# lec15 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 16.323 Lecture 15 Signals and System Norms H ∞ Synthesis Different type of optimal controller SP Skogestad and Postlethwaite(1996) Multivariable Feedback Control Wiley. JB Burl (2000). Linear Optimal Control Addison-Wesley. ZDG Zhou, Doyle, and Glover (1996). Robust and Optimal Control Prentice Hall. MAC Maciejowski (1989) Multivariable Feedback Design Addison Wesley. Spr 2008 16.323 15–1 Mathematical Background • Signal norms we use norms to measure the size of a signal. – Three key properties of a norm: 1. u ≥ , and u = 0 iff u = 0 2. αu = | α | u ∀ scalars α 3. u + v ≤ u + v • Key signal norms – 2-norm of u ( t ) – Energy of the signal 1 / 2 ∞ u ( t ) 2 ≡ u 2 ( t ) dt −∞ – ∞-norm of u ( t ) – maximum value over time u ( t ) ∞ = max u ( t ) t | | – Other useful measures include the Average power T 1 / 2 pow( u ( t )) = lim 1 u 2 ( t ) dt T →∞ 2 T − T u ( t ) is called a power signal if pow( u ( t )) < ∞ June 18, 2008 Spr 2008 16.323 15–2 • System norms Consider the system with dynamics y = G ( s ) u – Assume G ( s ) stable, LTI transfer function matrix – g ( t ) is the associated impulse response matrix (causal). • H 2 norm for the system: (LQG problem) 1 / 2 G 2 = 2 1 π ∞ trace [ G H (j ω ) G (j ω )] dω −∞ 1 / 2 = ∞ trace [ g T ( τ ) g ( τ )] dτ Two interpretations: – For SISO: energy in the output y ( t ) for a unit impulse input u ( t ) . – For MIMO 27 : apply an impulsive input separately to each actuator and measure the response z i , then = G 2 2 z i 2 2 i – Can also interpret as the expected RMS value of the output in response to unit-intensity white noise input excitation. • Key point: Can show that 1 / 2 1 ∞ G 2 = σ i 2 [ G (j ω )] dω 2 π −∞ i – Where σ i [ G (j ω )] is the i th singular value 28 29 of the system G ( s ) evaluated at s = j ω – H 2 norm concerned with overall performance ( σ i 2 ) over all i frequencies 27 ZDG114 28 http://mathworld.wolfram.com/SingularValueDecomposition.html 29 http://en.wikipedia.org/wiki/Singular_value_decomposition June 18, 2008 Spr 2008 16.323 15–3 • H ∞ norm for the system: G ( s ) ∞ = sup σ [ G (j ω )] ω Interpretation: – G ( s ) ∞ is the “energy gain” from the input u to output y ∞ y T ( t ) y ( t ) dt G ( s ) ∞ = max u ( t )=0 ∞ u T ( t ) u ( t ) dt – Achieve this maximum gain using a worst case input signal that is essentially a sinusoid at frequency ω with input direction that yields σ [ G (j ω )] as the amplification....
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lec15 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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