Models

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Benchmark Model Overview

This page outlines the types of models that are used as benchmark systems. For this general summary we assume an input $u : \mathbb{R} \to \mathbb{R}^M$, a state $x : \mathbb{R} \to \mathbb{R}^N$ and an output $y : \mathbb{R} \to \mathbb{R}^Q$.

Linear Time-Invariant System

$\begin{align} E\dot{x}(t) &= Ax(t) + Bu(t),\\ y(t) &= Cx(t), \end{align}$

with:

$E \in \mathbb{R}^{N \times N}$, $A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times M}$, $C \in \mathbb{R}^{Q \times N}$.

Linear Time-Varying System

$\begin{align} E(t)\dot{x}(t) &= A(t)x(t) + B(t)u(t),\\ y(t) &= C(t)x(t), \end{align}$

with:

$E : \mathbb{R} \to \mathbb{R}^{N \times N}$, $A : \mathbb{R} \to \mathbb{R}^{N \times N}$, $B : \mathbb{R} \to \mathbb{R}^{N \times M}$, $C : \mathbb{R} \to \mathbb{R}^{Q \times N}$.

Quadratic-Bilinear System

$\begin{align} E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{i=1}^M x(t) u_i(t) + B u(t), \\ y(t) &= Cx(t), \end{align}$

with:

$E \in \mathbb{R}^{N \times N}$, $A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times M}$, $H \in \mathbb{R}^{N \times N^2}$, $N_i \in \mathbb{R}^{N \times N}$, $C \in \mathbb{R}^{Q \times N}$.

Nonlinear Time-Invariant System

$\begin{align} E\dot{x}(t) &= Ax(t) + f(x(t),u(t)) + Bu(t),\\ y(t) &= Cx(t), \end{align}$

with:

$E \in \mathbb{R}^{N \times N}$, $A \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times M}$, $C \in \mathbb{R}^{Q \times N}$, $f : \mathbb{R}^N \times \mathbb{R}^M \to \mathbb{R}^N$.

Affine Parametric Linear Time-Invariant System

$\begin{align} (E_0 + \sum_{i=1}^{P_E} p^E_i E_i)\dot{x}(t) &= (A_0 \sum_{j=1}^{P_A} p^A_j A_j) x(t) + Bu(t),\\ y(t) &= Cx(t), \end{align}$

with:

$E_0 \in \mathbb{R}^{N \times N}$, $E_i \in \mathbb{R}^{N \times N}$, $A_0 \in \mathbb{R}^{N \times N}$, $A_j \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times M}$, $C \in \mathbb{R}^{Q \times N}$.

Second-Order System

$\begin{align} M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\ y(t) &= C x(t), \end{align}$

with:

$M \in \mathbb{R}^{N \times N}$, $E \in \mathbb{R}^{N \times N}$, $K \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times M}$, $C \in \mathbb{R}^{Q \times N}$.

Nonlinear Second-Order System

$\begin{align} M \ddot{x}(t) + D \dot{x}(t) + K x(t) &= B u(t) + f(x(t),u(t)), \\ y(t) &= C x(t), \end{align}$

with:

$M \in \mathbb{R}^{N \times N}$, $D \in \mathbb{R}^{N \times N}$, $K \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times M}$, $C \in \mathbb{R}^{Q \times N}$, $f : \mathbb{R}^N \times \mathbb{R}^M \to \mathbb{R}^N$.

Affine Parametric Second-Order System

$\begin{align} (M_0 + \sum_{i=1}^{P_M} p^M_i M_i)\ddot{x}(t) + (D_0 + \sum_{i=1}^{P_D} p^D_i D_i)\dot{x}(t) + (K_0 + \sum_{i=1}^{P_K} p^K_i K_i)x(t) &= B u(t), \\ y(t) &= C x(t), \end{align}$

with:

$M_0 \in \mathbb{R}^{N \times N}$, $M_i \in \mathbb{R}^{N \times N}$, $D_0 \in \mathbb{R}^{N \times N}$, $D_j \in \mathbb{R}^{N \times N}$, $K_0 \in \mathbb{R}^{N \times N}$, $K_k \in \mathbb{R}^{N \times N}$, $B \in \mathbb{R}^{N \times M}$, $C \in \mathbb{R}^{Q \times N}$.