Lecture-24 - MPSP - II...Title Microsoft PowerPoint - Lecture-24 - MPSP - II Author Radhakant Padhi Created Date 5/24/2013 7:59:48 PM

Lecture Lecture Lecture Lecture –––– 24242424

MPSP for Optimal Missile GuidanceMPSP for Optimal Missile GuidanceMPSP for Optimal Missile GuidanceMPSP for Optimal Missile Guidance

Prof. Prof. Prof. Prof. RadhakantRadhakantRadhakantRadhakant PadhiPadhiPadhiPadhi

Dept. of Aerospace Engineering

Indian Institute of Science - Bangalore

OPTIMAL CONTROL, GUIDANCE AND ESTIMATION

Prof. Radhakant Padhi, AE Dept., IISc-Bangalore2

Outline

� Principle of Missile Guidance

� Optimal Missile Guidance using MPSP• Ballistic Missile Guidance

• Tactical Missile Guidance with Impact Angle Constraint

� A quick glimpse of other problems solved



Fundamental Problem of

Strategic Missile Guidance

Guidance

phase



Fundamental Problem of

Tactical Missile Guidance

PN Guidance: M Ma NV λ= ɺ

Ma

LOS

MV

TV

λM

θ

Tθ

M

T



Missile Guidance Laws

� Many classical missile guidance laws are inspired from “observing nature”(e.g. Proportional Navigation (PN) guidance law is based on ensuring “collision triangle”)

� Control theoretic based guidance laws are usually based on “kinematics” and/or “linearized dynamics”. Hence, they are usually not very effective!

� Nonlinear optimal control theory is a “natural tool” to obtain effective missile guidance laws



Point-mass Missile Model with

Flat and Non-Rotating Earth

( )

( )

cos

sin

1cos sin

1sin cos

x V

h V

V T D mgm

T L mgmV

γ

γ

α γ

γ α γ

=

=

= − −

= + −

ɺ

ɺ

ɺ

ɺ



Point-mass Missile Model with

Spherical and Non-Rotating Earth

( )

sin

cos

1cos sin

21sin cos ,

cos

r V

V

r

V T D mgm

mVT L mg

mV R

rR

γ

γθ

α γ

γ α γγ

=

=

= − −

= + − +

=

ɺ

ɺ

ɺ

ɺ

MPSP for Ballistic Missile GuidanceMPSP for Ballistic Missile GuidanceMPSP for Ballistic Missile GuidanceMPSP for Ballistic Missile Guidance(with solid motors)(with solid motors)(with solid motors)(with solid motors)






Introduction:

Liquid Engines vs. Solid Motors

Liquid Engines

� Quick firing is not possible

� Sloshing and TWD effect

� Higher cost

� Thrust cut-off facility

� Burnout time is certain

� Manipulative T-t curve

� Guidance is easier

Solid Motors

� Quick firing is possible!

� No sloshing and TWD effects

� Lower cost

� No thrust cut-off facility

� Burnout time is uncertain

� Non-manipulative T-t curve

� Guidance is difficult..!



System Dynamics

: Local radius

: Velocity

: Flt. path angle

: Range angle

: Shear angle

(guidance parameter)

δ



Free-Flight and Hit Equation

FF Equation Hit Equation

Reference:

P.Zarchan, Tactical and Strategic Missile Guidance, 5th Edition, AIAA, 2007.

2

bo bo

bo

r v

GMλ

=



Guidance Design Using MPSP: Problem Specific Equations

Discretize System Dynamics:

Discretize Output Equation:



Guidance Design: Some implementation issues

� Discretization of state equation for control computation: (Euler method: Faster computation)

� Discretization of state equation for Simulation studies: (RK-4 method: Higher accuracy)

� Step size (update interval):

� GLC: 1 sec after burnout of first stage

� All derivatives were carried out symbolically

� Iteration unfolding has been implemented

100 sect m∆ =



0 20 40 60 80 100 120 140 160-2

0

2

4

6

8

10

12

14

Time (sec)

She

ar

an

gle

(de

g)

Range 4500 km

Range 4000 km

Range 3500 km

0 500 1000 1500 2000 2500 3000 3500 4000 45000

500

1000

1500

2000

2500

3000

3500

Range 3500 km

Range 4000 km

Range 4500 km

Numerical Result:With Nominal Thrust



Numerical Result:With Uncertainties in Motor Performance

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105

Time(sec)

Thru

st(

N)

Nominal performance

Over performance

Under performance



0 500 1000 1500 2000 2500 3000 3500 40000

500

1000

1500

2000

2500

3000

Down range (km)

Heig

ht

(km

)

Trajectory for nominal performance

Trajectory for over performance

Trajectory for under performance

0 20 40 60 80 100 120 140 160 1800

5

10

15

Time (sec)

Thru

st

defl

ecti

on

ang

le (

deg)

Nominal performance

Over performance

Under performance

Numerical Result:With Uncertainties in Motor Performance



Control Computation Time

� PC Used• Pentium 4 (2.4 GHz), 512 MB RAM

� Programming platform used• MATLAB 7.0

� First control update: 1.755 sec• Further control updates require lesser time

� Code in low-level language will require much lesser computational time• Suitable for online implementation



Comparison Between MPSP and

Nonlinear Programming

1.6843

1.6689

NLP

MPSP

J

J

=

=

Cost function computation

0 20 40 60 80 100 120 140 1600

5

10

15

Time (sec)

Thru

st d

eflect

ion a

ngle

(deg)

NLP

MPSP



A Hybrid Design For Energy-

Insensitive Guidance

� Step 1: Assume that the motor is guaranteed to burn up to a certain duration of predicted burnout time (say 90%). Design the MPSP Guidance

� Step 2: Switch over to Dynamic inversionguidance, which assures that the free flight equation is satisfied for the remaining time continuously.

� Motivation: To eliminate VTP requirement



5 10 15 20 25 30 35 40 45 50 55-5000

-4000

-3000

-2000

-1000

0

1000

Time (sec)

Heig

ht

err

or

at

targ

et

(Km

)

Range 1300 km

Range 1400 km

Range 1500 km

Range 1600 km

Range 1700 km

Numerical Results:MPSP + Dynamic Inversion

Height Error at the Target



Computational Time

� Machine specification • Pentium 4 (3 GHz), 512 MB RAM (future on-board computers

are expected to have better configurations)

� Language used• MATLAB 7.0 (a very high-level computationally inefficient

language)

� First control update: 0.47 sec (averaged over ten simulations, which had very small variation)

� Code in ‘C’ language will require much lesser computational time (suitable for “real time” applications!)



Conclusions:

Ballistic Missile Guidance Problem

� The new MPSP guidance algorithm successfully works

• Tested for various ranges (2000 km - 4500 km)

� It leads to high accuracy ( height error)

• With under/over performance of solid motors

� Computationally efficient algorithm

• Can be implemented online in future

� Comparison with optimal control formulation (NLP solution) shows close match

1 m±



Conclusions:Strategic Missile Guidance Problem

� A new energy-insensitive guidance algorithm is designed by blending the MPSP technique with dynamic inversion.

� The composite guidance algorithm meets the specification and requirements of an energy insensitivity design

� The composite guidance design can lead to elimination of Velocity Trimming package (VTP) requirement. However, it works only for “shallow trajectories”.

� The concept is equally valid for launch vehicle guidance

MPSP for Tactical Missile Guidance MPSP for Tactical Missile Guidance MPSP for Tactical Missile Guidance MPSP for Tactical Missile Guidance

with Impact Angle Constraintwith Impact Angle Constraintwith Impact Angle Constraintwith Impact Angle Constraint






Why Impact Angle Constrained

Guidance?

� Enhancement of warhead lethality (e.g. front-attack)

� Terminal trajectory shaping for attacking weak locations of targets (e.g. top attacks for tanks)

� Mission demand (e.g. bunker buster mission, terrorist hideouts in urban areas, water reservoir attack etc.)

� Coordinated attack by multiple munitions

� Countermeasure by enhanced stealthiness

� Range enhancement (indirectly)

� increase in observability of the target



Existing Methods of 3D Guidance

� Fewer references for 3D impact angle constraints

� Existing solutions render laws which are either non-optimal or based on kinematics only

� Latest “guaranteed 3D capture guidance” does not consider impact angular constraints.

Observations:

Literature:

� 3D PN and Augmented PN Laws

� Nonlinear 3D guidance for guaranteed capture

� Sliding Mode Control based 3D impact angular guidance



Motivations

� PN guidance laws are usually adequate to assure small miss distance, but are silent on impact angle.

� Optimal control theory based techniques are available, but they rely on “linearizedkinematics”…need to do better than that!

� The aim here is to develop a nonlinear optimal guidance law with 3-D impact angle constraints, using nonlinear point-mass dynamic models.



Challenges

� Strong (nonlinear) coupling between elevation angle and azimuth angle dynamics should be accounted for.

� Zero/Near-zero miss distance requirement cannot be compromised.

� Impact angle constraints in 3D (i.e. two angle constraints at the same time) must be ensured.

� Latax demand has to be as minimum as possible.



Problem Objectives

To design a 3-D optimal guidance law for maneuvering, moving and stationary targets,

(i) with thrust force (with autopilot delay)

(ii) without thrust force (with autopilot delay)

Aim: To obtain negligible miss distance as well as the desired azimuth and elevation impact angles simultaneously!



3D Engagement Geometry



Missile and Target Dynamics

Missile Model

[ ]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

State

[ ]T

m m m m m mX V x y zγ ψ=

Control

[ ]T

z yU a a=

Target Model

co s ( )

s in ( )

ty

t

m

t t t

t t t

a

m

x V

y V

ψ

ψ

ψ

=

=

=

ɺ

ɺ

ɺ



Assumptions about target

� Speed Vt is constant

� Target moves with a latax command ayt normal to its velocity Vt, which is known.

� Coordinates of target in inertial frame (x, y, z) are available

� ayt can be:

� Zero (straight line movement)

� Constant (constant g maneuvers)

� Sinusoidal (periodic maneuvers)



Normalized Dynamics

� Missile Model*

* *

*

* *

* * *

* * *

*

* * *

*

* *

*

sin( )[ ]

cos( )

cos( )

cos( )cos( )

cos( )sin( )

sin( )

n

n

n n

n

n

n

n

n n

n n n

n

n n n

n

n n

n

m mm m

m

m m m

z m m

m

m m m

y

m

m m m m m

m m m m m m

m

m

m m m m m m

m

m

m m m m

m

m

gT DV

m V V

a g g

V V

a g

V V

V Vx

x

V Vy

y

V Vz

z

γ γ

γ γγ

γ

ψγ γ ψ

γ γ ψ ψ

γ γ ψ ψ

γ γ

−= −

− −=

=

=

=

=

ɺ

ɺ

ɺ

ɺ

ɺ

ɺ

�Target Model

* *

* *

*

* *

*

cos( )

sin( )

n

n

n

n n

n

n n

n

yt

t

t m m

t t t m

t

t

t t t m

t

t

a g

V V

V Vx

x

V Vy

y

ψψ

ψ ψ

ψ ψ

=

=

=

ɺ

ɺ

ɺ

� Subscript n:

Normalized value

� Superscript * : Normalizing value



MPSP Guidance: Discretization

and Output Selection

1

1

6 6

( , ) ( , )k k k k k k

k

k k k

n k n n n k n n

n k k

n n n

X F X U X t f X U

X F fI t

X X X

+

+

×

= = + ∆

∂ ∂ ∂= = + ∆

∂ ∂ ∂

*

*

0

N N N N N

f f N N

T

N m m m m m

T

N m m t t

N N

Y x y z

Y x y

Y Y

γ ψ

γ ψ

=

=

→

Subscript f: Terminal Impact angles



Stationary Targets

MPSP Vs. APN: A Comparison

0

0

90

10

20

f

o

m

o

m

o

m

γ

γ

ψ

= −

=

=

Constraint in

single angle



Stationary Targets

MPSP Vs. APN: A Comparison

Latax CommandsConstraint in

single angle



Stationary Targets:

Same initial conditions & Different

Terminal Constraints

0

0

10

10

o

m

o

m

γ

ψ

=

=

Various constraints

in both angles



Stationary Targets:

Different initial conditions for Same

Terminal Constraints

20

20

f

f

o

m

o

m

γ

ψ

= −

=

Initial condition

perturbation with

same terminal

constraint



Stationary Targets:

Perturbation in initial conditions

(Angle Histories)



Zero Effort Miss (ZEM) Plot

(Sinusoidal Maneuver)



Missile for missile (air) defense:

A typical engagement scenario



Challenges

� Very high speed targets

• Very less engagement time

• Very high line-of-sight rate

� Zero/Near-zero miss distance is desired

� Impact/Aspect angle constraint

� Directional warhead

� Latax saturation (due to less dynamic pressure) should be avoided



Guidance command( generation)

Body rate generation

Fin deflectiongeneration

MissileDynamics

Target states

Translational Dynamics

Position and Veloc ity Components

p yrδ δδ

Rotational Dynamics

Three Loop Conventional Design Partial IGC Design

Philosophy of Partial Integrated

Guidance & Control (PIGC)

� Exploits the inherent time scale separation property between

faster rotational and slower translational dynamics

� Operates in a Two Loop Structure:

� Commanded body rates generated in outer loop – MPSP

� Commanded deflections generated in inner loop – Dyn. Inv.



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 world-wide acceptance.



References: Journal Publications

� Harshal Oza and Radhakant Padhi, Impact Angle Constrained Suboptimal MPSP Guidance of Air-to-Ground Missiles, AIAA Journal of Guidance, Control and Dynamics, Vol.35, No.1, 2012, pp.153-164.

� P. N. Dwivedi, A. Bhattacharya andRadhakant 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. 860-977.



References: Journal Publications

� Mangal Kothari and Radhakant Padhi, A Nonlinear Suboptimal Robust Guidance Scheme for Long Range Flight Vehicles with Solid Motors, Automatic Control in Aerospace (online journal), Vol. 3, No. 1, May 2010.

� 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.



Thanks for the Attention….!!

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Lecture-24 - MPSP - II...Title Microsoft PowerPoint - Lecture-24 - MPSP - II Author Radhakant Padhi Created Date 5/24/2013 7:59:48 PM