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Design of Power System Stabilizer using Power Rate Reaching Law based Sliding
Mode Control Technique
Vitthal Bandal, Student Member, IEEE, B. Bandyopadhyay, Member, IEEE, and A. M. Kulkarni,
Gp (s)
Ks
ωB
Tw
T1 , T2 , T3 , T4
Gf (s)
Eq
Td
xd
xd
Ef d
δ
ω
Sm
Sm0
id
H
D
KE
TE
xe
Pg0
Pe
xq
Vref
Vs
A
B
C
x
y
t
u
T
s
LIST OF SYMBOLS
generic power system stabilizer transfer function
stabilizer gain
system base frequency ( 377 rad/sec at 60 Hz)
washout time constant
lead/lag time constant
filtering in stabilizer
voltage proportional to the field flux linkages
of machine
d-axis transient open circuit time constant
d-axis synchronous reactance of machine
d-axis transient reactance of machine
generator field voltage
machine shaft angular displacement (degree)
rotor speed ( rad./sec.)
machine slip
nominal slip of the machine
direct axis armature current (pu)
inertia constant(sec.)
damping coefficient
AVR gain
AVR time constant(sec.)
line reactance (pu)
mechanical power on the shaft of machine
electrical power output of machine
q-axis synchronous reactance of machine
reference input voltage
correction voltage
state (plant) matrix of the system
control input matrix
output matrix
state vectors
output vectors
time
stabilizing signal
transpose
switching function
Abstract— The paper presents a new method for design
of power system stabilizer (PSS) using discrete time power
rate reaching law based sliding mode control technique. The
control objective is to enhance the stability and to improve
the dynamic response of a single machine infinite bus (SMIB)
system, operating in different conditions. The control rules are
constructed using discrete time power rate reaching law based
sliding mode control. We apply this controller to design power
system stabilizer for demonstrating the efficacy of the proposed
approach.
Vitthal Bandal is a Research Scholar with Systems and Control Engineering, Indian Institute of Technology Bombay, Mumbai-400076, INDIA
(e-mail:[email protected])
Prof. B. Bandyopadhyay is with Systems and Control Engineering,
Indian Institute of Technology Bombay, Mumbai-400076, INDIA (email:[email protected])(corresponding author)
Prof. A. M. Kulkarni is with Electrical Engineering department,
Indian Institute of Technology Bombay, Mumbai-400076, INDIA (email:[email protected])
I. I NTRODUCTION
Power system stabilizer (PSS) units have long been regarded as an effective way to enhance the damping of electromechanical oscillations in power system [1]. The action of
PSS is to extend the angular stability limits of a power system
by providing supplemental damping to the oscillation of
synchronous machine rotors through the generator excitation
[2]. This damping is provided by an electric torque applied
to the rotor that is in phase with the speed variation. Once
the oscillations are damped, the thermal limits of the tie-lines
in the system may then be approached. This supplementary
signal is very useful during large power transfers and line
outages [3].
Over the past four decades, various control methods have
been proposed for PSS design to improve overall system
performance. Among these, conventional PSS of the leadlag compensation type [1], [4], [5] have been adopted by
most utility companies because of their simple structure,
flexibility and ease of implementation. However, the performance of these stabilizers can be considerably degraded
with the changes in the operating condition during normal operation. Since power systems are highly nonlinear,
conventional fixed-parameter PSSs cannot cope with great
changes in the operating conditions. There are two main
approaches to stabilizing a power system over a wide range
of operating conditions, namely adaptive control and robust control [6]. Adaptive control is based on the idea of
continuously updating the controller parameters according
to recent measurements. However, adaptive controllers have
generally poor performance during the learning phase, unless
they are properly initialized. Successful operating of adaptive controllers requires the measurements to satisfy strict
persistent excitation conditions. Otherwise the adjustment of
the controller’s parameters fails. Robust control provides an
effective approach to dealing with uncertainties introduced
by variations of operating conditions.
Among many techniques available in the control literature,
H∞ and variable structure have received considerable attention in the design of PSSs. The H∞ approach is applied
by Chen [6] to PSS design for a single machine infinite
bus system. The basic idea is to carry out a search over
all possible operating points to obtain a frequency bound on
the system transfer function. Then a controller is designed
so that the worst-case frequency response of the closed
loop system lies within prespecified frequency bounds. It
is noted that the H∞ design requires an exhaustive search
and results in a high order controller. On the other hand the
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variable structure control is designed to drive the system to a
sliding surface on which the error decays to zero [7]. Perfect
performance is achieved even if parameter uncertainties are
present. However, such performance is obtained at the cost
of high control activities (chattering) [8].
In this paper a PSS design for SMIB system using discrete
time power rate reaching law based sliding mode control
technique is proposed. In the sliding mode controller a
switching surface is designed. When the sliding mode occurs,
the system dynamic behaves as a robust state feedback
control system. A discrete time power rate reaching law
based sliding mode controller is investigated, which is used
to minimize the chattering. Simulations results for single
machine infinite bus (SMIB) system are presented to show
the effectiveness of the proposed control strategy in damping
the oscillation modes.
The paper is organized as follows. Section II presents
basics of power system stabilizer and power system analysis.
Section III presents the review on multirate output feedback.
Section IV presents discrete time power rate reaching law
based sliding mode controller design; the same is used
for PSS design of SMIB system as discussed in section
V. Conclusions are drawn in Section VI. The controller is
validated using non-linear model simulation.
II. P OWER S YSTEM S TABILIZER
A. Basic concept
The basic function of a power system stabilizer is to extend
stability limits by modulating generator excitation to provide
damping to the oscillation of synchronous machine rotors
relative to one another. The oscillations of concern typically
occur in the frequency range of approximately 0.2 to 3.0
Hz, and insufficient damping of these oscillations may limit
ability to transmit power. To provide damping, the stabilizer
must produce a component of electrical torque, which is in
phase with the speed changes. The implementation details
differ, depending upon the stabilizer input signal employed.
However, for any input signal, the transfer function of
the stabilizer must compensate for the gain and phase of
excitation system, the generator and the power system,
which collectively determines the transfer function from the
stabilizer output to the component of electrical torque which
can be modulated via excitation system [4].
B. Classical Stabilizer implementation procedure
Implementation of a power system stabilizer implies adjustment of its frequency characteristic and gain to produce
the desired damping of the system oscillations in the frequency range of 0.2 to 3.0 Hz. The transfer function of a
generic power system stabilizer may be expressed as
Gp (s) = Ks
Tw s (1 + sT1 ) (1 + sT3 )
Gf (s)
(1 + Tw s) (1 + sT2 ) (1 + sT4 )
where Ks represents stabilizer gain and Gf (s) represents
combined transfer function of torsional filter (if required) and
input signal transducer. The stabilizer frequency characteristic is adjusted by varying the time constant Tw , T1 , T2 , T3
and T4 . A torsional filter may not be necessary with signals
like power or delta-P-omega signal [9].
A power system stabilizer can be most effectively applied
if it is tuned with an understanding of the associated power
characteristics and the function to be performed by the stabilizer. Knowledge of the modes of power system oscillation
to which the stabilizer is to provide damping establishes the
range of frequencies over which the stabilizer must operate.
Simple analytical models, such as that of a single machine
infinite bus (SMIB) systems, can be useful in determining the
frequencies of local mode oscillations during the planning
stage of a new plant. It is also desirable to establish the
weak power system conditions and associated loading for
which stable operation is expected, as the adequacy of the
power system stabilizer application will be determined under
these performance conditions. Since the limiting gain of the
some stabilizers, viz., those having input signal from speed
or power, occurs with a strong transmission system, it is
necessary to establish the strongest credible system as the
“tuning condition” for these stabilizers. Experience suggest
that designing a stabilizer for satisfactory operation with an
external system reactance ranging from 20% to 80% on the
unit rating will ensure robust performance [10].
C. Power System Analysis
Analysis of practical power system involves the simultaneous solution of equations consisting of synchronous
machines ,associated excitation system , prime movers, interconnecting transmission network, static and dynamic (
motor ) loads, and other devices such as HVDC converters,
static var compensator. The dynamics of the machine rotor
circuits, excitation systems, prime mover and other devices
are represented by differential equations. This results in
the complete system model consisting of large number of
ordinary differential and algebraic equations [9].
1) Generator Equations: The machine equations ( for
jth machine ) are
dEqj
dt
dδj
dt
dSmj
dt
=
−1 [Eqj − (xdj − xdj )idj − Ef dj ],
Td0j
= ωB (Smj − Smj0 ),
=
(1)
(2)
−1
[Dj (Smj − Smj0 ) − Pmj + Pej ]. (3)
2H
Model 1.0 is assumed for synchronous machines by neglecting the damper windings. In addition, the following
assumptions are made for simplicity [11].
1. The loads are represented by constant impedances.
2. Transients saliency is ignored by considering xq = xd .
3. Mechanical power is assumed to be constant.
4. Ef d is single time constant AVR.
2) State space model of power system (Machine model
1.0): The state space model of a SMIB power system, the
block diagram of which is shown in Fig. 1 can be obtained
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K1
∆ Te1
∆Τm
+
∆Sm
−
Σ
∆ Te2
IV. D ISCRETE T IME P OWER R ATE R EACHING L AW
BASED S LIDING M ODE C ONTROL
1
2Hs
∆δ
ωB
s
Consider a SISO plant described by a continuous time
linear model
K4
K2
ẋ
y
K5
∆Εq
_
K5
1 + s Td0 K3
Σ
KE
+
∆Εfd
1 + s TE
∆ vref
_
Σ
(6)
Where x ∈ Rn , u ∈ R, y ∈ R and the matrices A, B and
C are of appropriate dimensions.
Let ( Φτ , Γτ , C) be the system given by Eqn.(6) sampled
at sampling interval τ seconds and is represented as,
+
K6
Fig. 1.
= Ax + Bu,
= Cx.
Block diagram of a Single Machine Infinite Bus (SMIB) system
using generator, transformer, network and loadflow data as
given below [11],
.
x= Ax + B (Vref + Vs ) ,
(4)
y = Cx,
(5)
where
x denotes the states of the machine and are given as
x = [Sm , δ, Ef d , Eq ]. Similarly, y = Sm denotes the output
equation of the machine and C is the output matrix.( C =
[1, 0, 0, 0] ).
Where Sm is machine slip and is given by,
Sm =
(ω − ωB )
,
(ωB )
δ is machine shaft angular displacement in degrees, Ef d is
generator field voltage in pu and Eq is voltage proportional
to field flux linkages of machine in p.u.
The elements of matrix A are dependent on the operating
condition.
III. R EVIEW O N M ULTIRATE O UTPUT F EEDBACK
In the following, multirate output feedback is briefly
reviewed.
Multirate Output Feedback is the concept of sampling the
control input and sensor output of a system at different rates.
It was found that multirate output feedback can guarantee
closed loop stability, a feature not assured by static output
feedback [12] while retaining the structural simplicity of
static output feedback. Much research has been performed in
this field [13]–[17]. In multirate output feedback, the control
input [15], [17] or the sensor output [16] is sampled at a
faster rate than the other. In this paper, the term multirate
output feedback is used to refer the situation wherein the
system output is sampled at a faster rate as compared to the
control input.
It was found that state feedback based control laws of
any structure may be realized by the use of multirate output
feedback, by representing the system states in terms of the
past control inputs and multirate sampled system output [18],
[19].
x(k + 1) = Φτ x(k) + Γτ u(k),
y(k) = Cx(k).
(7)
(8)
In [20] a power rate reaching law approach for continuous
time systems had been proposed. The discrete power rate
reaching law can be directly obtained from the continuous
power rate reaching law as,
s(k + 1) − s(k)
= −kτ |s(k)|α sgn(s(k))
(9)
where τ > 0, is the sampling period, 0 < kτ < 1 and
0 < α < 1. s(k) is the switching function defined as a
function of system states as,
s(k)
= cT x(k)
(10)
= cT x(k + 1)
(11)
Hence,
s(k + 1)
So, from Eqns. (7), (9) and (11)
s(k + 1) − s(k) = cT [Φτ − I]x(k) + cT Γτ u(k)
(12)
Comparing the Eqns.(9) and (12), the control law is
obtained as follows [21],
u(k) = −(cT Γτ )−1 [cT Φδ x(k) + ρ|s(k)|α sgn(s(k))] (13)
where, Φδ = Φτ − I and ρ = kτ . Thus, a discrete time
sliding mode control based on power rate reaching law is
obtained. This control law is designed using the states of the
system. Switching gain c can be obtained using the procedure
given in [22]
As, in practice all states of the system are not available
for measurement and therefore control derived with the help
of only output information of the system will be more useful
from practical point of view. A generalized expression for the
switching surface and the control using output information
only has been derived and is given as [21],
x(k) = Φτ C0−1 yk + [Γτ − Φτ C0−1 D0 ]u(k − 1),
s(k) = cT (Φτ C0−1 yk + [Γτ − Φτ C0−1 D0 ]u(k − 1))
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(14)
(15)
Single Machine Infinte Bus System
1.3
= −(cT Γτ )−1 [cT Φδ Φτ C0−1 yk
−(cT Γτ )−1 [cT Φδ (Γτ − Φτ C0−1 D0 )u(k − 1)
−(cT Γτ )−1 ρ|s(k)|α sgn(s(k))]
(16)
Thus, it can be seen from the Eqns. (15) and (16) that the
states of the system are needed neither for switching function
evaluation nor for the feedback purpose.
POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS
1.28
1.26
1.24
Delta (radians)
u(k)
1.22
1.2
1.18
1.16
1.14
V. C ASE S TUDY
1.12
A single machine infinite bus power system is considered
here for PSS design using power rate reaching law based
sliding mode control technique.
The classical power system stabilizer (PSS) is designed in
the following way.
The eigenvalue analysis of a power system is carried out
and the participation ratio of the machine towards instability
in the network, is estimated. Power system stabilizer using
phase compensation technique is designed according to the
participation ratio of the machine towards instability, till
satisfactory closed loop performance of the power system
is achieved. The above design of classical power system
stabilizer (PSS) is iterative in nature and optimal tuning of
parameters is based on the experience. If the power characteristics of the system changes, then the whole procedure of
PSS design has to be repeated. So, design of classical power
system stabilizers cannot be considered robust in nature for
all operating points. The proposed power rate reaching law
based sliding mode control technique used for power system
stabilizer design is robust in nature for all the models and
is not iterative .
C. Design of PSS using discrete time power rate reaching
law based sliding mode control for Single Machine Infinite
Bus (SMIB) system
The single machine infinite bus power system data is
considered for designing PSS using power rate reaching law
based sliding mode control. The block diagram of the system
is shown in Fig. 1.
The following parameters are used for simulation of the
single machine infinite bus system model [11]:
H = 5 sec., D = 0, Tdo = 6 sec., KE = 100, TE = 0.02
sec., xe = 0.2 p.u.
2
3
−3
4
B. Classical power system stabilizer design for a power
system
1
4
5
Time in sec.
6
7
8
9
10
(a) Delta responses
A. Linearization of power system
Single Machine Infinte Bus System
x 10
POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS
3
2
Slip
The Nonlinear differential equations governing the behavior power system can be linearized about a particular
operating point to obtain a linear model which represents
the small signal oscillatory response of a power system. A
SIMULINK based block diagram including all the nonlinear
blocks can also be used to generate the linear state space
model of the system.
0
1
0
−1
−2
0
1
2
3
4
5
Time in sec.
6
7
8
9
10
(b) Slip responses
Fig. 2. Delta and slip responses with classical PSS and PSS using power
rate reaching law based sliding mode control technique for Pg0=0.5 pu,
Vref=1.0 pu and Xe=0.25 pu
As discussed in the previous section, the SISO linearized
model of entire system obtained at nominal operating condition is obtained, which is represented by Eqn. (6). The power
rate reaching law based sliding mode control given by Eqn.
(16) is then applied to the actual nonlinear system to carry
out simulations.
D. Simulation with Non-linear model
The slip of the machine is taken as output. This output
signal of the controller and a limiter is added to Vref signal.
This is used to damp out the small signal disturbances
via modulating the generator excitation. The disturbance
considered here is a self clearing fault which is cleared after
0.1 second. The limits of PSS output are taken as ±0.1.
Simulation results for SMIB system for various operating
conditions, with power rate reaching law based sliding mode
controller and classical controller are shown in Fig. 2 to Fig.
5.
As shown in plots, the proposed controller is able to damp
out the oscillations in 1 to 2 seconds after clearing the fault.
Even in some cases where, classical PSS cannot damp out
the oscillations, the proposed controller is able to damp out
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Single Machine Infinte Bus System
Single Machine Infinte Bus System
1.5
1.5
POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS
1.4
1.4
1.3
1.3
Delta (radians)
Delta (radians)
POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS
1.2
1.2
1.1
1.1
1
1
0.9
0
1
2
3
4
5
Time in sec.
6
7
8
9
0.9
10
0
1
2
(a) Delta responses
−3
8
4
5
Time in sec.
6
7
8
9
10
(a) Delta responses
−3
Single Machine Infinte Bus System
x 10
3
10
Single Machine Infinte Bus System
x 10
POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS
POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS
8
6
6
4
4
Slip
Slip
2
2
0
0
−2
−2
−4
−6
−4
0
1
2
3
4
5
Time in sec.
6
7
8
9
10
−6
0
1
2
3
4
5
Time in sec.
6
7
8
9
10
(b) Slip responses
(b) Slip responses
Fig. 3. Delta and slip responses with classical PSS and PSS using power
rate reaching law based sliding mode control technique for Pg0=1.0 pu,
Vref=1.0 pu and Xe=0.25 pu
Fig. 4. Delta and slip responses with classical PSS and PSS using power
rate reaching law based sliding mode control technique for Pg0=1.25 pu,
Vref=1.0 pu and Xe=0.25 pu
R EFERENCES
the oscillations.
VI. C ONCLUSION
This paper proposes, the design of PSS for single machine
infinite bus (SMIB) system by power rate reaching law based
sliding mode control technique. It is found that designed
controller provides good damping enhancement for various
operating points of SMIB power system. The proposed
controller results in a better response behavior to damp out
the oscillations. The conventional power system stabilizer is
dynamic in nature and is required to be tuned according to
power characteristics where as, the proposed controller is
non-dynamic in nature and a single controller structure is
able to damp out the oscillations for all models.
Simulation results from a nonlinear power system are
given to demonstrate the applicability and effectiveness of
the proposed approach.
The proposed approach can be extended to a multimachine
system by considering multimachine power system as a set
of interconnected single machine systems.
[1] F. DeMello and C.Concordia, “Concepts of synchronous machine
stability as affected by excitation control,” IEEE Trans. on Power
Apparatus and Systems, vol. PAS-88, pp. 316–329, 1969.
[2] H.Jiang, H. Cai, J. F. Dorsey, and Z. Qu, “Towards a globally
robust decentralized control for large scale power systems,” IEEE
Transactions on Control Systems Technology, vol. 5, no. 3, pp. 309–
319, May 1997.
[3] Y. Wang, G. Guo, and D. J. Hill, “Robust decentralized nonlinear
controller design for multi-machine power systems,” Automatica,
vol. 33, no. 9, pp. 1725–1733, September 1997.
[4] E.V.Larsen and D.A.Swann, “Applying power system stabilizers parti: General concepts,” IEEE Trans. on Power Apparatus and Systems,
vol. PAS-100, no. 6, pp. 3017–3024, June 1981.
[5] ——, “Applying power system stabilizers part- ii: General concepts,”
IEEE Trans. on Power Apparatus and Systems, vol. PAS-100, no. 6,
pp. 3025–3033, June 1981.
[6] S.Chen and O.P.Malik, “H∞ optimization- based power system stabilizer design,” in IEE proceedings Part-C, vol. 142, no. 2, March 1995,
pp. 179–184.
[7] M. L. Kothari, K. Bhattacharya, and J. Nanda, “Adaptive power system
stabilizer based on pole shifting technique,” in IEE proceedings-C, vol.
143, 1996, pp. 96–98.
[8] F. Rashidi, M. Rashidi, and H. Amiri, “An adaptive fuzzy sliding
mode control for power systems stabilizer,” in Industrial Electronics
Society,2003, IECON’03. The 29th Annual Conference of the IEEE,
November 2003, pp. 626–630.
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on December 4, 2008 at 04:01 from IEEE Xplore. Restrictions apply.
[19] B.Bandyopadhyay, V.K.Thakar, C.M.Saaj, and S. Janardhanan, “Algorithm for computing sliding mode control and switching surface
from output samples,” in Proc.8th IEEE Variable Structure Systems
Workshop, September 2004, p. Paper No.4.
[20] W. Gao and J. C. Hung, “Variable structure control of nonlinear systems: A new approach,” IEEE transactions on Industrial Electronics,
vol. 40, no. 1, pp. 45–5, February 1993.
[21] V. K. Thakar, “Power rate reaching law based discrete sliding mode
control,” Annual Progress Seminar 2005.
[22] W. Gao, Y. Wang, and A. Homaifa, “Discrete-time variable structure
control systems,” IEEE transactions on Industrial Electronics, vol. 42,
no. 2, pp. 117–122, April 1995.
Single Machine Infinte Bus System
1.6
POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS
1.5
Delta (radians)
1.4
1.3
1.2
1.1
1
L IST OF F IGURES
0.9
0
1
2
3
4
5
Time in sec.
6
7
8
9
10
2
(a) Delta responses
−3
10
1
Single Machine Infinte Bus System
x 10
POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS
3
8
6
4
Slip
2
4
0
−2
−4
−6
−8
5
0
1
2
3
4
5
Time in sec.
6
7
8
9
10
(b) Slip responses
Block diagram of a Single Machine Infinite Bus
(SMIB) system . . . . . . . . . . . . . . . . . .
Delta and slip responses with classical PSS
and PSS using power rate reaching law based
sliding mode control technique for Pg0=0.5 pu,
Vref=1.0 pu and Xe=0.25 pu . . . . . . . . . .
Delta and slip responses with classical PSS
and PSS using power rate reaching law based
sliding mode control technique for Pg0=1.0 pu,
Vref=1.0 pu and Xe=0.25 pu . . . . . . . . . .
Delta and slip responses with classical PSS
and PSS using power rate reaching law based
sliding mode control technique for Pg0=1.25
pu, Vref=1.0 pu and Xe=0.25 pu . . . . . . . .
Delta and slip responses with classical PSS
and PSS using power rate reaching law based
sliding mode control technique for Pg0=1.35
pu, Vref=1.0 pu and Xe=0.25 pu . . . . . . . .
Fig. 5. Delta and slip responses with classical PSS and PSS using power
rate reaching law based sliding mode control technique for Pg0=1.35 pu,
Vref=1.0 pu and Xe=0.25 pu
[9] P. Kundur, Power System Stability and Control. McGraw-Hill, Inc.
Newyork, 1993.
[10] E.V.Larsen and D.A.Swann, “Applying power system stabilizers partiii: General concepts,” IEEE Trans. on Power Apparatus and Systems,
vol. PAS-100, no. 6, pp. 3034–3046, June 1981.
[11] K.R.Padiyar, Power System Dynamics Stability and Control. Interline
publishing private Ltd. Bangalore, 1996.
[12] V. L. Syrmos, C. T. Abdallah, P.Dorato, and K.Grigoriadis, “Static
output feedback -a survey,” Automatica, vol. 33, no. 2, pp. 125–137,
February 1997.
[13] G. M. Kranc, “Input-output analysis of multirate feedback systems,”
IEEE transactions on Automatic Control, vol. 3, no. 1, pp. 21–28,
November 1957.
[14] E. Jury, “A note on multirate sampled-data systems,” IEEE transactions on Automatic Control, vol. 12, no. 3, pp. 319–320, June 1967.
[15] P. T. Kabamba, “Control of linear systems using generalized sampleddata hold functions,” IEEE transactions on Automatic Control, vol. 32,
no. 9, pp. 772–783, September 1987.
[16] T. Hagiwara and M. Araki, “Design of a stable state feedback
controller based on the multirate sampling of plant output,” IEEE
transactions on Automatic Control, vol. 33, no. 9, pp. 812–819,
September 1988.
[17] A. Chammas and C. Leondes., “Pole assignment by piecewise constant
output feedback,” International Journal of Control, vol. 29, no. 1, pp.
31–38, 1979.
[18] M. E. Polites, “Ideal state reconstructor for deterministic digital control
systems,” International Journal of Control, vol. 49, no. 6, pp. 2001–
2011, 1989.
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on December 4, 2008 at 04:01 from IEEE Xplore. Restrictions apply.
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