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. This model is used for truss structures in space environments i.e. the COFS-1 (Control of Flexible Structures) mast flight experiment.

Model

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

$\ddot{\nu }(t)=(2\xi \circ \omega )\circ \dot{\nu }(t)+(\omega \circ \omega )\circ \nu =Bu(t)$

$y(t)=C_r\dot{\nu }(t)+C_d\nu (t)$

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:

$\dot{x}(t)=Ax(t)+Bu(t)$

$y(t)=Cx(t)$

with the matrices:

$A:=\left(\begin{array}{ccc}{A}_1& & \\ & \ddots & \\ & & A_K\end{array}\right),B:=\left(\begin{array}{c}{B}_1\\ ⋮\\ B_K\end{array}\right),C:=\left(\begin{array}{ccc}{C}_1& \dots & C_K\end{array}\right),$

and their components:

$A_k:=\left(\begin{array}{cc}-2{\xi }_k{\omega }_k& -{\omega }_k\\ {\omega }_k& 0\end{array}\right),B_k:=\left(\begin{array}{c}{b}_k\\ 0\end{array}\right),C_k:=\left(\begin{array}{cc}{c}_{rk}& \frac{c_{dk}}{{\omega }_k}\end{array}\right),$

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 $\omega$ are sampled from a uniform random distributions ${{𝒰}}_{[0,\frac{1}{1000}]}^K$ and ${{𝒰}}_{[0,100]}^K$ 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 $K$, actuators (inputs) $M$ and sensors (outputs) $Q$.

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

    rand('seed',1009);
    xi = rand(1,K)*0.001;	% Sample damping ratio
    omega = rand(1,K)*100.0;	% 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