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Lecture Lecture – ––– 25 225525 Model Predictive Spread ... · Prof. Radhakant Padhi, AE...
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Lecture Lecture Lecture Lecture –––– 25252525
Model Predictive Spread Control (MPSC) and Model Predictive Spread Control (MPSC) and Model Predictive Spread Control (MPSC) and Model Predictive Spread Control (MPSC) and
Generalized MPSP (GGeneralized MPSP (GGeneralized MPSP (GGeneralized MPSP (G----MPSP) DesignsMPSP) DesignsMPSP) DesignsMPSP) Designs
Prof. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
Optimal Control, Guidance and Estimation
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore2
Outline
� Motivation
� MPSC Design
• Mathematical Development
• Alignment Angle Constrained Midcourse Guidance of a Tactical Missile
� G-MPSP Design
• Mathematical Development
• Tactical Missile Guidance with 3-D Impact Angle Constraint
� Concluding Remarks
Model Predictive Spread Control (MPSC)Model Predictive Spread Control (MPSC)Model Predictive Spread Control (MPSC)Model Predictive Spread Control (MPSC)
Prof. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore4
Motivations
� High computational efficiency: Real-time online solution (better than MPSP??)
� Terminal conditions should be met as “hard constraints” (in missile guidance problems, this leads to high accuracy)
� No approximation of system dynamics
� Minimum control usage (without compromising on output accuracy)
� Control Smoothness (by enforcement)
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore5
System dynamics:
MPSP Design: An Overview
Discretized
Goal: with additional (optimal) objective(s)*
N NY Y→
( )
( )
,X f X U
Y h X
=
=
ɺ ( )
( )1
,k k k k
k k
X F X U
Y h X
+ =
=
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore6
MPSP Design: An Overview
Philosophy:
• Guess a control history
• Simulate the system dynamics
• Compute the “error in the output” at k = N
• Update the control history optimally utilizing this error information
• Iterate the control history until convergence
( )* 0N N NY Y Y∆ − →Objective : ≜
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore7
MPSP Design: An Overview
1 1k k N N NB dU B dU dY
− −+ + =⋯
1 1
1 1
1 1
1 2 2 1
2 2
1 2 2 1
N
N N N
N
N N N
N N
N N N
N N N N N N
N N
N N N N N N
YY dY dX
X
Y F FdX dU
X X U
Y F F F Y FdX dU
X X X U X U
− −
− −
− −
− − − −− −
− − − −
∂∆ ≈ =
∂
∂ ∂ ∂ = +
∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ = + +
∂ ∂ ∂ ∂ ∂ ∂ 1
1 1 1
1
1 1 1
N
N N k N N k N N
k k N
N N k N N k N N
dU
Y F F Y F F Y FdX dU dU
X X X X X U X U
−
− − −−
− − −
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
⋮
⋯ ⋯ ⋯⋯
0
kB
1NB
−
(small error approximation)
The sensitivity matrices can be
computed “recursively”.
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore8
� Parameterize control as a linear function of
� Carryout a sensitivity analysis of the output error with respect to the error in the control history
got
MPSC with Linear
Parameterization of Control
( ) ( )� ( )�0 0
0
0 0
0
,k k
k
k go k go
k k k go
a a b b
U a t b U at b
dU U U a t b
− −
= + = +
= − = ∆ + ∆
( ) ( )1 1
1 1 1 1
1 1 1 1
1 1Note: can be computed recursively !
N
yy
N N N
go N go N
DC
y y N
dY B dU B dU
B t B t a B B b
C a D b B B
−
− −
− −
−
= + +
= + + ∆ + + + + ∆
= ∆ + ∆
⋯⋯
⋯ ⋯ ⋯����������������
⋯⋯
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore9
MPSC with Linear
Parameterization of Control
( )
( )
1 2
0 0
1Optimize subject to
2
T T
y y N y y y
J a R a b R b
C a D b dY C a D b K
= +
+ = − + + ≜
� Formulate an optimization problem
� Solve this optimization problem in closed form
( )
1
1
1
2
11 1
1 2where
T
y
T
y
T T
y y y y y
a R C
b R D
C R C D R D K
λ
λ
λ
−
−
−− −
= −
= −
= − +
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore1010
Control Parameterization
Error in control
Substituting for dUk for k = 1,.....,N-1 in
MPSC with Quadratic
Parameterization of Control
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore1111
one gets
MPSC with Quadratic
Parameterization of Control
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore12
� If number of equations is same as number of
unknowns, then
12
� if number of unknowns is greater than the number of
equations, then optimal solution can be obtained by minimizing the following objective (cost) function
MPSC with Quadratic
Parameterization of Control
( )1 2 3
1
2
T T TJ a R a b R b c R c= + +
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore13
Start
Guess a control history
(in parameter form)
Propagate system dynamics
Compute output
Converged control solution
Update the control history(parameters)
Compute the sensitivitymatrices recursively
Stop
Checkconvergence
Yes
No
MPSC
ALGORITHM
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore14
MPSC Design: Reasons for
Computational Efficiency
� Costate variable becomes “static”; i.e. only one time-independent (constant) costate vector is needed for the entire control history update!
� Dimension of costate vector is same as the dimension of the output vector (which is much lesser than the number of states)
� The costate vector is computed symbolically.
� Leads to closed form control history update.
� The computations needed include sensitivity matrices, which are computed “recursively”.
� If necessary, concepts like “iteration unfolding” can be incorporated to save computational time further.
Alignment Angle Constrained Alignment Angle Constrained Alignment Angle Constrained Alignment Angle Constrained
Midcourse Guidance using MPSCMidcourse Guidance using MPSCMidcourse Guidance using MPSCMidcourse Guidance using MPSC
Prof. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore16
OBJECTIVES
� Interceptor must have sufficient capability and proper
initial condition for terminal guidance phase .
� Mid course guidance to provide proper initial
condition to terminal guidance phase.
� Interceptor spends most of its time during mid
course phase and hence should be energy efficient
� Objective: Interceptor has to reach desired point(xd,
yd,zd) with desired heading angle (Φd) and flight path
angle (γd) using minimum acceleration ηΦ and ηγ.
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore1717
MID COURSE GUIDANCE WITH MPSC (Mathematical model)
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore1818
System Dynamics with Downrange as Independent Parameter
System Dynamics:
Control Parameterization:
Output Error at Final Time:
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore1919
RESULTS
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore20
20
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore21
Improvement with Iterations
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore22
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore23
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore24
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore25
Further Results
P. N. Dwivedi, A. Bhattacharya and Radhakant Padhi
Suboptimal Mid-course Guidance of Interceptors for High Speed Targets with Alignment Angle Constraint
AIAA Journal of Guidance, Control and Dynamics, Vol. 34, No. 3, 2011, pp. 860-877.
Reference:
Generalized Model Predictive Static Generalized Model Predictive Static Generalized Model Predictive Static Generalized Model Predictive Static
Programming (GProgramming (GProgramming (GProgramming (G----MPSP)MPSP)MPSP)MPSP)
Prof. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore27
Motivations
� High computational efficiency: Real-time online solution
� Terminal conditions should be met as “hard constraints” (in missile guidance problems, this leads to high accuracy)
� No approximation of system dynamics
� Minimum control usage (without compromising on output accuracy)
� Question: Can the discretized problem formulation be avoided?
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore28
System dynamics:
GMPSP Design: An Overview
Goal: with additional (optimal)
objective(s)( ) ( )*
f fY t Y t→
( )
( )
,X f X U
Y h X
=
=
ɺ
where, , ,n m pX U Y∈ℜ ∈ℜ ∈ℜ
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore29
GMPSP Design : An Overview
Philosophy:
• Guess a control history
• Simulate the system dynamics
• Compute the “error in the output” at t = tf
• Update the control history optimally utilizing this error information
• Iterate the control history until convergence
( ) ( ) ( )( )* 0f f fY t Y t Y tδ − →Objective : ≜
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore30
GMPSP Design: An Overview
( ) ( ) ( ) ( ) ( )( ),W t X t W t f X t U t=ɺ ( )where, p nW t ×∈ℜ
( ) ( ) ( ) ( ) ( )( )0 0
,f ft t
t tW t X t dt W t f X t U t dt = ∫ ∫ɺ
( )( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )0 0
,f ft t
f ft t
Y X t Y X t W t f X t U t dt W t X t dt = + − ∫ ∫ ɺ
� Multiplying both sides of the system dynamics by the matrix : ( )W t
� Integrating both sides from to : 0
tf
t
� Adding to both sides: ( )( )fY X t
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore31
GMPSP Design: An Overview
( ) ( ) ( ) ( )( )
( )00 0
f fft tt
tt t
dW tW t X t dt W t X t X t dt
dt
= −
∫ ∫ɺ
( )( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )0
0 0
,f
f f f f
t
t
Y X t Y X t W t X t W t X t
W t f X t U t W t X t dt
= − +
+ + ∫ ɺ
� Integrating by parts of the last term of the right hand side of last equation:
� Substituting above relation in last equation of previous slide:
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore32
GMPSP Design: An Overview
( )( )( )
( )( ) ( ) ( ) ( )
( )( ) ( )( )
( )( ) ( ) ( )
( ) ( )( )( )
( )0
0 0
( )
, ,
f
f
f
t t
t
t
B t
Y X tY t W t X t W t X t
X t
f X t U t f X t U tW t W t X t W t U t dt
X t U t
δ δ δ
δ δ
=
∂= − + ∂
∂ ∂ + + + ∂ ∂
∫ ɺ
����������
0
( ) ( ) ( )0
ft
ft
Y t B t U t dtδ δ= ∫
� Taking the variation of the both sides and re-arranging terms:
0
0
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore33
Recursive Relation for Computation
of Sensitivity Matrices
� General formula for Recursive Computation:
( ) ( )( ) ( )( )
( )
,f X t U tB t W t
U t
∂=
∂
( )( )( )
( )f
f
f
Y X tW t
X t
∂=
∂
( ) ( )( ) ( )( )
( )
,f X t U tW t W t
X t
∂= −
∂
ɺ
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore34
Augmented Cost Function:
GMPSP Design: Mathematical Formulation
Minimize:
Subject to: ( ) ( ) ( )0
ft
ft
Y t B t U t dtδ δ= ∫
( ) ( )( ) ( ) ( ) ( )( )0
0 01
2
ft T
ct
J U t U t R t U t U t dtδ δ = − − ∫
( ) ( )( ) ( ) ( ) ( )( )
( ) ( ) ( )
0
0
0 01
2
f
f
t T
ct
tT
ft
J U t U t R t U t U t dt
Y t B t U t dt
δ δ
λ δ δ
= − −
+ −
∫
∫
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore35
Necessary Conditions of Optimality:
GMPSP Design: Mathematical Formulation
( ) ( ) ( )0
0 ft
c
ft
JY t B t U t dtδ δ
λ
∂= ⇒ = ∂ ∫
( )( )( ) ( ) ( )( ) ( )( )0 0
TcJ
R t U t U t B tU t
δ λδ
∂= − − − =
∂
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore36
Control Solution:
GMPSP Design: Mathematical Formulation
( ) ( )( ) ( )( ) ( )1 0T
U t R t B t U tδ λ−
= +
( ) ( ) ( )0
ft
ft
Y t B t U t dtδ δ= ∫
( ) ( )1
fA Y t bλ λλ δ− = −
( ) ( )( ) ( )
( ) ( )
0
0
1
0
f
f
tT
t
t
t
A B t R t B t dt
b B t U t dt
λ
λ
−
∫
∫
≜
≜
where
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore37
Control Update:
GMPSP Design: Mathematical Formulation
( ) ( ) ( )
( ) ( )( ) ( )( ) ( ) ( ){ } ( )
( )( ) ( )( ) ( ) ( ){ }
0
1 10 0
1 1
=
=
T
f
T
f
U t U t U t
U t R t B t A Y t b U t
R t B t A Y t b
λ λ
λ λ
δ
δ
δ
− −
− −
= −
− − −
− −
where ( ) ( )1
fA Y t bλ λλ δ− = −
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore38
Start
Guess a control history
Propagate system dynamics
Compute output
Converged control solution
Update the static costateand the control history
Compute the weighting matrixby Backward integration
Stop
Checkconvergence
Yes
No
G-MPSP
ALGORITHM
Tactical Missile Guidance with 3Tactical Missile Guidance with 3Tactical Missile Guidance with 3Tactical Missile Guidance with 3----D D D D
Impact Angle ConstraintImpact Angle ConstraintImpact Angle ConstraintImpact Angle Constraint
Prof. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore40
Motivation
� Is it possible to achieve impact angles in 3D simultaneously in some optimal manner?
� Can terminal constraints in both the angles Azimuth-Angle (Direction of heading), Elevation-Angle (Pitch or Top) be dictated?
� Can the above objective be achieved for stationary, moving and maneuvering ground
targets?
� Can this be achieved with minimum latax demand?
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore41
Challenges
� Strong (nonlinear) coupling between elevation angle and azimuth angle dynamics should be accounted for
� Zero/Near-zero miss distance is desired
� Impact angle constraints in 3D are desired
� Latax demand has to be as minimum as possible.
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore42
A Typical Engagement Scenario
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore43
[ ]sin( )
cos( )
cos( )
cos( )cos( )
cos( )sin( )
sin( )
m m
m m
m
z m
m
m
y
m
m m
m m m m
m m m m
m m m
T DV g
m
a g
V
a
V
x V
y V
z V
γ
γγ
ψγ
γ ψ
γ ψ
γ
−= −
− −=
=
=
=
=
ɺ
ɺ
ɺ
ɺ
ɺ
ɺ
[ ]T
m m m m m mX V x y zγ ψ=
Missile System Dynamics
State Vector:Model:
[ ]T
z yU a a=
Control (Guidance Commands):
Note:
• Autopilot delays for both ay and az have also been considered while evaluating the guidance laws.
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore44
Target Model
Assumptions:
• A point mass model is assumed
• Measurement of coordinates (xt, yt) are available.
• Target velocity Vt is constant (includes stationary targets too)
• Moving targets can do one of the followings:
• No maneuver (straight line path)
• Constant g maneuvers
• Sinusoidal maneuvers
cos( )
sin( )
ty
t
t
t t t
t t t
a
V
x V
y V
ψ
ψ
ψ
=
=
=
ɺ
ɺ
ɺ
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore45
Problem Formulation in G-MPSP
Goal:
( ) ( ) ( ) ( ) ( ) ( )T
f m f m f m f m f m fY t t t x t y t z tγ ψ =
( ) ( )*
f fY t Y t→
Define:
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore46
Guess History: Augmented PN
2
1y z z y x
z x x z y
x y y x z
r r r r
r r r rr
r r r r
σ
σ σ
σ
− = − = −
ɺ ɺ ɺ
ɺ ɺ ɺ ɺ
ɺ ɺ ɺ
Line-of-Sight Rate:
Generation of Yaw and Pitch plane Line of sight rates
sin( ) cos( )
sin( )[cos( ) sin( ) ] cos( )
p m x m y
y m m x m y m z
σ ψ σ ψ σ
σ γ ψ σ ψ σ γ σ
= − +
= − + +
ɺ ɺ ɺ
ɺ ɺ ɺ ɺ
Generation of Yaw and Pitch plane latax commands using closing
velocity Vc
, cos( ),c c
x x y y z z
c z e c z m y e c y
r r r r r rV a N V g a N V
rσ γ σ
+ += − = + =
ɺ ɺ ɺɺ ɺ
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore47
Stationary Targets:
Same initial conditions & Different
Terminal Constraints
0
0
10
10
o
m
o
m
γ
ψ
=
=
Various Constraints in
both the angles
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore48
Stationary Targets:
Same initial conditions & Different
Terminal Constraints (Latax)
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore49
Stationary Targets:
Perturbation in initial conditions
20
20
f
f
o
m
o
m
γ
ψ
= −
=
Initial Condition perturbation with same terminal constraint
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore50
Stationary Targets:
Perturbation in initial conditions
(Angle Histories)
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore51
Stationary Targets:
Perturbation in initial conditions
(Latax)
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore52
Maneuvering Targets
GMPSP Vs APN: A Comparison
0
0
80
20
0
20
f
f
o
m
o
m
o
m
o
m
γ
ψ
γ
ψ
= −
=
=
=
Constraint in
both the angles
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore53
Maneuvering Targets
GMPSP Vs APN: A Comparison
0
0
80
20
0
20
f
f
o
m
o
m
o
m
o
m
γ
ψ
γ
ψ
= −
=
=
=
Constraint in
both the angles
Time histories of Azimuth and Elevation Angle
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore54
Time histories of 3D Angles
Moving/Maneuvering Targets:
Straight line, Constant g & sinusoidal maneuvers
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore55
Zero Effort Miss (ZEM) Plot
(Sinusoidal Maneuver)
Modified
Definition of
ZEM:
Vt Non Zero- Target
allowed to maneuver
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore56
Concluding Remarks: G-MPSP
� In G-MPSP formulation, discretization of system dynamics is not required in problem formulation.
� In G-MPSP, any higher-order integration technique can be used (e.g. forth-order Runge-Kutta scheme).
� MPSP is a special case of the G-MPSP
� 3-D impact angle constrained guidance problem has been resolved using G-MPSP.
� Results are pretty similar to MPSP results; i.e. Superior results have been obtained as compared to an “Augmented PN law” (especially for maneuvering targets).
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore57
Thanks for the Attention….!!