1. Satellite Guidance & Control I Southeast Satellite Operations Bidder’s Conference March 2005

2. Disclosure I have been hired by Southeast Satellite Operations to cover this basic material. Engineering Competence is demonstrated by –Knowing this material completely –Going beyond this material demonstrates competence relative to other competitors What goes on in this conference stays in this conference (feel free to ask questions)

3. Understanding Dimensions Knowledge –Extent to which unschooled perspectives have been transcended Methods –Process knowledge to produce additional knowledge Purposes –Explain, reinterpret, operate on world Forms –Process by which understanding is communicated Levels Master –Integrative, creative, thinking Apprentice –Disciplinary knowledge and modes of thinking Novice –Rituals and mechanisms of testing in school Naïve –directly available in the world

4. Simple Satellite Model Know these models. Know how to get from one to any other. Know what everything means.

5. Alternative SS Models OCF CCF Where does J go? –options SIDF All have the same transfer function How are the SS models related?

6. Perspective Friction? Spring constant?

7. Recall Solution to SS Models It is assumed that the SS model is committed to memory. In addition it is assumed that the solution and the derivation of the solution is committed to memory.

8. Guidance Problem Find an input function that will “guide” the state from its initial state, to a desired final state in specified time. –This is a design problem It involves making choices The answer is not unique Some choices are better than others An engineer is expected to evaluate options and choose best option for client/boss

9. A Few Example Inputs Use your imagination to generate others.

10. Example Guidance Problem Expected to know how to derive e At. Expected to know how to verify e At. Begin with solution to state equations.

11. Choose Parameters for first Input (1)

12. Choose Parameters for first Input (2)

13. Choose Parameters for first Input (3) What step could have caused failure?

14. Matlab functions for 1 st Input function xdot = AxpB1(t,x,A,B,m,b) u=m*t+b; %u1 is defined here xdot=A*x+B*u; function [m,b] = u1parms(x0,xf,tf,A) % compute parameters m and b for the input function % u(t)=mt+b % Syntax [m,b] = u1(x0,xf,tf,A) % x0 = initial state; xf = final state; tf = final time % A is 2 by 2 A matrix in state equations % assumptions: n = 2, CCF M=[-tf^3/3 -tf^2/2 tf^2/2 tf ]; % Should check for invertibility of M x = M^(-1)*(expm(-A*tf)*xf-x0); m = x(1); b = x(2);

15. Results >> [t,x]=ode45(@AxpBu1,[0 2],x0,[],A,B,m,b); >> plot(t,x)

16. Choose Parameters for 4th Input (1)

17. Choose Parameters for 4 th Input (2) Failure?

18. Matlab functions for 4 th Input function v=u4parms(x0,xf,tf,A) %need comments, syntax, definitions, etc W=[tf^3/3 -tf^2/2 -tf^2/2 tf]; v=W^(-1)*(expm(-A*tf)*xf-x0); function xprime = AxpBu4(t,x,A,B,v) u=B'*expm(-A'*t)*v; % u4 here xprime=A*x+B*u;

19. >> [t,x]=ode45(@AxpBu4,[0 2],x0,[],A,B,v); >> plot(t,x)

20. Choose Parameters for Additional Inputs Left for the entertainment of potential bidders Reduce to Computer code. –This is one way to clearly demonstrate to SSO that you have an algorithm with specified inputs that produces the specified outputs. Test/verify algorithm (computer code)

21. Evaluate Performance for various inputs Use one of the ODE? functions to solve the DE –ode23, ode45, ode113, ode15s, ode23s, ode23t, ode23tb –Define Matlab functions for the input functions –Define Matlab functions for the DE with specified input function –Plots –Balance performance and fuel consumption –Exercise judgement

22. Compare inputs Energy used?

23. What could cause failure? How can you guarantee success? Can you easily predict failure? CONTROLLABILITY or CONTROLLABLE Matrices that had to be inverted involved e -At B and/or its transpose Inverses fail to exist when ranks are too low Rank of integral of [(e -At B)u(t)] =? Cayley-Hamilton theorem

24. Loose Ends to tie up Many state space models can share the same transfer function. –How are these related? Under what conditions can an input be found that guides an arbitrary initial state to an arbitrary final state?