lecture lecture – ––– 25 225525 model predictive spread ... · prof. radhakant padhi, ae...

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)






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)



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

+ =

=




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 : ≜




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”.



� 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

−

− −

− −

−

= + +

= + + ∆ + + + + ∆

= ∆ + ∆

⋯⋯

⋯ ⋯ ⋯��

⋯⋯



MPSC with Linear


( )

( )

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

−

−

−− −

= −

= −

= − +



Control Parameterization

Error in control

Substituting for dUk for k = 1,.....,N-1 in

MPSC with Quadratic




one gets

MPSC with Quadratic




� 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


( )1 2 3

1

2

T T TJ a R a b R b c R c= + +



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



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






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.



MID COURSE GUIDANCE WITH MPSC (Mathematical model)



System Dynamics with Downrange as Independent Parameter

System Dynamics:

Control Parameterization:

Output Error at Final Time:



RESULTS



20



Improvement with Iterations





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)






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?



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



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 : ≜




( ) ( ) ( ) ( ) ( )( ),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




( ) ( ) ( ) ( )( )

( )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:




( )( )( )

( )( ) ( ) ( ) ( )

( )( ) ( )( )

( )( ) ( ) ( )

( ) ( )( )( )

( )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



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



Augmented Cost Function:

GMPSP Design: Mathematical Formulation

Minimize:

Subject to: ( ) ( ) ( )0

ft

ft


( ) ( )( ) ( ) ( ) ( )( )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

= − −

+ −

∫

∫



Necessary Conditions of Optimality:


( ) ( ) ( )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

∂= − − − =

∂



Control Solution:


( ) ( )( ) ( )( ) ( )1 0T

U t R t B t U tδ λ−

= +

( ) ( ) ( )0

ft

ft


( ) ( )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



Control Update:


( ) ( ) ( )

( ) ( )( ) ( )( ) ( ) ( ){ } ( )

( )( ) ( )( ) ( ) ( ){ }

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− = −



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






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?



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.



A Typical Engagement Scenario



[ ]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.



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



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:



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



Stationary Targets:

Same initial conditions & Different

Terminal Constraints

0

0

10

10

o

m

o

m

γ

ψ

=

=

Various Constraints in

both the angles



Stationary Targets:

Same initial conditions & Different

Terminal Constraints (Latax)



Stationary Targets:

Perturbation in initial conditions

20

20

f

f

o

m

o

m

γ

ψ

= −

=

Initial Condition perturbation with same terminal constraint



Stationary Targets:


(Angle Histories)



Stationary Targets:


(Latax)



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



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



Time histories of 3D Angles

Moving/Maneuvering Targets:

Straight line, Constant g & sinusoidal maneuvers



Zero Effort Miss (ZEM) Plot

(Sinusoidal Maneuver)

Modified

Definition of

ZEM:

Vt Non Zero- Target

allowed to maneuver



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).



Thanks for the Attention….!!

lecture lecture – ––– 25 225525 model predictive spread ... · prof. radhakant padhi, ae...

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