Presentation on theme: "SSO Bidder’s Conference II"— Presentation transcript:
1SSO Bidder’s Conference II Luck: When opportunity meets preparation.“Start by doing what’s necessary;then do what’s possible andsuddenly you are doing the impossible.”-- Francis of Assisi
2Controllability73, 90, 91, , ,BPH test, 226, 227Grammian, 223Need another way to compute eAtNeed additional ways to compute characteristic polynomial
3eAt AgainClosed form formulas for functions of a square matrix are availableThanks to Cayley Hamilton TheoremLet f(A) be a function of square matrixin our case f(A) = eAtWe look for a “polynomial” function g()Uses eigenvalues, uses C-H theorem
8eAt (6)Unproven Claim:Claim verification:Use properties of eAt
9GeneralizationThis expression for eAt is used in the discussion of CONTROLLABILITY and OBSERVABILITY for CT LTI systems.
10Controllability A SS system is controllable provided Given any initial state, x0Given any final state, xfIt 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 xfThe engineer must find the input function, u(t)To prove controllability of a system, no assumptions can be made about x0, and xf
13CT LTI Controllability (3) To solve for v,gamma sub c,its transpose andthe matrix in the square bracketsMust each have full rank (= n).Gamma sub c is the critical matrixThe matrix in ’s is always full rank.
14CT LTI Controllability (4) Theorem. If the controllability matrix is full rank, the system is controllable.Proof: Preceeding discussion.
15Stability (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 LHPEVs of A are roots of characteristic poly.
16Pole Placement via SVFB DrawPictureCan K be found that will place the poles wherever we want?Yes, if original system is controllable.Hence, controllability implies stabilizability.
17Determining K In CCF Not in CCF Solve det(sI-A+BK) = desired char poly Similarity transform to CCF, place poles, transform backMinimization approach
18Where 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.
19ObservabilityA 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
29Computation of (sI-A)-1 Some known mathematical Theorems (no proofs necessary before use)Cayley Hamilton Theoremtr(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