Matrix analysis
Linear stochastic systems
Balancing for nonlinear systems
Systems & Control Letters
Mathematical control theory: deterministic finite dimensional systems (2nd ed.)
Mathematical control theory: deterministic finite dimensional systems (2nd ed.)
A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations
SIAM Journal on Scientific Computing
Low Rank Solution of Lyapunov Equations
SIAM Journal on Matrix Analysis and Applications
Piecewise polynomial nonlinear model reduction
Proceedings of the 40th annual Design Automation Conference
Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control)
Model Reduction of Uncertain Systems with Multiplicative Noise Based on Balancing
SIAM Journal on Control and Optimization
On H2 model reduction of bilinear systems
Automatica (Journal of IFAC)
Projection-based approaches for model reduction of weakly nonlinear, time-varying systems
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
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We discuss the relation of a certain type of generalized Lyapunov equations to Gramians of stochastic and bilinear systems together with the corresponding energy functionals. While Gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems, the relation of Gramians and energy functionals for bilinear systems is less obvious. We discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic Gramians satisfying generalized Lyapunov equations. In any of the considered cases, the definition of algebraic Gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems. We illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems.