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

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$U(s)$</tex-math></inline-formula> with specific properties. Existing methods for finding <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$U(s)$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \prod _{i} \left(\frac{s+a_{i}}{s+b_{i}}\right) ^{m_{i}}$</tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$a_{i}, b_{i} &gt; 0$</tex-math></inline-formula>. While our final interpolant has integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula>’s, we allow noninteger or real <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula>’s in intermediate calculations. This allows our search to be continuous instead of discrete. Repeated right half plane zeros of the plant are accommodated easily. Real <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula>’s are obtained whenever the plant satisfies the PIP, and integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula>’s are obtained for suitably chosen <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$a$</tex-math></inline-formula>’s and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$b$</tex-math></inline-formula>’s. With numerical optimization of parameters, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula>’s take moderate integer values. We close with some numerical examples.