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**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

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Controllability 73, 90, 91, , , BPH test, 226, 227 Grammian, 223 Need another way to compute eAt Need additional ways to compute characteristic polynomial

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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

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eAt (2)

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eAt (3)

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**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.

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eAt (5)

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

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

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**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

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**CT LTI Controllability**

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**CT LTI Controllability (2)**

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**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.

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**CT LTI Controllability (4)**

Theorem. If the controllability matrix is full rank, the system is controllable. Proof: Preceeding discussion.

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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.

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**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.

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**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

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**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.

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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

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**Observability Grammian**

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Observability Matrix Follow work for Controllability Matrix

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**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

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**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

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**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

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**Determining L OCF Not in OCF Transform to OCF, design, untransform**

Solve det(sI-A+LC) = desired char poly Minimization approach

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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

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**More complex Satellite model**

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**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

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**Leverrier-Faddeev Algorithm**

See Hou’s paper.

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**DT LTI Controllability**

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**Minimization approach**

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