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# The generalised singular perturbation approximation for bounded real and positive real control systems

• The generalised singular perturbation approximation (GSPA) is considered as a model reduction scheme for bounded real and positive real linear control systems. The GSPA is a state-space approach to truncation with the defining property that the transfer function of the approximation interpolates the original transfer function at a prescribed point in the closed right half complex plane. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at infinity and at zero, respectively. Suitably modified, we show that the GSPA preserves classical dissipativity properties of the truncations, and existing a priori error bounds for these balanced truncation schemes are satisfied as well.

Mathematics Subject Classification: Primary: 34K07, 34K26, 93B11, 93C05, 93C70.

 Citation: • • Figure 5.1.  Semi-log plot of combined errors on the real axis for the bounded real GSPA from Example 5.1, with $r = 2$. The lines numbered 1-4 correspond to $\xi_1 = 0.1$, $\xi_2 = 1$, $\xi_3 = 10$ and $\xi_4 = 100$, respectively. Note the interpolation properties (2.7) and (3.7) hold and are highlighted with vertical dotted lines. The dashed dotted line is the bound (3.3)

Figure 5.2.  Semi-log plot of combined errors on the real axis for the bounded real GSPA from Example 5.1, with $r = 1$. The lines numbered 1-4 correspond to $\xi_1 = 0.1$, $\xi_2 = 1$, $\xi_3 = 10$ and $\xi_4 = 100$, respectively. Note the interpolation properties (2.7) and (3.7) hold and are highlighted with vertical dotted lines. The dashed dotted line is the error bound (3.3)

Figure 5.3.  Plots of errors on the imaginary axis for the bounded real GSPA from Example 5.1, with $r = 1$ and $r = 2$ in panels (a) and (b), respectively. The lines numbered 1-4 correspond to $\xi_1 = 0.1$, $\xi_2 = 1$, $\xi_3 = 10$ and $\xi_4 = 100$, respectively, and are symmetric around $\omega = 0$. The dashed dotted lines are the bounds (3.3)

Figure 5.4.  Semi-log plot of combined errors on the real axis for the positive real GSPA from Example 5.2, with $\xi = 10$. The lines numbered 1-3 correspond to $r \in \{1, 2, 3\}$ respectively. Note the interpolation property (2.7) holds

Figure 5.5.  Semi-log plot of gap metric error $\hat \delta( \mathbf G , \mathbf G _r^\xi)$ (crosses) and error bounds (4.4) (circles) for extended circuit model from Example 5.2. Here $\xi = 10$

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