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**Accuracy-enhancing methods for balancing-related frequency-weighted model and controller reduction.**
*(English)*
Zbl 1045.93010

For a given \(n\)th-order original continuous-time-space model
\[
{\mathbf G}= (A,B,C,D),
\]
let \({\mathbf G}_r= (A_r, B_r, C_r, D_r)\) be an \(r\)th-order approximation of the original model \((r< n)\). Thus, the methods for frequency-weighted model reduction try to minimize a weighted approximation error in the \(L\)-infinity norm and involving weights \(W_0\) and \(W_i\) which are suitably chosen to respectively weight the output and input transfer-function matrices.

On the other hand, in a controller reduction problem, for a given stabilizing controller of order \(n_c\), say \({\mathbf K}= (A_c, B_c, C_c, D_c)\) one wants to find \({\mathbf K}_r\), an \(r_c\)th-order approximation with the same number of unstable poles as \({\mathbf K}\), such that a similar weighted error is needed to be minimized.

Here, the square-root and balancing-free accuracy-enhancing techniques are extended to this kind of frequency-weighted problems. For this purpose, solving reduced-order Lyapunov equations is required by directly computing the grammians in terms of their Cholesky factors.

On the other hand, in a controller reduction problem, for a given stabilizing controller of order \(n_c\), say \({\mathbf K}= (A_c, B_c, C_c, D_c)\) one wants to find \({\mathbf K}_r\), an \(r_c\)th-order approximation with the same number of unstable poles as \({\mathbf K}\), such that a similar weighted error is needed to be minimized.

Here, the square-root and balancing-free accuracy-enhancing techniques are extended to this kind of frequency-weighted problems. For this purpose, solving reduced-order Lyapunov equations is required by directly computing the grammians in terms of their Cholesky factors.

Reviewer: Pablo Gonzalez-Vera (La Laguna)

### MSC:

93B11 | System structure simplification |

93B40 | Computational methods in systems theory (MSC2010) |

93B20 | Minimal systems representations |

### Keywords:

model reduction; balancing; frequency-weighting; controller reduction; numerical algorithms; square-root accuracy-enhancing techniques; reduced order Lyapunov equations### Software:

SLICOT
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\textit{A. Varga} and \textit{B. D. O. Anderson}, Automatica 39, No. 5, 919--927 (2003; Zbl 1045.93010)

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### References:

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