1 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

2 Outline OBJECTIVE OF MID COURSE GUIDANCE MODEL PREDICTIVE QUADRATIC CONTROL(MPQC) DESIGN MID COURSE GUIDANCE WITH MPQC RESULTS CONCLUSION

3 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(x d, y d,z d ) with desired heading angle (Φ d ) and flight path angle (γ d ) using minimum acceleration η Φ and η γ.

4 System dynamics: MPQC Design: Mathematical Development Discretized Goal: with additional (optimal) objective(s)

5 MPQC Design: Mathematical Formulation 0 (small error approximation)

6 Recursive Relation for Error Coefficient Computation General formula Recursive computation:

7 MPQC Design: Mathematical Formulation Now the acceleration can be approximated as straight line error in control can be given as Substituting for dU k for k = 1,.....,N-1 in

8 We get MPQC Design: Mathematical Formulation

 If no of eq is same as no of unknown 9  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

10 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

11 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

12 MID COURSE GUIDANCE WITH MPQC (Mathematical model)

13 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).

14 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 dY N is define as

15 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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22 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.

23 Thanks for the Attention….!!