The poles and zeros of a linear time-varying system
The author considers linear time-varying systems of the form \[ (z^ n+\sum^{n-1}_{i=0}a_ i(k)z^ i)y(k)=(\sum^{n}_{i=0}b_ i(k)z^ i)u(k) \] where \(z: f(k)\mapsto f(k+1)\) denotes the shift operator and \(a_ i(k)\), \(b_ i(k)\in {\mathbb{C}}\), resp. for the continuous case \[ (D^ n+\sum^{n-1}_{i=0}a_ i(t)D^ i)y(t)=(\sum^{n}_{i=0}b_ i(t)D^ i)u(t) \] where D: f(t)\(\mapsto \dot f(t)\) denotes the differential operator and \(a_ i(\cdot)\), \(b_ i(\cdot)\) are differentialble a suitable number of times. The polynomials above are defined over certain skew polynomial rings which are naturally induced by the shift resp. differential operator. Via a facorization of the polynomials Kamen presents a new approach to define poles and zeros of time-varying systems. He shows that the poles can be computed by solving certain nonlinear equations. The theory is applied to study (uniform) asymptotic stability.
scientific article; zbMATH DE number 4031495