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Unformatted text preview: 6.241 Dynamic Systems and Control Lecture 25: H ∞ Synthesis Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology May 11, 2011 E. Frazzoli (MIT) Lecture 25: H∞ Synthesis May 11, 2011 1 / 12 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 25: H∞ Synthesis May 11, 2011 2 / 12 Optimal H ∞ synthesis? In principle, we would like to find a controller K such that minimizes the energy ( L 2 ) gain of the closedloop system, i.e., that minimizes T zw H ∞ = sup z L 2 . w =0 w L 2 However, the optimal controller(s) are such that σ max ( T zw ( j ω ) is a constant over all frequencies, the response does not roll off at high frequencies, and the controller is not strictly proper. (The optimal controller is not unique.)unique....
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This note was uploaded on 01/12/2012 for the course ECE 6.241j taught by Professor Prof.emiliofrazzoli during the Spring '11 term at MIT.
 Spring '11
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