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:

\[A := \begin{pmatrix} A_1 & & \\ & \ddots & \\ & & A_K \end{pmatrix}, \; B := \begin{pmatrix} B_1 \\ \vdots \\ B_K \end{pmatrix}, \; C := \begin{pmatrix} C_1 & \dots & C_K \end{pmatrix}, \]

and their components:

\[A_k := \begin{pmatrix} -2\xi_k\omega_k & -\omega_k \\ \omega_k & 0 \end{pmatrix}, \; B_k := \begin{pmatrix} b_k \\ 0 \end{pmatrix}, \; C_k := \begin{pmatrix} c_{rk} & \frac{c_{dk}}{\omega_k} \end{pmatrix},\]

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 \(\omega\) are sampled from a uniform random distributions \(\mathcal{U}_{[0,\frac{1}{1000}]}^K\) and \(\mathcal{U}_{[0,100]}^K\) respectively. The components of the input matrix \(b_k\) are sampled form a uniform random distribution \(\mathcal{U}_{[0,1]}\), while the output matrix \(C\) is sampled from a uniform random distribution \(\mathcal{U}_{[0,10]}\) 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