Lecture Lecture ––––23223323 Model Predictive...
Transcript of Lecture Lecture ––––23223323 Model Predictive...
Lecture Lecture Lecture Lecture –––– 23232323
Model Predictive Static Programming (MPSP) Model Predictive Static Programming (MPSP) Model Predictive Static Programming (MPSP) Model Predictive Static Programming (MPSP)
and Optimal Guidance of Aerospace Vehiclesand Optimal Guidance of Aerospace Vehiclesand Optimal Guidance of Aerospace Vehiclesand Optimal Guidance of Aerospace Vehicles
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
MPSP Design: Mathematical Details
Reentry Guidance of a Reusable
Launch Vehicle (RLV) using MPSP
References
MPSP Design: MotivationMPSP Design: MotivationMPSP Design: MotivationMPSP Design: Motivation
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
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)
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore5
Model Predictive Static
Programming (MPSP)
MPSP Features
Model Predictive Control(output dynamics replace state
dynamics in a TPBVP)
Approximate Dynamic Programming(discrete formulation that avoids the HJB
equation)
Reference on MPSP:Radhakant Padhi and Mangal Kothari, Model Predictive Static
Programming: A Computationally Efficient Technique for Suboptimal Control Design, International Journal of Innovative Computing, Information and Control, Vol.5, No.2, Feb 2009.
MPSP Design: Mathematical DetailsMPSP Design: Mathematical DetailsMPSP Design: Mathematical DetailsMPSP Design: Mathematical Details
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-Bangalore7
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-Bangalore8
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-Bangalore9
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-Bangalore10
Recursive Relation for Computation
of Sensitivity Matrices
General formula
Recursive computation:
( ) ( )1 , 2 , ,k N N k= − −ɶ ⋯
1 1
1 1
N N k k
k
N N k k
F FY FB
X X X U
− +
− +
∂ ∂ ∂ ∂=
∂ ∂ ∂ ∂
ɶ ɶ
ɶ
ɶ ɶ
⋯
0
1N
N
N
YB
X−
∂=
∂
0 0 1
1
1
k
k k
k
FB B
X
+
+
+
∂=
∂
ɶ
ɶ ɶ
ɶ
0 k
k k
k
FB B
U
∂=
∂
ɶ
ɶ ɶ
ɶ
( ) ( )2 , 1 , ,k N N k= − −ɶ ⋯
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore11
Augmented Cost Function:
MPSP Design: Mathematical Formulation
Minimize:
Subject to: 1 1k k N N NB dU B dU dY
− −+ + =⋯
( ) ( )1
0 01
2
NT
k k k k k kk k
J U dU R U dU−
=
= − −∑ ɶ ɶ ɶ ɶ ɶɶ
( ) ( )1 1
0 01
2
N NT
T
k Nk k k k k k kk k k k
J U dU R U dU B dU dYλ− −
= =
= − − + −
∑ ∑ɶ ɶ ɶ ɶ ɶ ɶ ɶɶ ɶ
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore12
Necessary Conditions of Optimality:
MPSP Design: Mathematical Formulation
( )0
ˆ ˆ ˆ ˆ
ˆ
0
ˆfor , ( 1), , ( 1)
Tk
k k k k
k
JR U dU B
dU
k k k N
λ∂
= − − + =∂
= + −⋯
1
0 N
k
N k kk k
JdY B dU
λ
−
=
∂= ⇒ =
∂∑ ɶ ɶɶ
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore13
Control Solution:
MPSP Design: Mathematical Formulation
1 0
1 0
1 1 1 1
T
k k k k
T
N N N N
dU R B U
dU R B U
λ
λ
−
−
− − − −
= − +
= − +
⋮
1 1k k N N NB dU B dU dY− −+ + =⋯
( )1
NA dY bλ λλ −= −
( )
( )
1 1
1 1 1
0 0
1 1
T T
k k k N N N
k k N N
A B R B B R B
b B U B U
λ
λ
− −
− − −
− −
− + +
+ +
≜ ⋯
≜ ⋯
where
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore14
Control Update:
MPSP Design: Mathematical Formulation
( )
( )
0 1
0 1
1 1 1 1 1
T
k k k k k
T
N N N N N
U U dU R B
U U dU R B
λ
λ
−
−
− − − − −
= − =
= − =
⋮
where ( )1
NA dY bλ λλ −= −
Iteration unfolding: Update the remaining control history “only once”
at time step k and go to k+1
OPTIMAL CONTROL, GUIDANCE
AND ESTIMATION Prof. Radhakant
Padhi, AE Dept., IISc-Bangalore
15
Start
Guess a control history
Propagate system dynamics
Compute output
Converged control solution
Update the control history
Compute the sensitivitymatrices recursively
Stop
Checkconvergence
Yes
No
MPSP
ALGORITHM
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore16
MPSP 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.
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore17
Important Extensions
Model Predictive Spread Control
(MPSC): This is a version with control
parameterization
• Further improvement of computational time
• Smoothness of control history (by
enforcement)
Generalized MPSP (G-MPSP)
• MPSP in a continuous-time framework
MPSP for ReMPSP for ReMPSP for ReMPSP for Re----entry Guidance of aentry Guidance of aentry Guidance of aentry Guidance of a
ReReReRe----usable Launch Vehicle (RLV)usable Launch Vehicle (RLV)usable Launch Vehicle (RLV)usable Launch Vehicle (RLV)
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-Bangalore19
Typical trajectory of a RLV
Reentry
Segment
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore20
Objective and Challenges
Objective• To develop advanced nonlinear and optimal guidance
for a reusable launch vehicles (RLV) in the descent phase, with special emphasis on the critical re-entry segment.
Challenges• Path constraints: Structural load, Thermal load, Angle
of attack boundary• Terminal constraints: Final position and velocity• Optimal online trajectory generation• Robustness wrt. uncertainties in parameters • Real-time computability, Smoothness in guidance
command etc.
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore21
RLV Guidance using MPSP:
System Dynamics (Spherical Rotating Earth)Ref: Vinh et al., “Hypersonic and Planetary Entry Mechanics”, 1980
cos sin
cos cos
cos
sin
V
rV
r
r V
γ ψ
γ ψθ
φ
γ
φ =
=
=
ɺ
ɺ
ɺ
Reentry point
End Point
Curvilinear Abscissa
Kinematic Equations over Spherical Rotating Earth
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore2217 February 2010 22
RLV Guidance using MPSP:
System Dynamics (Spherical Rotating Earth)Ref: Vinh et al., “Hypersonic and Planetary Entry Mechanics”, 1980
2
2
2
sin cos sin cos cos sin sin
coscos cos2 cos cos
cos (cos cos sin sin sin )
sincos cos tan 2 (tan cos sin sin
cos
sin cos coscos
e
e
e
e
e
DV g r
m
gL V
mV V r
r
V
L V
mV r
r
V
γ φ γ φ γ φ ψ
γσ γγ ϕ ψ
φ γ φ γ φ ψ
σψ γ ψ φ γ φ ψ φγ
φ φ ψγ
= − − + Ω ( − )
= − + + Ω
Ω+ +
= − + Ω − )
Ω−
ɺ
ɺ
ɺ
Dynamic Equations over Spherical Rotating Earth
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore24
Reentry Constraints
Normal Load
cos sin3
L Dg
m
α α+<
0.5 3.15
2
n
11030 r60
COsl
ρ V
R ρ V
Heat Flux
W
cm
≤
Dynamic Pressure
25q kpa≤
Fligh t P ath A ngle
f
γ γ ∗=
Terminal Altitude
f
r r∗=
Terminal Velocity
f
V V∗=
min max( ) ( ) ( )M M Mα α α< <
Constraints
Path Constraints Terminal Constraints AOA Bound
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore25
RLV Guidance using MPSP:
Overview of Previous Guidance Design
Normalized State Vector
Control Vector
Output Vector
Goals
[ ]T
Z V γ=
U α=
Y Z=
( )
( ) ( )
( )
1 12 2
1 1
12
1
1
0
1 1min ( ) min min
2 2
1 + min
2
N N N
N N
k k NL k
k k
Np
smooth k k
k
Y Y Y
J R d R d
R
α α
α α
∗
− −
= =
−
−=
∆ = − →
= +
−
∑ ∑
∑
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore26
Comparison with Reference profile
200 250 300 350 400 450 500 5500
5
10
15
20
25
30
35
40
Time (s)
αα αα (
de
g)
reference
MPSP
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore27
200 300 400 500 6000
5
10
15
20
25
30
35
40
Time (s)
αα αα (
deg
)
Perturbation Study: Path Constraints
480 simulations has been carried out at 120 different middle points in the trajectory for a range of perturbations
of 4% and promising results have been obtained.
200 250 300 350 400 450 500 5500
0.5
1
1.5
2
2.5
3
3.5
Time (s)
No
rma
l L
oa
d (
g)
Only in
9 cases the
normal load goes above 3g
Maximum =3.19g
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore28
200 300 400 500 6001
2
3
4
5
6x 10
4
Time (s)
Alt
itu
de
(m
)
200 300 400 500 600500
1000
1500
2000
Time (s)
Ve
locit
y (
m/s
)
Terminal conditions are
satisfied within 2%
error bound
Perturbation Study: Terminal Constraints
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore30
RLV Guidance using MPSP:
Problem Formulation
Normalized State Vector
Control Vector
Output Vector
Goals
[ ]T
Z V γ φ θ=
[ ]T
U α ψ=
[ ]T
Y V γ φ θ=
( )1. 0
2. Normal Load Minimization
3. Control Deviation Minimization
4. Control Smoothness
N N NY Y Y
∗∆ = − →
Terminal
Coordinates
Intermediate
Control for
Bank Angle
Profile
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore31
RLV Guidance using MPSP:
Problem Formulation
Normalization of State Variables
Absolute values of state variables are not of the same order of magnitude
May create disparity during the control and state
update process
State variables are normalized using the desired final values
*n
VV
V=
*n
γγ
γ=
*n
φφ
φ=
*n
θθ
θ=
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore32
RLV Guidance using MPSP:
Performance Index Selection
1 2 3min( )J J J J= + +
1
2
1
1
2
NT
k d k
k
J dU R dU−
=
= ∑
1
1
1
1
2
p
Zk
k
NBN
Z
k
J Ae N−
=
= ∑
“Soft” Path Constraints where
Minimize normal load only if value close to bound
Exponential weight of nominal NL profile to achieve this
Minimize deviation (error) of updated profile from nominal one
Maintain control profile in the vicinity of nominal profile
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore33
RLV Guidance using MPSP:
Performance Index Selection
( ) ( )1
3 1 1
2
1
2
NT
p p
k k sm k k
k
J U U R U U−
− −=
= − −∑
For smoothness, minimize distance CD
Equivalent to minimizing distances AB, BD and AD
AB is nominal profile
Min (BD) = min (J2)
Min (AD) = min (J3)
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore3434
RLV Guidance using MPSP:Nominal Trajectory with 8 Initial Conditions
Smooth Profile
Trim Bounds not
ViolatedStable Lateral Profile
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore3535
Guidance Simulation Results:8 Cases of Initial Condition Perturbations
Smooth Trajectories,
Terminal Constraints Met
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore3617 February 2010 36
Guidance Simulation Results:8 Cases of Initial Condition Perturbations
Path Constraints Met
1. Normal Load < 3g
2. Heat Flux < 60 W/cm^2
3. Dynamic Pressure < 25 kPa
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore37
Conclusions
MPSP technique is very promising for optimal missile guidance (trajectory optimization philosophy is brought into guidance design).
Various challenging strategic/tactical missile guidance problems have been (and are being) solved.
MPSP has also been successfully demonstrated for Re-entry guidance of a Re-usable Launch Vehicle.
An important extension of the MPSP is the MPSC design with control parameterization. It has additional desirable characteristics like control smoothness, faster computation over MPSP etc.
MPSP has found good word-wide acceptance.
OPTIMAL CONTROL, GUIDANCE AND ESTIMATION
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore38
References
Omkar Halbe and Radhakant Padhi, Energy Based Suboptimal Reentry Guidance of a Reusable Launch Vehicle Using Model Predictive Static Programming, AIAA Guidance, Navigation and Control Conference, 2010, Toronto, Canada (Journal version is under preparation and will be submitted soon).
Charu Chawla, Pranjit Sarmah and Radhakant Padhi, Suboptimal Reentry Guidance of Reusable Launch Vehicles Using Pitch Plane Maneuver, Aerospace Science and Tech., Vol.14, No.6, 2010, pp.377-386.
Radhakant Padhi and Mangal Kothari, Model Predictive Static Programming: A Computationally Efficient Technique for Suboptimal Control Design, International Journal of Innovative Computing, Information and Control, Vol. 5, No.2, Feb 2009, pp.399-411.