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Unformatted text preview: 6.241 Dynamic Systems and Control Lecture 24: 2 Synthesis H Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology May 4, 2011 E. Frazzoli (MIT) Lecture 24: H 2 Synthesis May 4, 2011 1 / 27 Standard setup Consider the following system, for t R : x ( t ) = Ax ( t ) + B w w ( t ) + B u u ( t ) , x (0) = x z ( t ) = C z x ( t ) + D zw w ( t ) + D zu u ( t ) y ( t ) = C y x ( t ) + D yw w ( t ) + D yu u ( t ) , where w is an exogenous disturbance input (also reference, noise, etc.) u is a control input, computed by the controller K z is the performance output . This is a virtual output used only for design. y is the measured output . This is what is available to the controller K It is desired to synthesize a controller K (itself a dynamical system), with input y and output u , such that the closed loop is stabilized, and the performance output is minimized, given a class of disturbance inputs. In particular, we will look at controller synthesis with H 2 and H criteria. E. Frazzoli (MIT) Lecture 24: H 2 Synthesis May 4, 2011 2 / 27 Interpretation of the H 2 norm deterministic Consider a stable, causal CT LTI system with statespace model ( A , B , C , D ), transfer function G ( s ), and impulse response G ( t ). The H 2 norm of G measures: A) The energy of the impulse response: + + G L 2 2 := g ij ( t ) 2 2 dt = G ( t ) 2 F dt i j = Tr + G ( t ) G ( t ) dt = 2 1 Tr + G ( j ) G ( j ) d =: G H 2 2 . B) The energy of the response to initial conditions, of the form x (0) = Bu , for u = (1 , 1 , . . . , 1) . Set u ( t ) = u ( t ) to see this. Clearly, in order for G L 2 = G H 2 to be finite, it is necessary that lim G ( j ) = 0, i.e., that the system is strictly causal D = 0. E. Frazzoli (MIT) Lecture 24: H 2 Synthesis May 4, 2011 3 / 27 Interpretation of the 2 norm stochastic H Consider a stable, strictly causal CT LTI system with statespace model ( A , B , C , 0), transfer function G ( s ), and impulse response G ( t ). Consider a hypothetical stochastic input signal u such that E [ u ( t )] = 0, and E [ u ( t ) u ( t + ) ] = I ( ). This is called white noise, and is just a mathematical abstraction, since it is a signal with infinite power. The H 2 norm of G measures: C) The (expected) power of the response to white noise: T E lim 1 Tr y ( t ) y ( t ) dt T + T 1 T t t = lim Tr E G ( t 1 ) u ( 1 ) u ( 2 ) G ( t 2 ) d 1 d 2 dt T + T T t 1 = lim Tr G ( t ) G ( t ) d dt T + T T lim Tr G ( T ) G ( T ) d ( T ) 2 2 . = T + = G L 2 = G H 2 E. Frazzoli (MIT) Lecture 24: H 2 Synthesis May 4, 2011 4 / 27 Computation of the 2 norm H Computation of the H 2 norm is easy through statespace methods. In fact, + + G H 2 2 = Tr G ( t ) G ( t ) dt = Tr G ( t ) G ( t ) dt ....
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 Spring '11
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