lecture – 8 state space representation of dynamical systems 9.pdf · state space representation...
TRANSCRIPT
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Lecture – 8State Space Representation of Dynamical Systems
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
2
A Practical Control System
Temperature control system in a car
Ref.: K. Ogata,Modern Control Engineering3rd Ed., Prentice Hall, 1999.
(bias noise)
Window opening/closing(random noise)
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
3
Another Practical Control System
Water level control in an overhead tank
Ref.: K. Ogata,Modern Control Engineering3rd Ed., Prentice Hall, 1999.
noise
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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State Space Representation
Input variable:• Manipulative (control)• Non-manipulative (noise)
Output variable:Variables of interest that can be either be measured or calculated
State variable:Minimum set of parameters which completely summarize the system’s status.
System
noisecontrol
*Y Y→
Z
Controller
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Definitions
0
0
The state of a dynamic system is the smallestnumber of variables (called )such that the knowledge of these variables at ,together with the knowledge of the input for , co
t tt t==
State :state variables
0
mpletely determine the behaviour of the system for any time .
State variables need not be physically measurableor observable quantities. This gives extra flexibility.
t t≥
Note :
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Definitions
1 2
A - dimensional vector whos componentsare state variables that describe the systemcompletely.The - dimensional space whose co-ordinateaxes consist of the axis, axis,
nn
nx x
State Vector :
State Space :, axis
is called a state space.
For any dynamical system, the state space remainsunique, but the state variables are not unique.
nx
Note :
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Critical Considerations while Selecting State Variables
Minimum number of variables• Minimum number of first-order differential equations
needed to describe the system dynamics completely• Lesser number of variables: won’t be possible to
describe the system dynamics• Larger number of variables:
• Computational complexity• Loss of either controllability, or observability or both.
Linear independence. If not, it may result in:• Bad: May not be possible to solve for all other system
variables• Worst: May not be possible to write the complete state
equations
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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State Variable Selection
Typically, the number of state variables (i.e.the order of the system) is equal to the number of independent energy storage elements. However, there are exceptions!
Is there a restriction on the selection of the state variables ?YES! All state variables should be linearly independent and they must collectively describe the system completely.
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
9
Advantages of State Space Representation
Systematic analysis and synthesis of higher order systems without truncation of system dynamicsConvenient tool for MIMO systemsUniform platform for representing time-invariant systems, time-varying systems, linear systems as well as nonlinear systemsCan describe the dynamics in almost all systems (mechanical systems, electrical systems, biological systems, economic systems, social systems etc.)Note: Transfer function representations are valid for only for linear time invariant (LTI) systems
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Generic State Space Representation
[ ] [ ]
( )
( )( )
1 1
1 1 1 1
1 1
, ,
,
, ,
,
, ,
T Tn mn m
n m
n n n m
X f t X U
X x x U u u
x f t x x u u
t
x f t x x u u
+
∈ ∈
⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ = ∈⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦
R R
R
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
11
Generic State Space Representation
( )
( )( )
1
1 1 1 1
1 1
, ,
, ,
,
, ,
T pp
n m
p p n m
Y h t X U
Y y y
y h t x x u u
t
y h t x x u u
+
⎡ ⎤ ∈⎣ ⎦⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ = ∈⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦
R
R
( )( )
, , : A set of differential equations
, , : A set of algebraic equations
X f t X U
Y h t X U
=
=
Summary:
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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State Space Representation(noise free systems)
Nonlinear System
Linear System A - System matrix- n x n
- Input matrix- n x m
- Output matrix- p x n
- Feed forward matrix – p x m
BCD
X AX BUY CX DU= += +
p
mn
RYRURX
∈
∈∈ ,( , )( , )
X f X UY h X U==
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Block diagram representation of linear systems
Ref.: K. Ogata,Modern Control Engineering3rd Ed., Prentice Hall, 1999.
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Writing Differential Equations in First Companion Form(Phase variable form/Controllable canonical form)
1
1 01
11
22
2 2
1
1
Choose output ( ) and its ( 1) derivatives as state variables
differentiating
n n
nn n
nn
n nn n
d y d ya a y udt dt
y t ndyxx ydt
dyx d yxdt dt
d yx d yxdt dt
−
− −
−
−
+ + + =
−
⎡ ⎤==⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢⎢ ⎥= ⎢ =⎢ ⎥ ⎢⎯⎯→ ⎯⎯→⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢=⎢ ⎥ ⎢ =⎣ ⎦
⎣ ⎦
⎥⎥⎥⎥⎥⎥⎥
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
15
First Companion Form(Controllable Canonical Form)
[ ]
1 1
2 2
3 3
1 1
0 1 2 1
1
2
3
1
0 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 0
00 0 0 0 0 0 0 1 0
1
1 0 0 0
n n
n n n
n
n
x xx xx x
u
x xx a a a a x
xxx
y
xx
− −
−
−
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
− − − − ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡
= [ ]0 u
⎤⎢ ⎥⎢ ⎥⎢ ⎥
+⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
16
Example – 1Dynamical system
State variables:
xyuxxx
==++ 23
1 2
1 2 2 1 2
,, 2 3
x x x xx x x x x u= =− − +
[ ] [ ]
1 1
2 2
1
2
0 1 02 3 1
1 0 0
BXX A
C D
x xu
x x
xy u
x
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
⎡ ⎤= +⎢ ⎥
⎣ ⎦
Hence
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Example – 2(spring-mass-damper system)
System dynamics
State variables
State dynamics
my cy ky bu+ + =
1 2,x y x y
( ) ( )2
1
2 1 21 1
y xxb bx ky cy u kx cx u
m m m m
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − + − − +⎣ ⎦ ⎣ ⎦ ⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
18
Example – 2(spring-mass-damper system)
State dynamics
Output equation
1 1
2 2
0 1 0x xuk c bx x
m m m
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= +− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
[ ] [ ]1
1 0 0y xy X u= +
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Example – 3: Translational Mechanical System
( )
( ) ( )
21 1
1 1 22
22
2 2 12
0d x dxM D K x xdt dt
d xM K x x f tdt
+ + − =
+ − =
Ref: N. S. Nise: Control Systems Engineering, 4th Ed., Wiley, 2004
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
20
Example – 3: Translational Mechanical System
Define
System equations
State space equations in standard form
1 21 2,dx dxv v
dt dt
( )
11 1 2
1 1 1
21 2
2 2 2
1
dv K D Kx v xdt M M Mdv K Kx x f tdt M M M
= − − +
= − +
( )
1 1
1 1 11 1
2 2
2 22
2 2
0 1 0 00
0 00
0 0 0 11
0 0
x xK D KM M Mv v
f tx xv vK K
MM M
⎡ ⎤⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥− −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥= + ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎢ ⎥−⎢ ⎥ ⎣ ⎦
⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
21
Example – 4:Nonlinear spring in Example – 3
Dynamic equations
State space equation
( )
( ) ( )
231 1
1 1 22
232
2 2 12
0d x dxM D K x xdt dt
d xM K x x f tdt
+ + − =
+ − =
( )
( ) ( )
1 1
31 1 2 1
1 1
2 2
32 2 1
2 2
1
x vK Dv x x vM M
x vKv x x f t
M M
=
= − − −
=
= − − +
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
22
Example – 5The Ball and Beam System
The beam can rotate by applying a torque at the centre of rotation, and ball can move freely along the beam
Moment of Inertia of beam:Mass, moment of inertia and radius of ball:
J, ,bm J R
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
23
Example – 5The Ball and Beam System
State Space Model
1 22
3 1 42
2
3 4
1 3 1 2 44 2
1
sin
cos 2
b
b
x x
mg x m x xx JmR
x xmg x x m x x xx
m x J Jτ
=
− +=
+
=
− −=
+ +
θθ ==== 4321 ,,, xxrxrx
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
24
Example – 6: Van-der Pol’s Oscillator (Limit cycle behaviour)
Equation
State variables
State Space Equation
( ) { }22 1 0 , 0M x c x x k x c k+ − + = >
1 2,x x x x
( )( )
21
22 1 2 1
: Homogeneous nonlinear system2 1X
F X
xxc kx x x x
m m
⎡ ⎤⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
25
Example – 7: Spinning Body Dynamics(Satellite dynamics)
Dynamics: 2 31 2 3 1
1 1
3 12 3 1 2
2 2
1 23 1 2 3
3 3
1
1
1
I II I
I II I
I II I
ω ω ω τ
ω ω ω τ
ω ωω τ
⎛ ⎞ ⎛ ⎞−= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞−
= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞−
= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
1 2 3
1 2 3
1 2 3
, , : MI about principal axes, , : Angular velocities about principal axes, , : Torques about principal axes
I I Iω ω ωτ τ τ
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
26
Example – 8: Airplane Dynamics, Six Degree-of-Freedom Nonlinear ModelRef: Roskam J., Airplane Flight Dynamics and Automatic Controls, 1995
( ) ( )( ) ( )
( ) ( )
1 2 3 A 4 A
2 25 6 7
8 2 4 9
= c PQ+c L +c
= c
= c
T T
A T
A T A T
P QR c L N N
Q PR c P R c M M
R PQ c QR c L L c N N
+ + +
− − + +
− + + + +
( )( )( )
sin /
sin cos /
cos cos /
X X
Y Y
Z Z
A T
A T
A T
U VR WQ g F F m
V WP UR g F F m
W UQ VP g F F m
= − − Θ+ +
= − + Φ Θ+ +
= − + Φ Θ+ +
( )
= sin tan cos tan = cos sin = sin cos sec
P Q RQ R
Q R
Φ + Φ Θ+ Φ Θ
Θ Φ − Φ
Ψ Φ+ Φ Θ
cos sin 0 cos 0 sin 1 0 0sin cos 0 0 1 0 0 cos sin
0 0 1 sin 0 cos 0 sin cos
X UY VZ W
′⎡ ⎤ Ψ − Ψ Θ Θ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥′ = Ψ Ψ Φ − Φ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥′ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− Θ Θ Φ Φ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦
Note: h Z⎡ ⎤′= −⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
27
Example – 8: Airplane Dynamics, Six Degree-of-Freedom Nonlinear ModelRef: Roskam J., Airplane Flight Dynamics and Automatic Controls, 1995
1
1
1
cos cos
cos sin
sin
x i i
y i i
z i
N
T i T TiN
T i T Ti
N
T i Ti
F T
F T
F T
=
=
=
= Φ Ψ
= Φ Ψ
= − Φ
∑
∑
∑
( ) ( )
( ) ( )
( ) ( )
1 1
1 1
1 1
cos sin sin
cos cos sin
cos cos cos sin
i i i i i
i i i i i
i i i i i i
N N
T i T T T i T Ti iN N
T i T T T i T Ti i
N N
T i T T T i T T Ti i
L T z T y
M T z T x
N T y T x
= =
= =
= =
= − Φ Ψ − Φ
= Φ Ψ + Φ
= − Φ Ψ + Φ Ψ
∑ ∑
∑ ∑
∑ ∑
( ) ( )( )1
X O is hX E
Z Z O is h E
A D D D DAE
A A L L L Lh
F C C C CFT T qS
F F C C C Ci
α δ
α δ
α α α δ⎛ ⎞⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎜ ⎟⎢ ⎥= = − +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎜ ⎟⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎣ ⎦⎝ ⎠
( ) ( )S A R
S A R
A l llA A
RA A n n n
L C CCLT T qSb
N N C C Cδ δβ
β δ δ
δα α β
δ
⎛ ⎞⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤⎜ ⎟= = +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎝ ⎠
Y A R
AA Y Y Y Y
R
F qS C qS C C Cβ δ δ
δβ
δ⎛ ⎞⎡ ⎤⎡ ⎤= = +⎜ ⎟⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠
[ ]1o ih E
TA m m m m h m EM qSc C qSc C C C i C
α δα δ⎡ ⎤= = +⎣ ⎦
( )cos sinsin cos
Tα α
αα α
−⎡ ⎤⎢ ⎥⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Advantages of State Space Representation
Systematic analysis and synthesis of higher order systems without truncation of system dynamicsConvenient tool for MIMO systemsUniform platform for representing time-invariant systems, time-varying systems, linear systems as well as nonlinear systemsCan describe the dynamics in almost all systems (mechanical systems, electrical systems, biological systems, economic systems, social systems etc.)Note: Transfer function representations are valid for only for linear time invariant (LTI) systems
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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