ModelRed

# ModelRed - Model Reduction of State Space Systems Keith...

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Model Reduction of State Space Systems Keith Glover CDS 212, Caltech Nov. 2010 Cambridge University Engineering Department

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References: Glover, K. (1989) A tutorial on Hankel-norm approximation In: Willems, J.C., (ed.) From Data to Model. Springer-Verlag, Berlin, Germany, pp. 26-48. ISBN 3540515712 Zhou, K., Doyle, J.C. and Glover, K. (1996) Robust and optimal control Prentice Hall International, London, UK. ISBN 0134565673 and the references therein! 2/23
Outline Hankel Operators Balanced Realisation and Truncation Hankel norm approximation Frequency Weighted Model Reduction 3/23

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Let the Hankel operator, Γ G , corresponding to the stable system ˙ x ( t ) = Ax ( t )+ Bu ( t ); y ( t ) = Cx ( t ) with transfer function, G ( s ) = C ( sI - A ) - 1 B and impulse response h ( t ) = Ce At B be deﬁned as Γ G : L 2 ( - , 0) L 2 (0 , ) : y ( t ) = Z 0 - h ( t - τ ) u ( τ ) d τ , t > 0 = Ce At Z 0 - e - A τ Bu ( τ ) d τ = Ce At x (0) with adjoint Γ * G : L 2 (0 , ) L 2 ( - , 0) : u ( t ) = Z 0 h ( - t + τ ) 0 y ( τ ) d τ , t < 0 = B 0 e - A 0 t Z 0 e A 0 τ C 0 y ( τ ) d τ .
Since the Hankel operator acts via the state at t = 0 its rank must be the state dimension. The Hankel operator is a restriction of the Toeplitz (convolution) operator so that k G k ≥ k Γ G k =: k G k H In addition if F RH , - (i.e. anti-causal) then we still have k G +

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ModelRed - Model Reduction of State Space Systems Keith...

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