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Controller reduction by \(H_\infty \) balanced truncation for infinite-dimensional, discrete-time systems

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Abstract

We consider Lyapunov and \(H_\infty \) balancing as well as model reduction by balanced truncation for infinite-dimensional, discrete-time linear systems. A functional analytic approach to state space transformation leading to balanced realization is presented. Furthermore, we show that a finite-dimensional \(H_\infty \) controller designed for a \(H_\infty \) balanced and truncated system stabilizes the original infinite-dimensional system, provided the error made by this model reduction procedure is small enough.

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Abbreviations

\({\mathbb {N}}, {\mathbb {Z}}_+\) :

\({\mathbb {Z}}_+:={\mathbb {N}}:=\{0,1,2,\ldots \}\)

\({\mathbb {Z}}_-\) :

\({\mathbb {Z}}_-:={\mathbb {Z}}{\setminus }{\mathbb {Z}}_+=\{-1,-2,-3,\ldots \}\)

\(\mathcal {X}^I\) :

The set of all mappings from the set \(I\) to \(\mathcal {X}\)

\(\mathcal {X}^{I}_\mathrm{c}\) :

The set of sequences \((x_n):I\subset {\mathbb {Z}}\rightarrow \mathcal {X}\) whose support is bounded to the left

\(c_0({\mathbb {N}})\) :

The set of sequences in \((x_n):{\mathbb {N}}\rightarrow {\mathbb {C}}\) that converge to zero

\(\ell ^p_r(I;\mathcal {X})\) :

The space of sequences \((x_n): I\subset {\mathbb {Z}}\rightarrow \mathcal {X}\) with \(\displaystyle {\Vert (x_n)\Vert _{\ell ^p_r(S;\mathcal {X})}:=\left( \sum _{n\in I}r^{-n}\Vert x_n\Vert _{\mathcal {X}}^p\right) ^{\frac{1}{p}}<\infty }\)

\(\displaystyle {\ell ^p(I;\mathcal {X})}\) :

\(:= \ell ^p_1(I;\mathcal {X})\)

\(\ell ^p\) :

\(:=\ell ^p_1({\mathbb {N}};{\mathbb {C}})\)

\(e_i\) :

The sequence \(e_i:{\mathbb {N}}\rightarrow {\mathbb {C}}\) with \((e_i)_n=\delta _{i,n}\)

\(\mathcal {B}(\mathcal {X};\mathcal {Y}), \mathcal {B}(\mathcal {X})\) :

The set of bounded linear operators from \(\mathcal {X}\) to a Hilbert space \(\mathcal {Y}\) or to \(\mathcal {X},\) respectively

\({{\mathrm{diag}}}(\sigma _n):X^{\mathbb {N}}\rightarrow X^{\mathbb {N}}\) :

The mapping \((x_n)\mapsto (\sigma _n x_n)\) for \((\sigma _n) \in {\mathbb {C}}^{\mathbb {N}}\)

\({{\mathrm{I}}}\) :

The identity operator

\(\pi _{Z}\) :

The orthogonal projection onto a closed subspace \(Z\)

\({{\mathrm{dom}}}A, {{\mathrm{ran}}}A\) :

The domain and the range of a linear operator \(A\)

\(\overline{A}\) :

The closure of a linear operator \(A\)

\(A|_Z\) :

The restriction of a linear operator \(A\) to the subspace \(Z\)

\(A^{-*}:=(A^{-1})^*\) :

The adjoint of the inverse of a closed operator \(A,\) note \(A^{-*}=(A^*)^{-1}\)

\(\sigma (A), \rho (A)\) :

The spectrum and resolvent set of a linear operator \(A\)

\(r(A)\) :

The spectral radius of a linear operator \(A\)

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Acknowledgments

This work was supported by the Klaus Tschira Stiftung. The author would like to thank Prof. Achim Ilchmann and Prof. Timo Reis for their helpful advice.

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Correspondence to Tilman Selig.

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Selig, T. Controller reduction by \(H_\infty \) balanced truncation for infinite-dimensional, discrete-time systems. Math. Control Signals Syst. 27, 111–147 (2015). https://doi.org/10.1007/s00498-014-0137-7

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