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Nonlinear Sub-optimal Mid Course Guidance with

Desired Alinement using MPQCP. N. Dwivedi, Dr. A.Bhattacharya, Scientist, DRDO, Hyderabad-,INDIA

Dr. Radhakant PadhiAsst. Professor, IISC, Banglore,INDIA

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Outline OBJECTIVE OF MID COURSE

GUIDANCE

MODEL PREDICTIVE QUADRATIC

CONTROL(MPQC) DESIGN

MID COURSE GUIDANCE WITH MPQC

RESULTS

CONCLUSION

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OBJECTIVE OF MID COURSE GUIDANCE

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 Hence should be energy efficient

Hence Objective is:

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.

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System dynamics:

MPQC Design: Mathematical Development

Discretized

Goal: with additional (optimal) objective(s)*N NY Y

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MPQC Design: Mathematical Formulation

1 11 1

1 1

1 2 2 12 2

1 2 2 1

NN N

N

N N NN N

N N N

N N N N N NN N N

N N N N N N

YdY dX

X

Y F FdX dU

X X U

Y F F F Y FdX dU dU

X X X U X U

1

1 1 1 11

1 1 1 1

N N k N N k N N Nk k N

N N k N N k N N N

Y F F Y F F Y F FdX dU dU

X X X X X U X X U

0kB 1NB

1 1k k N N NB dU B dU dY

(small error approximation)

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Recursive Relation for Error Coefficient Computation

General formula

Recursive computation:

2 , 1 , ,k N N k

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MPQC Design: Mathematical Formulation

Now the acceleration can be approximated as straight line

error in control can be given as

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

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We get

MPQC Design: Mathematical Formulation

If no of eq is same as no of unknown

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if number of unknowns is greater than the number of equations, the optimal solution can be obtained by minimizing the following objective (cost) function,

MPQC Design: Mathematical Formulation

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Start

Guess a control history

Propagate system dynamics

Compute Output

Converged control Solution

Update the control history

Compute sensitivity matrices

Stop

Check Convergence

Yes

No

MPQC algorithm

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MPQC Design: Features Advantages

• Closed form control update

• Computationally very efficient and can be implemented online

Limitations

• Finite time formulation

• Performance index is a function of control variable only

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MID COURSE GUIDANCE WITH MPQC (Mathematical model)

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MID COURSE GUIDANCE WITH MPQC

In state equation of the interceptor, time is used as an independent variable.

Hence if we want to propagate state, we must have knowledge of final time which is quite difficult .

So instead of time, x can be used as independent variable as final position of x is known (because Missile has to reach at particular point(desired) after mid course).

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MID COURSE GUIDANCE WITH MPQC

For this purpose missile model can be modified as where X’ represent the derivative of state with respect to position x.

For MPQC design, state model has to be in discreet form as

•And dYN is define as

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RESULTS

To show the capability of guidance the initial position of missile and 2 different case for different final condition has been chosen as given in table.

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CONCLUSION A newly developed MPQC( MODEL PREDICTIVE

QUADRATIC CONTROL) is utilized to solve optimal mid-course guidance problem for a homing interceptor.

Acceleration demand has been minimized for reaching desired position with desired velocity vector.

This technique is computationally efficient and can be applied online for getting closed form sub-optimal solution of mid course guidance problem.

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Thanks for the Attention….!!