Denominator Assignment, Invariants, and Canonical Forms Under Dynamic Feedback Compensation in Linear Multivariable Nonsquare Systems
Antonis I. G. Vardulakis, Aristotelis Yannakoudakis, Cui Wei, Tianyou Chai · IEEE Transactions on Automatic Control · 2020
In this article, we generalize previously reported results for linear, time-invariant, stabilizable multivariable systems described by astrictlyproper transfer function matrix$P(s)$with number of outputs greater than or equal to the number of inputs. By making use of a special kind of a left generalized inverse$P(s)_{\alpha }^{\oplus }$of$P(s)$, we define and examine the equivalent relation$\mathcal {R}$relating$P(s)$with the members of the equivalence class$[P(s)]_{R}$of the closed loop-transfer function matrices$P_{C}(s)$obtainable from$P(s)$by the use of a proper compensator$C(s)$in the feedback path. For$\mathcal {R}$, we establish a set of complete invariants and a canonical form. These results give rise to a simple algorithmic procedure for the computation ofproperinternally stabilizing and denominator assigning compensators$C(s)$for the class of plants with$p=m$and having no zeros in the closed right half complex plane:$\mathbb {C}^{+}$and in the case when$p>m$plants characterized by right polynomial matrix fraction descriptions with a polynomial matrix numerator having at least one subset of$m$rows that give rise to a nonsingular polynomial matrix with no zeros in$\mathbb {C}^{+}$.