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SSO Bidder’s Conference II

SSO Bidder’s Conference II. Luck: When opportunity meets preparation . “ Start by doing what’s necessary; then do what’s possible and suddenly you are doing the impossible .” -- Francis of Assisi. Controllability. 73, 90, 91, 212-218, 220-223, 226-230 BPH test, 226, 227 Grammian, 223

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SSO Bidder’s Conference II

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  1. SSO Bidder’s Conference II Luck: When opportunity meets preparation. “Start by doing what’s necessary; then do what’s possible and suddenly you are doing the impossible.” -- Francis of Assisi

  2. Controllability • 73, 90, 91, 212-218, 220-223, 226-230 • BPH test, 226, 227 • Grammian, 223 • Need another way to compute eAt • Need additional ways to compute characteristic polynomial

  3. eAt Again • Closed form formulas for functions of a square matrix are available • Thanks to Cayley Hamilton Theorem • Let f(A) be a function of square matrix • in our case f(A) = eAt • We look for a “polynomial” function g() • Uses eigenvalues, uses C-H theorem

  4. eAt (2)

  5. eAt (3)

  6. eAt (4) • 1 is a repeated eigenvalue. • We need to use different equation. Not repeat the previous one. • We differentiate wrt lambda to generate a new equation.

  7. eAt (5)

  8. eAt (6) Unproven Claim: Claim verification: Use properties of eAt

  9. Generalization This expression for eAt is used in the discussion of CONTROLLABILITY andOBSERVABILITYfor CT LTI systems.

  10. Controllability • A SS system is controllable provided • Given any initial state, x0 • Given any final state, xf • It is possible to find an input function, u(t), that will take the initial state to the final state. • The SSO will specify x0 and xf • The engineer must find the input function, u(t) • To prove controllability of a system, no assumptions can be made about x0, and xf

  11. CT LTI Controllability

  12. CT LTI Controllability (2)

  13. CT LTI Controllability (3) • To solve for v, • gamma sub c, • its transpose and • the matrix in the square brackets • Must each have full rank (= n). • Gamma sub c is the critical matrix • The matrix in []’s is always full rank.

  14. CT LTI Controllability (4) Theorem. If the controllability matrix is full rank, the system is controllable. Proof: Preceeding discussion.

  15. Stability (initial) • A SS system is stable, provided the free response of any (and all) initial state(s) decays to zero. • Nature determines the initial state, not the engineer. • The eigenvalues of A must be in LHP. • Compare with roots of characteristic poly must be in LHP • EVs of A are roots of characteristic poly.

  16. Pole Placement via SVFB Draw Picture • Can K be found that will place the poles wherever we want? • Yes, if original system is controllable. • Hence, controllability implies stabilizability.

  17. Determining K • In CCF • Not in CCF • Solve det(sI-A+BK) = desired char poly • Similarity transform to CCF, place poles, transform back • Minimization approach

  18. Where do the X’s come from? • We have tacitly assumed that we can actually measure ALL components of the state vector and use them in SVFB. • This was wishful thinking at best. • Sensors are the most expensive components per lb. • We need to COMPUTE the state based on measurements of the input to the plant and the output from the plant. • We must design an OBSERVER.

  19. Observability • A system is observable, if the INITIAL value of the state can be determined (i.e. computed) from the (measured) input and the (measured) output. • Picture

  20. Observability Grammian

  21. Observability Matrix • Follow work for Controllability Matrix

  22. Observability Theorem • The matrix in square brackets is always invertible • The observability grammian is invertible provided the observability matrix (gamma-sub-O) is full rank

  23. Luenberger Observers (1) Choose (DESIGN) G, H, L so that x-hat approaches x as t goes to infinity. Choose B=H; G=A-LC

  24. Luenberger Observers (2) • Choose L so that the error goes to zero • A-LC must have eigenvalues in LHP • Eigenvalues of A-LC must be to the left of A-BK • Observer must be faster than controller. • Picture

  25. Determining L • OCF • Not in OCF • Transform to OCF, design, untransform • Solve det(sI-A+LC) = desired char poly • Minimization approach

  26. Separation Principle • The controller (K) and the observer (L) can be designed separately provided • eVERY EV of A-CL is to the left of every EV of A-BK

  27. More complex Satellite model

  28. Computation of (sI-A)-1 • Some known mathematical Theorems (no proofs necessary before use) • Cayley Hamilton Theorem • tr(PM) = tr(MP) • Any square matrix is similar to an upper triangular matrix (use row operations). The diagonal elements are the eigenvalues. • Any square matrix is similar to a matrix built from companion form matrices, i.e. PMP-1 = diag(A1, …,Am) where each Ai is in companion form

  29. Leverrier-Faddeev Algorithm See Hou’s paper.

  30. DT LTI Controllability

  31. Minimization approach

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