Sparsity Preserving Discretization With Error Bounds
For control engineers dealing with distributed systems, this method enables efficient controller design by maintaining sparsity after discretization, addressing a key bottleneck in discrete-time distributed control.
The paper proposes a discretization method that preserves sparsity of continuous-time models, outperforming simple truncation in approximating the ground truth, and provides error bounds. It also shows that distributed controllers designed on the sparse model can stabilize the dense ground truth model.
Typically when designing distributed controllers it is assumed that the state-space model of the plant consists of sparse matrices. However, in the discrete-time setting, if one begins with a continuous-time model, the discretization process annihilates any sparsity in the model. In this work we propose a discretization procedure that maintains the sparsity of the continuous-time model. We show that this discretization out-performs a simple truncation method in terms of its ability to approximate the "ground truth" model. Leveraging results from numerical analysis we are also able to upper-bound the error between the dense discretization and our method. Furthermore, we show that in a robust control setting we can design a distributed controller on the approximate (sparse) model that stabilizes the dense ground truth model.