27 - controllability27 filled

27 - controllability27 filled - ME 575 Handouts Example of...

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Unformatted text preview: ME 575 Handouts Example of Forced Response 1⎤ ⎡0 ⎡0 ⎤ x=⎢ x + ⎢ ⎥u ⎥ ⎣ −2 −3 ⎦ ⎣ 1⎦ find x(t) for u(t) a unit step and x(0) = 0: t x(t) = e At x(0) + ∫ e A( t −τ ) B u( τ)dτ 0 ME575 Session 27: Controllability School of Mechanical Engineering Purdue University Slide 1 Example (cont.) ⎡ ⎡e−( t −τ ) − 1 e−2( t −τ ) ⎤ t ⎤ 2 ⎣ ⎦0 ⎥ x(t) = ⎢ t⎥ ⎢ ⎡ −e−( t −τ ) + e−2( t −τ ) ⎤ ⎥ ⎢⎣ ⎦0 ⎦ ⎣ ME575 Session 27: Controllability School of Mechanical Engineering Purdue University Slide 2 1 ME 575 Handouts Controllability (Kalman - 1960) Complete State Controllability: The system x = A x +Bu is completely state controllable if a control u exists that will transfer the state of the system from any x(to) = xo to any x(t1) = x1 in a finite time t1−to. School of Mechanical Engineering Purdue University ME575 Session 27: Controllability Slide 3 Example - Inverted Pendulum d(t) g m = 1 kg θ =1m u(t) −10 V +10 V vpot M = 10 kg Is it possible to achieve zero position of both the cart and the rod with only a single control input u? ME575 Session 27: Controllability School of Mechanical Engineering Purdue University Slide 4 2 ME 575 Handouts Simpler Example 1⎤ ⎡0 ⎡0 ⎤ x=⎢ ⎥ x + ⎢ 1⎥ u ⎣ −2 −3 ⎦ ⎣⎦ Is it completely state controllable? Transform to diagonal form: ME575 Session 27: Controllability School of Mechanical Engineering Purdue University Slide 5 Controllability Note that controllability depends not only on B, but on the combination of A (through T) and B. Can we generalize this? YES The system x = A x +Bu is completely state controllable iff the column vectors of the controllability matrix n−1 W C = [B A B A B A B] span the n-dimensional space (i.e., WC has rank n). 2 ME575 Session 27: Controllability School of Mechanical Engineering Purdue University Slide 6 3 ME 575 Handouts Proof of Controllability Theorem Assume u is a scalar: t x(t) = e At x(0) + ∫ e A( t −τ ) B u( τ)dτ 0 let x(0) = 0, x(t1) = x1 t1 x(t1 ) = x1 = ∫ e A( t1−τ ) B u( τ)dτ but 0 1 e A( t1−τ ) = I + A(t1 − τ) + 2! A (t1 − τ)2 + 2 t1 t1 0 0 x1 = B ∫ u( τ)dτ + A B ∫ (t1 − τ)u( τ)dτ + + A B∫ 2 ME575 Session 27: Controllability t1 1 02 (t1 − τ)2 u( τ)dτ + School of Mechanical Engineering Purdue University Slide 7 Controllability Proof (cont.) If only linearly independent vectors exist, then linearly A B = α0 B + α1 A B + α 2 A B + 2 +1 A B = α 0 A B + α1 A B + α 2 A B + 2 3 −1 + α −1 A B + α −1 A B Thus, AkB is linearly dependent on B, . . ., A −1B, for all k ≥ . rank[WC] is the order of the largest nonzero minor in WC. This works even if WC is nonsquare. ME575 Session 27: Controllability School of Mechanical Engineering Purdue University Slide 8 4 ME 575 Handouts Return to Inverted Pendulum Define state vector: ⎡0 ⎢10.79 x=⎢ ⎢0 ⎢ ⎣ −0.98 ME575 Session 27: Controllability 1 0 0 0 x = [θ 0 0 0 0 θ x x]T 0⎤ ⎡0⎤ ⎢ −0.1⎥ 0⎥ ⎥x+ ⎢ ⎥u 1⎥ ⎢0⎥ ⎥ ⎢ ⎥ 0⎦ ⎣ 0.1 ⎦ School of Mechanical Engineering Purdue University Slide 9 Check Controllability Controllability Matrix: ⎡0 ⎢ −0.1 WC = ⎢ ⎢0 ⎢ ⎣ 0.1 ME575 Session 27: Controllability −0.1 0 0.1 0 0 −1.08 0 0.1 −1.08 ⎤ 0⎥ ⎥ 0.1 ⎥ ⎥ 0⎦ School of Mechanical Engineering Purdue University Slide 10 5 ME 575 Handouts Another Example ⎡ −2 ⎢ A=⎢ 0 ⎢0 ⎣ ME575 Session 27: Controllability 1 −2 0 0⎤ ⎡0 ⎤ ⎥ 0⎥ , B = ⎢b ⎥ ⎢⎥ ⎥ a⎦ ⎢ 1⎥ ⎣⎦ School of Mechanical Engineering Purdue University Slide 11 Output Controllability The system x = A x +Bu y = C x +Du is output controllable if a control u exists that will transfer the output of the system from any y(to) to any y(t1) in a finite time t1−to. This is true iff rank[C B CAB ME575 Session 27: Controllability 2 CA B School of Mechanical Engineering Purdue University n−1 CA B D] = m Slide 12 6 ...
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