Models

Note: This page has not been verified by our editors.

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

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

with:

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

Nonlinear Time-Invariant System

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

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

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

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

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

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

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

with:

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