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Dynamical model approximation, some issues...

Model approximation/reduction, what for?

The objective of the dynamical model approximation is to

  • speed up the dynamical system simulation time,
  • simplify memory management,
  • enhance controller and observer design (optimal, $\mathcal{H}_\infty$, $\mathcal{H}_2$, MPC...),
  • enhance performance and robustness evaluation ($\mathcal{H}_\infty$, $\mu$, ...),
  • enhance passive optimization steps.

Diagram

The linear dynamical model reduction case (MORE framework)

Given a continuous $n_u$ inputs $n_y$ outputs MIMO linear time invariant dynamical model of order $n$ (with $n$ potentially large), described as $H(s)\in \mathbb{C}^{n_y \times n_u}$, the approximation problem consists of seeking for $\hat{H}(s)=\hat{C}(sE-\hat{A})^{-1}\hat{B}+\hat{D} \in \mathbb{C}^{n_y \times n_u}$ a $r$th order model ($r\ll n$), with realization
$$
\hat{\mathbf H}: \left\{
\begin{array}{rcl}
\hat{E}\dot{\hat{x}}(t) &=& \hat{A}\hat{x}(t) + \hat{B}u(t) \\
\hat{y}(t)&=& \hat{C}\hat{x}(t) + \hat{D}u(t)
\end{array}
\right.
$$
where $\hat{E}\in\mathbb{R}^{r \times r}$, $\hat{A}\in\mathbb{R}^{r \times r}$, $\hat{B}\in \mathbb{R}^{r\times n_u}$, $\hat{C} \in \mathbb{R}^{n_y \times r}$, $\hat{D} \in \mathbb{R}^{n_y \times n_u}$, such that

  • the approximation overall error and/or global error bound is small ($|\hat{y}(t)-y(t)|$ or equivalently $||H-\hat{H}||$ is small according to a given metric),
  • some properties (such as stability, structure, positiveness...) are preserved, and
  • the numerical procedure is robust, stable, fast and applicable to large-scale problems.

How to measure the approximation?

Usually, within the (linear) model approximation domain, the $\mathcal{H}_2$-norm is used to measure the quality of the approximated model. Indeed, measuring the $\mathcal{H}_2$-norm of the error between the original $H(s)$ and the obtained reduced-order $\hat{H}(s)$ model provides an information of the error over the entire frequency response. This error is also known as the mismatch error.
That is why many approximation techniques seek for minimizing the $\mathcal{H}_2$-norm of the error system, i.e.
$$
\hat{H}(s) := \arg \min_{G(s)} ||H - G||_{\mathcal{H}_2}
$$
In a similar philosophy, as in many engineering applications the initial model is not very accurate over the entire spectrum, it is sometimes desirable to seek for an approximate model that minimizes the mismatch error over a limited frequency range. To this end, the $\mathcal{H}_{2,\Omega}$-norm is preferred, and the problem is then
$$
\hat{H}(s) := \arg \min_{G(s)} ||H - G||_{\mathcal{H}_{2,\Omega}}
$$
It is to be noticed that both problem are continuous but non-convex.