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7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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TUTORIALS ON SLIDING MODECONTROL
September 20, 2010
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMOESO
Advanced State Observers
Observers-A Survey Classical observers such as the Kalman Filter and
Luenberger Observer depend on accuratemathematical representation of the plant.
These state observers are useful in system monitoringand regulation as well as detecting and identifyingfailures in dynamical systems.
The presence of disturbances, dynamic uncertainties
and non-linearities pose a great challenge in practicalapplication of these observers.
D Viswanath Sliding Mode Control Sep 2010 2/ 19
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Observers
CLO
HGO
SMOESO
Advanced State Observers
Observers-A Survey Classical observers such as the Kalman Filter and
Luenberger Observer depend on accuratemathematical representation of the plant.
These state observers are useful in system monitoringand regulation as well as detecting and identifyingfailures in dynamical systems.
The presence of disturbances, dynamic uncertainties
and non-linearities pose a great challenge in practicalapplication of these observers.
D Viswanath Sliding Mode Control Sep 2010 2/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
4/45
Observers
CLO
HGO
SMOESO
Advanced State Observers
Observers-A Survey Classical observers such as the Kalman Filter and
Luenberger Observer depend on accuratemathematical representation of the plant.
These state observers are useful in system monitoringand regulation as well as detecting and identifyingfailures in dynamical systems.
The presence of disturbances, dynamic uncertainties
and non-linearities pose a great challenge in practicalapplication of these observers.
D Viswanath Sliding Mode Control Sep 2010 2/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
5/45
Observers
CLO
HGO
SMOESO
Advanced State Observers
Observers-A Survey The design of a robust observer which overcomes the
above challenge has been attempted by manyresearchers and several advanced observer designs havebeen proposed. Some are:-
High Gain Observer proposed by Khalil [1] and
Esfandiari [2] for the design of output feedbackcontrollers.
Sliding Mode Observer proposed by Slotine [3] andUtkin [?].
A class of non-linear extended state observers (NESO)proposed by J.Han [4].
Sliding mode control with perturbationestimator(SMCPE) based on time delay control byElmali and Olgac [5]
D Viswanath Sliding Mode Control Sep 2010 3/ 19
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Observers
CLO
HGO
SMOESO
Advanced State Observers
Observers-A Survey The design of a robust observer which overcomes the
above challenge has been attempted by manyresearchers and several advanced observer designs havebeen proposed. Some are:-
High Gain Observer proposed by Khalil [1] and
Esfandiari [2] for the design of output feedbackcontrollers.
Sliding Mode Observer proposed by Slotine [3] andUtkin [?].
A class of non-linear extended state observers (NESO)proposed by J.Han [4].
Sliding mode control with perturbationestimator(SMCPE) based on time delay control byElmali and Olgac [5]
D Viswanath Sliding Mode Control Sep 2010 3/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
7/45
Observers
CLO
HGO
SMOESO
Advanced State Observers
Observers-A Survey The design of a robust observer which overcomes the
above challenge has been attempted by manyresearchers and several advanced observer designs havebeen proposed. Some are:-
High Gain Observer proposed by Khalil [1] and
Esfandiari [2] for the design of output feedbackcontrollers.
Sliding Mode Observer proposed by Slotine [3] andUtkin [?].
A class of non-linear extended state observers (NESO)proposed by J.Han [4].
Sliding mode control with perturbationestimator(SMCPE) based on time delay control byElmali and Olgac [5]
D Viswanath Sliding Mode Control Sep 2010 3/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
8/45
Observers
CLO
HGO
SMOESO
Advanced State Observers
Observers-A Survey The design of a robust observer which overcomes the
above challenge has been attempted by manyresearchers and several advanced observer designs havebeen proposed. Some are:-
High Gain Observer proposed by Khalil [1] and
Esfandiari [2] for the design of output feedbackcontrollers.
Sliding Mode Observer proposed by Slotine [3] andUtkin [?].
A class of non-linear extended state observers (NESO)proposed by J.Han [4].
Sliding mode control with perturbationestimator(SMCPE) based on time delay control byElmali and Olgac [5]
D Viswanath Sliding Mode Control Sep 2010 3/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
9/45
Observers
CLO
HGO
SMOESO
Advanced State Observers
Observers-A Survey The design of a robust observer which overcomes the
above challenge has been attempted by manyresearchers and several advanced observer designs havebeen proposed. Some are:-
High Gain Observer proposed by Khalil [1] and
Esfandiari [2] for the design of output feedbackcontrollers.
Sliding Mode Observer proposed by Slotine [3] andUtkin [?].
A class of non-linear extended state observers (NESO)proposed by J.Han [4].
Sliding mode control with perturbationestimator(SMCPE) based on time delay control byElmali and Olgac [5]
D Viswanath Sliding Mode Control Sep 2010 3/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
10/45
Observers
CLO
HGO
SMOESO
Advanced State Observers
Observers-A Survey Sliding mode state and perturbation observer
(SMSPO) by Olgac [6].
Sliding mode state and perturbation observer(SMSPO) by Jiang [7].
Wang and Gao [8] carried out a comparison study offirst three advanced state observers.
D Viswanath Sliding Mode Control Sep 2010 4/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Advanced State Observers
Observers-A Survey Sliding mode state and perturbation observer
(SMSPO) by Olgac [6].
Sliding mode state and perturbation observer(SMSPO) by Jiang [7].
Wang and Gao [8] carried out a comparison study offirst three advanced state observers.
D Viswanath Sliding Mode Control Sep 2010 4/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Advanced State Observers
Observers-A Survey Sliding mode state and perturbation observer
(SMSPO) by Olgac [6].
Sliding mode state and perturbation observer(SMSPO) by Jiang [7].
Wang and Gao [8] carried out a comparison study offirst three advanced state observers.
D Viswanath Sliding Mode Control Sep 2010 4/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
Classical Luenberger Observer
Consider a linear, time invariant continuous timedynamical system given by
x = Ax + Bu (1)
y = Cx
where the matrices A,B and C are parameters of thestate space model.
The Luenberger observer for the above plant is given as
x = Ax + Bu + L(y
Cx) (2)
where L is the observer gain matrix which can befound using pole placement.
D Viswanath Sliding Mode Control Sep 2010 5/ 19
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
Classical Luenberger Observer
Consider a linear, time invariant continuous timedynamical system given by
x = Ax + Bu (1)
y = Cx
where the matrices A,B and C are parameters of thestate space model.
The Luenberger observer for the above plant is given as
x = Ax + Bu + L(y
Cx) (2)
where L is the observer gain matrix which can befound using pole placement.
D Viswanath Sliding Mode Control Sep 2010 5/ 19
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
Classical Luenberger Observer The estimation error is given as
e = x x (3)
Differentiating the above equation, the error dynamicsis arrived at as
e = (A LC)e (4)
The estimation error will converge to zero if (A LC)has all its eigen values in the left half plane.
D Viswanath Sliding Mode Control Sep 2010 6/ 19
Ob A S
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
Classical Luenberger Observer The estimation error is given as
e = x x (3)
Differentiating the above equation, the error dynamicsis arrived at as
e = (A LC)e (4)
The estimation error will converge to zero if (A LC)has all its eigen values in the left half plane.
D Viswanath Sliding Mode Control Sep 2010 6/ 19
Ob A S
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
Classical Luenberger Observer The estimation error is given as
e = x x (3)
Differentiating the above equation, the error dynamicsis arrived at as
e = (A LC)e (4)
The estimation error will converge to zero if (A LC)has all its eigen values in the left half plane.
D Viswanath Sliding Mode Control Sep 2010 6/ 19
Ob A S
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
Classical Luenberger Observer Considering a second order dynamic system,
x1 = x2 (5)
x2 = a1x1 a2x2 + b0u
Assuming that the output variable (or state) is theonly state available for measurement i.e.,y = x1, theLuenberger observer for the above plant is given as
x1 = x2 + l1(x1
x1) (6)x2 = a1x1 a2x2 + b0u + l2(x1 x1)
where L = [l1 l2]T is the observer gain matrix which
can be found using pole placement.
D Viswanath Sliding Mode Control Sep 2010 7/ 19
Ob A S
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
Classical Luenberger Observer Considering a second order dynamic system,
x1 = x2 (5)
x2 = a1x1 a2x2 + b0u
Assuming that the output variable (or state) is theonly state available for measurement i.e.,y = x1, theLuenberger observer for the above plant is given as
x1 = x2 + l1(x1
x1) (6)x2 = a1x1 a2x2 + b0u + l2(x1 x1)
where L = [l1 l2]T is the observer gain matrix which
can be found using pole placement.
D Viswanath Sliding Mode Control Sep 2010 7/ 19
Observers A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
Classical Luenberger Observer Defining estimation error x = x x, differentiating and
substituting the above equations gives
x1 = x2 l1(x1) (7)
x2 = a1x1 a2x2 l2(x1)
Hence the estimation error dynamics can be given as
x1x2 =
l1 1
a1
l2
a2
x1
x2 (8)
or
x1x2
=
0 1a1 a2
x1x2
+
l1 0l2 0
x1x2
(9)
D Viswanath Sliding Mode Control Sep 2010 8/ 19
Observers A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
Classical Luenberger Observer Defining estimation error x = x x, differentiating and
substituting the above equations gives
x1 = x2 l1(x1) (7)
x2 = a1x1 a2x2 l2(x1)
Hence the estimation error dynamics can be given as
x1x2 =
l1 1
a1
l2
a2
x1
x2 (8)
or
x1x2
=
0 1a1 a2
x1x2
+
l1 0l2 0
x1x2
(9)
D Viswanath Sliding Mode Control Sep 2010 8/ 19
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Observers-A Survey
High Gain Observer The high gain observer (HGO) has an error dynamics
structure which is the same as the LuenbergerObserver.
The difference is in the selection of the observer gains. In case of Luenberger Observer, these gains are
calculated using pole placement.
In the case of HGO, the observer gains calculated
using pole placement are divided by a quantity suchthat 0 < < 1.
D Viswanath Sliding Mode Control Sep 2010 10/ 19
Observers-A Survey
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Observers
CLO
HGO
SMO
ESO
Observers A Survey
High Gain Observer The high gain observer (HGO) has an error dynamics
structure which is the same as the LuenbergerObserver.
The difference is in the selection of the observer gains. In case of Luenberger Observer, these gains are
calculated using pole placement.
In the case of HGO, the observer gains calculated
using pole placement are divided by a quantity suchthat 0 < < 1.
D Viswanath Sliding Mode Control Sep 2010 10/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Observers A Survey
High Gain Observer The high gain observer (HGO) has an error dynamics
structure which is the same as the LuenbergerObserver.
The difference is in the selection of the observer gains. In case of Luenberger Observer, these gains are
calculated using pole placement.
In the case of HGO, the observer gains calculated
using pole placement are divided by a quantity suchthat 0 < < 1.
D Viswanath Sliding Mode Control Sep 2010 10/ 19
Observers-A Survey
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Observers
CLO
HGO
SMO
ESO
Observers A Survey
High Gain Observer The high gain observer (HGO) has an error dynamics
structure which is the same as the LuenbergerObserver.
The difference is in the selection of the observer gains. In case of Luenberger Observer, these gains are
calculated using pole placement.
In the case of HGO, the observer gains calculated
using pole placement are divided by a quantity suchthat 0 < < 1.
D Viswanath Sliding Mode Control Sep 2010 10/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
O s s S y
High Gain Observer Considering a second order dynamic system,
x1 = x2 (11)
x2 = f(x) + b0u
The High Gain Observer for the above plant is given as
x1 = x2 + h1(x1 x1) (12)
x2 = f0(x) + b0u + h2(x1 x1)
where H = [h1 h2]T is the observer gain matrix which
can be found by dividing the values calculated usingpole placement by the quantity such that 0 < < 1i.e., h1 =
l1
and h2 =l2
2 .
D Viswanath Sliding Mode Control Sep 2010 11/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
y
High Gain Observer Considering a second order dynamic system,
x1 = x2 (11)
x2 = f(x) + b0u
The High Gain Observer for the above plant is given as
x1 = x2 + h1(x1 x1) (12)
x2 = f0(x) + b0u + h2(x1 x1)
where H = [h1 h2]T is the observer gain matrix which
can be found by dividing the values calculated usingpole placement by the quantity such that 0 < < 1i.e., h1 =
l1
and h2 =l2
2 .
D Viswanath Sliding Mode Control Sep 2010 11/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
29/45
Observers
CLO
HGO
SMO
ESO
y
High Gain Observer Defining estimation error x = x
x, differentiating and
substituting the above equations gives
x1 = x2 h1(x1) (13)
x2 = f (x)
f0(x)
h2(x1)x2 = (x) h2(x1)
Hence the estimation error dynamics can be given as
x1x2
=
h1 1h2 0
x1x2
+
01
(x) (14)
D Viswanath Sliding Mode Control Sep 2010 12/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
y
High Gain Observer Defining estimation error x = x
x, differentiating and
substituting the above equations gives
x1 = x2 h1(x1) (13)
x2 = f (x)
f0(x)
h2(x1)x2 = (x) h2(x1)
Hence the estimation error dynamics can be given as
x1x2
=
h1 1h2 0
x1x2
+
01
(x) (14)
D Viswanath Sliding Mode Control Sep 2010 12/ 19
EXERCISE
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Observers
CLO
HGO
SMO
ESO
High Gain Observer-Exercise What is the physical significance of dividing the
observer gains calculated using pole placement by aquantity such that 0 < < 1 in the case of HGO?
What are the disadvantages of HGO?
D Viswanath Sliding Mode Control Sep 2010 13/ 19
EXERCISE
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Observers
CLO
HGO
SMO
ESO
High Gain Observer-Exercise What is the physical significance of dividing the
observer gains calculated using pole placement by aquantity such that 0 < < 1 in the case of HGO?
What are the disadvantages of HGO?
D Viswanath Sliding Mode Control Sep 2010 13/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Sliding Mode Observer Considering a second order dynamic system,
x1 = x2 (15)
x2 = f(x) + b0u
The Sliding Mode Observer (SMO) for the above plantwith y = x1 is given as
x1 = x2 + l1(y x1) + k1 sgn(y x1) (16)
x2 = f0(x) + b0u + l2(y
x1) + k2 sgn(y
x1)
where L = [l1 l2]T is the observer gain matrix which
can be calculated using pole placement andK = [k1 k2]
T > 0.
D Viswanath Sliding Mode Control Sep 2010 14/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Sliding Mode Observer Considering a second order dynamic system,
x1 = x2 (15)
x2 = f(x) + b0u
The Sliding Mode Observer (SMO) for the above plantwith y = x1 is given as
x1 = x2 + l1(y x1) + k1 sgn(y x1) (16)
x2 = f0(x) + b0u + l2(y
x1) + k2 sgn(y
x1)
where L = [l1 l2]T is the observer gain matrix which
can be calculated using pole placement andK = [k1 k2]
T > 0.
D Viswanath Sliding Mode Control Sep 2010 14/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Extended State Observer
Consider a second order dynamic system,
x1 = x2 (17)
x2 = f(x) + b0u
The non-linear function f(x) is now considered as anextended state x3 and the above set of equations canbe modified as
x1 = x2 (18)
x2 = x3 + b0u
x3 = h
D Viswanath Sliding Mode Control Sep 2010 15/ 19
Observers-A Survey
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Observers
CLO
HGO
SMO
ESO
Extended State Observer
Consider a second order dynamic system,
x1 = x2 (17)
x2 = f(x) + b0u
The non-linear function f(x) is now considered as anextended state x3 and the above set of equations canbe modified as
x1 = x2 (18)
x2 = x3 + b0u
x3 = h
D Viswanath Sliding Mode Control Sep 2010 15/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Extended State Observer The Extended State Observer (ESO) for the above
plant with y = x1 is given as
x1 = x2 + 1(y x1) (19)
x2 = x3 + b0u + 2(y x1)
x3 = 3(y x1)
where = [1 2 3]T is the observer gain matrix
which can be calculated using pole placement in case
of Linear ESO. How are the gains selected in case of Non-linear ESO
(NESO)?
D Viswanath Sliding Mode Control Sep 2010 16/ 19
Observers-A Survey
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Observers
CLO
HGO
SMO
ESO
Sliding Mode Control Perturbation Estimator
D Viswanath Sliding Mode Control Sep 2010 17/ 19
Observers-A Survey
7/23/2019 Sliding Mode Control Tutorials-17 Ctd
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Observers
CLO
HGO
SMO
ESO
Sliding Mode Control Perturbation Estimator
D Viswanath Sliding Mode Control Sep 2010 17/ 19
Observers-A Survey
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Observers
CLO
HGO
SMO
ESO
Sliding Mode State and PerturbationObserver - Olgac
D Viswanath Sliding Mode Control Sep 2010 18/ 19
Observers-A Survey
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Observers
CLO
HGO
SMO
ESO
Sliding Mode State and PerturbationObserver - Olgac
D Viswanath Sliding Mode Control Sep 2010 18/ 19
Observers-A Survey
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Observers
CLO
HGO
SMO
ESO
Sliding Mode State and PerturbationObserver - Jiang
D Viswanath Sliding Mode Control Sep 2010 19/ 19
Observers-A Survey
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Observers
CLO
HGO
SMO
ESO
Sliding Mode State and PerturbationObserver - Jiang
D Viswanath Sliding Mode Control Sep 2010 19/ 19
Khalil, H., High Gain Observers in Nonlinear FeedbackControl:New Directions in Nonlinear Observer Design
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Observers
CLO
HGO
SMO
ESO
Control:New Directions in Nonlinear Observer Design,Lecture Notes in Control and Information Sciences,Vol. 24, No. 4, 1999, pp. 249268.
Esfandiari and Khalil, Output feedback stabilisation offully linearisable systems, International Journal ofControl, Vol. 56, 1992, pp. 10071037.
Slotine, J. J. E. and Misawa, E. A., On Sliding Mode
Observers for Nonlinear Systems, Journal of DynamicSystems, Measurement and Control, Vol. 109, 1987,pp. 245252.
Elmali, H. and Olgac, N., Sliding Mode Control withPerturbation Estimation (SMCPE): A New Approach,International Journal of Control, Vol. 56, No. 4, 1992,pp. 923941.
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