Flexible Space Structures

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Description

The flexible space structure benchmark is a procedural modal model which represents structural dynamics with a selectable number actuators and sensors.

Model

In modal form the flexible space structure model for $K$ modes, $M$ actuators and $Q$ sensors is of second order and given by:

Failed to parse (syntax error): {\displaystyle \ddot{\nu}(t) &= (2 \xi \circ \omega) \circ \dot{\nu}(t) + (\omega \circ \omega) \circ \nu = Bu(t) \\ y(t) &= C_r\dot{\nu}(t) + C_d\nu(t) }

with the parameters $\xi \in {\mathbb{ℝ}}_{>0}^K$ (damping ratio), $\omega \in {\mathbb{ℝ}}_{>0}^K$ (natural frequency) and using the Hadamard product $\circ$. The first order representation follows for $x(t)=(\dot{nu}(t),{\omega }_1{\nu }_1,\dots ,{\omega }_K{\nu }_K)$ by:

Failed to parse (syntax error): {\displaystyle \dot{x}(t) &= Ax(t) + Bu(t) \\ y(t) &= Cx(t) }

with the matrices:

Failed to parse (syntax error): {\displaystyle A := \begin{pmatrix} A_1 & & \\ & \ddots & \\ & & A_K \end{pmatrix}, \\ B := \begin{pmatrix} B_1 \\ \vdots \\ B_K \end{pmatrix}, \\ C := \begin{pmatrix} C_1 & \dots & C_K \end{pmatrix}, }

and their components:

Failed to parse (syntax error): {\displaystyle A_k := \begin{pmatrix} -2\xi_k\omega_k & -\omega_k \\ \omega_k & 0 \end{pmatrix}, \\ B_k := \begin{pmatrix} b_k \\ 0 \end{pmatrix}, \\ C_k := \begin{pmatrix} c_{rk} & \frac{c_{dk}}{\omega_k} \end{pmatrix}, }

where $b_k\in {\mathbb{ℝ}}^{1\times M}$ and $c_{rk},c_{dk}\in {\mathbb{ℝ}}^{Q\times 1}$.

Benchmark Specifics

For this benchmark the system matrix is block diagonal and thus chosen to be sparse. The parameters $\xi$ and math>\omega</math> are sampled from a uniform random distributions Failed to parse (syntax error): {\displaystyle \mathcal{U}_[0,\frac{1}{1000}]}^K} and Failed to parse (syntax error): {\displaystyle \mathcal{U}_[0,100]}^K} respectively. The components of the input matrix $b_k$ are sampled form a uniform random distribution ${{𝒰}}_{[0,1]}$, while the output matrix $C$ is sampled from a uniform random distribution ${{𝒰}}_{[}^{}0,10]$ completely w.l.o.g, since if the components of $C_d$ are random their scaling can be ignored.

Data

The following Matlab code assembles the above described $A$, $B$ and $C$ matrix for a given number of modes $K$.

function [A,B,C] = fss(K,M,Q)

    rand('seed',1009);
    xi = rand(1,K)*0.001;	% Sample damping ratio
    omega = rand(1,K)*100;	% Sample natural frequencies

    A_k = cellfun(@(p) sparse([-2.0*p(1)*p(2),-p(2);p(2),0]), ...
                  num2cell([xi;omega],1),'UniformOutput',0);

    A = blkdiag(A_k{:});

    B = kron(rand(K,M),[1;0]);

    C = 10.0*rand(Q,2*K);
end

Contact

Christian Himpe