Deconvolution filter design via l1 optimization technique

A new l₁ optimal deconvolution filter design approach for systems with uncertain (or unknown)-but-bounded inputs and external noises is proposed. The purpose of this deconvolution filter is to minimize the peak gain from the input signal and noise to the error by the viewpoint of the time domain. The solution consists of two steps. In the first step, the l₁ norm minimization problem is transferred to an equivalent A-norm minimization problem, and the minimum value of the peak gain is calculated. In the second step, based on the minimum peak gain, the l₁ optimal deconvolution filter is constructed by solving a set of constrained linear equations. Some techniques of inner-outer factorization, polynominal spectral factorization, linear programming, and some optimization theorems found in a book by Luenberger are applied to treat the l₁ optimal deconvolution filter design problem. Although the analysis of the algorithm seems complicated, the calculation of the proposed design algorithm for actual systems is simple. Finally, one numerical example is given to illustrate the proposed design approach. Several simulation results have confirmed that the proposed l₁ optimal deconvolution filter has more robustness than the l₂ optimal deconvolution filter under uncertain driving signals and noises