33 - observability33 filled

33 - observability33 filled - ME 575 Handouts Obtaining...

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Unformatted text preview: ME 575 Handouts Obtaining State Feedback In our design example, we had to have: y, y, y, y In general, not all of these measurements are available. Simple finite difference approximations to the derivative are sensitive to noise. We wish to reconstruct all the states from available measurements. This requires observability. ME575 Session 33: Observability School of Mechanical Engineering Purdue University Slide 1 Observability The system x = A x +Bu y = C x +Du is completely observable if every state x(to) can be determined from the observations of y(t) over a finite time interval, to ≤ t ≤ t1. ME575 Session 33: Observability School of Mechanical Engineering Purdue University Slide 2 1 ME 575 Handouts Example (HW #8, Problem 2) ⎡ −3 x=⎢ 2 ⎢ ⎢0 ⎣ ⎡1 y = ⎢0 ⎢ ⎢0 ⎣ ME575 Session 33: Observability 0⎤ ⎡1 ⎥ x + ⎢0 −3 2 ⎥ ⎢ 1 −3 ⎥ ⎢0 ⎦ ⎣ 0 0⎤ 1 0⎥ x ⎥ 0 1⎥ ⎦ 1 0 ⎤ ⎡ u1 ⎤ 0 ⎥ ⎢u2 ⎥ ⎥⎢ ⎥ 1⎥ ⎢u3 ⎥ ⎦⎣ ⎦ 0 1 0 School of Mechanical Engineering Purdue University Slide 3 Example (cont.) put in diagonal form: ⎡1 T = ⎢2 ⎢ ⎢1 ⎣ 1 0 −1 1⎤ −2⎥ ⎥ 1⎥ ⎦ x = A x + B u x =T z z = T −1 A T z + T −1B u y=Cx ⇒ y=CTz 1 1⎤ ⎡ z1 ⎤ ⎡1 ⎢2 y= 0 −2⎥ ⎢ z 2 ⎥ ⎢ ⎥⎢ ⎥ 1⎥ ⎢ z3 ⎥ ⎢ 1 −1 ⎣ ⎦⎣ ⎦ ME575 Session 33: Observability School of Mechanical Engineering Purdue University Slide 4 2 ME 575 Handouts Example (Mode Shapes) School of Mechanical Engineering Purdue University ME575 Session 33: Observability Slide 5 Observability Let’s take a closer look: y1(t) = z1(t) + z2 (t) + z3 (t) = e − t z1(0) + e−3t z2 (0) + e −5t z3 (0) ME575 Session 33: Observability School of Mechanical Engineering Purdue University Slide 6 3 ME 575 Handouts Observability Alternatively, past history can be captured by looking at n−1 derivatives of a given output: y=Cx y=Cx=CA x y=Cx=CA x 2 y(n−1) = C A n−1 x ME575 Session 33: Observability School of Mechanical Engineering Purdue University Slide 7 Observability The system x = A x +Bu y=Cx is completely observable iff the row vectors of the observability matrix ⎡C ⎤ ⎢C A ⎥ ⎢ ⎥ 2 W O = ⎢C A ⎥ ⎢ ⎥ ⎢ ⎥ ⎢C A n−1 ⎥ ⎣ ⎦ span the n-dimensional space (i.e., WO has rank n). ME575 Session 33: Observability School of Mechanical Engineering Purdue University Slide 8 4 ME 575 Handouts Observability of Diagonal System ⎡λ1 ˆ A = ⎢0 ⎢ ⎢0 ⎣ ˆ C = [c 1 ME575 Session 33: Observability 0 λ2 0 0⎤ 0⎥ ⎥ λ3 ⎥ ⎦ c2 c3 ] School of Mechanical Engineering Purdue University Slide 9 Return to Three-Tank Example 1 0⎤ ⎡ −3 A = ⎢ 2 −3 2⎥ ⎢ ⎥ 1 −3 ⎥ ⎢0 ⎣ ⎦ ⎡1 0 0⎤ C = ⎢0 1 0 ⎥ ⎢ ⎥ ⎢ 0 0 1⎦ ⎥ ⎣ pick second output: ME575 Session 33: Observability School of Mechanical Engineering Purdue University Slide 10 5 ME 575 Handouts Observability of Jordan Form ⎡ −2 ⎢ A=⎢ 0 ⎢0 ⎣ C = [c ME575 Session 33: Observability 1 −2 0 1] 0 0⎤ ⎥ 0⎥ a⎥ ⎦ School of Mechanical Engineering Purdue University Slide 11 Observable Canonical Form ⎡0 AO = ⎢1 ⎢ ⎢0 ⎣ 0 1 −ao ⎤ T −a1 ⎥ = A C ⎥ − a2 ⎥ ⎦ CO = [ 0 0 1] = BC ME575 Session 33: Observability 0 T School of Mechanical Engineering Purdue University Slide 12 6 ...
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This note was uploaded on 12/09/2011 for the course ME 575 taught by Professor Meckl during the Fall '10 term at Purdue.

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