Structure-preserving model reduction for spatially interconnected systems with experimental validation on an actuated beam

ABSTRACT

A technique for model reduction of exponentially stable spatially interconnected systems is presented, where the order of the reduced model is determined by the number of truncated small generalised singular values of the structured solutions to a pair of Lyapunov inequalities. For parameter-invariant spatially interconnected systems, the technique is based on solving a pair of Lyapunov inequalities in continuous-time and -space domain with a rank constraint. Using log-det and cone complementarity methods, an improved error bound can be obtained. The approach is extended to spatially parameter-varying systems, and a balanced truncation approach using parameter-dependent Gramians is proposed to reduce the conservatism caused by the use of constant Gramians. This is done by considering two important operators, which can be used to represent multidimensional systems (temporal- and spatial-linear parameter varying interconnected systems). The results are illustrated with their application to an experimentally identified spatially interconnected model of an actuated beam; the experimentally obtained response to an excitation signal is compared with the response predicted by a reduced model.

Keywords:

• Linear fractional representation (LFR)
• linear parameter varying systems (LPV)
• model order reduction
• multidimensional systems
• spatially interconnected systems

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Exact reduction here refers to the reduction of a state space model to a minimal realization, i.e. the removal of uncontrollable or unobservable modes.

2. The generalization of Hankel singular values (Beck, 1997).

3. For the LPV case, we use the bold letters to denote operators on l^•₂.

4. According to the definition of hyperdiagonal operator, note that A_TT is hyperdiagonal operator contains the blocks A_TT(δ_T, ρ_S) (in (23)) varying with respect to t, s along its diagonal; the same is true for the others as well.

5. Note that ΛA Y A*Λ* − Y < 0 is equivalent to AYA* − Λ*YΛ < 0, since Λ*Λ = I, but ΛΛ* ≠ I.

6. Remark: Note that according to the definition of hyperdiagonal operators, the conditions of Theorem 6.1 and (27(27) $$\begin{array}{ccc}& & \mathbf{A}\mathbf{X}{\mathbf{A}}^{*}-\left({\Lambda }^{*}\mathbf{X}\Lambda \right)+{\mathbf{B}\mathbf{B}}^{*}<0,\hfill \\ & & {\mathbf{A}}^{*}\left({\Lambda }^{*}\mathbf{Y}\Lambda \right)\mathbf{A}-\mathbf{Y}+{\mathbf{C}}^{*}\mathbf{C}<0\hfill \end{array}$$(27) )–(28(28) $$\begin{array}{c}\hfill I{n}_{-}\left({\Lambda }^{*}\mathbf{X}\Lambda \right)=I{n}_{-}\left(\mathbf{X}\right),I{n}_{-}\left({\Lambda }^{*}\mathbf{Y}\Lambda \right)=I{n}_{-}\left(\mathbf{Y}\right)\end{array}$$(28) ) are defined for all ${\delta }_t,{\rho }_s\in {\mathcal{F}}_{\delta }\times {\mathcal{F}}_{\rho }$.

7. As mentioned before, our results are based on gridding the parameter range, here ρ_s has values defined in (32(32) $$\begin{array}{ccc}\hfill s& =& [0.125,0.425,1.025,1.625,1.925,2.225,\hfill \\ & & 3.125,3.725,4.025,4.625]\mathrm{m}.\hfill \end{array}$$(32) ), which is a grid of ten values.

8. Note, we present the differential and the integral of v₊(t, s), v₋(t, s) respectively, as w₊(t, s), w₋(t, s); see (11(11) $$\Delta =\left[\begin{array}{ccc}\hfill \int dt{I}_{n_1}& \hfill 0& \hfill 0\\ \hfill 0& \hfill \int ds{I}_{n_2}& \hfill 0\\ \hfill 0& \hfill 0& \hfill \frac{d}{ds}{I}_{n_3}\end{array}\right]$$(11) ).