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

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

with:

$E \in \mathbb{R}^{n \times n}$, $A \in \mathbb{R}^{n \times n}$, $Q \in \mathbb{R}^{n \times n^2}$, $N_i \in \mathbb{R}^{n \times n}$, $u_i: \mathbb{R} \to \mathbb{R}$, $B \in \mathbb{R}^{n \times m}$, $C \in \mathbb{R}^{q \times n}$.

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

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

with:

$E_0 \in \mathbb{R}^{n \times n}$, $E_j \in \mathbb{R}^{n \times n}$, $A_0 \in \mathbb{R}^{n \times n}$, $A_i \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $C \in \mathbb{R}^{q \times n}$.

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

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

with:

$M \in \mathbb{R}^{n \times n}$, $E \in \mathbb{R}^{n \times n}$, $K \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $C \in \mathbb{R}^{q \times n}$, $f : \mathbb{R}^N \times \mathbb{R}^M \to \mathbb{R}^N$.

Affine Parametric Second-Order System

$\begin{align} (M_0 + \sum_{i=1}^{P_M} p^M_i M_i)\ddot{x}(t) + (E_0 + \sum_{j=1}^{P_E} p^E_j E_j)\dot{x}(t) + (K_0 + \sum_{k=1}^{P_K} p^K_k K_k)x(t) &= B u(t), \\ y(t) &= C x(t), \end{align}$

with:

$M_0 \in \mathbb{R}^{n \times n}$, $M_i \in \mathbb{R}^{n \times n}$, $E_0 \in \mathbb{R}^{n \times n}$, $E_j \in \mathbb{R}^{n \times n}$, $K_0 \in \mathbb{R}^{n \times n}$, $K_k \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $C \in \mathbb{R}^{q \times n}$.