Mechatronics 15 (2005) 1005–1024
Design of an enhanced nonlinear
PID controller
Y.X. Su
a,b,*
, Dong Sun a, B.Y. Duan
b
a
b
Department of Manufacturing Engineering and Engineering Management,
City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
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 performance
than the conventional linear fixed-gain PID controller is proposed in this paper, by incorporating a sector-bounded nonlinear gain in cascade with a conventional PID control architecture. To achieve the high robustness against noise, two nonlinear tracking differentiators are
used to select high-quality differential signal in the presence of measurement noise. The criterion to determine the nonlinear gain to retain the stability of the proposed EN-PID control
system is addressed, by using the Popov stability criterion. The main advantages of the proposed 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 better
performance of the developed EN-PID controller than the conventional fixed-gain PID
controller.
Ó 2005 Published by Elsevier Ltd.
Keywords: PID control; Nonlinear PID control; Tracking filters; Measurement noise; Robotic control
*
Corresponding author. Address: School of Electro-Mechanical Engineering, Xidian University, Xian
710071, China. Tel.: +86 29 8820 2262; fax: +86 29 8821 3676.
E-mail address: yxsu@mail.xidian.edu.cn (Y.X. Su).
0957-4158/$ - see front matter Ó 2005 Published by Elsevier Ltd.
doi:10.1016/j.mechatronics.2005.03.003
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Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
1. Introduction
Proportional-integral-derivative (PID) controllers have been the most popular and
the most commonly used industrial controllers in the past years. The popularity and
widespread use of PID controllers are attributed primarily to their simplicity and performance characteristics. PID controllers have been utilized for control of diverse
dynamical systems ranged from industrial process to aircraft and ship dynamics [1–4].
Although linear fixed-gain PID controllers are often adequate for controlling
a nominal physical process, the requirements for high-performance control with
changes in operating conditions or environmental parameters are often beyond the
capabilities of simple PID controllers. In order to enhance the performance of linear
PID controllers, many approaches have been developed to improve the adaptability
and robustness by adopting the self-tuning method, general predictive control, fuzzy
logic and neural networks strategy, and other methods [5–23].
Amongst these approaches, nonlinear PID (N-PID) control is viewed as one of
the most effective and simple method for industrial applications [12–23]. Nonlinear
PID control may be any control structure of the following form:
Z
uðtÞ ¼ k p ðÞeðtÞ þ k i ðÞ eðsÞ ds þ k d ðÞ_eðtÞ
ð1Þ
where kp(Æ), ki(Æ) and kd(Æ) are time-varying controller gains, which may depend on
system state, input, or other variables, and u(t) and e(t) are the system input and
error, 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 nonlinearity, 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 not
achievable by a linear PID control, such as increased damping, reduced rise
time for step or rapid inputs, improved tracking accuracy, and friction compensation [16–23].
For linear systems, two broad categories of N-PID control are found [16]: those
with gains modulated according to the magnitude of the state [22,23], and those with
gains modulated according to the phase of the state [16–21]. The former category of
N-PID will be used in this paper. According to the magnitude of the state, the
enhancement of the controller is achieved by adapting its response based on the performance of the closed-loop control system. When the error between the commanded
and actual values of the controlled variable is large, the gain amplifies the error substantially 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 oscillations and large overshoots in the response. Because of this automatic gain adjustment, the N-PID controllers enjoy the advantage of high initial gain to obtain a fast
response, 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 controller is proposed by incorporating a sector-bounded nonlinear gain into a linear fixedgain PID control architecture. To achieve the high robustness against noise, two
nonlinear tracking differentiators are used to select the high-quality differential signal
in the presence of measurement noise.
The paper is organized as follows. A nonlinear tracking differentiator is developed
in 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 ENPID controller is analyzed. To verify the performance enhancement, simulations
are performed on a two-link revolute robot in Section 5. Finally, some remarking
conclusions are summarized in Section 6.
2. Nonlinear tracking differentiator (TD) design
In practice, optical encoder is still the most popular accurate position sensor used
in the industrial fields because of its simple detection circuit, high resolution, high
accuracy, and relative ease of adaptation in digital control systems. Conversely,
the velocity measurement, obtained by means of tachometers, is often contaminated
by noise. Furthermore, the use of sensors in the control architecture inherently increases the possibility of failure and increases the cost of the control system due
to additional hardware. Therefore, it is necessary to numerically reconstruct velocity
signal from position measurement with noise [15,24–28].
The so-called nonlinear tracking differentiator (TD) [15,27,28] is referred to as the
following system: given a reference signal r(t), the system provides two signals r1(t)
and r2(t), such that r1(t) = r(t) and r2 ðtÞ ¼ r_ ðtÞ.
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 satisfies
lim zðtÞ ¼ 0. If r(t) = z(Rt), R > 0, then for an arbitrary given T > 0, the following
t!1
expression holds
Z T
jrðtÞjdt ¼ 0
ð2Þ
lim
R!1
0
Proof. Based on the mean value theorem, there is a s between 0 and T such that
Z T
jrðtÞjdt ¼ T jrðsÞj ¼ T jzðRsÞj
ð3Þ
0
Since lim zðtÞ ¼ 0, it is straightforward that
t!1
Z T
jrðtÞjdt ¼ lim ðT jzðRsÞjÞ ¼ 0
lim
R!1
0
R!1
Thus Lemma 1 is just established.
h
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Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
Lemma 2. If the solutions to the system (4) hold that z1(t) ! 0 and z2(t) ! 0 as t ! 1
z_ 1 ¼ z2
ð4Þ
z_ 2 ¼ f ðz1 ; z2 Þ
then for an arbitrarily constant c and T > 0, the solution r1(t) to the system
(
r_ 1 ¼ r2 r2 r_ 2 ¼ R2 f r1 c;
R
makes the following expression hold
Z T
jr1 ðtÞ cjdt ¼ 0
lim
R!1
ð5Þ
ð6Þ
0
Proof. Firstly, change the variable of integration as follows:
8
t
>
<s ¼
R
r ðsÞ ¼ z1 ðtÞ þ c
>
: 1
r2 ðsÞ ¼ Rz2 ðtÞ
ð7Þ
Based on the mean value theorem, there is a s between 0 and T such that
Z T
jr1 ðtÞ cjdt ¼ T jr1 ðsÞ cj ¼ T jz1 ðtÞj
ð8Þ
0
Using Lemma 1, it follows that
Z T
jr1 ðtÞ cjdt ¼ lim ðT jz1 ðtÞjÞ ¼ T lim jz1 ðRsÞj ¼ 0
lim
R!1
R!1
0
Therefore, Lemma 2 is right justified.
R!1
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
(
r_ 1 ¼ r2
r2 ð9Þ
r_ 2 ¼ R2 f r1 r;
R
satisfies
Z
lim
R!1
T
jr1 ðtÞ rðtÞjdt ¼ 0
0
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.
ð10Þ
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
1009
Case 2: If r(t) (t 2 [0, T]) is a bounded integral function, then it is an element of
L1[0, T], where L1[0, T] denotes the set of all first-integrable function in
the range of [0, T]. For an arbitrarily given e > 0, there exists a simple series
un(t) (n = 1, 2, . . .) that uniformly converges to a continuous function
u(t) 2 C[0, T] such that [27]
Z T
e
ð11Þ
jrðtÞ uðtÞjdt <
2
0
Thus, there exists an integer N0 such that juðtÞ uM ðtÞj < 4Te for all M > N0. Consequently, the following inequality holds
Z
T
jrðtÞ uM ðtÞjdt 6
0
Z
T
jrðtÞ uðtÞjdt þ
Z
0
T
juðtÞ uM ðtÞjdt <
0
e
2
ð12Þ
Since u(t) is a continuous function, the simple series uM(t) that partitions the
range [0, T] into some bounded intervals denoted by li (i = 1, 2, . . . , m). Selecting
uM(t) to be a deterministic constant in each bounded interval, and based on Lemma
2, there exists R0 > 0 such that
Z
e
; i ¼ 1; 2; . . . ; m
ð13Þ
jr1 ðtÞ uM ðtÞjdt <
2m
li
for all R > R0. Then,
Z T
e
jr1 ðtÞ uM ðtÞjdt <
2
0
ð14Þ
Thereby, the following inequality holds
Z T
Z T
Z
jr1 ðtÞ rðtÞjdt <
jr1 ðtÞ uM ðtÞjdt þ
0
0
T
juM ðtÞ rðtÞjdt < e
ð15Þ
0
for all R > R0.
The proof is right justified. h
Theorem 1 shows that r1(t) averagely converges to r(t). If the bounded integrable
function r(t) is viewed as a generalized function, then r2(t) weakly converges to the
generalized derivative of r(t). Therefore, system (9) can be used as a nonlinear tracking differentiator to provide a smooth approach to the original generalized function
and its generalized derivative in the sense of average convergence and weak convergence, respectively.
One feasible second-order TD for the reference signal can be expressed as [15,27]
8
< z_ 1 ¼ z2
ð16Þ
z2 j z2 j
: z_ 2 ¼ R sat z1 r þ
;d
2R
where R and d are two positive design parameters, and sat(A, d) is the following nonlinear saturation function:
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Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
8
< sgnðAÞ; jAj > d
satðA; dÞ ¼ A
: ; jAj 6 d
d
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 transition
and tracking, and large d is helpful to cancel out noise. In general, for the reference
signal corrupted with maximum amplitude of 0.01 white noise component, R can be
chosen from the range of [2.5 50]; for the reference signal with large noise magnitude
than 0.01, R will be increased to preserve a better tracking performance at the sacrifice of differential signal obtained. There is a tradeoff between the tracking and
filtering.
The better performance of TD is shown in Fig. 1. The reference input is
rðtÞ ¼ sinðtÞ mm and it is perturbed by an additive white noise component with
the maximum amplitude of 0.01 mm. For comparison, the differential signal obtained by the conventional backward differentiator is also shown in Fig. 2. The simulations were programmed in Matlab with a fourth-order Runge–Kutta and run on
a PC Pentium III-860. The simulation step is determined as h = 0.0025, and the initial values of z1 and z2 is z1(0) = 0 and z2(0) = 0. The parameters of the TD are determined as R = 20, and d = 0.001. It can be seen that the differential tracking
Reference
0.03
Estimate
1.0
0.02
0.01
Position error (rad)
Position (rad)
0.5
0.0
-0.5
0.00
-0.01
-0.02
-0.03
-1.0
-0.04
0
5
10
15
0
20
5
10
15
20
15
20
Time (s)
Time (s)
1.5
1.2
Estimate
Reference
1.0
1.0
0.8
Velocity error (rad/s)
Velocity (rad/s)
0.5
0.0
-0.5
0.6
0.4
0.2
0.0
-1.0
-0.2
-1.5
0
5
-0.4
10
15
20
0
5
10
Time (s)
Time (s)
Fig. 1. Performance of TD with noise.
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
1011
6
Reference
Obtained by backward difference
4
4
2
Velocity error (rad/s)
Velocity (rad/s)
2
0
-2
0
-2
-4
-4
-6
0
5
10
15
20
0
10
5
15
20
Time (s)
Time (s)
Fig. 2. Performance of backward differentiator with noise.
performance of TD is much better over that of the conventional backward difference
method.
3. Enhanced nonlinear PID (EN-PID) controller
The proposed enhanced nonlinear PID (EN-PID) controller consists of a sectorbounded nonlinear gain k(e), a linear fixed-gain PID controller expressed by
K(s) = kp + ki/s + kds, and two nonlinear tracking differentiators (TDs), as shown
in Fig. 3. kp, ki, and kd are proportional, integral, and derivative gains, which can
be determined by the Ziegler–Nichols criterion [1]. The nonlinear gain k(e) is a
sector-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 by
the developed TDs, we have the following enhanced nonlinear N-PID control law:
Z t
d
uðtÞ ¼ k p þ k i
dt þ k d
f ðeÞ
dt
0
Z t
ð18Þ
¼ k p ½kðeÞ eðtÞ þ k i ½kðeÞ eðtÞdt þ k d ½kðeÞ cðtÞ
0
where e(t) and c(t) are expressed as, respectively,
r
TD
(I)
z2 +
N-PID controller
c
z1 -+ e
-
K ( s) = k p +
k
z3
z4
ki
+ kd s
s
u
TD (II )
Fig. 3. Block diagram of the EN-PID control system.
Plant
y
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Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
2.4
k 0 = 0.125
2.2
k 0 = 0.15
Nonlinear gain
2.0
1.8
1.6
1.4
1.2
1.0
-10
-5
0
5
10
Error
Fig. 4. Variation of the nonlinear gain k(e) with the error e.
eðtÞ ¼ z1 ðtÞ z3 ðtÞ
ð19Þ
cðtÞ ¼ z2 ðtÞ z4 ðtÞ
ð20Þ
The nonlinear gain k(e) represents any general nonlinear function of the error e
which is bounded in the sector 0 6 k(e) 6 kmax. There is a broad range of options
available for the nonlinear gain k(e) [21]. One simple form of the nonlinear gain function can be expressed as
expðk 0 eÞ þ expðk 0 eÞ
kðeÞ ¼ chðk 0 eÞ ¼
2
e
jej 6 emax
e¼
emax sgnðeÞ jej > emax
ð21Þ
where k0 and emax are user-defined positive constants. The nonlinear gain k(e) is lowerbounded 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 of
k(e). Fig. 4 illustrates a typical variation of k(e) with respect to e when k0 = 0.125
and k0 = 0.15, with emax = 10. It can be seen that the gain k(e) is an even function of
e, that is, k(e) = k(e).
As a result, the proposed EN-PID controller can be realized by cascading this sector-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 adequately modeled by a second-order differential equation. Even when the system
dynamics is of high order, the response of the system is often largely dependent
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
1013
on the location of a pair of dominant complex poles, which can be embodied in a
second-order model [21,29]. For these systems, the second-order transfer function
relating the system output y(t) to the control action u(t) can be expressed as
GðsÞ ¼
yðsÞ
x2n k
c
¼ 2
¼
uðsÞ s þ 2nxn s þ x2n s2 þ as þ b
ð22Þ
where n, xn, and k are the damping ratio, natural frequency, and DC gain of the controlled system, respectively, a = 2nxn, b ¼ x2n , and c ¼ x2n k.
For such a control system, it has been proven that there exists a suitable choice of
the PID controller parameters so that the overall closed-loop system is globally
asymptotically stable for the position (set-point) control [30–32]. Since the convergence of the developed nonlinear tracking differentiator (TD) has just been investigated, in this section, the criterion to determine the nonlinear gains to retain the
stability of the proposed EN-PID control system that incorporates a sector-bounded
nonlinear gain in cascade with a stable linear fixed-gain PID control system is
addressed, following the idea presented by Seraji [21] using the Popov stability
criterion. The Popov stability criterion can be stated graphically as follows: A sufficient condition for the closed-loop system to be global asymptotically stable for all
nonlinear gains in the sector 0 6 k(e) 6 k(e)max is that the Popov plot of W(jx) lies
entirely to the right of a straight line with a nonnegative slope passing through the
point 1/kmax + j0 [33,34].
For some applications, it is desirable to introduce the sector-bounded nonlinear
gain k(e) on some terms of the linear fixed-gain PID controller, instead of equally
on all three terms. In this section, we will investigate the criterion to determine the
nonlinear gains for the three complex cases: the enhanced nonlinear PI, PD, and
PID controllers, by using the Popov stability criterion, respectively. The criterion
to determine the nonlinear gains for other partially enhanced nonlinear P, I, and
D 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 following linear third-order transfer function W(s):
W ðsÞ ¼ KðsÞGðsÞ ¼
cðk p s þ k i Þ
sðs2 þ as þ bÞ
ð23Þ
which describes the controlled plant (22) and the linear fixed-gain PI controller cascades with the sector-bounded nonlinear gain k(e). kp and ki are positive constant
proportional 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 that
the sector-bounded nonlinear gain k(e) can be assumed while retaining the closedloop stability. From (23), the real and imaginary parts of W(jx) can be calculated
[21,34]
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Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
Re W ðjxÞ ¼
cðk p x2 þ ak i bk p Þ
ð24Þ
a2 x2 þ ðb x2 Þ2
x Im W ðjxÞ ¼
c½ðak p k i Þx2 þ bk i a2 x2 þ ðb x2 Þ
ð25Þ
2
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, depending on the relative values of kp and ki.
Case 1: ki 6 akp. In this case, x Im W(jx) is always negative for all x, that is, the
Popov plot of W(jx) remains entirely in the third and fourth quadrants and does not
cross the real axis. This implies that one can construct a straight line with a nonnegative slope passing through the origin such that the Popov plot is entirely to the right
of this line. Therefore, according to the Popov stability criterion [33,34], the range of
the allowable nonlinear gain k(e) is (0, 1). A typical Popov plot for a = 3, b = 2, and
c = 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 be
found by solving x Im W(jx) = 0 to yield [21,34]
x20 ¼
bk i
k i ak p
ð26Þ
As a result, the value of W(jx) at the crossover is
Re W ðjx0 Þ ¼
cðak p k i Þ
ab
ð27Þ
It indicates that the Popov plot crosses the negative real axis. So, the maximum
allowable nonlinear gain can be calculated as [21]
kðeÞmax ¼ 1
ab
¼
Re W ðjx0 Þ cðak p k i Þ
ð28Þ
Then a straight line with a nonnegative slope passing through the point 1/kmax + j0
can be constructed such that the Popov plot of W(jx) is entirely to the right of this
0.0
0.5
a
b
-0.5
0.0
-1.0
-0.5
-1.5
-1.0
-2.0
-2.5
-1.5
-3.0
-2.0
-3.5
-4.0
-1.0
-0.5
0.0
0.5
1.0
-2.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
Fig. 5. Popov plot for the EN-PI control system: (a) Case 1 and (b) Case 2.
-0.5
0.0
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
1015
line. The range of the allowable nonlinear gain k(e) is (0, k(e)max). A typical Popov
plot 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 and
the plant (22), that is,
cðk p þ k d sÞ
s2 þ as þ b
W ðsÞ ¼ KðsÞGðsÞ ¼
ð29Þ
From (29), the following expressions of Popov plot W(jx) can be obtained [21,34]
Re W ðjxÞ ¼
c½ðak d k p Þx2 þ bk p ð30Þ
2
x Im W ðjxÞ ¼
a2 x2 þ ðb x2 Þ
cx2 ðk d x2 þ ak p bk d Þ
a2 x2 þ ðb x2 Þ
ð31Þ
2
It can be seen that, the Popov plot of W(jx) starts at the point P(ckp/b, 0) for x = 0
and terminates at the point Q(0, ckd) for x = 1. Two distinct cases are now possible, 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 Im W(jx) is always
negative for all nonzero x, that is, the Popov plot of W(jx) remains entirely in the
third and fourth quadrants and does not cross the real axis. Therefore, according to
the 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, with
kp = 1 and kd = 0.5.
Case 2: bkd > akp. The crossover frequency x0 of W(jx) to the real axis is found
by solving x Im W(jx) = 0 to yield
bk d ak p
kd
x20 ¼
0
ð32Þ
a
1
-1
b
0
-1
-2
-2
-3
-3
-4
-4
-5
-5
0
1
2
3
4
5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fig. 6. Popov plot for the EN-PD control system: (a) Case 1 and (b) Case 2.
1.6
1.8
1016
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
and the value of W(jx0) is then found to be
Re W ðjx0 Þ ¼
ck d
a
ð33Þ
which indicates that the Popov plot crosses the positive real axis. The general shape
of 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 construct a straight line with a positive slope passing through the origin such that the
Popov plot is entirely to the right of this line. Hence, the range of the allowable nonlinear gain k(e) is (0, 1).
It can be summarized that, in these two cases, the stability of the enhanced nonlinear PD controlled system can be always guaranteed with the unbounded nonlinear
gain 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 function W(s) cascades a nonlinear gain k(e). The third-order transfer function W(s) can
be expressed as
W ðsÞ ¼ KðsÞGðsÞ ¼
cðk d s2 þ k p s þ k i Þ
sðs2 þ as þ bÞ
ð34Þ
where kp, ki, and kd are the positive constant proportional, integral, and derivative
gains, respectively.
From (34), the real parts and imaginary parts of W(jx) can be expressed as [21,34]
Re W ðjxÞ ¼
c½ðak d k p Þx2 þ bk p ak i x Im W ðjxÞ ¼
2
a2 x2 þ ðb x2 Þ
c½k d x4 þ ðak p bk d k i Þx2 þ bk i a2 x2 þ ðb x2 Þ
2
ð35Þ
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 criterion, the crossing of the Popov plot of W(jx) with the real axis must be determined.
From (36), it is clear that when (akp bkd ki) P 0, or
bk d þ k i 6 ak p
ð37Þ
then x Im W(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 values
of k(e).
When bkd + ki > akp, the closed-loop system may become unstable for some values of k(e). These values of k(e) correspond to the cases where the Popov plot crosses
the real axis, that is, x Im W(jx) = 0. Therefore, two distinct cases are possible,
depending on the relative values of kp, ki and kd.
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
1017
pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi pffiffiffiffi
Case 1: ak p 6 j bk d k i j. In this case, the two crossover frequencies of the
Popov plot of W(jx) to the real axis, x21 and x22 , where x21 < x22 . These frequencies
are the roots of the following equation [21]:
ðk d x2 k i Þðx2 bÞ þ ak p x2 ¼ 0
ð38Þ
The values of W(jx) at the two crossovers are then found from (35) as
Re W ðjxn Þ ¼
c½ðak d x2n k i Þ k p ðx2n bÞ
a2 x2n þ ðb x2n Þ2
n ¼ 1; 2
ð39Þ
Substituting for ðx2n bÞ from (38) into (39) and simplifying the result yields
Re W ðjxn Þ ¼
ck p
b x2n
n ¼ 1; 2
ð40Þ
Now for the Popov plot to cross the negative axis, we need to find the condition
under which b < x2n . Consider the following polynomial:
gðx2 Þ ¼ k d x4 þ ðak p bk d k i Þx2 þ bk i
2
2
ð41Þ
2
where the plot of g(x ) versus x is a parabola that cross the x -axis at
x22 .
2
Since kd > 0, for any value of x ‘‘inside’’ the parabola, the expression g(x ) is negative, whereas for all values of x2 ‘‘outside’’ the parabola (including the origin
x2 = 0), the expression g(x2) is positive. For x2 = b, we have g(b) = abkp > 0, hence
x2 = b is located outside the parabola, that is, either b < x21 < x22 or x21 < x22 < b.
To find out the condition needed for the former case to occur, we only need to compare the location of the midpoint x20 ¼ ðx21 þ x22 Þ=2 relative to b. For b < x2i , we
need b < x20 . Using the sum-of-roots relationship for (38) yields [21]
b<
ak p k i bk d
2k d
x21 and
2
ð42Þ
which can be simplified to
ak p þ bk d < k i
ð43Þ
pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi pffiffiffiffi
Therefore, we can conclude that when ak p 6 j bk d k i j and akp + bkd < ki, the
Popov plot of W(jx) crosses the negative real axis [Re W(jxi) < 0], and the nonlinear
gain k(e) must be upper-bounded by
kðeÞmax ¼ 1
x2 b
¼ 1
Re W ðjx1 Þ
ck p
ð44Þ
to ensure closed-loop stability, that is, 0 6 k(e) 6 k(e)max. Notice that since x21 < x20 ,
from (44), the maximum nonlinear gain is bounded by
k max <
k i ak p bk d
2ck p k d
ð45Þ
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.
1018
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
-2
a
0
b
-3
-4
-2
-5
-6
-4
-7
-8
-6
-9
-8
-6
-4
-2
0
2
-10
4
-3
-2
-1
0
1
2
Fig. 7. Popov plot for the EN-PID control system: (a) Case 1 and (b) Case 2.
pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi pffiffiffiffi
On the other hand, when ak p 6j bk d k i j but akp + bkd P ki, the Popov
plot of W(jx) crosses the positive real axis [Re W(jxi) > 0], and the general shape
of the Popov plot is shown in Fig. 7a. It can be seen that it is possible to construct
a straight line with a positive slope passing through the origin such that the Popov
plot is entirely to the right of this line. Hence from the Popov stability criterion, the
nonlinear gain
is ffiffiffiffiffiffiffi
[0, +1).
pffiffiffiffiffiffiffik(e) p
pffiffiffiffi
Case 2: ak p > j bk d k i j. 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 entirely
in the third and fourth quadrants. Therefore, according to the Popov stability criterion [33,34], the range of the allowable nonlinear gain k(e) is (0, 1). Fig. 7b illustrates 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 the
derivative 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 m2 l22 ð€
q1 þ €
q2 Þ þ m2 l1 l2 c2 ð2€
q1 þ €
q2 Þ þ ðm1 þ m2 Þl21 €q1 2m2 l1 l2 s2 q_ 1 q_ 2
m2 l1 l2 s2 q_ 22 þ ðm1 þ m2 Þl1 gc1 þ m2 l2 gc12 þ f1 ðq_ 1 Þ þ T d
s2 ¼ m2 l1 l2 c2 €
q1 þ
m2 l22 ð€
q1
þ€
q2 Þ þ
m2 l1 l2 s2 q_ 21
ð46Þ
þ m2 l2 gc12 þ f2 ðq_ 2 Þ þ T d
where s2 ¼ sinðq2 Þ, c2 ¼ cosðq2 Þ, c1 ¼ cosðq1 Þ, c12 ¼ cosðq1 þ q2 Þ, q1 and q2 are link
angles, m1 and m2 are link masses, l1 and l2 are link lengths, and s1 and s2 are the
output torques of motors reflected to the joint axes, respectively. f1 ðq_ 1 Þ, f2 ðq_ 2 Þ,
and Td are the viscous friction and disturbed torque, respectively. They can be expressed as
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
1019
y
l2
q2
l1
q1
x
Fig. 8. Two-link revolute robot.
f1 ðq_ 1 Þ ¼ 0.5 sgnðq_ 1 Þ;
f 2 ðq_ 2 Þ ¼ 0.5 sgnðq_ 2 Þ;
T d ¼ 5 cosð5tÞ
ð47Þ
Assume 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 controller
and the conventional fixed-gain PD controller. For the two PD controllers, the following identical gains are used: kp = 5500, kd = 60. The design parameters for the
sector-bounded nonlinear gain k(e) is chosen as k0 = 200 and emax = 0.006. The
2.5
0.010
Actual
Reference
Position error of first link (rad)
Position of first link (rad)
0.008
2.0
1.5
1.0
0.5
0.006
0.004
0.002
0.000
-0.002
-0.004
0.0
0
5
10
15
-0.006
0
20
5
Time (s)
2.5
15
20
15
20
0.004
Actual
Position error of second link (rad)
Reference
Position of second link (rad)
10
Time (s)
2.0
1.5
1.0
0.5
0.003
0.002
0.001
0.000
-0.001
0.0
0
5
-0.002
10
Time (s)
15
20
0
5
10
Time (s)
Fig. 9. Simulation result of the conventional PD controller.
1020
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
parameters of TDs are determined as the same as that in Section 2. The sampling
period 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 from
comparison of Fig. 9 with Fig. 10, that the tracking error of EN-PD controller is
much less than that obtained by the conventional fixed-gain PD controller. The variation of the nonlinear gain is shown in Fig. 11. It is clear that when there are large
errors, the nonlinear gain rises from 1 to 1.67 and 1.17 for the two EN-PD controllers, respectively, so that a large corrective action can be generated to rapidly drive
the system output to its goal. Therefore, it can be concluded that the favorable result
mainly comes from the nonlinear gain variation of the proposed EN-PD control.
To validate the high robustness against noise and the contribution of the developed nonlinear tracking differentiator (TD), an additive noise with the magnitude
of 0.001 rad is added to the feedback position signal. For limitation of space, only
the position tracking error is presented in Figs. 12–14, respectively, by using the conventional PD controller, the nonlinear PD (N-PD) controller without the developed
TD, and the EN-PD controller. From the comparison results, it can be seen that the
respond speed of the developed EN-PD controller is much faster than that of the
conventional fixed-gain PD controller, and lower tracking error is obtained. It can
conclude that the developed EN-PD controller has better robustness against noise
2.5
0.005
Reference
Actual
Position error of first link (rad)
Position of first link (rad)
0.004
2.0
1.5
1.0
0.5
0.003
0.002
0.001
0.000
-0.001
-0.002
0.0
0
5
10
15
-0.003
20
0
5
Time (s)
2.5
15
20
0.003
Actual
Position error of second link (rad)
Reference
Position of second link (rad)
10
Time (s)
2.0
1.5
1.0
0.5
0.002
0.001
0.000
-0.001
0.0
0
5
10
Time (s)
15
20
-0.002
0
10
5
Time (s)
Fig. 10. Simulation result of the EN-PD controller.
15
20
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
1.8
1.20
1.7
1.18
1021
1.16
1.6
1.14
1.5
1.12
k2
k1
1.4
1.10
1.08
1.3
1.06
1.2
1.04
1.1
1.02
1.0
1.00
0.98
0.9
5
0
10
15
5
0
20
10
15
20
Time (s)
Time (s)
0.08
0.04
0.06
0.03
Position error of second link (rad)
Position error of first link (rad)
Fig. 11. Variation of nonlinear gain k.
0.04
0.02
0.00
-0.02
-0.04
-0.06
0.02
0.01
0.00
-0.01
-0.02
-0.03
-0.08
-0.04
5
0
10
15
20
0
5
Time (s)
10
15
20
Time (s)
0.08
0.04
0.06
0.03
Position error of second link (rad)
Position error of first link (rad)
Fig. 12. Position tracking errors of the conventional PD controller with noise.
0.04
0.02
0.00
-0.02
-0.04
0.02
0.01
0.00
-0.01
-0.02
-0.03
-0.06
-0.04
-0.08
0
5
10
Time (s)
15
20
0
5
10
15
20
Time (s)
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 a
solid base for the better performance.
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
0.08
0.04
0.06
0.03
Position error of second link (rad)
Position error of first link (rad)
1022
0.04
0.02
0.00
-0.02
-0.04
-0.06
0.02
0.01
0.00
-0.01
-0.02
-0.03
-0.04
-0.08
0
5
10
Time (s)
15
20
0
5
10
15
20
Time (s)
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 a
sector-bounded nonlinear gain function and two nonlinear tracking differentiators
(TDs) into the available fixed-gain PID control architecture. The nonlinear gain is
a function of the error, and acts on the error to produce the ‘‘scaled’’ error for fast
response. The proposed nonlinear tracking differentiators select a high-quality differential signal in the presence of measurement noise. The EN-PID controller can be
implemented by cascading a nonlinear gain with an available fixed-gain PID controller without re-determination of the parameters of the PID controller. The stability of
the proposed EN-PID controllers is analyzed using the Popov criterion, and the
appreciate selection of the nonlinear gain to guarantee the stability and performance
enhancement is addressed. The comparison of the proposed EN-PD controller with
a conventional fixed PD controller on a two-link revolute-joint manipulator is performed to verify the effectiveness. The major significance of the proposed EN-PID
controller lies in its high robustness against noise and superior performance over
the conventional fixed-gain PID controller, and easy of engineering implementation,
which explores a convenient engineering method to improve the performance of the
available conventional linear fixed-gain PID control system.
Acknowledgments
The authors are much grateful to the anonymous reviewers for the numerous
comments and suggestions which have lead to significant improvements of this
paper.
The authors would like to express their appreciation for the financial support of
the Shaanxi Natural Science Foundation, under Grant 2000C22, and the Robotics
Laboratory at Shenyang Institute of Automation, Chinese Academy of Sciences,
China, under Grant RL200104.
Y.X. Su et al. / Mechatronics 15 (2005) 1005–1024
1023
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