Integrator Back-stepping; Linear Quadratic (LQ) Observer
Transcript of Integrator Back-stepping; Linear Quadratic (LQ) Observer
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Lecture – 39
Integrator Back-stepping;Linear Quadratic (LQ) Observer
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
Philosophy of Nonlinear Control Design Using Lyapunov Theory
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
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≤ − >
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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⇒ − + ≤ −
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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)
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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
= +
−
− + ≤ − −
+ ≤ −
+ = − = − −
≤
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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
=
= −
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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"
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Control Design UsingIntegrator Back-stepping
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
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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
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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.
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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.
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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
= + + −
= + + −⎡ ⎤⎣ ⎦
= + +
→ = +
→
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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∂ ∂⎛ ⎞ ⎛ ⎞= = +⎡ ⎤⎜ ⎟ ⎜ ⎟ ⎣ ⎦∂ ∂⎝ ⎠ ⎝ ⎠
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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Back-stepping:Conceptual Block Diagram
Back-stepping
Ref: H. J. Marquez, Nonlinear Control Systems: Analysis and Design,Wiley, 2003.
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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
∂⎛ ⎞= − − >⎜ ⎟∂⎝ ⎠≤ − − <
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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⎡ ⎤⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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+
−
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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
−
= +
⎛ ⎞ ⎛ ⎞∂ ∂⎡ ⎤= + = − + + +⎜ ⎟ ⎜ ⎟⎣ ⎦∂ ∂⎝ ⎠ ⎝ ⎠
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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 >
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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
= +
= + −⎡ ⎤⎣ ⎦
= + + +
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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
= +
=
=
⎡ ⎤⎣ ⎦
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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.
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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+ −⎡ ⎤⎣ ⎦
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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= + +
− − >⎡ ⎤⎣ ⎦
− = − + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦
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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∀
+
≠
… …
… …
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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
= −⎡ ⎤⎣ ⎦
≠ ∀
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Linear Quadratic (LQ) Observer
Dr. Radhakant PadhiAsst. Professor
Dept. of Aerospace EngineeringIndian Institute of Science - Bangalore
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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.
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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
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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 :
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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
∵
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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= + + −
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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!
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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
⎡ ⎤− = −⎣ ⎦
= −
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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= −
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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)
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Continuous-time Kalman Filter Design for Linear Time Invariant (LTI) 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
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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⎡ ⎤ = >⎣ ⎦
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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:
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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= + + −
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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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
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ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore
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