A new estimation of the lower error bound in balanced truncation method

For a single-input/single-output (SISO) linear time-invariant dynamical system, the standard H ∞ -norm lower error bound of balanced truncation method is ‖ G ( s ) − G r ( s ) ‖ H ∞ ≥ σ r + 1 , where σ i , i = 1 , … , n , are the Hankel singular values of system in decreasing order. In this paper we provide a new estimation of the lower error, namely ‖ G ( s ) − G r ( s ) ‖ H ∞ ≥ max { σ d , 2 | ∑ i ∉ J s i σ i | } , where s i is the sign associated with the Hankel singular value σ i in Ober’s canonical form. The subset J and the index d in the above inequality will be introduced in the paper. We show by means of an example that the new bound may be relevant in deciding which states need to be kept in the balanced truncation method, and that using the standard result does not always yield the best approximation.