New method for SISO strong stabilization with advantages over existing methods
Abdul Hannan Faruqi, Anindya Chatterjee
TL;DR
This paper tackles strong stabilization of SISO LTI plants in the Laplace domain by proposing Real to Integers (RTI), a simple method valid for plants with relative degree $<3$. RTI builds a stable, invertible $U(s)$ in the form $U(s)=\prod_{k=1}^r \left(\frac{s+a_{2k-1}}{s+a_{2k}}\right)^{m_k}$, turning the controller design into a tractable interpolation-like problem for the real exponents $m_k$. Real exponents serve as intermediate degrees of freedom, and a trigonometric trick together with an implicit-function theorem ensures the existence of integer $m_k$ values after suitable parameter adjustment; a two-stage numerical procedure then yields small-size integer solutions. Ten numerical examples across relative degrees $0$, $1$, and $2$ demonstrate stable stabilization even with multiple RHP zeros, illustrating RTI's practical viability and simplicity compared to Nevanlinna-Pick based methods. The approach promises a broadly accessible alternative for robust strong stabilization in applications where a stable controller is preferred and high mathematical sophistication is undesirable.
Abstract
We address stabilization of linear time-invariant (LTI), single-input single-output (SISO) systems in the Laplace domain, with a stable controller in a single feedback loop. Such stabilization is called strong. Plants that satisfy a parity interlacing property are known to be strongly stabilizable. Finding such controllers is a well known difficult problem. Existing general methods are based on either manual search or a clever use of Nevanlinna-Pick interpolation with polynomials of possibly high integer order. Here we present a new, simple, and general method for strongly stabilizing systems of relative degree less than 3. We call our method Real to Integers (RTI). Our theoretical contributions constitute proposing the functional form used, which involves a product of several terms of the form $\displaystyle \left ( \frac{s+a}{s+b} \right )^m$, showing that real $m$'s will arise whenever the plant is strongly stabilizable, and proving that integer $m$'s can be obtained by continuously varying free parameters (i.e., the $a$'s and $b$'s). Our practical contributions include demonstrating a simple way, based on a trigonometric trick, to adjust the fractional powers until they take reasonable integer values. We include brief but necessary associated discussion to make the paper accessible to a broad audience. We also present ten numerical examples of successful control design with varying levels of difficulty, including plants whose transfer functions have relative degrees of 0, 1 or 2; and with right half plane zeros of multiplicity possibly exceeding one.