Nonlinear Sub-optimal Mid Course Guidance with Desired Alinement using MPQC

1. Nonlinear Sub-optimal Mid Course Guidance withDesired Alinement using MPQC P. N. Dwivedi, Dr. A.Bhattacharya, Scientist, DRDO, Hyderabad-,INDIA Dr. RadhakantPadhi 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(xd, yd,zd) with desired heading angle (Φd) and flight path angle (γd) using minimum acceleration ηΦand ηγ.

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

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

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

7. 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

8. MPQC Design: Mathematical Formulation We get

9. MPQC Design: Mathematical Formulation • If no of eq is same as no of unknown • if number of unknowns is greater than the number of equations, the optimal solution can be obtained by minimizing the following objective (cost) function,

10. MPQC algorithm Start Guess a control history Update the control history Propagate system dynamics Compute Output Check Convergence No Compute sensitivity matrices Yes Converged control Solution Stop

11. MPQC Design: Features • Advantages • Closed form control update • Computationally very efficient and can be implemented online • Limitations • Finite time formulation • Performance index isa 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 dYN 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.

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….!! Questions ... ??