nonlinear sub-optimal mid course guidance with desired alinement using mpqc
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Nonlinear Sub-optimal Mid Course Guidance with Desired Alinement using MPQC. P. N. Dwivedi, Dr. A.Bhattacharya , Scientist, DRDO, Hyderabad-,INDIA Dr. Radhakant Padhi Asst. Professor, IISC, Banglore,INDIA. Outline. OBJECTIVE OF MID COURSE GUIDANCE - PowerPoint PPT PresentationTRANSCRIPT
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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
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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….!!