Design of ℓ1 New Suboptimal Fractional Delays Controller for Discrete Non-Minimum Phase System under Unknown-but-Bounded Disturbance

• Published in: Mathematics (Basel) 2022-01, Vol.10 (1), p.69
• Main Authors: Ivanov, Dmitrii, Granichin, Oleg, Pankov, Vikentii, Volkovich, Zeev
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Combinatorial analysis; Control algorithms; Control theory; Controllers; Data compression; Difference equations; Food science; fractional delays; Image processing; Mathematical analysis; Noise; Noise (mathematics); non-minimum phase system; Object recognition; Optimization; Partial differential equations; Pattern recognition; Regularization; Regularization methods; Response time; Rounding; Sampling; Signal processing; stabilizing controller; unknown-but-bounded noise
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10010069

ℓ1-regularization methodologies have appeared recently in many pattern recognition and image processing tasks frequently connected to ℓ1-optimization in the control theory. We consider the problem of optimal stabilizing controller synthesis for a discrete non-minimum phase dynamic plant described by a linear difference equation with an additive unknown-but-bounded noise. Under considering the “worst” case of noise, the solving of these optimization problem has a combinatorial complexity. The choosing of an appropriate sufficiently high sampling rate allows to achieve an arbitrarily small level of suboptimality using a noncombinatorial algorithm. In this paper, we suggest to use fractional delays to achieve a small level of suboptimality without increasing the sampling rate so much. We propose an approximation of the fractional lag with a combination of rounding and a first-order fractional lag filter. The suggested approximation of the fractional delay is illustrated via a simulation example with a non-minimum phase second-order plant. The proposed methodology appears to be suitable to be used in various pattern recognition approaches.