Sliding Mode Control Tutorials-17 Ctd

7/23/2019 Sliding Mode Control Tutorials-17 Ctd

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TUTORIALS ON SLIDING MODECONTROL

September 20, 2010


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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-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]



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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.



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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.



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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.



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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.


Ob A S


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Observers-A Survey


e = x x (3)


e = (A LC)e (4)



Ob A S


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Observers-A Survey


e = x x (3)


e = (A LC)e (4)



Ob A S


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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.


Ob A S


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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)


can be found using pole placement.


Observers A Survey


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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)


Observers A Survey


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Observers-A Survey

Classical Luenberger Observer Defining estimation error x = x x, differentiating and


x1 = x2 l1(x1) (7)

x2 = a1x1 a2x2 l2(x1)


x1x2 =

l1 1

a1

l2

a2

x1

x2 (8)

or

x1x2

=

0 1a1 a2

x1x2

+

l1 0l2 0

x1x2

(9)



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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.


Observers-A Survey


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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 .


Observers-A Survey


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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 .


Observers-A Survey


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y

High Gain Observer Defining estimation error x = x

x, differentiating and


x1 = x2 h1(x1) (13)

x2 = f (x)

f0(x)

h2(x1)x2 = (x) h2(x1)


x1x2

=

h1 1h2 0

x1x2

+

01

(x) (14)


Observers-A Survey


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y

High Gain Observer Defining estimation error x = x

x, differentiating and


x1 = x2 h1(x1) (13)

x2 = f (x)

f0(x)

h2(x1)x2 = (x) h2(x1)


x1x2

=

h1 1h2 0

x1x2

+

01

(x) (14)


EXERCISE


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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?


EXERCISE


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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?


Observers-A Survey


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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)


can be calculated using pole placement andK = [k1 k2]

T > 0.


Observers-A Survey


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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)


can be calculated using pole placement andK = [k1 k2]

T > 0.


Observers-A Survey


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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


Observers-A Survey


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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


Observers-A Survey


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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)?


Observers-A Survey


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Sliding Mode Control Perturbation Estimator


Observers-A Survey


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Sliding Mode Control Perturbation Estimator


Observers-A Survey


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Sliding Mode State and PerturbationObserver - Olgac


Observers-A Survey


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Sliding Mode State and PerturbationObserver - Olgac


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Sliding Mode State and PerturbationObserver - Jiang


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Sliding Mode State and PerturbationObserver - Jiang


Khalil, H., High Gain Observers in Nonlinear FeedbackControl:New Directions in Nonlinear Observer Design


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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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Sliding Mode Control Tutorials-17 Ctd