  The subspace method is one of the mainstream system identification method of
linear systems, and its basic idea is to estimate the system parameter matrices
by projecting them into a subspace related to input and output. However, most
of the existing subspace methods cannot have the statistic performance
guaranteed since the lack of closed-form expression of the estimation.
Meanwhile, traditional subspace methods cannot deal with the uncertainty of the
noise, and thus stable identification results cannot be obtained. In this
paper, we propose a novel improved subspace method from the perspective of
input design, which guarantees the consistent and stable identification results
with the minimum variance. Specifically, we first obtain a closed-form
estimation of the system matrix, then analyze the statistic performance by
deriving the maximum identification deviation. This identification deviation
maximization problem is non-convex, and is solved by splitting it into two
sub-problems with the optimal solution guaranteed. Next, an input design method
is proposed to deal with the uncertainty and obtain stable identification
results by minimizing the variance. This problem is formulated as a constrained
min-max optimization problem. The optimal solution is obtained from
transforming the cost function into a convex function while ensuring the safety
constraints through the method of predictive control. We prove the consistency
and the convergence of the proposed method. Simulation demonstrates the
effectiveness of our method.
