Massspringdamper system
For a stochasticallyforced massspringdamper system with masses on a line with statespace representation
the state vector is determined by and contains the position and velocity of all masses. The state and input matrices are
where and are zero and identity matrices of suitable sizes, and is a symmetric tridiagonal Toeplitz matrix with on the main diagonal and on the first upper and lower subdiagonals. Stochastic disturbances are generated by a lowpass filter
where represents a zeromean unit variance white process. The steadystate covariance of the of the stochasticallyforced massspringdamper system can be extracted from
the steadystate covariance of the cascade of filter and system dynamics, and is partitioned as
Performance comparison
The following table compares solve times of CVX and the customized algorithms discussed in the paper. Each method stops when an iterate achieves a certain distance from optimality, i.e., and . The choice of , guarantees that the primal objective is within of . For and , CVX ran out of memory. Clearly, for large problems, AMA with BB stepsize initialization significantly outperforms both regular AMA and ADMM.
Quality of completion and verification in linear stochastic simulations
For , the spectrum of contains positive and negative eigenvalues. Based on Proposition of the paper, can be decomposed into , where has independent columns. In other words, the identified can be explained by driving the statespace model with stochastic inputs . Based on this, we construct the optimal filter that generates the suitable stochastic input and conduct linear stochastic simulations of the filter dynamics with zeromean unit variance input to validate our modeling approach.

Time evolution of the variance of the massspringdamper system's state vector for different realizations of whiteintime forcing to the filter dynamics. Since proper comparison requires ensembleaveraging, we have conducted twenty stochastic simulations with different realizations of the stochastic input. The variance averaged over all simulations is marked by the thick black line. Even though the responses of individual simulations differ from each other, the average of twenty sample sets asymptotically approaches the correct steadystate variance.

