Models

Benchmark Model Templates

This page specifies templates for the types of models used as benchmark systems. In particular, the naming schemes established here are used in the corresponding data sets for all benchmarks. For example, $A$ always serves as the name of the component matrix applied to the state $x(t)$ in a linear time-invariant system. For all models we assume an input $u:\mathbb{ℝ}\to {\mathbb{ℝ}}^m$, with components $u_j,j=1,\dots ,m$, a state $x:\mathbb{ℝ}\to {\mathbb{ℝ}}^n$, and an output $y:\mathbb{ℝ}\to {\mathbb{ℝ}}^q$. For all parametric models, we assume each component has $\ell$ parameters; in cases where a component has fewer than $\ell$ parameters, the extras are treated as $0$. Some benchmarks (e.g., Bone Model) have a constant forcing term, in which case, it is assumed that $u(t)$ is identically $1$.

Linear Time-Invariant First-Order System (LTI-FOS)

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

with

$E\in {\mathbb{ℝ}}^{n\times n}$, $A\in {\mathbb{ℝ}}^{n\times n}$, $B\in {\mathbb{ℝ}}^{n\times m}$, $C\in {\mathbb{ℝ}}^{q\times n}$, $D\in {\mathbb{ℝ}}^{q\times m}$.

Linear Time-Varying First-Order System (LTV-FOS)

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

with

$E:\mathbb{ℝ}\to {\mathbb{ℝ}}^{n\times n}$, $A:\mathbb{ℝ}\to {\mathbb{ℝ}}^{n\times n}$, $B:\mathbb{ℝ}\to {\mathbb{ℝ}}^{n\times m}$, $C:\mathbb{ℝ}\to {\mathbb{ℝ}}^{q\times n}$, $D:\mathbb{ℝ}\to {\mathbb{ℝ}}^{q\times m}$.

Affine Parametric LTI-FOS (AP-LTI-FOS)

$\begin{array}{rl}(E+\sum _{i=1}^{\ell }{p}_i^E{E}_i)\dot{x}(t)& =(A+\sum _{i=1}^{\ell }{p}_i^A{A}_i)x(t)+(B+\sum _{i=1}^{\ell }{p}_i^B{B}_i)u(t),\\ y(t)& =(C+\sum _{i=1}^{\ell }{p}_i^C{C}_i)x(t),\end{array}$

with

$E,E_i\in {\mathbb{ℝ}}^{n\times n}$; $A,A_i\in {\mathbb{ℝ}}^{n\times n}$; $B,B_i\in {\mathbb{ℝ}}^{n\times m}$; and $C,C_i\in {\mathbb{ℝ}}^{q\times n}$, for all $i=1,\dots ,\ell$.

Linear Time-Invariant Second-Order System (LTI-SOS)

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

with

$M\in {\mathbb{ℝ}}^{n\times n}$, $E\in {\mathbb{ℝ}}^{n\times n}$, $K\in {\mathbb{ℝ}}^{n\times n}$, $B\in {\mathbb{ℝ}}^{n\times m}$, $C_p,C_v\in {\mathbb{ℝ}}^{q\times n}$, $D\in {\mathbb{ℝ}}^{q\times m}$.

When $C_v=0$, we denote $C=C_p$.

Affine Parametric LTI-SOS (AP-LTI-SOS)

$\begin{array}{rl}(M+\sum _{i=1}^{\ell }{p}_i^M{M}_i)\ddot{x}(t)+(E+\sum _{i=1}^{\ell }{p}_i^E{E}_i)\dot{x}(t)+(K+\sum _{i=1}^{\ell }{p}_i^K{K}_i)x(t)& =(B+\sum _{i=1}^{\ell }{p}_i^B{B}_i)u(t),\\ y(t)& =(C+\sum _{i=1}^{\ell }{p}_i^C{C}_i)x(t),\end{array}$

with

$M,M_i\in {\mathbb{ℝ}}^{n\times n}$; $E,E_i\in {\mathbb{ℝ}}^{n\times n}$; $K,K_i\in {\mathbb{ℝ}}^{n\times n}$; $B,B_i\in {\mathbb{ℝ}}^{n\times m}$; and $C,C_i\in {\mathbb{ℝ}}^{q\times n}$, for all $i=1,\dots ,\ell$.

Quadratic-Bilinear System (QBS)

$\begin{array}{rl}E\dot{x}(t)& =Ax(t)+Hx(t)\otimes x(t)+\sum _{j=1}^m{N}_{j}x(t)u_j(t)+Bu(t),\\ y(t)& =Cx(t)+Du(t),\end{array}$

with

$E\in {\mathbb{ℝ}}^{n\times n}$, $A\in {\mathbb{ℝ}}^{n\times n}$, $H\in {\mathbb{ℝ}}^{n\times n^2}$, $N_j\in {\mathbb{ℝ}}^{n\times n}$, $B\in {\mathbb{ℝ}}^{n\times m}$, $C\in {\mathbb{ℝ}}^{q\times n}$, $D\in {\mathbb{ℝ}}^{q\times m}$.

Nonlinear Time-Invariant First-Order System (NLTI-FOS)

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

with

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

Nonlinear Time-Invariant Second-Order System (NLTI-SOS)

$\begin{array}{rl}M\ddot{x}(t)+E\dot{x}(t)+Kx(t)& =Bu(t)+Ff(x(t),u(t)),\\ y(t)& =C_{p}x(t)+C_v\dot{x}(t)+Du(t),\end{array}$

with

$M\in {\mathbb{ℝ}}^{n\times n}$, $E\in {\mathbb{ℝ}}^{n\times n}$, $K\in {\mathbb{ℝ}}^{n\times n}$, $B\in {\mathbb{ℝ}}^{n\times m}$, $F\in {\mathbb{ℝ}}^{n\times n}$, $C_p,C_v\in {\mathbb{ℝ}}^{q\times n}$, $D\in {\mathbb{ℝ}}^{q\times m}$, $f:{\mathbb{ℝ}}^n\times {\mathbb{ℝ}}^m\to {\mathbb{ℝ}}^n$.

When $C_v=0$, we denote $C=C_p$.

Affine-parametric and time-varying versions of nonlinear systems are clearly also possible by combining patterns of the above models.