Lecture Lecture ––––23223323 Model Predictive...

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Lecture Lecture Lecture Lecture – 23 23 23 23 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 Vehicles and Optimal Guidance of Aerospace Vehicles and Optimal Guidance of Aerospace Vehicles and Optimal Guidance of Aerospace Vehicles Prof. Radhakant Padhi Prof. Radhakant Padhi Prof. Radhakant Padhi Prof. 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-Bangalore 2 Outline Motivation MPSP Design: Mathematical Details Reentry Guidance of a Reusable Launch Vehicle (RLV) using MPSP References

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.

OPTIMAL CONTROL, GUIDANCE AND ESTIMATION

Prof. Radhakant Padhi, AE Dept., IISc-Bangalore39

Thanks for the Attention….!!