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

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

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

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

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

$\begin{array}{rl}E\dot{x}(t)& =Ax(t)+Hx(t)\otimes x(t)+\sum _{i=1}^{M}x(t)u_i(t)+Bu(t),\\ y(t)& =Cx(t),\end{array}$

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

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

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

$\begin{array}{rl}(E_0+\sum _{i=1}^{P_E}{p}_i^E{E}_i)\dot{x}(t)& =(A_0\sum _{j=1}^{P_A}{p}_j^A{A}_j)x(t)+Bu(t),\\ y(t)& =Cx(t),\end{array}$

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

$\begin{array}{rl}M\ddot{x}(t)+E\dot{x}(t)+Kx(t)& =Bu(t),\\ y(t)& =Cx(t),\end{array}$

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

$\begin{array}{rl}M\ddot{x}(t)+E\dot{x}(t)+Kx(t)& =Bu(t)+f(x(t),u(t)),\\ y(t)& =Cx(t),\end{array}$

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

$\begin{array}{rl}(M_0+\sum _{i=1}^{P_M}{p}_i^M{M}_i)\ddot{x}(t)+(E_0+\sum _{i=1}^{P_E}{p}_i^E{E}_i)\dot{x}(t)+(K_0+\sum _{i=1}^{P_K}{p}_i^K{K}_i)x(t)& =Bu(t),\\ y(t)& =Cx(t),\end{array}$

with:

$M_0\in {\mathbb{ℝ}}^{N\times N}$, $M_i\in {\mathbb{ℝ}}^{N\times N}$, $E_0\in {\mathbb{ℝ}}^{N\times N}$, $E_j\in {\mathbb{ℝ}}^{N\times N}$, $K_0\in {\mathbb{ℝ}}^{N\times N}$, $K_k\in {\mathbb{ℝ}}^{N\times N}$, $B\in {\mathbb{ℝ}}^{N\times M}$, $C\in {\mathbb{ℝ}}^{Q\times N}$.