terms in its Taylor series. Theauthor in Luus (1999) introduces a method to optimally

choose the coecients for a second-order reduced model,the transfer function for which is given by (3). The optimiza-tions are performed in the frequency domain, using a multi-

pass optimization technique, and the resulting second-orderreduced models can closely follow the Nyquist plots of theoriginal fth- and eighth-order systems. Last, but not least,plant models of the type in (3) can be the result of an identi-

cation procedure. The authors in Wang and Chen (2009) usesubspace system identication methods to determine thetransfer function matrices for the Multi-Input Multi-Output

dynamics of a proton exchange membrane fuel cell (PEMFC)system. The resulting transfer functions, after being convertedinto continuous-time by zero-order-hold, are all in the

form of (3).According to (2), the characteristic polynomial can be cal-

culated as

Ps s3 c akps2 d aki bkps bki: 6

The Hurwitz stability criterion for the characteristic polyno-

mial in (6) is satised if and only if every root of P(s) lies inthe left half plane (LHP). In other words, the imaginary axisand the origin are the only places where the stability shift of

the system will occur (Fang et al., 2009). However, the posi-tions of roots of P(s) will change continuously as long as itscoecients are continuous functions of the plant or controller

parameters and those parameters are changed continuously.Thus, a stable polynomial, P(s), whose roots all lie in theLHP, becomes unstable if and only if at least one root crossesthe imaginary axis. Accordingly, the set of coecients that

would lead to a stable closed-loop system can be denoted asstability domains in the parameter space of the characteristicpolynomial P(s). These stability domains are determined by

the following three boundaries, which describe the rootscrossing from the LHP to the RHP and vice versa(Hohenbichler and Ackermann, 2003b; Hamamci, 2008;

Hamamci and Koksal, 2010):

Denition 1. The Three Root Boundaries:

(a) Real Root Boundary (RRB): A real root crosses over theimaginary axis at the origin (s 0), and for P(s) given by(6) it can be determined as

bki 0! ki 0, 7

i.e. by setting the constant term to zero.

(b) Complex Root Boundary (CRB): A pair of complex rootswill cross over the imaginary axis at s jv, and their cor-responding locations in the parameter space are obtained by

substituting s jv in P(s) and setting its real and imaginaryparts to zero.

(c) Innite Root Boundary (IRB): The last possibility is for aroot to cross the imaginary axis at innity (|s|!), and itcan be determined by setting the coecient for the termwith the largest power of s in (6) to zero. Here, since theterm with the largest power corresponds to s3 and its coe-

cient cannot be zero, the IRB does not exist.

Once the coecients of (6) are described in terms of con-

troller parameters, the three boundaries given by Denition 1will translate into stability boundaries in the parameter spaceof the PI controller, i.e. {kp, ki}. These stability boundaries

separate dierent regions in the parameter space, and so, todetermine the stability of a given region, it suces to checkthe stability of one test point within that region (e.g. by theNyquist criterion or through investigation of the roots of the

corresponding characteristic polynomial).Next, substituting s jv in (6) and equating the real and

imaginary parts to zero results in the following two equations,

respectively:

bki c akpv2, 8

v2 d bkp aki: 9

Eliminating the parameter v between the two equations in (8)

and (9) yields

ki cd cb ad kp abkp2

b ca a2kp : 10

In light of the previous discussion, the stability boundaries for

the control system described by Figure 1 and equations (3)and (4) are given by equations (7) and (10), which specify theRRB and CRB, respectively.

Lastly, in order to determine the stability regions in thecontroller parameter space, we should test the stability ofsingle test points in each of the regions generated from theintersection of the boundaries given by (7) and (10).

Example 1. An unstable non-minimum-phase second-orderrelative-degree-one linear time invariant (LTI) system.

In order to demonstrate the aforementioned procedure,the following unstable non-minimum-phase second-orderrelative-degree-one LTI system, which is taken from Roup

and Bernstein (2003: Example 1), is considered:

G1s s 5s2 15s 5 : 11

The sample plant in (11) can be derived from (3) by setting

a 1, b5, c 15 and d5, and it is used throughout therest of the paper to demonstrate the applicability of the pro-posed method. Having an unstable pole at p 0.3262 and anon-minimum phase zero at z 5, the system in (11) is a

Rahimian and Tavazoei 489


typical hard-to-control second-order plant (Astrom, 2000).The presence of the RHP zero imposes a fundamental limita-tion on control, and high controller gains will induce closed-

loop instability (Skogestad and Postlethwaite, 1996). The sta-bility region for (11) can be computed using the boundaries in(7) and (10), and is depicted in Figure 2. The closed andconvex shape of the stability region in Figure 2, which occu-

pies a limited portion of the plane, is a further indication thatthe underlying closed-loop system is not well-behaved and ishard to control.

In the next section we nd the centroid of the stabilityregion depicted in Figure 2 analytically, and derive a set ofalgebraic formulas that can be used as a suitable choice for

the unknown controller parameters, kp and ki. Such a choicehas the advantage that it lies at the farthest possible distancefrom the boundaries of the stability region and would there-

fore ensure the robust stability of the resulting closed-loopcontrol system.

Centroid stable point and the proposedtuning formulas

This section focuses on the calculation of stability region cen-troids. This is motivated by the fact that the geometric centreof an objects shape can be a best option if the aim is to avoid

its boundaries and exterior as much as possible. This is trueprovided that the underlying object has a closed convexshape. Accordingly, prior to the calculation of centroidstable points, a set of constraints on the plant parameters a,

b, c and d are needed to ensure that the stability regions haveindeed a closed convex shape, as was the case with Figure 2,for which the stability region was an upward semi-parabola,

capped at the top by the kp axis. These conditions ensure that

calculation of the centroid for the stable region is justiedfrom the perspective of closed-loop system stability. Suchsystems, for which the stability region is bounded, would in

eect be hard to control, and examples often include planttransfer functions that are unstable or non-minimum phase.

Peering into (10), it will dawn on us that the CRB crossesthe kp axis at the following two points:

kpr1 c

a, 12

kpr2 d

b, 13

and it has a vertical asymptote at

kpa b ac

a2: 14

Depending on the relative locations of the two roots in

(12) and (13) and the asymptote in (14), several cases mayarise, each of which is realized under a particular set of con-ditions on the plant parameters. Among the many possibili-

ties, those specied in Tables 1 to 4 culminate in stabilityregions with a closed convex shape. The conditions inTable 1 and Table 2 yield a downward semi-parabola that

is capped at the bottom by the kp axis, while those inTable 3 and Table 4 yield an upward semi-parabola that iscapped at the top by the kp axis. Table 1 and Table 3 corre-spond to the cases where ab< 0, while Table 2 and Table 4 listthe conditions for the cases where ab> 0. The plant numera-tor dynamics in the former case include a non-minimumphase zero, leading to closed-loop systems that are particu-

larly hard to control. Each table lists the additional

40 30 20 10 0 10 20400

350

300

250

200

150

100

50

0

50

100

kp

k i

CRBRRB

Stability region

Figure 2 The stability region is plotted for the control system of Figure 1, where G(s) and C(s) are given by (11) and (4), respectively.



constraints that should be satised for various combinationsof signs of the parameters c and d.

The analytical formulas derived for the CRB in the previoussection provide us with the possibility of selecting a point inside

the stable region that is at the farthest possible distance fromevery point on the boundary. The coordinates (xc, yc) for thecentroid of a graph in the (x, y) plane are given by

xc R R

sx, yx dxdyM

, 15

yc R R

sx, yy dydxM

, 16

where M is given by

M Z Z

sx, y dydx, 17

and s(x, y) denotes the surface density function (Thomas andFinney, 1999). The double integrals in (15), (16) and (17) are

calculated over a stable region such as the one in Figure 2.Under the assumption that no information is available as tothe range or type of variations against which the closed-loop

system should exhibit robustness, the surface density s(x, y) isset to be a constant number and is thus eliminated from thenumerators and denominators of (15) and (16). Using (7) and

(10) for the boundaries of the stability region, and assumingthat one of the condition sets in Tables 1 to 4 are satised, thecentroid stable point for the closed-loop system of Figure 1,with the plant and controller given by (3) and (4) respectively,

can be calculated analytically. The results are the followingalgebraic tuning rules:

kpc I1

Mc, 18

kic I2

Mc, 19

where I1, I2 and Mc are given by

I1 S33a4b2

PS22a5b2

RS1a6

ac ba2

Q, 20

I2 S36a5b

PS22a6b

P2 3b2RS1

2a7b R b2

a3Q, 21

Mc S22a3b

S1Pa4b

Q, 22

and P, Q, R and Sn are dened as

P a2d b2, 23

Q bRa5

ln Rb2

, 24

R P abc, 25

Sn andn bncn, n 2 N: 26

The algebraic tuning formulas introduced in (18) and (19) can

be applied to Example 1 and the corresponding centroidstable point can be computed as

kpc1 8:8929,

kic1 8:9291: 27

Figure 3 denotes the centroid stable point along with thestability region for the system in (11).

Here it should be noted that the proposed choice of thestability region centroid as the tuning rule for setting control-ler parameters is inherently conservative. The centroid stable

point tuning rules in (18) and (19) are, in fact, a sucient butnot necessary condition for closed-loop system stability, andusing them as the basis for design is justied only when sta-

bilization is the rst and foremost criterion that is expected tobe satised by the control system. Such conservatism in

Table 3 Constraints on the plant parameters for the case where a> 0

and b< 0

Signs of parameters c and d Additional conditions

c< 0, d> 0 baca2\ db \ ca or ca \ db

c> 0, d< 0 baca2\ db \ ca or ca \ db

c< 0, d< 0 baca2\ db

c> 0, d> 0 None

Table 1 Constraints on the plant parameters for the case where a< 0

and b> 0


c< 0, d> 0 ca \ db \ baca2 or db \ cac> 0, d< 0 ca \ db \ baca2 or db \ cac< 0, d< 0 db \ baca2c> 0, d> 0 None

Table 4 Constraints on the plant parameters for the case where a> 0

and b> 0


c> 0, d> 0 ca \ db \ baca2 or db \ cac< 0, d< 0 ca \ db \ baca2 or db \ cac> 0, d< 0 db \ baca2c< 0, d> 0 None

Table 2 Constraints on the plant parameters for the case where a< 0

and b< 0


c< 0, d< 0 baca2\ db \ ca or db \ ca

c> 0, d> 0 baca2\ db \ ca or db \ ca

c> 0, d< 0 baca2\ db

c< 0, d> 0 None



design is best justied in the case of highly uncertain systems,where plant parameters can only be determined with a limitedprecision and are always subject to variations. Lack of infor-mation with regard to the nature and range of such variations

is a second factor which favours the use of the stability regioncentroid as the tuning point. Another class of systems forwhich the tuning rules of (18) and (19) are particularly

useful is the class of highly unstable systems with narrowstability regions, where it becomes critically important toavoid the boundaries of the stability region. Being situated

away from the boundaries of the stability region in the con-troller parameter space, controllers designed based on thecentroid method are both robust and non-fragile.

Accordingly, such controllers can tolerate certain variationsin the parameters of both the controller and the plant (Keeland Bhattacharyya, 1997; Makila et al., 1998). The parameteruncertainties can be due to either inherent physical properties

or modelling diculties, and having control systems whichremain operational in the face of parameter variations isdesirable in both cases.

In the next section, the eects of the choice of surfacedensity function on the calculation of the centroid stablepoint in (15) to (17) is investigated. It is further demonstrated

through an example that the choice of surface density func-tion can, in fact, be leveraged to produce desirable optimalityor robustness properties.

Choice of surface density for robust andoptimal tuning

In the previous section it was pointed out that in the absenceof any extra information the surface density function in (15),

(16) and (17) is considered to be a constant number, which is

then cancelled out in the calculations. The uniform density,in eect, corresponds to the case where robust stability is tobe guaranteed and no extra information is available withregard to variations of plant parameters and coecient

uncertainty bounds. This, however, need not necessarily bethe case. For instance, if we know how the shape of thestability region is aected by variations in the plant para-

meters, then we might be able to place the centroid so that itis properly distanced from sensitive locations. Accordingly,the undesirable locations are discriminated against by being

assigned lower weights when designating the surface densityfunction across the stability region. The following para-graphs elaborate on the choice of surface density function

and how it can be modied to satisfy various designspecications.

By and large, any tuning formula can be thought of as acentroid stable point which is derived through (15), (16) and

(17) with an appropriate choice of surface density function s.In general, the surface density function will be adjusted sothat less weight is given to the undesirable regions and the

centroid stable point is drawn toward more favourableregions because of their higher weights. To clarify thispoint, an example is given in which the surface density func-

tion is designed to provide the tuned control system withoptimal disturbance rejection characteristics.

Tuning methods based on an optimization approach havereceived considerable attention in the literature

(Panagopoulos et al., 2002). The problem of designing anoptimal PI controller, C*(s), given by (4), can be formulatedas determining the optimal parameters kp

and ki so that aset of constraints are satised and a cost function J(kp, ki) isminimized. In general, the parameter optimization designprocess consists of searching the space of variable controller

parameters as a function of some performance index J to

40 30 20 10 0 10 2030

20

10

0

10

20

30

40

50

60

kp

k iCRBCentroid stable pointRRB

Figure 3 The centroid stable point is depicted for the control system given by Figure 1 and equations (11) and (4).



determine where the performance index is minimized (Ogata,2009).

The occurrence of eective constraints which necessitate theoptimal solution to reside at the boundaries of the admissibleparameter domain would mean that there is no guarantee forthe necessary control requirements to remain satised under

various parameter variations and coecient uncertainties.Using the centroid approach, however, the centroid point isguaranteed to maintain a safe distance from admissible

region boundaries, while the optimality properties are moreor less preserved through the appropriate choice of surfacedensity function s(kp, ki). Such a choice of s(kp, ki) would

have to be a positive decreasing function of the cost J so thatthe regions with higher values of J are discriminated against.The positive s would ensure that the centroid is within the

admissible region, provided that it has a closed convex shape.In eect, it suces fors to not change sign across the region forwhich the centroid is being calculated. This is due to the factthat the actual positive or negative sign of a uni-sign function s

is cancelled out from the numerator and denominator of both(15) and (16), and hence it does not aect the nal result.

The freedom in choosing the function s can be a valuableasset in the management of the important trade-o betweenperformance and robustness in control systems (Kristiansson

and Lennartson, 2002). On the one hand, aggressive choicesof s, which decrease drastically with increasing J, would cul-minate in centroids that are closer to the truly optimal solu-

tions. Such aggressive choices, on the other hand, maycompromise the closed-loop system robustness by being situ-ated too close to the admissible region boundaries. In otherwords, with the choice of s, the designer can decide to move

away from a conservative design and closer to an optimalone, or vice versa.

Examples of decreasing functions which may be used todescribe the surface density function in terms of a positivecost function J include

s 1JN

, N 1, 2, 3, 4, . . . , 28

where lower values of N correspond to more conservativechoices, and higher values of N are chosen when aggressively

optimal solutions that are near the truly optimal one arepreferred. Depending on the relative importance of robust-ness or optimality properties, the power of J in the denomi-

nator of (28) may be decreased or increased. In the followingparagraphs an optimal tuning scenario is considered, in whichs is chosen as

s 1J: 29

Once the cost function J in (29) is described in terms of con-troller parameters kp and ki, the function ss*, given by(29), can be used in the context of (15) to (17) to calculate

the corresponding centroid stable point. Integrated Error (IE)is an integral performance index, which is dened (Astromand Hagglund, 1995) as

IE Z 0

et dt, 30

where e(t) is the error to a step input function. If the closed-loop system output is denoted by y(t), then e(t) is given by

et 1 yt, 8t>0: 31

Here, the cost function to be minimized is chosen as follows:

J jIEj: 32

The integral performance index in (30) can be used to evalu-ate the performance of a designed control system or, as here,

for optimal tuning of xed structure controllers. In the lattercase, the parameters of the control system are optimized byminimizing the integral performance index given by (32). In

Astrom and Hagglund (1995) it is shown that for a stableclosed-loop system, if the error is initially zero and a unitstep disturbance is applied at the plant input, then

IE 1ki: 33

Accordingly, the closed-loop system disturbance rejection canbe optimized by minimizing the IE criterion, and it is a nat-ural choice for the control of quality variables for processes

where the product is sent to a mixing tank (Astrom andHagglund, 1995). Several PID tuning rules based on minimi-zation of the IE criterion have been presented, and some

incorporate gain and phase margin specications to ensurestability and robustness of the controlled systems (Astromet al., 1998). The authors in Hwang and Hsiao (2002), forinstance, present an eective solution to the non-convex opti-

mization problem, which arises in the design of stabilizing PIand PID controllers based on the minimization of the integralof the error caused by a step disturbance and subject to con-

straints on maximum sensitivity Ms.

Using the results in (29), (32) and (33), we can write

skp, ki 1J jkij: 34

Next, supposing that one of the condition sets in Tables 1 to 4

is satised, the surface density function given in (34) can beused in the context of (15) to (17) to calculate the centroidstable point for the stability boundaries given by (7) and (10).The resultant centroid stable point is given by

kp I1

I2, 35

ki I2

I2, 36

where I2 is given by (21). I1 and I2 in (35) and (36) are

given by

I1 f1 kpr2

f1 kpr1 , 37I2 f2 kpr2

f2 kpr1 , 38



where kpr1 and kp

r2 are given by (12) and (13), and the func-tions f1() and f2() are dened in the Appendix.

The algebraic tuning formulas introduced in (35) to (38)

can be applied to Example 1, and the corresponding optimalcentroid stable point can be computed as

kp1 9:2808,

ki1 12:8080: 39

Figure 4 denotes the optimal centroid stable point given by

(39), along with the centroid stable point in (27) computed forthe system in Example 1.

In the next section the two choices of controllers given by

the two centroid points computed in this and the previoussections for the sample plant (11) are simulated, and the per-formance of the corresponding closed-loop systems are com-

pared. A second example compares the performance of twoPI controllers: one is designed using the proposed centroidscheme, the other is tuned using the ZieglerNichols fre-quency domain method.

Simulation results and discussion

The output responses of the closed-loop system of Figure 1with the sample system in (11) and the two controllers givenby (27) and (39) are plotted in Figure 5. The results show that,

being situated at a farther distance from the stability regionboundaries, the controller in (27) yields a less oscillatoryresponse to the unit step input applied at t 0. The controllerin (39) is, however, more eective in the rejection of the unit

step disturbance which is applied to the plant input at t 5.The Integrated Error (IE) is a common performance index,

which is dened as in (30). The value of IE for a unit stepdisturbance has been calculated in Astrom and Hagglund(1995) and is given by (33). Accordingly, the corresponding

values for the controllers in (27) and (39) are 0.1120 and0.0781. This conrms that the integral of the error causedby the unit disturbance at t 5 has a lower absolute value forthe controller in (39). This is to be expected, since the con-

troller in (39) has been designed for an optimal performancewith regard to the IE performance index, and for its distur-bance rejection.

Next, in order to compare the robust stability of the twosystems, uncertainties are introduced in plant parameters aand c and their eects are investigated. First, a is increased

from its nominal value of a 1 until the corresponding systemis no longer stable. The closed-loop system with the controllerin (27) destabilizes at a 1.5, whereas the one with the sub-optimal controller given by (39) becomes unstable at a 1.34.This conrms the superior robustness of the centroid stablepoint as compared to its optimal version. Repeating theexperiment by decreasing c from its nominal value of c 15results in a similar observation. The closed-loop systems withthe controllers in (27) and (39) destabilize at c 10.37 andc 11.51 respectively. It is worth highlighting that the trueoptimal solution, which minimizes the cost function in (32)subject to the stability condition, corresponds to the base ofthe downward semi-parabola in Figure 4, and thus lies on the

boundary of the stability region.Last, but not least, it is worth mentioning that the presence

of a non-minimum phase zero at z 5 for the plant in (11)imposes inherent restrictions on the disturbance rejection

properties of the closed-loop system (Astrom, 2000). Due tothe so-called push-pull eect, low-frequency non-minimum

40 30 20 10 0 10 2030

20

10

0

10

20

30

40

50

60

kp

k i

Centroid stable points

s = constant

s = 1J

= |ki|

Figure 4 The two centroid stable points, derived using a constant and an optimal s, are depicted above.



phase zeros inhibit superior disturbance rejection at low fre-quencies (Freudenberg and Looze, 1998). Accordingly,increasing the parameter a from its nominal value of a 1causes the zero at z 5a to move closer to the origin, thusfurther degrading the disturbance rejection at low frequen-cies. The next example compares the performance of the

proposed tuning rules with classical ZieglerNichols rulesfor an inverse response process.

Example 2. PI-control of an inverse response process.Here, Skogestad and Postlethwaite (1996: Example 2.3) is

considered and the performance of the proposed centroid

0 20 40 60 80 100 1200.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t (s)

y(t)

Unit step responses

Ziegler and NicholsCentroid stable point

Figure 6 The responses for the closed-loop system of Figure 1, with (40) as the plant and the two PI controllers specified by (41) and (42).

0 2 4 6 8 101

0.5

0

0.5

1

1.5

2

2.5

t

y(t)

Unit step and disturbance responses

s = constant

s = 1J = |k i|

Figure 5 The responses for the closed-loop system of Figure 1, with (11) as the plant and the two PI controllers specified by (27) and (39).



stable point controller is compared with a PI controller that isdesigned according to the classical tuning rules of Ziegler andNichols. The plant model (time in seconds) is given by

G2s 32s 15s 110s 1 : 40

The ZieglerNichols PI controller parameters, which arederived from the ultimate gain and ultimate period accordingto the frequency response method (Astrom and Hagglund,

1995), are set as follows (Skogestad and Postlethwaite, 1996):

kpZN 1:1400,

kiZN 0:0898: 41

The algebraic tuning formulas introduced in (18) and (19) canbe applied to the plant in (40) and the corresponding centroidstable point can be computed as

kpc2 1:1563,

kic2 0:0730: 42

The output responses and control signals for the closed-loopsystem of Figure 1 with the sample system in (40) and the twocontrollers given by (41) and (42) are plotted in Figures 6 and

7 respectively. It is important for the control signals not toviolate the actuator and plant dynamics due to the presence ofnonlinearities such as saturation.

The presence of integral action in form of a PI controllerremoves the steady-state oset in the step response of theclosed-loop system. Table 5 compares various performance

measures for the two control systems and their correspondingstep responses. The results conrm that the ZieglerNicholstuning rules are somewhat aggressive, culminating in a closed-loop system with smaller stability margins and a more oscil-

latory response (Skogestad and Postlethwaite, 1996). On theother hand, the conservative nature of the centroid tuningrules leads to a closed-loop system with larger stability mar-

gins and a superior transient response. The conservative rules,proposed in (18) and (19), are particularly useful for thetuning of systems with narrow stability regions, such as the

one in Example 2, for which it is critically important to avoidthe boundaries.

Conclusion

This paper oered a set of algebraic tuning formulas thatwould ensure robust stability of the closed-loop system without

0 20 40 60 80 100 1200.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t (s)

u(t)

Control signals for a unit step reference input

Ziegler and NicholsCentroid stable point

Figure 7 The control signals for the closed-loop system of Figure 1, with (40) as the plant and the two PI controllers specified by (41) and (42).

Table 5 Comparison of the performance measures for centroid stable

point and frequency domain ZieglerNichols tuning rules

Performance measure Centroid ZN

Phase margin 22.93558 19.30198

Gain margin 1.7250 1.6298

Phase crossover frequency 0.353 rad/s 0.3375 rad/s

Gain crossover frequency 0.2363 rad/s 0.2370 rad/s

Rise time (0 90%) 8.0669 s 7.9916 sSettling time (65%) 52.7526 s 65.2532 s

Overshoot 53.5915% 63.0855%

Decay ratio 0.3046 0.3602

Maximum sensitivity (Ms) 3.4579 3.9373



exploiting any prior knowledge as to the nature or range ofparameter variations in the plant or controller. The controllerparameters are chosen so that the designed system will lie at

the farthest possible distance from every point on the bound-ary of the stability region in the controller parameter space.Analytical formulas for the stability regions of a generalsecond-order plant were derived and the desired controller

parameters were selected as the centroid of the stabilityregions. Moreover, a set of conditions were introducedwhich will ensure that the stability regions have a closed

convex shape, and the calculation of centroids is thereforemeaningful stability-wise. Unlike classical robust stabilizationtechniques, the stability region centroid approach proposed in

this paper does not require the coecient uncertainty boundsto be known or satisfy any inequality constraints. The con-servative nature of the proposed tuning rules can prove par-

ticularly useful in the control of closed-loop systems withnarrow stability regions or highly uncertain parameters. Thedevised scheme, which is developed for PI controllers and aclass of second-order relative-degree-one LTI plants, can be

further extended to PD and PID controllers as well as to thegeneral case of controllers for which the stability region has aclosed convex shape. In the case of PID and other three-para-

meter controllers, however, the design parameter spaceinvolves three unknowns (kp, ki, kd) and the resulting calcula-tions, which involve the centre of mass for a three-dimen-

sional gure, are analytically cumbersome.

Acknowledgment

The authors would like to thank Samira Rahimian for her help with

calculation of centroids and derivation of conditions in Tables 1 to 4.

Funding

This research received no specic grant from any funding agency in

the public, commercial or not-for-prot sectors.

Conflict of interest

None declared.

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Appendix

The centroid stable point with s*(kp, ki) |ki|Functions f1() and f2(), which appear in the formulation ofthe optimal centroid stable point in (37) and (38), are denedas follows:

f1x b2x4

8a2 P2x2

4a6 x33a4

bR2

2a10gx ac b

l x

hx bPRa6

kx,

f2x b3x4

12a3 P3x

3a9 bR

3

6a11l2x b2Px3

3a5 bP2x2

2a7

bP2Rgxa11

b3R2a11

gx ac bl x

bahx

b2PR2a11l x

2b2PRa7

kx,

where P and R are dened in (23) and (25), and l(x), g(x), h(x)and k(x) are given by

l x a2x ca b,gx ln l x ,hx b2R

a10a2c2 b2 2abc gx 0:5x2a4 xa2ac b ,

kx xa2 ac b

a4gx:



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