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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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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DOI: https://doi.org/10.1007/s00498-014-0137-7