Model order reduction for linear dynamical systems with quadratic outputs

Roland Pulch (Universität Greifswald), Akil Narayan

We consider initial value problems for linear time-invariant systems consisting of ordinary differential equations \[\begin{array}{rcl} E \dot{x}(t) & = & A x(t) + B u(t), \qquad x(t_0) = x_0 \\[0.5ex] y(t) & = & x(t)^{\top} M x(t) \\ \end{array}\] with state variables \(x\) and inputs \(u\). The quadratic output \(y\) represents a quantity of interest defined by a symmetric matrix \(M\) of rank \(k\). We investigate model order reduction (MOR) for systems of high dimension. The system can be transformed into a linear dynamical system with \(k\) linear outputs, see . However, many MOR methods for linear dynamical systems become inefficient or even infeasible in the case of large numbers \(k\). Alternatively, we transform the system into a quadratic-bilinear (QB) form with a single linear output. The properties of this QB system are analyzed. We apply the MOR technique of balanced truncation from  to the QB system, where a stabilization is required. The solution of quadratic Lyapunov equations is traced back to the solution of linear Lyapunov equations. We present numerical results for a relevant example including a high rank \(k\), where the two MOR approaches are compared.

References

[1] R. Van Beeumen, K. Van Nimmen, G. Lombaert, K. Meerbergen: Model reduction for dynamical systems with quadratic output. Int. J. Numer. Meth. Engng. 91:3 (2012) 229–248.

[2] P. Benner, P. Goyal: Balanced truncation model order reduction for quadratic-bilinear control systems. arXiv :1705.00160v1, April 29, 2017.