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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
\[ \begin{align} E\dot{x}(t) &= Ax(t) + Bu(t),\\ y(t) &= Cx(t), \end{align} \]
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
\[ \begin{align} E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{i=1}^M x(t) u_i(t) + B u(t), \\ y(t) &= Cx(t), \end{align} \]
with\[E \in \mathbb{R}^{N \times N}\], \(A \in \mathbb{R}^{N \times N}\), \(B \in \mathbb{R}^{N \times M}\), \(H \in \mathbb{R}^{N \times N^2}\), \(N_i \in \mathbb{R}^{N \times N}\), \(C \in \mathbb{R}^{Q \times N}\).
Nonlinear Time-Invariant System
\[ \begin{align} E\dot{x}(t) &= Ax(t) + f(x(t),u(t)) + Bu(t),\\ y(t) &= Cx(t), \end{align} \]
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
\[ \begin{align} (E_0 + \sum_{i=1}^{P_E} p^E_i E_i)\dot{x}(t) &= (A_0 \sum_{j=1}^{P_A} p^A_j A_j) x(t) + Bu(t),\\ y(t) &= Cx(t), \end{align} \]
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
\[ \begin{align} M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\ y(t) &= C x(t), \end{align} \]
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
\[ \begin{align} M \ddot{x}(t) + D \dot{x}(t) + K x(t) &= B u(t) + f(x(t),u(t)), \\ y(t) &= C x(t), \end{align} \]
with\[M \in \mathbb{R}^{N \times N}\], \(D \in \mathbb{R}^{N \times N}\), \(K \in \mathbb{R}^{N \times N}\), \(B \in \mathbb{R}^{N \times M}\), \(C \in \mathbb{R}^{Q \times N}\), \(f : \mathbb{R}^N \times \mathbb{R}^M \to \mathbb{R}^N\).
Affine Parametric Second-Order System
\[ \begin{align} (M_0 + \sum_{i=1}^{P_M} p^M_i M_i)\ddot{x}(t) + (D_0 + \sum_{i=1}^{P_D} p^D_i D_i)\dot{x}(t) + (K_0 + \sum_{i=1}^{P_K} p^K_i K_i)x(t) &= B u(t), \\ y(t) &= C x(t), \end{align} \]
with\[M_0 \in \mathbb{R}^{N \times N}\], \(M_i \in \mathbb{R}^{N \times N}\), \(D_0 \in \mathbb{R}^{N \times N}\), \(D_j \in \mathbb{R}^{N \times N}\), \(K_0 \in \mathbb{R}^{N \times N}\), \(K_k \in \mathbb{R}^{N \times N}\), \(B \in \mathbb{R}^{N \times M}\), \(C \in \mathbb{R}^{Q \times N}\).