Bilinear PMOR method

The model reduction method we present here is applicable to linear parameter-varying (LPV) systems of the form

$\dot{x} (t) = A x (t) + \sum_{i = 1}^{d} p_{i} (t) A_{i} x (t) + B_{0} u_{0} (t), y (t) = C x (t),$

where $A, A_{i} \in ℝ^{n \times n}, B_{0} \in ℝ^{n \times m}$ and $C \in ℝ^{p \times n} .$

The main idea is that the structure of the above type of systems is quite similar to so-called bilinear control systems. Although belonging to the class of nonlinear control systems, the latter exhibit many features of linear time-invariant systems. In more detail, a bilinear control system is given as follows

$\dot{x} (t) = A x (t) + \sum_{i = 1}^{m} N_{i} x (t) u_{i} (t) + B u (t),$

where $A, N_{i} \in ℝ^{n \times n}, B \in ℝ^{n \times m}$ and $C \in ℝ^{p \times n} .$

As one can see, there seems to be a close connection between LPV and bilinear systems which may be advantageous. To be more precise, following [2], we can interpret LPV systems as special bilinear system by simply setting

$\tilde{A} = A, {\tilde{N}}_{i} = 0, i = 1, \dots, m, {\tilde{N}}_{i} = A_{i}, i = m + 1, \dots, m + d, \tilde{B} = [\begin{matrix} B_{0} & 0 \end{matrix}], \tilde{C} = C, \tilde{u} = [\begin{matrix} u_{0} \\ p_{1} (t) \\ ⋮ \\ p_{d} (t) \end{matrix}] .$

It is important to note that in this way the parameter dependency has been hidden in the structure of a bilinear system and if we use a bilinear model reduction method, we automatically end up with a structure-preserving model reduction method for LPV systems as well. Due to the above mentioned similarity of bilinear and linear control systems, one can generalize some useful and well-known linear concepts. For example, it is possible to extend the method of balanced truncation for bilinear systems. Here, we have to solve generalized Lyapunov equations of the form

$A X + X A^{T} + \sum_{i = 1}^{m} N_{i} X N_{i}^{T} + B B^{T} = 0, A^{T} Y + Y A + \sum_{i = 1}^{m} N_{i}^{T} Y N_{i} + C^{T} C = 0 .$

For a detailed analysis of this approach, we refer to e.g. [3]. Another possibility is to aim at a model reduction method which is optimal with respect to a certain system norm. In [1], the authors investigate a possible extension of the linear $H_{2}$ -norm and propose an algorithm which extends the well-known iterative rational Krylov algorithm (IRKA) to the bilinear setting specified above.

Both methods have been tested on several LPV control systems and seem to be a reasonable alternative to existing parametric model reduction methods.

References

[1] P. Benner and T. Breiten, Interpolation-based H2-model reduction of bilinear control systems, 2011, Preprint MPIMD/11-02.

[2] P. Benner and T. Breiten, On H2-model reduction of linear parameter-varying systems, In Proceedings in Applied Mathematics and Mechanics. Wiley InterScience, 2011.

[3] P. Benner and T. Damm, Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems, SIAM J. Cont. Optim., 49 (2011), pp. 686-711.