Journal of Applied Mathematics and Physics, 2013, 1, 85-92
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15013
Open Access JAMP
Adaptive P iecewise Linear Controller for Servo
Mechanical Control Systems
Tain-Sou Tsay
Department of Aeronautical Engineering, National Formosa University, Yunlin, Taiwan
Email: tstsay@nfu.edu.tw
Received September 18, 2013; revised October 18, 2013; accepted October 27, 2013
Copyright © 2013 Tain-Sou Tsay. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, an adaptive piecewise linear control scheme is proposed for improving the performance and response time
of servo mechanical control systems. It is a gain stabilized control technique. No large phase lead compensations or
pole zero cancellations are needed for performance improvement. Large gain is used for large tracking error to get fast
response. Small gain is used between large and small tracking error for good performance. Large gain is used again for
small tracking error to cope with disturbance. It gives an almost command independent response. It can speed up the
rise time while keeping robustness unchanged. The proposed control scheme is applied to a servo system with large
time lag and a complicated electro-hydraulic velocity/position servo system. Time responses show that the proposed
method gives significant improvements for response time and performance.
Keywords: Piecewise Linear Controller; Nonlinear Controller; Adaptive Gain; Servo System
1. Introduction
This template Gain and phase stabilized are two conven-
tional design methods for feedback control systems.
They can be analyzed and designed in gain-phase plots to
get wanted gain margin (GM) and phase margin (PM) or
gain crossover frequency
cg
selected for switching. An adaptive switching algorithm
is used. There is no discontinuous connection between
two systems. Therefore, there is no chattering problem.
Gain scheduling has been used successfully to control
nonlinear systems for many decades and in many differ-
ent applications, such as autopilots and chemical proc-
esses [8-10]. It consisted of many linear controllers for
operating points to cope with large parameter variations.
This concept will be expanded for response time and
performance. Operating points are replaced by fast re-
sponse and good performance conditions and interpola-
tion for gain evaluation is replaced by an adaptive
switching point. It is determined by the filtered command
tracking errors. Nonlinear controllers syntheses using
inverse describing function for use with hard nonlinear
system have been developed for several researchers
[11-14]. They are complicated but effective for nonlinear
systems. In this paper, a simple three segments piecewise
linear controller is proposed. It is easy to analyse and
design. Furthermore, it gives an almost reference input
independent response.
and phase crossover
frequency
cp
[1,2]. The gain crossover frequency is
closely related to the system bandwidth (or rise time).
The phase margin is closely related to performance (or
peak overshoot). In general, fast response time and good
performance can not be obtained simultaneously for
some feedback control systems. For example, the altitude
control system of the airframe with altitude and altitude
rate feedbacks needs large altitude loop gain for fast re-
sponse time and low altitude loop gain for good robust-
ness. It is in conflict with another. A simple and effective
way to solve this problem and better results for those of
linear controllers is generally expected. This is the mo-
tivation of this paper. Variable structure control is a
switching control method for feedback control systems
[3-7]. It gives good performance and robustness for cop-
ing with system uncertainty. But it suffered from chat-
tering problem and state measurements. In this paper, a
fast response system and a good performance system are
The proposed control scheme is applied to a servo
system with large transportation lag and a complicated
electro-hydraulic velocity/position servo system. Time
responses show that the proposed method gives signifi-
T.-S. TSAY
86
cant improvements for response time and performance.
2. The Adaptive Piecewise Linear Controller
2.1. Piecewise Linear Nonlinearity
Figure 1(a) shows piecewise linear description of the
symmetrical nonlinearity. Piecewise linear segments
 
ii
yy

are in the form of

1
1
yK
x
1
x
1
(1)


11
2
;
i
ii jjj
j
yKx KKDi

 
(2)

1
1
yK
(3)


11
2
;
i
ii jjj
j
yKx KKDi

 
(4)
Now, the problem is to determine the values of switch
points i and gains i
D
K
between i and 1i
D D
for
the wanted responses time and performance. For illus-
trating purpose, two switching points 1, 1
DD
and
two gains 1
K
, 2
K
will be used to illustrate the advan-
tage of the proposed piecewise linear controller; i.e.,
three segments are discussed. In this work, switching
points 1 and 1 are not fixed and will be deter-
mined by the absolute value of the command tracking
error of feedback control systems. The control configura-
tion of the industry process using the piecewise linear
nonlinearity and PID controller is shown in Figure 1(b).
The finding of will be discussed in the next subsec-
tion.
DD
1
D
+D1
-D1 +D2
-D2 +S1
-S1
+S2
-S2
k1
k3
k2
+S3
-S3
-D3 +D3
X
y
k2
k3
k1
(a)
(b)
Figure 1. (a) Piecewise linear description of an adaptive
gain; (b) Control configuration of the industry process us-
ing PID controller.
2.2. Gain Adapting Using the Piecewise Linear
Nonlinearity
The loop gain of the closed-loop system can be adapted
by the piecewise linear linearity. Considers a second or-
der numerical example described by
 
1
2
Gs ss
(5)
It is closed with a loop gain K. Then the closed-loop
transfer function is

22
K
Ts
sK
 (6)
Poles locations and natural frequency

n
for two
loop gains
12
,
K
K are given below:
10.500;poles :0.2929,1.7071K
;
210.00;poles:1.03.0;3.1623;
n
Kj

They are an over-damped and an under-damped sys-
tems. Time responses are shown in Figure 2 for 1
K
K
(small-dot-line) and 2
K
K
(large-dot-line) in which R
represents the reference input and C represents the plant
output.
The strategy for gain switching is (1) large gain
2
K
for large tracking error to get fast response and (2) small
gain
1
K
for small tracking error (E) to get good per-
formance. It is a variable structure system and can be
achieved by selecting a proper switching point 1 of
the piecewise linear controller shown in Figure 1(a). For
example, the optimal switching point 1 is selected as
0.525 for R = 1 to get both fast response and good per-
formance. Large gain
D
D
2
K
is used for 1
DE and
small gain
1
K
is used for 1
ED. Step response is
shown in Figure 2 (solid-line) also for R = 1. It shows
that adaptive gain can give a good result for fast response
and good performance.
However, it is not true for R is equal to 5, 10 and 50,
respectively. Those step responses are shown in Figure 3.
Naturally, another switching point for R = 5, 10 and
1
D
Figure 2. Time responses for K1 = 0.5, K2 = 10 and adaptive
gain of the illustrating example.
Open Access JAMP
T.-S. TSAY 87
50 can be selected for getting good performance. They
are 2.625, 5.250 and 26.250 for R = 5, 10 and 50, respec-
tively. They are true for step responses from zeros to 5,
10 and 50 only. Another possible way for the switching
point can be dependent on the tracking error (E). A pos-
sible switching rule for 1 is found as D10.925DE
for good performance. Figure 3 shows time responses
for R = 1, 5, 10 and 50, respectively. It can be seen that
the switching rule gives an input command (R) inde-
pendent results. However, they are slower than results
shown in Figures 2 and 4.
One possible way to speed up the time response is
enlarging the large gain phase in the beginning. A
low-pass filter

1
nn
DsK Ts for the absolute
tracking error (E) to get is used. Figure 5 shows
faster response is get for
1
D
n
K1.0445
and 1
nn
T
.
The switching point 1 is shaped for speed up the re-
sponses while keeping performance unchanged. Figure 6
shows input independent responses for R = 1, 5, 10 and
50. Note that the natural frequency
D

n
for 2
K
K
is used to find . Therefore, it is needed to find
n
Tn
K
only.
The design procedures for the proposed method using
the adaptive piecewise linear controller can be deduced
as:
Step 1: Selecting two loop gains for fast response and
good performance, respectively.
Figure 3. Time responses for R = 1, 5, 10, 50 using
D
E
10.925 of the illustrating example.
Figure 4. Time responses for R = 1, 5, 10, 50 using D1 =
0.525 of the illustrating example.
In general, high loop gain
2
K
K for fast re-
sponses and low gain
1
K
K for good performance.
The rise time
c
T of the system using high gain meets
the design specification. The peak overshoot of the sys-
tem with low gain meet the design specification.
Step 2: determining parameters of low-pass filter
1
nn
DsK Ts
to find the optimal switching point
1. The natural frequency D

n
for the high gain sys-
tem
2
K
K is used to find n
T. The natural fre-
quency
n
is close related to the rise time. Another
parameter n can be found by the optimization method
using performance index formulated by integration of the
absolute error (IAE) and integration of the square error
(ISE) or on-line parameterized method [15,16]. The it-
eration rule for finding
T
n
K
is formulated as




1
j
nn
GkTT GkTMpcMps
 ;
(7)
nn
K
GkTT
(8)
where
M
ps is the specification of the Peak point;
M
pc is the peak point found using
nn
K
GkT; T is
simulation period of one step response; and k is the
step responses.
th
k
The proposed control scheme will be applied to a
servo system with large transportation lag and a compli-
Figure 5. Time responses for R = 1 using
D
s =
1
nn
KTs
of the illustrating example.
Figure 6. Time responses for R = 1, 5, 10 and 50 of the illus-
trating example.
Open Access JAMP
T.-S. TSAY
88
cated electro-hydraulic velocity/position servo system.
3. Numerical Example
Example 1: Consider a stable plant has the transfer
function [16,17]
 
2
e
1
s
Gs s
(9)
It is a second order dynamic plus a pure time delay
(SOPDT). In this example, a PID controller with pa-
rameters
1.1953;0.5942;0.7338;
pi d
KKK 
is designed first. And then low gain 1 is se-
lected and high gain 2 is selected for the sys-
tem is just in the sustaining oscillating condition. The
oscillation frequency is
0.50K
2.587K
1.57
n08 rads
. Time re-
sponses using low gain
1
K
2.587;
0.5
1.1385;
nn
K
and high gain
are shown in Figure 7. They show an
over-damped system and a zero-damped system. Now,
applying the proposed control scheme to the system us-
ing
22.587K
0.5
12
The
000; 0.6366;KK T
n
K
is found by following on-line computing
rule:


2
0.9 0.1
nn
GkTT GkTMpcMps 
;
(10)
nn
K
GkTT
(11)
with , and

00.
n
G525 secondsT1.001Mps
.
The found are
n
GkT
 
 
0 0.5;1.0794;21.1365;
31.1385; 41.1385;
nn n
nn
GGTGT
GT GT
 


n
GkT
is converged to be 1.1385 within three pe-
riod simulations. The time response is shown in Figure 7
also. It can be seen that the proposed method can give
fast response and good performance simultaneously. It is
Figure 7. Step responses for constant gains (K = 0.5 & 2.587 )
and adaptive gain with D1 of Example 1.
the combination of over-damped and zero-damped sys-
tems with 1. Zero-damped system is used for fast re-
sponses and over-damped system is used for good per-
formance. Naturally, it is input command (R) independ-
ent also.
D
Simulation results of the proposed method and four
other methods are presented for comparisons. They are
Ziegler-Nichols method [18,19] for finding PI and PID
compensators, Tan et al. [20,21] for finding PID com-
pensator and Majhi [17] for finding PI compensator. Pa-
rameters of five found compensators are given below:
1) Proposed Method:
1.1953;0.5942;0.7338;
pi d
KKK

0.5000;2.587; 1.1385;
KKK
12 0.6366;
nn
T

1.240
K
2) ZN(PI): p
and . 0.251K
i
3) ZN(PID):
1.6367,0.4187 and 0.5972
pi d
KK K
.
4) Tans (PID):
0.620,0.5636 and 0.1705
pi d
KK K

0.864 and 0.3653
KK
.
5) Majhis (PI): pi
.
Time responses are shown in Figure 8. Gain/phase
margins, phase/gain crossover frequencies, Integral of
the Square Error (ISE), and Integral of the Absolute
Error (IAE) are given in Table 1. From Table 1 and
Figure 8, one can see that the proposed method gives
faster response, better performance, and better robustness
than those of other methods presented. Note that the
proposed mrthod can provide a simple way to improved
the system that has been controlled.
Example 2: Consider an electro-hydraulic velocity/po-
sition servo control system [22] shown in Figure 9. The
relation between the servo spool position v
X
and the
input voltage u is in the form of

22
21
v
v
vvv
XK
Gs
uss


v
(12)
where v
K
is the valve gain, v
is the damping ratio of
the servo valve and v
is the natural frequency of the
Figure 8. Comparisons with other methods for Example 1.
Open Access JAMP
T.-S. TSAY
Open Access JAMP
89
Table 1. The gain/phase margins, phase/gain crossover frequencies, ISE and IAE of Example 1 using different methods.
Method GM

rad s
CRP
degPM
rad s
CRG
ISE IAE
Proposed 3.941 1.653 78.45 0.322 1.359 1.687
ZN (PI) 1.986 1.214 72.96 0.572 2.268 4.011
ZN (PID) 1.830 1.459 56.25 0.792 1.770 2.876
Tan’s (PID) 2.418 0.929 38.30 0.488 2.247 3.073
Majhi’s (PI) 2.381 1.099 65.58 0.441 2.465 4.066
Figure 9. Block diagram of the ele ctro-hydraulic system .
servo valve. In general, Equation (12) can be approxi-
mated by vv
X
Ku for large v
. The relation between
the valve displacement V
X
and the load flow rate
L
Q
is governed by the well-known orifice law [22]
72
t
3.5 10Nm
o
 ;
52
3.3 10mrad
t
V
 ;
11 2
t
2.310msN
tp
C
 ;
53
1.6 10mrad
m
D
 ;
32
5.810Kg m sJ

; 0.864 Kgmsrad
m
B
;

L
VJSV LVs
QXKPsignXPXK  (13)
0.4;
v
628 rads.
v
where
j
K
is a constant for specific hydraulic motor;
is the supply pressure;
S
P
L
P is the load pressure and;
s
K
is the valve flow gain which varies at different oper-
ating points. The following continuity property of the
servo valve and motor chamber yields
The control configuration for velocity and position
servo control of the considered system is shown in Fi-
gure 11, in which inner loop and outer loop adaptive
nonlinear controllers are included.
Design results of the velocity control loop are dis-
cussed below:
4
L
mtpLto
QD CPVP
L
; (14)
1) Inner loop PI controller
where m is the volumetric displacement; tp is the
total leakage coefficient; t is the total volume of the
oil; o
DC
V
is the bulk modulus of the oil; and
is the
velocity of the motor shaft. The torque balance equation
for the motor is in the form of
The PI controller is first found by the optimization
toolbox of MATLAB for minimized the integration of
absolute errors (IAE), integration of square errors (ISE)
and zero peak overshoot. Parameters of the PI controller
are 3
1.127 10
p
K
 and . Time re-
sponses of the controlled system using the found PI con-
troller are shown in Figure 12.
3.9632
i
K
mLm L
DP JBT
 

; (15)
where m is the viscous damping coefficient and B
L
T
is the external load disturbance which is assumed to be
dependent upon the velocity of the shaft. The mathe-
matical model of the considered system is shown in Fig-
ure 10. System parameters are given below:
2) Parameters of inner loop adaptive nonlinear
controller
Low gain
11K
and high gain are
selected. The low gain case is the optimized result and
the high gain case is the controlled system in the sustain-
ing condition

29.223K

72
2.3 10ms
sSVL
KPsignXP
 ;
.71 rads312
n
. The
72
t
1. 410Nm
S
P ; 0.5 mv
v
K
;
312.71 rads
gives n
T. 0.0031822.28K
n
89
n
is found by Equations (8) and (9) using 1.001
ps
M
.
T.-S. TSAY
90
Figure 10. Mathematic model of the electro-hydraulic system.
Figure 11. Control configuration of velocity and position servo control system.
Figure 12. Time responses of velocity control system for low gain
K11
and high gain
K29.223 and adaptive gain.
Time responses for low gain, high gain and adaptive gain
are shown in Figure 12. Rise times of the optimization
method and the proposed method are 0.0202 sec and
0.0124 sec; respectively. It shows the proposed method
can give faster response than that of controlled by the
optimized method. The Gain gain/phase margins,
phase/gain crossover frequencies, and rise times are
given also in Table 2. It gives controlled system using
two methods have same robustness while Figure 12
shows the proposed method gives faster response.
Design results of the position control loop are dis-
cussed below:
1) Outer loop PI controller
The PI controller are first found by the optimizations
toolbox of MATLAB for minimized the integration of
absolute errors (IAE), integration of square errors(ISE)
and zero peak overshoot. Parameters of the PI controller
are 18.506
p
K
and 0.3666
i
K
. Time responses of
the controlled system using the found PI controller are
shown in Figure 13.
2) Parameters of outer adaptive nonlinear control-
ler
Low gain
11K
and high gain are
selected. The low gain case is the optimized result and
the high gain case is the controlled system in the sustain-
ing condition

27.877K
91.95 rads
n
. The
91.95 rads
gives n. 0.001T
n0875 13.5K
n
is
found by Equations (8) and (9) using ps
Me
responses for low gain, high gain, and adaptive gain are
shown in Figure 13. Rise times of the optimization
method and the proposed method are 0.0513 sec and
0.0334 sec; respectively. It shows the proposed method
can give faster response than that of controlled by the
optimized method. The Gain gain/phase margins, pha-
1.001 . Tim
Open Access JAMP
T.-S. TSAY 91
Table 2. Gain/phase margins, phase/gain cr ossover frequen-
cies and rise times.
Method GM

Hz
cp
PM
(deg.)

Hz
cg
Rise Time
(sec)
Optimization 9.05 50.03 69.35 8.13 0.0202
Adaptive Gain 9.19 49.73 69.35 8.13 0.0124
Figure 13. Time responses of position control system for
, and adaptive gain. K11K27.877
Table 3. Gain/phase margins, phase/gain crossover fre-
quencies and rise times.
Method GM

Hz
cp
PM
(deg.)

Hz
cg
Rise Time (sec)
Optimization 8.35 47.19 52.14 8.91 0.0513
Adaptive Gain 8.39 47.22 51.09 8.71 0.0334
se/gain crossover frequencies and rise times are given
also in Table 3. It gives controlled system using two
methods have same robustness while Figure 13 shows
the proposed method gives faster response.
4. Conclusions
The proposed adaptive piecewise linear controller has
been shown that provided controlled systems are refer-
ence input independent and both good performance and
fast response were obtained simultaneously. Three seg-
ments piecewise linear controller provided a switching
algorithm for low gain and high systems; i.e., low gain
for performance and high gain for response time. The
switching points were dependent on the command track-
ing errors. There are zero-damped ones used in Example
1 and 2 to get fast responses in large tracking error
phases.
Two servo control system examples were designed and
comparisons were made with famous on-line computing
and control methods and optimization method. They have
illustrated better performance and fast response of the
proposed method than those of other mentioned methods.
REFERENCES
[1] B. C. Kuo and F. Golnaraghi, “Automatic Control Sys-
tems,” 8th Edition, John Wiley & Sons, Inc., Hoboken,
2003.
[2] R. C. Dorf and R. H. Bisop, “Modern Control Systems,”
7th Edition, Pearson Education Singapore Pte, Ltd., Sin-
gapore, 2008.
[3] V. I. Utkin, “Variable Structure Systems with Sliding
Modes,” IEEE Transactions on Automatic Control, Vol.
22, No. 2, 1977, pp. 212-222.
http://dx.doi.org/10.1109/TAC.1977.1101446
[4] G. Bartolini, E. Punta and T. Zolezzi, “Simplex Methods
for Nonlinear Uncertain Sliding-Mode Control,” IEEE
Transactions on Automatic Control, Vol. 49, No. 6, 2004,
pp. 922-933. http://dx.doi.org/10.1109/TAC.2004.829617
[5] J. Y. Hung, W. Gao and J. C. Hung, “Variable Structure
Control: A Survey,” IEEE Transactions on Industry Elec-
tron, Vol. 40, No. 1, 1993, pp. 2-22.
http://dx.doi.org/10.1109/41.184817
[6] G. Bartolini, A. Ferrara, E. Usai and V. I. Utkin, “On
Multi-Input Chattering-Free Second Order Sliding Mode
Control,” IEEE Transactions on Automatic Control, Vol.
45, No. 9, 2000, pp. 1711-1717.
http://dx.doi.org/10.1109/9.880629
[7] S. R. Vadali, “Variable-Structure Control of Spacecraft
Large-Angle Maneuvers,” Journal of Guidance, Control,
and Dynamics, Vol. 9, No. 2, 1986, pp. 235-239.
http://dx.doi.org/10.2514/3.20095
[8] M. Corno, M. Tanelli, S. M. Savaresi and L. Fabbri, “De-
sign and Validation of a Gain-Scheduled Controller for
the Electronic Throttle Body in Ride-by-Wire Racing
Motorcycles,” IEEE Transactions on Control Systems Te-
chnology, Vol. 19, No. 1, 2011, pp. 18-30.
http://dx.doi.org/10.1109/TCST.2010.2066565
[9] R. A. Nichols, R. T. Reichert and W. J. Rugh, “Gain Sche-
duling for H-Infinity Controllers: A Flight Control Ex-
ample,” IEEE Transactions on Control Systems Technol-
ogy, Vol. 1, No. 2, 1993, pp. 69-79.
http://dx.doi.org/10.1109/87.238400
[10] T. A. Johansen, I. Petersen, J. Kalkkuhl and J. Ludemann,
“Gain-Scheduled Wheel Slip Control in Automotive Brake
Systems,” IEEE Transactions on Control Systems Tech-
nology, Vol. 11, No. 6, 2003, pp. 799-811.
http://dx.doi.org/10.1109/TCST.2003.815607
[11] J. H. Taylor and K. Strobel, “Nonlinear Compensator
Synthesis via Sinusoidal-Input Describing Functions,” Pro-
ceedings of American Control Conference, Boston, 1985,
pp. 1242-1247.
[12] R. D. Colgern and A. Jonckheere, “H Control of a Class
of Nonlinear Systems Using Describing Functions and
Simplicial Algorithms,” IEEE Transactions on Automatic
Control, Vol. 42, No. 5, 1997, pp. 707-712.
http://dx.doi.org/10.1109/9.580883
[13] A. Nassirharand and H. Karimi, “Controller Synthesis
Open Access JAMP
T.-S. TSAY
Open Access JAMP
92
Methodology for Multivariable Nonlinear Systems with
Application to Aerospace,” ASME Journal of Dynamic
and System Measurement Control, Vol. 126, No. 3, 2004,
pp. 595-604. http://dx.doi.org/10.1115/1.1789975
[14] A. Nassirharand and H. Karimi, “Nonlinear Controller
Synthesis Based on Inverse Describing Function Tech-
nique in the MATLAB Environment,” Advances in En-
gineering Software, Vol. 37, No. 6, 2006, pp. 370-374.
http://dx.doi.org/10.1016/j.advengsoft.2005.09.009
[15] W. K. Ho, T. H. Lee, H. P. Han and Y. Hong, “Self-Tun-
ing IMC-PID Controller with Gain and Phase Margins
Assignment,” IEEE Transactions on Control System Te ch-
nology, Vol. 9, No. 3, 2001, pp. 535-541.
[16] T. S. Tsay, “On-Line Computing of PI/Lead Compensa-
tors for Industry Processes with Gain and Phase Specifi-
cations,” Computers and Chemical Engineering, Vol. 33,
No. 9, 2009, pp. 1468-1474.
http://dx.doi.org/10.1016/j.compchemeng.2009.05.001
[17] S. Majhi, “On-Line PI Control of Stable Process,” Jour-
nal of Process Control, Vol. 15, No. 8, 2005, pp. 859-867.
http://dx.doi.org/10.1016/j.jprocont.2005.04.006
[18] J. G. Ziegler and N. B. Nichols, “Optimum Setting for
Automatic Controller,” Transactions of ASME, Vol. 65,
1942, pp. 759-768.
[19] K. J. Ǻström and T. Hägglund, “Revisting the Ziegler-
Nichols Step Responses Method for PID Control,” Jour-
nal of Process Control, Vol. 14, No. 6, 2004, pp. 635-650.
http://dx.doi.org/10.1016/j.jprocont.2004.01.002
[20] K. K. Tan, T. H Lee and X. Jiang, “Robust On-line Relay
Automatic Tuning of PID Control System,” ISA Transac-
tions, Vol. 39, 2000, pp. 219-232.
[21] K. K. Tan, T. H. Lee and X. Jiang, “On-Line Relay Iden-
tification, Assessment and Tuning of PID Controller,”
Journal of Process Control, Vol. 11, No. 5, 2001, pp. 483-
486. http://dx.doi.org/10.1016/S0959-1524(00)00012-3
[22] H. E. Merritt, “Hydraulic Control System,” John Wiley,
New York, 1967.