Mass-spring-damper system


For a stochastically-forced mass-spring-damper system with n masses on a line with state-space representation

 	dot{x} 	;=; 	A,x 	;+; 	B_u, u

the state vector is determined by x = [,p^T, ~, v^T ,]^T and contains the position and velocity of all masses. The state and input matrices are

 	A 	;=; 	left[ 	begin{array}{cc} 	O & I  -T & -I 	end{array} 	right],~~~~ 	B_u 	;=; 	left[ 	begin{array}{c} 	O  I 	end{array} 	right]

where O and I are zero and identity matrices of suitable sizes, and T is a symmetric tridiagonal Toeplitz matrix with 2 on the main diagonal and -1 on the first upper and lower sub-diagonals. Stochastic disturbances are generated by a low-pass filter

 	dot{u} 	;=; 	-u 	;+; 	d

where d represents a zero-mean unit variance white process. The steady-state covariance of the of the stochastically-forced mass-spring-damper system can be extracted from the steady-state covariance of the cascade of filter and system dynamics, and is partitioned as

 	Sigma_{xx} 	;=; 	left[ 		begin{array}{cc} 			Sigma_{pp} & Sigma_{pv} 			 			Sigma_{vp} & Sigma_{vv} 		end{array} 	right]


We assume knowledge of one-point correlations of the position and velocity of
masses, i.e., the diagonal elements of matrices Sigma_{pp}, Sigma_{vv}, and Sigma_{pv}, which are
shown as orange lines in the figure. It is desired to identify the suitable forcing
model which accounts for the available and completes the unavailable
second-order statistics in X.

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., ||X^k-X^star||_F/ ||X^star||_F < epsilon_1 and ||Z^k-Z^star||_F/ ||Z^star||_F < epsilon_2. The choice of epsilon_1, epsilon_2 = 0.01, guarantees that the primal objective is within 0.1% of J_p(X^star,Z^star). For n=50 and n=100, CVX ran out of memory. Clearly, for large problems, AMA with BB step-size initialization significantly outperforms both regular AMA and ADMM.

bf{n} bf{CVX} bf{ADMM} bf{AMA} bf{AMA_{BB}}
bf{10} 28.4 2 1.3 0.5
bf{20} 419.7 54.7 30.7 2.2
bf{50} - 3442.9 3796.7 52.7
bf{100} - 40754 34420 5429.8

Convergence curves showing the performance of the customized AMA
with (-) and without (triangle) BB step-size initialization and ADMM (circ)
for the mass-spring-damper system with 50 masses and gamma=2.2. Here,
J_d^star is the optimal dual objective.

Quality of completion and verification in linear stochastic simulations

For gamma=2.2, the spectrum of Z contains 50 positive and 12 negative eigenvalues. Based on Proposition 2 of the paper, Z can be decomposed into B H^*+ H B^*, where B has 50 independent columns. In other words, the identified X can be explained by driving the state-space model with 50 stochastic inputs u. Based on this, we construct the optimal filter that generates the suitable stochastic input and conduct linear stochastic simulations of the filter dynamics with zero-mean unit variance input to validate our modeling approach.


For gamma = 2.2, the matrix Z which results from solving problem (CC)
has a clear-cut in its singular values with 62 of them being nonzero.


Diagonals of position (left) and velocity
(right) covariances for the mass-spring-
damper system with 50 masses; Solid
black lines show diagonals of Sigma_{xx} and
red circles mark solutions of
optimization problem (CC).


The true covariance Sigma_{pp} of the
mass-spring-damper system (left) and
the recovered covariance matrix X_{pp}
resulting from the ensemble-averaged
simulations (right). We note that (i) only
diagonal elements of this matrix (marked
by the black line) are used as data in
optimization problem (CC), and that
(ii) the recovery of the off-diagonal
elements is remarkably consistent.


Time evolution of the variance of the mass-spring-damper system's
state vector for different realizations of white-in-time forcing to the
filter dynamics. Since proper comparison requires ensemble-averaging,
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
steady-state variance.