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
\[ \begin{align} E\dot{x}(t) &= Ax(t) + Bu(t),\\ y(t) &= Cx(t), \end{align} \]
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\[E \in \mathbb{R}^{N \times N}\], \(A \in \mathbb{R}^{N \times N}\), \(B \in \mathbb{R}^{N \times M}\), \(Q \in \mathbb{R}^{N \times N^2}\), \(N_i \in \mathbb{R}^{N \times N}\), \(C \in \mathbb{R}^{Q \times N}\).
Nonlinear Time-Invariant System
\[ \begin{align} E\dot{x}(t) &= Ax(t) + f(x(t),u(t)) + Bu(t),\\ y(t) &= Cx(t), \end{align} \]
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\[M_0 \in \mathbb{R}^{N \times N}\], \(M_i \in \mathbb{R}^{N \times N}\), \(E_0 \in \mathbb{R}^{N \times N}\), \(E_j \in \mathbb{R}^{N \times N}\), \(K_0 \in \mathbb{R}^{N \times N}\), \(K_k \in \mathbb{R}^{N \times N}\), \(B \in \mathbb{R}^{N \times M}\), \(C \in \mathbb{R}^{Q \times N}\).