presentation_VIT_final

Robust Control of Linear and Non-Linear DynamicalSystem by using Extended State Observer

Presented by: Kaliprasad Mahapatro Advisor: Prof Milind E. Rane

Department of Electronics & Telecommunication Engg.

Vishwakarma Institute of Technology, Pune

KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 1 / 48

Contents I

1 Robust Control

2 Extended State Observer

3 Plant Dynamics

4 Linear and Nonlinear Plants

5 Motion Control

6 Magnetic Levitation

7 Flexible Link


Contents II8 Inverted Pendulum

9 Applications

10 Conclusion

11 State of Art

12 Acknowledge


Robust Control Expectation and Problem

1 ExpectationHigh Precision Controlwith Minimum Convergence Time

2 ProblemUncertainty due to some unknown dynamics, parameter variations,external disturbances are prime parameters present in engineeringsystems.Due to their non-linear characteristic it becomes very difficult tocontrol this unknown parameter.


Robust Control Solution

The key solution to achieve robust control is to first estimate theuncertainty and then implement the appropriate control lawbased on estimation.

Estimation of state along with uncertainties will serve bettersolution for implementing control signal as dependency on

practical plant reduces


Extended State Observer Innovation

Extended State Observer(ESO)



1 Owing to the great advances in non-linear control theory, theobserver-based controller has become one of the most commonlyused schemes in industrial applications [WG03] [ARD08].

2 The extended state observer (ESO) has high efficiency inaccomplishing the non-linear dynamic estimation.

3 ESO serves the best estimation by extending the internal andexternal disturbance to a rank new state and then apply a specialnon-smooth nonlinear error feedback to achieve statetracking [ZGG12].












Plant Dynamics

Linear and Nonlinear Plants


Linear and Nonlinear Plants Miscellaneous Plants

Table: Dissertation Flow Structure

Plant Dynamics Application DescriptionMotion control setup Robotics vehicle Plant dynamics(2nd order system)

Magnetic levitation Bullet train Design of ESO(3rd order system) (n+1)th order

Flexible Joint system Robotics arm(4th order system) Controller DesignFlexible Link system Space robots(4th order system) ApplicationInverted Pendulum Humanoid Robot Classical(4th order system) launch pad of missile Control


Motion Control

ECP-220


Motion Control Mathematical Model

z1 = z2

z2 =−c∗dJ∗d

z2 +TD

J∗d︸︷︷︸α(x)

+1J∗d︸︷︷︸

β (x)

u (1)

J∗d = combined inertia c∗d = shaft friction coefficientTD = Torque disturbance u = control voltage

Figure: Plant dynamic model ECP220


Motion Control Feedback Linearization control law

[SL91] feedback control law can be designed as υ = α(x)+β (x)u

u =υ−α(x)

β (x)(2)

υ = υc + k1(υc− z1)+ k2(υc− z2) (3)

Figure: Feedback Linearization

u =υ−a0

b0where a0 = α(x)+d b0 = β (x)+d (4)


Motion Control Extended State Observer (ESO)

Extended State Observer(ESO) for ECP-220


Motion Control Innovation

Figure: Block diagram for FL+ESO

u =(

υ︷︸︸︷k1(υc− z1)− k2z2− z3−a0)

b0(5)


Motion Control Mathematical Interpretation of ESO

˙z1 = z2 +β1g1(e)˙z2 = z3 +β2g2(e)+b0u˙z3 = β3g3(e)−→ Lumped disturbancesy = z1

e = y− z1−→ error

(6)

βi is the observer gainwhere gi is

gi(e,αi,δ ) =

| e |αi , | e |> δ

eδ

1−αi, | e |≤ δ

(7)

forNonlinear ESO α = [1 0.5 0.25]Linear ESO α = [1 1 1]δ is a small number used to limit the gain


Motion Control Result Analysis of NESO and LESO

TD = 10% and b0 = 38 but actually it is b0 = 23.2/Kg−m2

Figure: Tracking for Step input


Magnetic Levitation

MagLev


Magnetic Levitation Problem Statement

Problem?

Use a voltage control electromagnet to suspend a ferromagnetic ball inair at an height of 3mm and maintain the desired trajectory in-spite ofuncertainty and disturbances by using Feedback Linearization andExtended State Observer.

Figure: MagLev modelQuanser- Canada



x1x2x3

=

−(RL )x1x3

g− 1m

x21

b0+b1y+b2y2+b3y3

+

1L00

u

Considering another space coordination zi as stated in [SL91]

z =

x2x3

ddt (g−

1m

x21

b0+b1y+b2y2+b3y3 )

[z]T =

x2x3

L3f h(x)︸︷︷︸α(x)

+Lg(L2f h(x))︸︷︷︸

β (x)

u

[z]T

Now as stated [SL91] considering a non linear feedback control lawυ = α(x)+β (x)u such that z = φ(x) and the new input υ satisfy a lineartime invariant relation



Figure: Block Diagram



Due to uncertainty and disturbances

uFL =k1(υc− z1)− k2z2− k3z3−a0−d

b0(8)

uESO =1b0

(−a0 + k1(υc− z1)− k2z2− k3z3− z4) (9)

(a) FL (b) ESO+FL

Figure: Estimation of states with 80% uncertainty


Flexible Link

Flexible Link


Flexible Link Distinguish between Flexible Joint and Link

Flexible Joint Flexible Link

Bulky Light weight manipulatorsSpring Tension No SpringLess Complex Lightweight slender manipulators,

increasing the complexity



θ

α

θ

α

=

0 0 1 00 0 0 10 Ksti f f

Jeq−A1 0

0 −Ksti f f+JarmJeqJarm

A1 0

θ

α

θ

α

+

00

B1−B1

Vm (10)

where A1 uηmηgKt KmK2

g+BeqRm

JeqRm

and B1 uηmηgKt Kg

JeqRm.

State space represented in (10) can be simplified in z domain as

z1 = z2

z2 = z3

z3 = z4

z4 = −Ksti f f A1

Jeqz2−

Ksti f f (Jeq + Jarm)

JarmJeqz3−A1z4︸︷︷︸

α

+Ksti f f B1

Jarm︸︷︷︸β

Vm

y = z1 (11)KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 24 / 48


Table: Numerical values of parameters of flexible link system

Symbols Meaning ValueKsti f f modeled stiffness 0.7883 N.m/radJeq gear inertia 0.000931 kg.m2

Jarm link inertia 0.0026 kg.m2

Beq viscous damping coeff 0.0015ηm motor efficiency 0.69ηg gearbox efficiency 0.9Kt torque constant 0.00767 N.mKg motor gear ratio 70Km back-EMF constant 0.00767 V/rpmRm armature resistance 2.6 Ω

ωc natural frequency 20.1 rad/sm mass of link 0.065 kgl length of link 0.3 m



Figure: Basic Feedback Linearization block

u =1β[....υ c + k1(υc− z1)+ k2(υc− z2)

+ k3(υc− z3)+ k4(...υ c− z4)−α] (12)



Figure: ESO and FL block for estimating states and uncertainty

u =1b0

[....υ c + k1(υc− z1)+ k2(υc− z2)

+ k3(υc− z3)+ k4(...υ c− z4)−a0− z5] (13)



Figure: States and its estimate in flexible link


Inverted Pendulum

Inverted Pendulum


Inverted Pendulum Problem Statement

To balance the rod at a 0 or at a 7

Figure: Inverted Pendulum Apparatus


Inverted Pendulum Nonlinear & Linear Mathematical Model

ddt

θ

θ

xx

=

z2

2m1z3z4z2−m2lcgsin(z1)−m1gz3cos(z1)−m1l0z3z22+F(t)l0

(m1l20−J0)

z4−J0m1

F(t)−J0z3z22−gJ0sin(z1)−2m1z3l0z4z2+l0(m1l0+m2lc)gsin(z1)+m1gl0z3cos(z1)

m1l20−J0

(14)

By inspection we get

z = Az+BF(t);

where z1 = θ , z2 = θ = z1, z3 = x, z4 = x = z3

z =

θ

θ

xx

;A =

0 1 0 0

m2lcgJ∗ 0 m1g

J∗ 00 0 0 1

(J∗−m2l0lc)gJ∗ 0 −m1l0g

J∗ 0

;B =1J∗

0−l00

Joem1

;


Inverted Pendulum Linear-quadratic (LQ) state-feedback regulator

uval = kp f cmdpos− k1enc1pos− k3enc2pos

−k2(enc1pos− pastpos1)− k4(enc2pos− pastpos2) (15)

kp f = hardware gain, k1,2,3,4= lqr constant gains

(a) Simulation (b) Hardware

Figure: Step response of an inverted pendulum by lqr technique


Applications

Practical Applications


Applications Plants and its Application

Figure: Flexible Joint

Figure: Flexible Joint Application



Figure: Flexible Link

Figure: Flexible Link Application



Figure: Inverted Pendulum in Military

Figure: Inverted Pendulum in Humanoid


Conclusion

Conclusion


Conclusion

1 Model dependent trajectory tracking control for linear andnonlinear plant is possible by using Extended State Observer(ESO).

2 If we assumed calibration error in position sensor by adjustingproper observer gains better estimation of position and otherrelevant states of plant can be done.

3 Lumped Uncertainty and Disturbances are well estimated byESO in model dependent plant

4 All the aforementioned details are proved mathematically and viasimulation to be stable and also validated experimentally onvarious plant. Which inherently proves the robustness.


Conclusion

Publication


Conclusion Published Journal

TITLE - A Novel Approach for Internet Congestion Control Using anExtended State Observer.JOURNAL - International Journal of Electronics and CommunicationEngineering & Technology

TITLE - Comparative Analysis of Linear and Non-linear ExtendedState Observer with Application to Motion ControlJOURNAL - IEEE Conference on Convergence of Technology


Conclusion Communicated Journal

TITLE - Estimating and Compensating Wide Range of Uncertainties inMagLev by using Extended State Observer.JOURNAL - Journal of Systems and Control

TITLE - Extended State Observer based Control of Flexible LinkManipulator in presence of Unknown Payload DynamicsJOURNAL - Transaction on Institute of Measurement and Control


State of Art

State of Art


State of Art

State of Art

1 SAGE Journal of Systems and Control Engineering

2 IEEE Transactions on Industrial Electronics

3 IEEE Transactions on Control Systems Technology

4 IEEE Transactions on Magnetics

5 American Control Conference

6 IEEE/ASME Transaction on Mechatronics7 Elsevier Automatica8 International Symposium on Magnetic Bearings9 ISA Transactions


State of Art

References I

[ARD08] B. X. S. Alexander, Richard Rarick, and Lili Dong.A novel application of an extended state observer for highperformance control of nasa’s hss flywheel and faultdetection.American Control Conference, pages 5216 – 5221, June2008.

[SL91] Jean Jacques E. Slotine and Weiping Li.Applied Nonlinear Control.Prentice Hall, New Jersey, U.S.A, 1st edition, 1991.

[WG03] Weiwen Wang and Zhiqiang Gao.A comparison study of advanced state observer designtechniques.In Proceeding of the American Control Conference, pages4754 – 4759, Denver, Colorado, 2003.


State of Art

References II

[ZGG12] Qing Zheng, Linda Q. Gao, and Zhiqiang Gao.On validation of extended state observer through analysisand experimentation.Journal of Dynamic Systems, Measurement, and Control,ASME, 134:024505–1 – 024505–6, March 2012.


Acknowledge

Profound Thanks

Prof. Milind E. RaneProf. S. R. BandewarProf. A. M. ChopdeStaff of Department of E&TC Engg for their kind support.


Acknowledge

Formal Thanks

MatlabSimulinkLATEX

Template- Beamer

Package- Cambridge-US


Acknowledge

Thanks

THANK YOUBy- KALIPRASAD A. MAHAPATRO

Gr.No.12M042


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