Lecture – 26
Classical Numerical Methods to Solve
Optimal Control Problems
Dr. Radhakant Padhi
Asst. Professor
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
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Classical Numerical, Slides of Aeronautical Engineering
Some concept of Advanced Control System Design for Aerospace Vehicles are Advanced Control System, Basic Principles, Calculus of Variations, Classical Control Three, Gain Scheduling and Dynamic Inversion. Main points of this lecture are: Classical Numerical Methods, State Equation, Optimality In Optimal Control, Necessary Conditions, Costate Equation, Boundary Condition, State and Costate Equations, Dynamic Equations, Formulation, Problems
Typology: Slides
2012/2013
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Lecture – 26
Classical Numerical Methods to Solve
Optimal Control Problems
Dr. Radhakant Padhi
Asst. Professor
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
Necessary Conditions ofOptimality in Optimal Control
z
State Equation
z
Costate Equation
z
Optimal ControlEquation
z
Boundary Condition
,^
,
H
X
f
t X U
λ ∂
=
=
∂
,^
,
H
g t X U
X
λ
∂ ⎛
⎞
= −
=
⎜
⎟
∂ ⎝
⎠
f
f ϕ X
λ
∂
=
∂
(
) 0
0
:Fixed
X t
X
0
,
H
U
X
U
ψ
λ
∂ ⎛
⎞
=
⇒
=
⎜
⎟
∂ ⎝
⎠
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
Classical Methods to SolveTPBVPs z
Gradient Method
z
Shooting Method
z
Quasi-Linearization Method
Gradient Method
Dr. Radhakant Padhi
Asst. Professor
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
Gradient Method
(
)
(
)
(
)
f 0 f 0 f 0
T
f^
f
f
t
T
t t
T
t t
T
t
J
X
X
H
X
dt
X H
U
dt
U
H
X
dt
∫ ∫ ∫
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
Gradient Method z
After satisfying the state & costate equationsand boundary conditions, we have
(
)
t^ f^0
T
t
H
J
U
dt
U
z
Select
( )
,^
0
H
U
t
U
δ
τ
τ
∂ ⎡
⎤
= −
⎢
⎥
∂ ⎣
⎦
t^ f^0
T
t
H
H
J
dt
U
U
z
This leads to
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
Gradient Method: Procedure z
Assume a control history (not a trivial task)
z
Integrate the state equation forward
z
Integrate the costate equation backward
z
Update the control solution
This can either be done at each step while integratingthe costate equation backward or after the integrationof the costate equation is complete
z
Repeat the procedure until convergence
0
(a pre-selected constant)
t^ f
T
t
H
H
dt
U
U
γ
∂
∂
⎡
⎤
⎡
⎤
≤
⎢
⎥
⎢
⎥
∂
∂
⎣
⎦
⎣
⎦
∫
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
Gradient Method: Selection of z
Select
so that it leads to a certain
percentage reduction of
z
Let the percentage be
z
Then
z
This leads to
J
α
0
100
t^ f
T
t
H
H
dt
J
U
U
α
τ
∂
∂
⎡
⎤
⎡
⎤
=
⎢
⎥
⎢
⎥
∂
∂
⎣
⎦
⎣
⎦
0
100
t^ f
T
t
J
H
H
dt
U
U
α
τ =
∂
∂
⎡
⎤
⎡
⎤
⎢
⎥
⎢
⎥
∂
∂
⎣
⎦
⎣
⎦
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
13
MATHEMATICAL PERSPECTIVE: • Minimum time optimization problem• Fixed initial conditions and free final time problem SYSTEM DYNAMICS: Equations of motion for a missile in vertical plane. The non-dimensional equationsof motion (point mass) in a vertical plane are:
2
2
'^
sin( )
cos(
)
1
'^
[
sin(
)
cos( )]
where prime denotes differentiation with respect to the non-dimensional time
w
D
w
w
L^
w
M
S M C
T
S M C
T
M
γ
α
γ
α
γ
τ
= −
−
=
−
A Real-Life Challenging Problem
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
2
The non-dimensional parameters are defined as follows:
;^
;^
;
2
where
flight Mach numberflight path angle
thrust
mass of the
w
w
g
T
a S
V
T
S
M
at
mg
mg
a
M
T
m
ρ
τ
γ
=
=
=
=
= =
=
=
missile
reference aerodynamic area
speed of the missile
lift coefficient
drag coefficient
the acceleration due to gr
L
D
S
V
C
C
g
=
=
=
=
=
avity
the local speed of sound
the atmospheric density
flight time after launch
NOTE:
,^
are usually functions of
&
(tabulated data)
L
D
a t
C
C
M
ρ α
=
=
=
A Real-Life Challenging Problem
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
(^
2
2 2
Choosing
as the independent variable the equations are reformulated as follows:
sin( )
cos(
)
cos( )
sin(
)
cos
w
D
w
w
L
w
w
L
S M C
T
M
dMd
S M C
T
dt
a M
d
g S M C
γ
γ
α
γ
γ
α
γ
−
−
=
−
=
−
(^
2
0
0
0
( )
sin(
)
and the transformed cost function is
cos( )
sin(
)
A difficult minimum-time problem has been converted to a relativelyeasier fix
f
w
t
w
L^
w
T
dt
a M
J
dt
d
d
d
g S M C
T
π
π
γ
α
γ
γ
γ
γ
α
−
−
=
=
=
−
∫
∫
∫
ed final-time problem (with hard constraint:
(
)
0.8)!
f
M
γ
⎛
⎞
⎜
⎟
=
⎝
⎠
A Real-Life Challenging Problem
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
Task
Solve the problem using gradient method. Assume
(0)
0.5 and
engagement height as 5 km. Next, generate the trajectories andtabulate the values of
for various
values.
Use the following system par
f
M
M
q
=
ameters
(typical for an air-to-air missile):
(^240) 0.
2
24,
0.53.
Use standard atmosphere chart for the atmospheric data.
D L m
kg
S
m
T
N
C C
= = =
= =
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
Necessary Conditions ofOptimality (TPBVP): A Summary
z
State Equation
z
Costate Equation
z
Optimal ControlEquation
z
Boundary Condition
,^
,
H
X
f
t X U
λ ∂
=
=
∂
,^
,^
,
H
g t X U
X
λ
λ
∂ ⎛
⎞
= −
=
⎜
⎟
∂ ⎝
⎠
0
H U ∂
=
∂
f
f ϕ X
λ
∂
=
∂
(
) 0
0
:Fixed
X t
X
ADVANCED CONTROL SYSTEM DESIGNDr. Radhakant Padhi, AE Dept., IISc-Bangalore
Shooting Method