design of an enhanced nonlinear pid controllerdownloads.deusm.com/designnews/1477-elsevier06.pdf ·...

Mechatronics 15 (2005) 1005–1024

Design of an enhanced nonlinearPID controller

Y.X. Su a,b,*, Dong Sun a, B.Y. Duan b

a Department of Manufacturing Engineering and Engineering Management,

City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kongb School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, China

Accepted 2 March 2005

Abstract

An enhanced nonlinear PID (EN-PID) controller that exhibits the improved performancethan the conventional linear fixed-gain PID controller is proposed in this paper, by incorpo-rating a sector-bounded nonlinear gain in cascade with a conventional PID control architec-ture. To achieve the high robustness against noise, two nonlinear tracking differentiators areused to select high-quality differential signal in the presence of measurement noise. The crite-rion to determine the nonlinear gain to retain the stability of the proposed EN-PID controlsystem is addressed, by using the Popov stability criterion. The main advantages of the pro-posed EN-PID controller lie in its high robustness against noise and easy of implementation.Simulation results performed on a robot manipulator are presented to demonstrate the betterperformance of the developed EN-PID controller than the conventional fixed-gain PIDcontroller.� 2005 Published by Elsevier Ltd.

Keywords: PID control; Nonlinear PID control; Tracking filters; Measurement noise; Robotic control

0957-4158/$ - see front matter � 2005 Published by Elsevier Ltd.doi:10.1016/j.mechatronics.2005.03.003

* Corresponding author. Address: School of Electro-Mechanical Engineering, Xidian University, Xi�an710071, China. Tel.: +86 29 8820 2262; fax: +86 29 8821 3676.

E-mail address: [email protected] (Y.X. Su).

mailto:[email protected]

1006 Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024

1. Introduction

Proportional-integral-derivative (PID) controllers have been the most popular andthe most commonly used industrial controllers in the past years. The popularity andwidespread use of PID controllers are attributed primarily to their simplicity and per-formance characteristics. PID controllers have been utilized for control of diversedynamical systems ranged from industrial process to aircraft and ship dynamics [1–4].

Although linear fixed-gain PID controllers are often adequate for controllinga nominal physical process, the requirements for high-performance control withchanges in operating conditions or environmental parameters are often beyond thecapabilities of simple PID controllers. In order to enhance the performance of linearPID controllers, many approaches have been developed to improve the adaptabilityand robustness by adopting the self-tuning method, general predictive control, fuzzylogic and neural networks strategy, and other methods [5–23].

Amongst these approaches, nonlinear PID (N-PID) control is viewed as one ofthe most effective and simple method for industrial applications [12–23]. NonlinearPID control may be any control structure of the following form:

uðtÞ ¼ kpð�ÞeðtÞ þ kið�ÞZ

eðsÞdsþ kdð�Þ _eðtÞ ð1Þ

where kp(Æ), ki(Æ) and kd(Æ) are time-varying controller gains, which may depend onsystem state, input, or other variables, and u(t) and e(t) are the system input anderror, respectively.

The nonlinear PID (N-PID) control has found two broad classes of applications:

(1) nonlinear systems, where N-PID control is used to accommodate the nonlin-earity, usually to achieve consistent response across a range of conditions[12–15]; and

(2) linear systems, where N-PID control is used to achieve performance notachievable by a linear PID control, such as increased damping, reduced risetime for step or rapid inputs, improved tracking accuracy, and friction com-pensation [16–23].

For linear systems, two broad categories of N-PID control are found [16]: thosewith gains modulated according to the magnitude of the state [22,23], and those withgains modulated according to the phase of the state [16–21]. The former category ofN-PID will be used in this paper. According to the magnitude of the state, theenhancement of the controller is achieved by adapting its response based on the per-formance of the closed-loop control system. When the error between the commandedand actual values of the controlled variable is large, the gain amplifies the error sub-stantially to generate a large correction to rapidly drive the system output to its goal.As the error diminishes, the gain is automatically reduced to prevent excessive oscil-lations and large overshoots in the response. Because of this automatic gain adjust-ment, the N-PID controllers enjoy the advantage of high initial gain to obtain a fastresponse, followed by a low gain to prevent an oscillatory behavior [16].

Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024 1007

In this study, a performance enhancement to the conventional linear PID control-ler is proposed by incorporating a sector-bounded nonlinear gain into a linear fixed-gain PID control architecture. To achieve the high robustness against noise, twononlinear tracking differentiators are used to select the high-quality differential signalin the presence of measurement noise.

The paper is organized as follows. A nonlinear tracking differentiator is developedin Section 2, which is followed by the development of an enhanced nonlinear PID(EN-PID) controller in Section 3. In Section 4, the stability of the proposed EN-PID controller is analyzed. To verify the performance enhancement, simulationsare performed on a two-link revolute robot in Section 5. Finally, some remarkingconclusions are summarized in Section 6.

2. Nonlinear tracking differentiator (TD) design

In practice, optical encoder is still the most popular accurate position sensor usedin the industrial fields because of its simple detection circuit, high resolution, highaccuracy, and relative ease of adaptation in digital control systems. Conversely,the velocity measurement, obtained by means of tachometers, is often contaminatedby noise. Furthermore, the use of sensors in the control architecture inherently in-creases the possibility of failure and increases the cost of the control system dueto additional hardware. Therefore, it is necessary to numerically reconstruct velocitysignal from position measurement with noise [15,24–28].

The so-called nonlinear tracking differentiator (TD) [15,27,28] is referred to as thefollowing system: given a reference signal r(t), the system provides two signals r1(t)and r2(t), such that r1(t) = r(t) and r2ðtÞ ¼ _rðtÞ.

For better interpretation of TD, the following lemmas are firstly given.

Lemma 1. Suppose z(t) is a continuous function defined in [0,1) that satisfieslimt!1

zðtÞ ¼ 0. If r(t) = z(Rt), R > 0, then for an arbitrary given T > 0, the following

expression holds

limR!1

Z T

0

jrðtÞjdt ¼ 0 ð2Þ

Proof. Based on the mean value theorem, there is a s between 0 and T such thatZ T

0

jrðtÞjdt ¼ T jrðsÞj ¼ T jzðRsÞj ð3Þ

Since limt!1

zðtÞ ¼ 0, it is straightforward that

limR!1

Z T

0

jrðtÞjdt ¼ limR!1

ðT jzðRsÞjÞ ¼ 0

Thus Lemma 1 is just established. h


Lemma 2. If the solutions to the system (4) hold that z1(t)! 0 and z2(t) ! 0 as t ! 1_z1 ¼ z2_z2 ¼ f ðz1; z2Þ

�ð4Þ

then for an arbitrarily constant c and T > 0, the solution r1(t) to the system

_r1 ¼ r2_r2 ¼ R2f r1 � c;

r2R

� �(ð5Þ

makes the following expression hold

limR!1

Z T

0

jr1ðtÞ � cjdt ¼ 0 ð6Þ

Proof. Firstly, change the variable of integration as follows:

s ¼ tR

r1ðsÞ ¼ z1ðtÞ þ cr2ðsÞ ¼ Rz2ðtÞ

8><>: ð7Þ

Based on the mean value theorem, there is a s between 0 and T such thatZ T

0

jr1ðtÞ � cjdt ¼ T jr1ðsÞ � cj ¼ T jz1ðtÞj ð8Þ

Using Lemma 1, it follows that

limR!1

Z T

0

jr1ðtÞ � cjdt ¼ limR!1

ðT jz1ðtÞjÞ ¼ T limR!1

jz1ðRsÞj ¼ 0

Therefore, Lemma 2 is right justified. h

Theorem 1 [27]. If the arbitrary solutions to system (4) satisfy z1(t) ! 0 and z2(t) ! 0

as t ! 1, then for any arbitrarily bounded integrable function r(t) and given constant

T > 0, the solution r1(t) to the system

_r1 ¼ r2

_r2 ¼ R2f r1 � r;r2R

� �(ð9Þ

satisfies

limR!1

Z T

0

jr1ðtÞ � rðtÞjdt ¼ 0 ð10Þ

Proof. The proof can be justified by dividing into the following two cases:

Case 1: If r(t) is a constant function, then Theorem 1 is Lemma 2.


Case 2: If r(t) (t 2 [0,T]) is a bounded integral function, then it is an element ofL1[0,T], where L1[0,T] denotes the set of all first-integrable function inthe range of [0,T]. For an arbitrarily given e > 0, there exists a simple seriesun(t) (n = 1,2, . . .) that uniformly converges to a continuous functionu(t) 2 C[0,T] such that [27]Z T

0

jrðtÞ � uðtÞjdt < e2

ð11Þ

Thus, there exists an integer N0 such that juðtÞ � uMðtÞj < e4T for all M > N0. Conse-

quently, the following inequality holds

Z T

0

jrðtÞ � uMðtÞjdt 6Z T

0

jrðtÞ � uðtÞjdt þZ T

0

juðtÞ � uMðtÞjdt <e2

ð12Þ

Since u(t) is a continuous function, the simple series u (t) that partitions the
M
range [0,T] into some bounded intervals denoted by li (i = 1,2, . . . ,m). SelectinguM(t) to be a deterministic constant in each bounded interval, and based on Lemma2, there exists R0 > 0 such thatZ

li

jr1ðtÞ � uMðtÞjdt <e2m

; i ¼ 1; 2; . . . ;m ð13Þ

for all R > R0. Then,Z T

0

jr1ðtÞ � uMðtÞjdt <e2

ð14Þ

Thereby, the following inequality holdsZ T

0

jr1ðtÞ � rðtÞjdt <Z T

0

jr1ðtÞ � uMðtÞjdt þZ T

0

juMðtÞ � rðtÞjdt < e ð15Þ

for all R > R0.The proof is right justified. h

Theorem 1 shows that r1(t) averagely converges to r(t). If the bounded integrablefunction r(t) is viewed as a generalized function, then r2(t) weakly converges to thegeneralized derivative of r(t). Therefore, system (9) can be used as a nonlinear track-ing differentiator to provide a smooth approach to the original generalized functionand its generalized derivative in the sense of average convergence and weak conver-gence, respectively.

One feasible second-order TD for the reference signal can be expressed as [15,27]

_z1 ¼ z2

_z2 ¼ �R sat z1 � r þ z2 j z2 j2R

; d

� �8<: ð16Þ

where R and d are two positive design parameters, and sat(A,d) is the following non-linear saturation function:


satðA; dÞ ¼sgnðAÞ; jAj > dAd; jAj 6 d

8<: ð17Þ

in which sgn(Æ) stands for a standard signum function.The parameters R and d can be determined empirically. R is named as ‘‘velocity

factor’’, and d ‘‘filtering factor’’, respectively. Large R is helpful to fast the transitionand tracking, and large d is helpful to cancel out noise. In general, for the referencesignal corrupted with maximum amplitude of 0.01 white noise component, R can bechosen from the range of [2.5 50]; for the reference signal with large noise magnitudethan 0.01, R will be increased to preserve a better tracking performance at the sac-rifice of differential signal obtained. There is a tradeoff between the tracking andfiltering.

The better performance of TD is shown in Fig. 1. The reference input isrðtÞ ¼ sinðtÞ mm and it is perturbed by an additive white noise component withthe maximum amplitude of 0.01 mm. For comparison, the differential signal ob-tained by the conventional backward differentiator is also shown in Fig. 2. The sim-ulations were programmed in Matlab with a fourth-order Runge–Kutta and run ona PC Pentium III-860. The simulation step is determined as h = 0.0025, and the ini-tial values of z1 and z2 is z1(0) = 0 and z2(0) = 0. The parameters of the TD are deter-mined as R = 20, and d = 0.001. It can be seen that the differential tracking

0 10 15 20

-1.0

-0.5

0.0

0.5

1.0

Pos

ition

(ra

d)

Time (s)

Reference Estimate

0 10 15 20-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

Pos

ition

err

or (

rad)

Time (s)

0 10 15 20-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Vel

ocity

(ra

d/s)

Time (s)

Reference Estimate

0 10 15 20-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Vel

ocity

err

or (

rad/

s)

Time (s)

5 5

5 5

Fig. 1. Performance of TD with noise.

0 10 15 20

-4

-2

0

2

4

Vel

ocity

err

or (

rad/

s)

Time (s)0 10 15 20

-6

-4

-2

0

2

4

6

Vel

ocity

(ra

d/s)

Time (s)

Reference Obtained by backward difference

5 5

Fig. 2. Performance of backward differentiator with noise.


performance of TD is much better over that of the conventional backward differencemethod.

3. Enhanced nonlinear PID (EN-PID) controller

The proposed enhanced nonlinear PID (EN-PID) controller consists of a sector-bounded nonlinear gain k(e), a linear fixed-gain PID controller expressed byK(s) = kp + ki/s + kds, and two nonlinear tracking differentiators (TDs), as shownin Fig. 3. kp, ki, and kd are proportional, integral, and derivative gains, which canbe determined by the Ziegler–Nichols criterion [1]. The nonlinear gain k(e) is asector-bounded function of the error e(t), and acts on the error to produce the‘‘scaled’’ error f(e) = k(e) Æ e(t). Using the high-quality differential signal selected bythe developed TDs, we have the following enhanced nonlinear N-PID control law:

uðtÞ ¼ kp þ ki

Z t

0

dt þ kdd

dt

� �� f ðeÞ

¼ kp½kðeÞ � eðtÞ� þ ki

Z t

0

½kðeÞ � eðtÞ�dt þ kd ½kðeÞ � cðtÞ� ð18Þ

where e(t) and c(t) are expressed as, respectively,

-+ur y

e sks

kksK d

ip ++=)(

c

4

3

2

1

-

+

z

z

zz

TD (II)

TD( I )

Plantk

N-PID controller

Fig. 3. Block diagram of the EN-PID control system.

0-10 -5 101.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Non

linea

r ga

in

Error

k0

= 0.125 k0

= 0.15

5

Fig. 4. Variation of the nonlinear gain k(e) with the error e.


eðtÞ ¼ z1ðtÞ � z3ðtÞ ð19ÞcðtÞ ¼ z2ðtÞ � z4ðtÞ ð20Þ

The nonlinear gain k(e) represents any general nonlinear function of the error ewhich is bounded in the sector 0 6 k(e) 6 kmax. There is a broad range of optionsavailable for the nonlinear gain k(e) [21]. One simple form of the nonlinear gain func-tion can be expressed as

kðeÞ ¼ chðk0eÞ ¼expðk0eÞ þ expð�k0eÞ

2

e ¼e jej 6 emax

emax sgnðeÞ jej > emax

� ð21Þ

where k0 and emax are user-defined positive constants. The nonlinear gain k(e) is lower-bounded by k(e)min = 1 when e = 0, and upper-bounded by k(e)max = ch(k0emax).Therefore, emax denotes the range of variation, and k0 defines the rate of variation ofk(e). Fig. 4 illustrates a typical variation of k(e) with respect to e when k0 = 0.125and k0 = 0.15, with emax = 10. It can be seen that the gain k(e) is an even function ofe, that is, k(�e) = k(e).

As a result, the proposed EN-PID controller can be realized by cascading this sec-tor-bounded nonlinear gain k(e) with the available linear fixed-gain PID controller,using the high-quality differential signal selected by the developed TDs.

4. Determination of nonlinear gain

In many applications, the dynamics of the system to be controlled can be ade-quately modeled by a second-order differential equation. Even when the systemdynamics is of high order, the response of the system is often largely dependent


on the location of a pair of dominant complex poles, which can be embodied in asecond-order model [21,29]. For these systems, the second-order transfer functionrelating the system output y(t) to the control action u(t) can be expressed as

GðsÞ ¼ yðsÞuðsÞ ¼

x2nk

s2 þ 2nxnsþ x2n

¼ cs2 þ asþ b

ð22Þ

where n, xn, and k are the damping ratio, natural frequency, and DC gain of the con-trolled system, respectively, a = 2nxn, b ¼ x2

n, and c ¼ x2nk.

For such a control system, it has been proven that there exists a suitable choice ofthe PID controller parameters so that the overall closed-loop system is globallyasymptotically stable for the position (set-point) control [30–32]. Since the conver-gence of the developed nonlinear tracking differentiator (TD) has just been investi-gated, in this section, the criterion to determine the nonlinear gains to retain thestability of the proposed EN-PID control system that incorporates a sector-boundednonlinear gain in cascade with a stable linear fixed-gain PID control system isaddressed, following the idea presented by Seraji [21] using the Popov stabilitycriterion. The Popov stability criterion can be stated graphically as follows: A suffi-cient condition for the closed-loop system to be global asymptotically stable for allnonlinear gains in the sector 0 6 k(e) 6 k(e)max is that the Popov plot of W(jx) liesentirely to the right of a straight line with a nonnegative slope passing through thepoint �1/kmax + j0 [33,34].

For some applications, it is desirable to introduce the sector-bounded nonlineargain k(e) on some terms of the linear fixed-gain PID controller, instead of equallyon all three terms. In this section, we will investigate the criterion to determine thenonlinear gains for the three complex cases: the enhanced nonlinear PI, PD, andPID controllers, by using the Popov stability criterion, respectively. The criterionto determine the nonlinear gains for other partially enhanced nonlinear P, I, andD controller can be analyzed in a similar way.

4.1. Determination of nonlinear gains for EN-PI controllers

In this case, the enhanced nonlinear PI control system can be viewed as the fol-lowing linear third-order transfer function W(s):

W ðsÞ ¼ KðsÞGðsÞ ¼ cðkpsþ kiÞsðs2 þ asþ bÞ ð23Þ

which describes the controlled plant (22) and the linear fixed-gain PI controller cas-cades with the sector-bounded nonlinear gain k(e). kp and ki are positive constantproportional and integral gains of the linear PI controller, respectively.

To apply the Popov stability criterion, the crossing of the Popov plot of W(jx)with the real axis should be computed. This plot reveals the range of values thatthe sector-bounded nonlinear gain k(e) can be assumed while retaining the closed-loop stability. From (23), the real and imaginary parts of W(jx) can be calculated[21,34]


ReW ðjxÞ ¼ �cðkpx2 þ aki � bkpÞa2x2 þ ðb� x2Þ2

ð24Þ

x ImW ðjxÞ ¼ �c½ðakp � kiÞx2 þ bki�a2x2 þ ðb� x2Þ2

ð25Þ

The Popov plot of W(jx) starts at the point P(�c(aki � bkp)/b2, �cki/b) for x = 0

and terminates at the point Q(0,0) for x = 1. Two cases are now possible, depend-ing on the relative values of kp and ki.

Case 1: ki 6 akp. In this case, x ImW(jx) is always negative for all x, that is, thePopov plot of W(jx) remains entirely in the third and fourth quadrants and does notcross the real axis. This implies that one can construct a straight line with a nonneg-ative slope passing through the origin such that the Popov plot is entirely to the rightof this line. Therefore, according to the Popov stability criterion [33,34], the range ofthe allowable nonlinear gain k(e) is (0,1). A typical Popov plot for a = 3, b = 2, andc = 10, with kp = 1 and ki = 0.5, is shown in Fig. 5a.

Case 2: ki > akp. The crossover frequency x0 of W(jx) to the real axis can befound by solving x ImW(jx) = 0 to yield [21,34]

x20 ¼

bkiki � akp

ð26Þ

As a result, the value of W(jx) at the crossover is

ReW ðjx0Þ ¼cðakp � kiÞ

abð27Þ

It indicates that the Popov plot crosses the negative real axis. So, the maximumallowable nonlinear gain can be calculated as [21]

kðeÞmax ¼ � 1

ReW ðjx0Þ¼ ab

cðakp � kiÞð28Þ

Then a straight line with a nonnegative slope passing through the point �1/kmax + j0can be constructed such that the Popov plot of W(jx) is entirely to the right of this

-1.0 -0.5 0.0 0.5 1.0-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5a b

Fig. 5. Popov plot for the EN-PI control system: (a) Case 1 and (b) Case 2.


line. The range of the allowable nonlinear gain k(e) is (0,k(e)max). A typical Popovplot for a = 3, b = 2, c = 10, with kp = 0 and ki = 0.5, is shown in Fig. 5b.

4.2. Determination of nonlinear gains for EN-PD controllers

In this case, the closed-loop system can be separated out the nonlinear gain k(e)and a second-order transfer function W(s) which consists of the PD controller andthe plant (22), that is,

W ðsÞ ¼ KðsÞGðsÞ ¼ cðkp þ kdsÞs2 þ asþ b

ð29Þ

From (29), the following expressions of Popov plot W(jx) can be obtained [21,34]

ReW ðjxÞ ¼ c½ðakd � kpÞx2 þ bkp�a2x2 þ ðb� x2Þ2

ð30Þ

x ImW ðjxÞ ¼ �cx2ðkdx2 þ akp � bkdÞa2x2 þ ðb� x2Þ2

ð31Þ

It can be seen that, the Popov plot of W(jx) starts at the point P(ckp/b, 0) for x = 0and terminates at the point Q(0,�ckd) for x = 1. Two distinct cases are now pos-sible, depending on the relative values of kp and kd.

Case 1: bkd 6 akp. In this case, from (31) it can be seen that x ImW(jx) is alwaysnegative for all nonzero x, that is, the Popov plot of W(jx) remains entirely in thethird and fourth quadrants and does not cross the real axis. Therefore, according tothe Popov stability criterion [33,34], the range of the allowable nonlinear gain k(e) is(0,1). Fig. 6a illustrates a typical Popov plot for a = 3, b = 2, and c = 10, withkp = 1 and kd = 0.5.

Case 2: bkd > akp. The crossover frequency x0 of W(jx) to the real axis is foundby solving x ImW(jx) = 0 to yield

x20 ¼

bkd � akpkd

ð32Þ

0 3-5

-4

-3

-2

-1

0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

-5

-4

-3

-2

-1

0

1 ba

1 2 4 5

Fig. 6. Popov plot for the EN-PD control system: (a) Case 1 and (b) Case 2.


and the value of W(jx0) is then found to be

ReW ðjx0Þ ¼ckda

ð33Þ

which indicates that the Popov plot crosses the positive real axis. The general shapeof the Popov plot can be seen from a typical case shown in Fig. 6b, where a = 3,b = 2, and c = 10, with kp = 0 and kd = 0.5. It can be seen that it is possible to con-struct a straight line with a positive slope passing through the origin such that thePopov plot is entirely to the right of this line. Hence, the range of the allowable non-linear gain k(e) is (0,1).

It can be summarized that, in these two cases, the stability of the enhanced non-linear PD controlled system can be always guaranteed with the unbounded nonlineargain k(e).

4.3. Determination of nonlinear gains for EN-PID controllers

In this case, the closed-loop system can be viewed as a third-order transfer func-tion W(s) cascades a nonlinear gain k(e). The third-order transfer function W(s) canbe expressed as

W ðsÞ ¼ KðsÞGðsÞ ¼ cðkds2 þ kpsþ kiÞsðs2 þ asþ bÞ ð34Þ

where kp, ki, and kd are the positive constant proportional, integral, and derivativegains, respectively.

From (34), the real parts and imaginary parts ofW(jx) can be expressed as [21,34]

ReW ðjxÞ ¼ c½ðakd � kpÞx2 þ bkp � aki�a2x2 þ ðb� x2Þ2

ð35Þ

x ImW ðjxÞ ¼ �c½kdx4 þ ðakp � bkd � kiÞx2 þ bki�a2x2 þ ðb� x2Þ2

ð36Þ

The Popov plot of W(jx) starts at the point P(�c(aki � bkp)/b2, cki/b) for x = 0

and terminates at the point Q(0,�ckd) for x = 1. To apply the Popov stability cri-terion, the crossing of the Popov plot ofW(jx) with the real axis must be determined.From (36), it is clear that when (akp � bkd � ki) P 0, or

bkd þ ki 6 akp ð37Þthen x ImW(jx) is negative for all x; thus the Popov plot does not cross the real axis.In this case, the range of the nonlinear gain k(e) for stability is (0,1). Hence (37)gives a sufficient, but not a necessary condition for closed-loop stability for all valuesof k(e).

When bkd + ki > akp, the closed-loop system may become unstable for some val-ues of k(e). These values of k(e) correspond to the cases where the Popov plot crossesthe real axis, that is, x ImW(jx) = 0. Therefore, two distinct cases are possible,depending on the relative values of kp, ki and kd.


Case 1:ffiffiffiffiffiffiffiakp

p6 j

ffiffiffiffiffiffiffibkd

p�

ffiffiffiffiki

pj. In this case, the two crossover frequencies of the

Popov plot of W(jx) to the real axis, x21 and x2

2, where x21 < x2

2. These frequenciesare the roots of the following equation [21]:

ðkdx2 � kiÞðx2 � bÞ þ akpx2 ¼ 0 ð38Þ

The values of W(jx) at the two crossovers are then found from (35) as

ReW ðjxnÞ ¼c½ðakdx2

n � kiÞ � kpðx2n � bÞ�

a2x2n þ ðb� x2

nÞ2

n ¼ 1; 2 ð39Þ

Substituting for ðx2n � bÞ from (38) into (39) and simplifying the result yields

ReW ðjxnÞ ¼ckp

b� x2n

n ¼ 1; 2 ð40Þ

Now for the Popov plot to cross the negative axis, we need to find the conditionunder which b < x2

n. Consider the following polynomial:

gðx2Þ ¼ kdx4 þ ðakp � bkd � kiÞx2 þ bki ð41Þwhere the plot of g(x2) versus x2 is a parabola that cross the x2-axis at x2

1 and x22.

Since kd > 0, for any value of x2 ‘‘inside’’ the parabola, the expression g(x2) is neg-ative, whereas for all values of x2 ‘‘outside’’ the parabola (including the originx2 = 0), the expression g(x2) is positive. For x2 = b, we have g(b) = abkp > 0, hencex2 = b is located outside the parabola, that is, either b < x2

1 < x22 or x2

1 < x22 < b.

To find out the condition needed for the former case to occur, we only need to com-pare the location of the midpoint x2

0 ¼ ðx21 þ x2

2Þ=2 relative to b. For b < x2i , we

need b < x20. Using the sum-of-roots relationship for (38) yields [21]

b < � akp � ki � bkd2kd

ð42Þ

which can be simplified to

akp þ bkd < ki ð43ÞTherefore, we can conclude that when

ffiffiffiffiffiffiffiakp

p6 j


p�

ffiffiffiffiki

pj and akp + bkd < ki, the

Popov plot of W(jx) crosses the negative real axis [ReW(jxi) < 0], and the nonlineargain k(e) must be upper-bounded by

kðeÞmax ¼ � 1

ReW ðjx1Þ¼ x2

1 � bckp

ð44Þ

to ensure closed-loop stability, that is, 0 6 k(e) 6 k(e)max. Notice that since x21 < x2

0,from (44), the maximum nonlinear gain is bounded by

kmax <ki � akp � bkd

2ckpkdð45Þ

For this case, a typical Popov plot when a = 3, b = 2, and c = 10, with kp = 0,ki = 1 and kd = 1, is illustrated in Fig. 7a.

a

-6 -4 -2 0 4-8

-6

-4

-2

0

-3 -2 -1 0 2-10

-9

-8

-7

-6

-5

-4

-3

-2b

2 1

Fig. 7. Popov plot for the EN-PID control system: (a) Case 1 and (b) Case 2.


On the other hand, whenffiffiffiffiffiffiffiakp

p6j


p�

ffiffiffiffiki

pj but akp + bkd P ki, the Popov

plot of W(jx) crosses the positive real axis [ReW(jxi) > 0], and the general shapeof the Popov plot is shown in Fig. 7a. It can be seen that it is possible to constructa straight line with a positive slope passing through the origin such that the Popovplot is entirely to the right of this line. Hence from the Popov stability criterion, thenonlinear gain k(e) is [0,+1).

Case 2:ffiffiffiffiffiffiffiakp

p> j


p�

ffiffiffiffiki

pj. In this case, (36) cannot have positive real roots for

x. Hence, the Popov plot of W(jx) does not cross the real axis and remains entirelyin the third and fourth quadrants. Therefore, according to the Popov stability crite-rion [33,34], the range of the allowable nonlinear gain k(e) is (0,1). Fig. 7b illus-trates a typical Popov plot in this case for a = 3, b = 2, and c = 10, with kp = 1,ki = 1 and kd = 1.

From the condition akp + bkd < ki, we conclude that the effect of increasing thederivative gain kd is to increase the range of the integral gain ki for stability.

5. Simulation results

Numerical simulations on a two-link revolute-joint manipulator shown in Fig. 8,were performed to validate the effectiveness of the developed EN-PID controller.

The dynamics of the two-link revolute-joint manipulator are given by [35]

s1 ¼ m2l22ð€q1 þ €q2Þ þ m2l1l2c2ð2€q1 þ €q2Þ þ ðm1 þ m2Þl21€q1 � 2m2l1l2s2 _q1 _q2

� m2l1l2s2 _q22 þ ðm1 þ m2Þl1gc1 þ m2l2gc12 þ f1ð _q1Þ þ T d

s2 ¼ m2l1l2c2€q1 þ m2l22ð€q1 þ €q2Þ þ m2l1l2s2 _q

21 þ m2l2gc12 þ f2ð _q2Þ þ T d

ð46Þ

where s2 ¼ sinðq2Þ, c2 ¼ cosðq2Þ, c1 ¼ cosðq1Þ, c12 ¼ cosðq1 þ q2Þ, q1 and q2 are linkangles, m1 and m2 are link masses, l1 and l2 are link lengths, and s1 and s2 are theoutput torques of motors reflected to the joint axes, respectively. f1ð _q1Þ, f2ð _q2Þ,and Td are the viscous friction and disturbed torque, respectively. They can be ex-pressed as

1q

2q

1l

2l

x

y

Fig. 8. Two-link revolute robot.


f1ð _q1Þ ¼ 0.5 sgnð _q1Þ; f 2ð _q2Þ ¼ 0.5 sgnð _q2Þ; T d ¼ 5 cosð5tÞ ð47ÞAssume that the parameters are known and given as follows. m1 = m2 = 1.0 kg,

and l1 = l2 = 1.0 m. We compare the performance of the proposed EN-PD controllerand the conventional fixed-gain PD controller. For the two PD controllers, the fol-lowing identical gains are used: kp = 5500, kd = 60. The design parameters for thesector-bounded nonlinear gain k(e) is chosen as k0 = 200 and emax = 0.006. The

Pos

ition

of f

irst l

ink

(rad

)P

ositi

on o

f sec

ond

link

(rad

)

Pos

ition

err

or o

f firs

t lin

k (r

ad)

Pos

ition

err

or o

f sec

ond

link

(rad

)

0 10 15 20

0.0

0.5

1.0

1.5

2.0

2.5

Time (s)

Reference Actual

5 0 10 15 20-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

Time (s)

5

0 10 15 20

0.0

0.5

1.0

1.5

2.0

2.5

Time (s)

Reference Actual

5 0 10 15 20-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

Time (s)

5

Fig. 9. Simulation result of the conventional PD controller.


parameters of TDs are determined as the same as that in Section 2. The samplingperiod is selected as h = 0.0025 s. The initial conditions are all chosen to be zero.The desired trajectory is q1d ¼ q2d ¼ ð1� cosðtÞÞ rad.

The link trajectory and its tracking error are shown in Figs. 9 and 10, respectively,using the conventional PD controller and EN-PD controller. It can be seen fromcomparison of Fig. 9 with Fig. 10, that the tracking error of EN-PD controller ismuch less than that obtained by the conventional fixed-gain PD controller. The var-iation of the nonlinear gain is shown in Fig. 11. It is clear that when there are largeerrors, the nonlinear gain rises from 1 to 1.67 and 1.17 for the two EN-PD control-lers, respectively, so that a large corrective action can be generated to rapidly drivethe system output to its goal. Therefore, it can be concluded that the favorable resultmainly comes from the nonlinear gain variation of the proposed EN-PD control.

To validate the high robustness against noise and the contribution of the devel-oped nonlinear tracking differentiator (TD), an additive noise with the magnitudeof 0.001 rad is added to the feedback position signal. For limitation of space, onlythe position tracking error is presented in Figs. 12–14, respectively, by using the con-ventional PD controller, the nonlinear PD (N-PD) controller without the developedTD, and the EN-PD controller. From the comparison results, it can be seen that therespond speed of the developed EN-PD controller is much faster than that of theconventional fixed-gain PD controller, and lower tracking error is obtained. It canconclude that the developed EN-PD controller has better robustness against noise

Pos

ition

err

or o

f firs

t lin

k (r

ad)

Pos

ition

of f

irst l

ink

(rad

)P

ositi

on o

f sec

ond

link

(rad

)

Pos

ition

err

or o

f sec

ond

link

(rad

)

0 10 15 20

0.0

0.5

1.0

1.5

2.0

2.5

Time (s)

Reference Actual

5 0 10 15 20-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

0.005

Time (s)

5

0 10 15 20-0.002

-0.001

0.000

0.001

0.002

0.003

Time (s)

50 10 15 20

0.0

0.5

1.0

1.5

2.0

2.5

Time (s)

Reference Actual

5

Fig. 10. Simulation result of the EN-PD controller.

k

0 10 15 200.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

k1

0 10 15 200.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.20

2

Time (s)

55

Time (s)

Fig. 11. Variation of nonlinear gain k.

0 10 15 20-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

Pos

ition

err

or o

f firs

t lin

k (r

ad)

Time (s)

0 10 15 20-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04P

ositi

on e

rror

of s

econ

d lin

k (r

ad)

Time (s)

5 5

Fig. 12. Position tracking errors of the conventional PD controller with noise.

0 10 15 20-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

Pos

ition

err

or o

f firs

t lin

k (r

ad)

Time (s)

0 10 15 20-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

Pos

ition

err

or o

f sec

ond

link

(rad

)

Time (s)

5 5

Fig. 13. Position tracking errors of the EN-PD controller without TD with noise.


over the conventional fixed-gain PD controller, and the developed TD has laid asolid base for the better performance.

0 10 15 20-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

Pos

ition

err

or o

f firs

t lin

k (r

ad)

Time (s)

50 10 15 20-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

Pos

ition

err

or o

f sec

ond

link

(rad

)

Time (s)

5

Fig. 14. Position tracking errors of the proposed EN-PD controller with noise.


6. Conclusions

To enhance the performance of a conventional fixed-gain PID controlled system,an enhanced nonlinear PID (EN-PID) controller is proposed, by incorporating asector-bounded nonlinear gain function and two nonlinear tracking differentiators(TDs) into the available fixed-gain PID control architecture. The nonlinear gain isa function of the error, and acts on the error to produce the ‘‘scaled’’ error for fastresponse. The proposed nonlinear tracking differentiators select a high-quality differ-ential signal in the presence of measurement noise. The EN-PID controller can beimplemented by cascading a nonlinear gain with an available fixed-gain PID control-ler without re-determination of the parameters of the PID controller. The stability ofthe proposed EN-PID controllers is analyzed using the Popov criterion, and theappreciate selection of the nonlinear gain to guarantee the stability and performanceenhancement is addressed. The comparison of the proposed EN-PD controller witha conventional fixed PD controller on a two-link revolute-joint manipulator is per-formed to verify the effectiveness. The major significance of the proposed EN-PIDcontroller lies in its high robustness against noise and superior performance overthe conventional fixed-gain PID controller, and easy of engineering implementation,which explores a convenient engineering method to improve the performance of theavailable conventional linear fixed-gain PID control system.

Acknowledgments

The authors are much grateful to the anonymous reviewers for the numerouscomments and suggestions which have lead to significant improvements of thispaper.

The authors would like to express their appreciation for the financial support ofthe Shaanxi Natural Science Foundation, under Grant 2000C22, and the RoboticsLaboratory at Shenyang Institute of Automation, Chinese Academy of Sciences,China, under Grant RL200104.


References

[1] Astrom KJ, Hagglund T. PID controllers: theory, design, and tuning. North Carolina: InstrumentSociety of America, Research Triangle Park; 1995.

[2] Silva GJ, Datta A, Bhattacharyya SP. New results on the synthesis of PID controllers. IEEE TransAutomat Control 2002;47(2):241–52.

[3] Kristiansson B, Lennartson B. Robust and optimal tuning of PI and PID controllers. IEE ProcControl Theory Appl 2002;149(1):17–25.

[4] Panagopoulos H, Astrom KJ, Hagglund T. Design of PID controllers based on constrainedoptimization. IEE Proc Control Theory Appl 2002;149(1):32–40.

[5] Cervantes I, Alvarez-Ramirez J. On the PID tracking control of robot manipulators. Syst ControlLett 2001;42(1):37–46.

[6] Wang G-J, Fong C-T, Chang KJ. Neural-network-based self-tuning PI controller for precise motioncontrol of PMAC motors. IEEE Trans Indust Electron 2001;48(2):408–15.

[7] Wang Q-G, Lee T-H, Fung H-W, Bi Q, Zhang Y. PID tuning for improved performance. IEEE TransControl Syst Technol 1999;7(4):457–65.

[8] Shieh M-Y, Li T-HS. Design and implementation of integrated fuzzy logic controller for aservomotor system. Mechatronics 1998;8(3):217–40.

[9] Jin HP, Su WS, Lee I-B. An enhanced PID control strategy for unstable processes. Automatica1998;34(6):751–6.

[10] Howell MN, Best MC. On-line PID tuning for engine idle-speed control using continuous actionreinforcement learning automata. Control Eng Practice 2000;8(2):147–54.

[11] Chen W-H, Balance DJ, Gawthrop PJ, Gribble JJ, O�Reilly J. Nonlinear PID predictive controller.IEE Proc Control Theory Appl 1999;146(6):603–11.

[12] Rugh WJ. Design of nonlinear PID controllers. AIChE J 1987;33(10):1738–42.[13] Taylor JH, Astrom KJ. Nonlinear PID autotuning algorithm. In: Proceedings of the American

Control Conference, Seattle, WA, 1986. p. 2118–23.[14] Han JQ. Nonlinear PID controller. Acta Automat Sin 1994;20(4):487–90 [in Chinese].[15] Su YX, Duan BY, Zheng CH. Nonlinear PID control of a six-DOF parallel manipulator. IEE Proc

Control Theory Appl 2004;151(1):95–102.[16] Armstrong B, Wade BA. Nonlinear PID control with partial state knowledge: Damping without

derivatives. Int J Robot Res 2000;19(8):715–31.[17] Armstrong B, Neevel D, Kusik T. New results in NPID control: Tracking, integral control, friction

compensation and experimental results. IEEE Trans Control Syst Technol 2001;9(2):399–406.[18] Bucklaew TP, Liu C-S. A new nonlinear gain structure for PD-type controllers in robotic

applications. J Robot Syst 1999;16(11):627–49.[19] Xu Y, Hollerbach JM, Ma D. A nonlinear PD controller for force and contact transient control.

IEEE Control Syst Mag 1995;15(1):15–21.[20] Xu Y. Hollerbach JM, Ma D. Force and contact transient control using nonlinear PD control. IEEE

Int Conf Robot Automat, Atlanta, GA 1994:924–30.[21] Seraji H. A new class of nonlinear PID controllers with robotic applications. J Robot Syst

1998;15(3):161–81.[22] Seraji H. Nonlinear and adaptive control of force and compliance in manipulators. Int J Robot Res

1998;17(5):467–84.[23] Shahruz SM, Schwartz AL. Nonlinear PI compensators that achieve high performance. J Dyn Syst,

Measure Control 1997;119(5):105–10.[24] Xu L, Yao B. Output feedback adaptive robust precision motion control of linear motors.

Automatica 2001;37(7):1029–39.[25] Martinez-Guerra R, Poznyak A, Gortcheva E, Diaz de Leon V. Robot angular link velocity

estimation in the presence of high-level mixed uncertainties. IEE Proc Control Theory Appl2000;147(5):515–22.

[26] Belanger PR, Dobrovolny P, Helmy A, Zhang X. Estimation of angular velocity and accelerationfrom shaft-encoder measurements. Int J Robot Res 1998;17(11):1225–33.


[27] Han JQ, Wang W. Nonlinear tracking-differentiator. J Syst Sci Math Sci 1994;14(2):177–83 [inChinese].

[28] Su YX, Duan BY, Zhang YF. Robust precision motion control for AC servo systems. In: The 4thWorld Congress on Intelligent Control and Automation, Shanghai, China, vol. 4, 2002. p. 3319–23.

[29] Ogata K. Modern control engineering. 3rd ed. Upper Saddle River, NJ: Prentice Hall; 1997.[30] Arimoto S. Control theory of non-linear mechanical systems: A passivity-based and circuit-theoretic

approach. Oxford: Clarendon Press; 1996.[31] Tomei P. Adaptive PD controller for robot manipulators. IEEE Trans Robot Automat

1991;7(4):565–70.[32] Kelly R. Comments on �Adaptive PD controller for robot manipulators�. IEEE Trans Robot Automat

1993;9(1):117–9.[33] Parks PC, Hahn V. Stability theory. New York: Prentice Hall; 1993.[34] Wu Q. Principle of automatic control. Beijing: Tsinghua University Press; 1994 [in Chinese].[35] Qu ZH, Dawson DM. Robust tracking control of robot manipulators. New York: IEEE Press; 1996.

design of an enhanced nonlinear pid controllerdownloads.deusm.com/designnews/1477-elsevier06.pdf ·...

Documents