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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 16.323 151 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. u = | | u scalars 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 16.323 152 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 trace [ G H (j ) G (j )] d 1 / 2 = trace [ g T ( ) g ( )] d 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 = i 2 [ G (j )] d 2 i Where i [ G (j )] is the i th singular value 28 29 of the system G ( s ) evaluated at s = j H 2 norm concerned with overall performance ( 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 16.323 153 H norm for the system: G ( s ) = sup [ G (j )] 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 with input direction that yields [ G (j )] as the amplification....
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lec15 - MIT OpenCourseWare http://ocw.mit.edu 16.323...

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