New Method for SISO Strong Stabilization With Advantages Over Nevanlinna–Pick Interpolation

Abstract

Linear time-invariant (LTI) single-input–single-output (SISO) systems which satisfy a parity interlacing property (PIP) can be stabilized with a stable controller in a single feedback loop. We consider such stabilization of plants with rational transfer functions of relative degree 0, 1, or 2. Finding such controllers requires an interpolant $U(s)$ with specific properties. Existing methods for finding $U(s)$ use an iterative manual calculation or, when the plant's right half plane zeros are simple, a matrix calculation based on the Nevanlinna–Pick interpolation. We present a new interpolant of the form $ \prod _{i} \left(\frac{s+a_{i}}{s+b_{i}}\right) ^{m_{i}}$, where $a_{i}, b_{i} > 0$. While our final interpolant has integer $m$’s, we allow noninteger or real $m$’s in intermediate calculations. This allows our search to be continuous instead of discrete. Repeated right half plane zeros of the plant are accommodated easily. Real $m$’s are obtained whenever the plant satisfies the PIP, and integer $m$’s are obtained for suitably chosen $a$’s and $b$’s. With numerical optimization of parameters, the $m$’s take moderate integer values. We close with some numerical examples.