Lecture – 39
Integrator Back-stepping;Linear Quadratic (LQ) Observer
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
2
Philosophy of Nonlinear Control Design Using Lyapunov Theory
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
3
Philosophy of Feedback Control Design Using Lyapunov Theory
( )( )
( )( )
( )( )
1
1
Motivation
Goal Design such that
is asymptotically stable
Design Idea:
Choose a pdf
Make
: ,
:
,
U
V X
V X
X f X U
X
X f X X
ϕ
ϕ
=
=
=
∗
∗ ( ) ( )2 2, where (pdf)0V X V X≤ − >
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
4
Feedback Control Design Using Lyapunov Theory: An Example
( )
( )
( )( ) ( )
2 3
21
2 31
3 4
22
1 2
Problem: Design a stabilizing controller for the following system
1
Solution: Let 2
Let us choose
x ax x u
V X x
V x x x ax x u
ax x xu
V X x
V X V X
= − +
=
= = − +
= − +
=
∴ ≤ −3 4 2 ax x xu x⇒ − + ≤ −
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
5
Philosophy of feedback control Design Using Lyapunov Theory
2 4 3
3 2
2 3 3 2
i.e
Analysis:
Advantage: The closed loop system is globally asymptotically stable.
Problem: The benefitial nonlinearity got cancell
xu x x ax
u x x ax
x ax x x x ax
x x
≤ − + −
= − + −
= − − + −
= −
ed. (which is not desirable)
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
6
Philosophy of feedback control Design Using Lyapunov Theory
( )
( )
2 42
1 2
3 4 2 4
3 2
2 2
Let us choose:
Then
leads to:
or
V X x x
V V X
ax x xu x x
ax xu x
ax u x u x ax
= +
−
− + ≤ − −
+ ≤ −
+ = − = − −
≤
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
7
Philosophy of feedback control Design Using Lyapunov Theory
2 3 2
3
Closed Loop system:
i.e. The destabilizing nonlinearity got cancelled,
but the benefitical nonlinearity is retained !
Another Problem: If
x ax x x ax
x x x
= − − −
= − −
⇒
( ) 22
, only if is accurate
If the actual parameter value is , then the feedback loop operates with
,V X x
x x a
a
=
= −
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
8
Philosophy of feedback control Design Using Lyapunov Theory
( )
( )
( )
is
Can be potentially destabilizing termif high
2
2
. . The global stability reduces to local stability.
This excits robustness issues!
However, if is made "Sufficiently powerful",
a a
x x a a x
i e
V X
−
= − + −
then the destabilizing effect can be minimized.
Hence, Lyapunov based designs can be "very robust"
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
9
Control Design UsingIntegrator Back-stepping
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
10
Integrator Back-stepping
( ) ( )
Design a state feedback asymptotically stabilizing controller for the following system
where , ,
Note:
n
X
X f X g X
u
X u
ξ
ξ
ξ
ξ
= +
=
∈ ∈ ∈
⎡ ⎤⎢⎣
Problem :
R R R
1: State of the system, Control input (single input) : n u+∈⎥⎦
R
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
11
Integrator Back-stepping
( )
Assumptions:
, : are smooth
0 0
Considering state as a "control input" of subsystem (1) we assume that a state feedback control law
nf g D
f
ξ
∗ →
∗ =
∗
∃
R
( ) ( )
( ) ( ) ( ) ( ) ( )
( )
1
11
of the from Moreover,
a Lyapunov function such that
where, is a pdf functi
, 0 0.
:
,
:
T
a
a
V D
V X
D
X
V f X g X X V X X DX
V X
ξ ϕ ϕ
ϕ
+
+
∃ →
→
= =
∂⎛ ⎞= + ≤ − ∀ ∈⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠
R
R on.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
12
Integrator Back-stepping
( ) ( )
( )
An important observation:
When 0, = &
(i.e. everything is nice)
However, when 0, and hence 0
in general. That is the core problem!
We need some algebraic manipulation as
0 0 0 0
X
X
X f
X f X
ξ ϕ
ξ
=
→ →
∴
= = =
=
follows.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
13
Integrator Back-stepping
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
Step - 1:
By this construction, when
which is asymptotically stable (i.e.
0,0)!
z
X f X g X g X X g X X
f X g X X g X X
f X g X X g X z
z X f X g X XX
ξ ϕ ϕ
ϕ ξ ϕ
ϕ
ϕ
= + + −
= + + −⎡ ⎤⎣ ⎦
= + +
→ = +
→
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
14
Integrator Back-stepping
( )
( ) ( ) ( ) ( )
This is backstepping, since is
stepped back by differentiation
So, we have
This system is equivalent to the original sys
v
z
Xu
X f X g X X g X zz v
ϕ
ξ ϕ
ϕ
ϕ
= −
⎡ ⎤= − ⎢ ⎥
⎣ ⎦
= + +
=
[ ]
( ) ( )
tem
Note: T T
X f X g XX Xϕ ϕϕ ξ∂ ∂⎛ ⎞ ⎛ ⎞= = +⎡ ⎤⎜ ⎟ ⎜ ⎟ ⎣ ⎦∂ ∂⎝ ⎠ ⎝ ⎠
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
15
Back-stepping:Conceptual Block Diagram
Back-stepping
Ref: H. J. Marquez, Nonlinear Control Systems: Analysis and Design,Wiley, 2003.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
16
Integrator Back-stepping
( )
( ) ( ) ( )
( )
( )
( ) ( )
21
1
1
Step-2: Let
Then
Let
1 , ( )2
a
T
V X
T
a
V X z V X z
VV f X g X X g X z z vX
VV X g X v zX
ϕ
≤−
= +
∂⎛ ⎞= + + +⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠
⎡ ⎤∂⎛ ⎞≤ − + +⎢ ⎥⎜ ⎟∂⎝ ⎠⎢ ⎥⎣ ⎦
( )
( )
1
2
ndf
Then (ndf)
, 0
0
T
anddf
Vv g X k z kX
V V X k z
∂⎛ ⎞= − − >⎜ ⎟∂⎝ ⎠≤ − − <
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
17
Integrator Back-stepping
( )
( ) ( )
( ) ( )
( )
1
1
Control Solution
where
Note: In the design, there is a need to design
:
, , 0
T
T
T
Vv u g X k zX
Vu g X k XX
f X g X kX
Xϕ
ϕ
ϕ ξ ϕ
ϕϕ ξ
∂⎛ ⎞= − = − −⎜ ⎟∂⎝ ⎠
∂⎛ ⎞= − − −⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠
∂⎛ ⎞= + >⎡ ⎤⎜ ⎟ ⎣ ⎦∂⎝ ⎠
first⎡ ⎤⎣ ⎦
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
18
Integrator Back-stepping:An Example
( ) ( )( )
( )
( )
2 31 1 1 2
2
2 31 1 1 1 2 1
1
21 1 1
2 31 1 1 1 1 1 2
Problem
Solution
To find
:
: , , , 1
:
1 2
x ax x xx u
X x f x ax x x g x
x
V x x
V x x x ax x x
ϕ
ξ
= − +=
= = − = =
=
= = − + ≤ ( )2 41 1
1a
x x
V x+
−
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
19
Integrator Back-stepping:An Example
3 41 1 ax x− 2 4
1 2 1 1x x x x+ ≤ − −
( )
( ) ( )
( ) ( )
2 21 1 2 1
21 2 1
22 1 1 1
2 31 1 1 1 2 1
2
Let
Modified system
:
z
x ax x x
ax x x
x ax x x
x ax x x x x
z v x
ϕ
ϕ ϕ
+ ≤ −
+ = −
⇒ = − −
= − + + −⎡ ⎤⎣ ⎦
= ( )( )( ) ( )
( )
1
21 1 1
2 31 11 1 1 1
1 1
Let1 ,2
ux
V x z V x z
V VV V z v ax x x v zx x
ϕ
ϕ
−
= +
⎛ ⎞ ⎛ ⎞∂ ∂⎡ ⎤= + = − + + +⎜ ⎟ ⎜ ⎟⎣ ⎦∂ ∂⎝ ⎠ ⎝ ⎠
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
20
Integrator Back-stepping:An Example
( )
( ) ( )
( )( )( )( ) ( )
1
1
1 2 1
2 3 21 1 2 1 2 1 1
1
2 3 21 1 1 2 1 2 1 1
2 3 21 1 1 2 1 1 2 1
Let
, 0
2 1
1 2
Vv k z kx
u x k x x
u ax x x x k x ax xx
ax ax x x x k x ax x
u ax ax x x x k x x ax
ϕ ϕ
ϕ
⎛ ⎞∂= − − >⎜ ⎟∂⎝ ⎠
− = − − −⎡ ⎤⎣ ⎦∂ ⎡ ⎤= − + − − − − −⎣ ⎦∂
⎡ ⎤= − − − + − − + +⎣ ⎦
= − + − + − − + +
where 0k >
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
21
Integrator Back-stepping:An Example
( )
( )
21
221 2 1
22 21 2 1 1
Note: The composite Lyapunov function is:1 2
1 1 2 21 1 2 2
V V z
x x x
x x x ax
ϕ
= +
= + −⎡ ⎤⎣ ⎦
= + + +
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
22
Integrator Back-stepping:More General Case
( ) ( ) 1
1 2
2
System Dynamics:
Idea : Successive iteration.
Note: The procedure for order system is entirely analogous
thn
X f X g X
u
ξ
ξ ξ
ξ
= +
=
=
⎡ ⎤⎣ ⎦
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
23
Integrator Back-stepping:More General Case
( ) ( )
( )( ) ( )
( )
1
1 2
1
1
1
Step-1 Consider the subsystem
Assumption:
is a stabiliting feedback law for
and is the coresponding Lyapunov fun
:
=
X f X g X
X
X f X g X
V X
ξ
ξ ξ
ξ ϕ
ξ
= +
=
= +
ction.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
24
Integrator Back-stepping:More General Case
( ) ( ) ( ) ( ) ( )
( )
12 1 1
1 1
2
By the result obtained before, we have
We also have
, ( 0)
T T
X
X Vf X g X g X k XX X
X k
V
ϕξ ξ ξ ϕ
ϕ ξ
∂⎛ ⎞ ∂⎛ ⎞= + − − −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦∂ ∂⎝ ⎠⎝ ⎠
>
= ( ) 21 1
12
V Xξ ϕ+ −⎡ ⎤⎣ ⎦
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
25
Integrator Back-stepping:More General Case
( ) ( )
( ) ( )
( ) ( )
1 11 1
11 2
1
2 11
1 21 1 1 1 2
1 1
Step - 2:
where,
Using the same idea,
0
10
g Xf X
T
X
X f X g XX
Xu
Vu f X g XX X
ξξ
ξ
ξξ
ϕ ξ
∴
⎡ ⎤ ⎡ + ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦
⎡ ⎤= ⎢ ⎥
⎣ ⎦
⎛ ⎞ ⎛ ⎞∂ ∂= + −⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦∂ ∂⎝ ⎠ ⎝ ⎠
( ) ( )
( ) ( ) ( )
1 1 1 2 1 1 1
2 2 22 2 1 1 1 1 2 1 1and
, 0
1 1 12 2 2
T
V V V
g X k X k
X X X
ξ ϕ
ξ ϕ ξ ϕ ξ ϕ= + +
− − >⎡ ⎤⎣ ⎦
− = − + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
26
Integrator Back-stepping for Strict Feedback Systems
( ) ( )( ) ( )( ) ( )
1
1 1 1 1 1 2
2 2 1, 2 2 1 2 3
System Dynamics
, ,
, , ,
,k k
X f X g X
f X g X
f X g X
f X
ξ
ξ ξ ξ ξ
ξ ξ ξ ξ ξ ξ
ξ
= +
= +
= +
= ( ) ( )
( ) ( ) ( )
1 1
1 1 2 1 2 1
Strong Assumption:
over the domain of interest
, , , , ,
, , , , , , , , , 0
k k k
k k
t
g X u
g X g X g X
ξ ξ ξ ξ
ξ ξ ξ ξ ξ∀
+
≠
… …
… …
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
27
Integrator Back-stepping for Strict Feedback Systems
( ) ( )( ) ( )
( ) ( )
Special Case:
Solution:
Define and carryout the design for as before.
Finally
, ,
, ,
. .
a a
a a
v
X f X g X
f X g X u
v
f X g X u v
i e
ξ
ξ ξ ξ
ξ
ξ ξ
= +
= +
=
+ =
( ) ( )
( ) Note: By assumption,
1 ,,
, 0
aa
a
u v f Xg X
g X t
ξξ
ξ
= −⎡ ⎤⎣ ⎦
≠ ∀
Linear Quadratic (LQ) Observer
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
29
Why Observers?State feedback control designs need the state information for control computationIn practice all the state variables are not available for feedback. Possible reasons are: • Non-availability of sensors
• Expensive sensors
• Quality of some sensors may not acceptable due to noise (its an issue in output feedback control design as well)
A state observer estimates the state variables based on the measurement of some of the output variables as well as the plant information.
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
30
Observer
An observer is a dynamic system whose output is an estimate of the state vector
Full-order Observer
Reduced-order Observer
• Observability condition must be satisfied for designing an observer (this is true for filter design as well)
X
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
31
Observer Design for Linear Systems
Plant
State observer( )
(sensor output vector)
ˆLet the observed state be and the be
ˆˆ ˆ ˆ
ˆ
e
X AX BUY CX
X
X AX BU K Y
X X X
= +=
= + +
−
Plant :
Observer dynamics
Error :
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
32
Observer Design for Linear Systems
( ) ( )
( ) ( )
ˆ
ˆ ˆ ˆ
Add and Substract and substitute ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ( ) ( ) ( )ˆ ˆ ˆ
1. Make the err
Error Dynamics:
Goals: o
e
e
e
e
X X X
AX BU AX BU K Y
AX Y CX
X AX AX AX AX BU BU K C X
A A X A X X B B U K C X
AX A A K C X B B U
= −
= + − + +
=
= − + − + − −
= − + − + − −
= + − − + −
r dynamics independent of
( may be large, even though may be small) 2. Eliminate the effect of from eror dynamics
X
X XU
∵
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
33
Observer Design for Linear SystemsThis can be done by enforcing ˆ 0
ˆand 0
This results in ˆ
ˆ
e
e
A A K C
B B
A A K C
B B
− − =
− =
= −
=
Observer dynamics:
Necessary and sufficient condition for the existence of Ke :
The system should be “observable”.
( )ˆ ˆ ˆeX AX BU K Y CX= + + −
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
34
Observer Design: Full Order Order of the observer is same as that of the system (i.e. all states are estimated, irrespective of whether they are measured or not).
Goal: Obtain gain Ke such that the error dynamics are asymptotically stable with sufficient speed of response.
This means that  = A – Ke C is Hurwitz (i.e. it has all eigenvalues strictly in the left half plane.
Note: ÂT =AT – CTKeT and the eigen values of both Â
and ÂT are same!
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
35
Comparison of Control and Observer Design Philosophies
CL Dynamics
Objective
CL Error Dynamics
Objective
Notice that
Control Design Observer Design
( )X A BK X= −
( ) 0, as X t t→ →∞ ( ) 0, as X t t→ →∞
( )ˆeX AX A K C X= = −
( ) ( )
( )
Te e
T T Te
A K C A K C
A C K
λ λ
λ
⎡ ⎤− = −⎣ ⎦
= −
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
36
Algebraic Riccati Equation (ARE)Based Observer Design
X AX BU= + T TZ A Z C V= +Y CX=
1nM B AB A B−⎡ ⎤= ⎣ ⎦1nT T T T TN C A C A C−⎡ ⎤= ⎣ ⎦
Tn B Z=
1nT T T T TM C A C A C−⎡ ⎤= ⎣ ⎦
1nN B AB A B−⎡ ⎤= ⎣ ⎦
System Dual System
LQR DesignU K X= −
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
37
ARE Based Observer Design
1 , 0TK R B P P−= >
1 0T TPA A P PBR B P Q−+ − + =
Error Dynamics
( )eX A K C X= −
( )T T T Te eA K C A C K− = −
Analogous1T
eK R CP−=where,
1 0T TPA AP PC R CP Q−+ − + =Observer Dynamics
ˆ ˆ ( )eX AX BU K Y CX= + + −
Acts like a controller
gainwhere,
( )0
X A BK XX as t= −
→ →∞
CL system (control design)
Continuous-time Kalman Filter Design for Linear Time Invariant (LTI) Systems
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
39
Problem Statement
( ) ( ) ( )0 0
System Dynamics: ( ) : Process noise vectorMeasured Output: ( ) : Sensor noise vector
Assumptions:
( ) (0) , , ( ) 0, and ( ) 0,
are "mu
X AX BU GW W tY CX V V t
i X X P W t Q V t R
= + += +
∼ ∼ ∼
[ ]tually orthogonal" (0) : initial condition for ( ) ( ) and ( ) are uncorrelated white noise
( ) ( ) ( ) ( ), 0 (psdf)
( ) (
T
T
X Xii W t V t
iii E W t W t Q Q
E V t V t
τ δ τ
τ
⎡ ⎤+ = ≥⎣ ⎦
+ ) ( ), 0 (pdf)R Rδ τ⎡ ⎤ = >⎣ ⎦
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
40
Problem Statement
( )ˆTo obtain an estimate of the state vector using the state dynamics as well as a "sequence of measurements"as accurate as possible.
ˆi.e., to make sure that the error ( ) ( ) ( ) becomes
v
X t
X t X t X t⎡ ⎤−⎣ ⎦
( )ery small ideally ( ) 0 as .X t t→ →∞
Objective:
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
41
Observer/Estimator/Filter Dynamics
( )( ) ( )( ) ( )
ˆwhere ( ) ( ) : Estimate of the state ˆ ( ) ( ) : Estimate of the output
( ) 0
i X E X X
ii Y E Y YE CX V
E CX E V
CE X E V
=
=
= +
= +
= =∵ˆ
( ) : Estimator/Filter/Kalman Gain How to design ?
e
e
CXiii K
K
=
Problem :
( )ˆ ˆ ˆ eX AX BU K Y Y= + + −
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
42
1
1
ˆ (i) Initialize (0) (ii) Solve for Riccati matrix from the Filter ARE: 0 (iii) Compute Kalman Gain: (iv) Propagate the Filter dynamics:
T T T
Te
XP
AP PA PC R CP GQG
K PC R
−
−
+ − + =
=
( )ˆ ˆ ˆ
where is the measurement vectoreX AX BU K Y CX
Y
= + + −
Solution: Summary
ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
43
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