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, first-order system. For all models we assume an input $u : \mathbb{R} \to \mathbb{R}^m$, with components $u_j, j = 1, \ldots, m$, 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)

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

with

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

By default $E = I$ and $D = 0$, unless explicitly provided.

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

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

with

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

By default $E = I$ and $D = 0$, unless explicitly provided.

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

$\begin{align} (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) &= (A + \sum_{i=1}^{\ell} p^A_i A_i) x(t) + (B + \sum_{i=1}^{\ell} p^B_i B_i)u(t),\\ y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i)x(t), \end{align}$

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 $i = 1, \ldots, \ell$.

By default $E = I, E_i = 0$, unless explicitly provided. If $A_i$ are provided without $A$, then it is assumed $A = 0$. Likewise for $B$, $C$, and $E$.

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

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

with

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

When $C_v = 0$, we denote $C = C_p$. By default $E = I$ and $D = 0$, unless explicitly provided.

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

$\begin{align} (M + \sum_{i=1}^{\ell} p^M_i M_i)\ddot{x}(t) + (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) + (K + \sum_{i=1}^{\ell} p^K_i K_i)x(t) &= (B + \sum_{i=1}^{\ell} p^B_i B_i) u(t), \\ y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i) x(t), \end{align}$

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 $i = 1, \ldots, \ell$.

By default $E = I, E_i = 0$, unless explicitly provided. If $M_i$ are provided without $M$, then it is assumed $M = 0$. Likewise for $K$, $B$, and $C$.

Quadratic-Bilinear System (QBS)

$\begin{align} E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^m N_j x(t) u_j(t) + B u(t), \\ y(t) &= Cx(t) + Du(t), \end{align}$

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)

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

with

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

By default $F = I$, $E = I$, $D = 0$, unless explicitly provided.

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

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

with

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

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

By default $F = I$, $E = I$, $D = 0$, unless explicitly provided.

Other System Classes

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