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# topic24 - Topic#24 16.31 Feedback Control Systems • H ∞...

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Unformatted text preview: Topic #24 16.31 Feedback Control Systems • H ∞ Synthesis 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. Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 24–1 Synthesis • For the synthesis problem, typically define a generalized version of the system dynamics (SISO or MIMO) P zw ( s ) P zu ( s ) P yw ( s ) P yu ( s ) G c w u z y Signals: z – Performance output w – Disturbance/ref inputs y – Sensor outputs u – Actuator inputs Generalized plant: P ( s ) = P zw ( s ) P zu ( s ) P yw ( s ) P yu ( s ) contains plant G ( s ) and all perfor- mance/uncertainty weights • With the loop closed ( u = G c y ), can show that z = P zw + P zu G c ( I − P yu G c ) − 1 P yw w ≡ F l ( P, G c ) w – Called a (lower) Linear Fractional Transformation (LFT). • Define the H ∞ norm for a TFM P ( s ) as P ( s ) ∞ = sup ω σ [ P ( j ω )] – Basically finds the peak amplification possible for the TFM over all frequencies (works for MIMO). December 9, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 24–2 • Design Objective: Find G c ( s ) to stabilize the closed-loop sys- tem and minimize F l ( P, G c ) ∞ . • Hard problem to solve, so typically consider suboptimal problem: – Find G c ( s ) to satisfy F l ( P, G c ) ∞ < γ – Then use bisection (called a γ iteration ) to find the smallest value ( γ opt ) for which F l ( P, G c ) ∞ < γ opt ⇒ hopefully get that G c approaches G opt c • Consider the suboptimal H ∞ synthesis problem: 1 Find G c ( s ) to satisfy F l ( P, G c ) ∞ < γ P ( s ) = P zw ( s ) P zu ( s ) P yw ( s ) P yu ( s ) := ⎡ ⎣ A B w B u C z D zu C y D yw ⎤ ⎦ where it is assumed that: 1. ( A, B u , C y ) is stabilizable/detectable (essential) 2. ( A, B w , C z ) is stabilizable/detectable (essential) 3. D T zu [ C z D zu ] = [ 0 I ] (simplify/essential) 4. B w D yw D T yw = I (simplify/essential) 1 SP367 December 9, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 24–3 • There exists a stabilizing G c ( s ) such that F l ( P, G c ) ∞ < γ iff (1) ∃ X ≥ that solves the ARE A T X + XA + C T z C z + X ( γ − 2 B w B T w − B u B T u ) X = 0 and R λ i A + ( γ − 2 B w B T w − B u B T u ) X < ∀ i (2) ∃ Y ≥ that solves the ARE AY + Y A T + B T w B w + Y ( γ −...
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topic24 - Topic#24 16.31 Feedback Control Systems • H ∞...

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