For g, n coprime integers, let `g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of `g(n) as n ≤ x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptot...

We consider the problem of reducing a model in a way that preserves a partition of the system states. This is motivated, for instance, in situations where state variables are associated with the topology of a networked system. In earlier work we proposed an LMI method based on block-structured generalized controllability and observability gramians; to make such strategy feasible, coprime factor...

We propose general computational procedures based on descriptor state-space realizations to compute coprime factorizations of rational matrices with minimum degree denominators. Enhanced recursive pole dislocation techniques are developed, which allow to successively place all poles of the factors into a given “good” domain of the complex plane. The resulting McMillan degree of the denominator ...

This paper is concerned with the order reduction of controllers produced by μ-synthesis controller design paradigm. A new proof has been proposed for the additive perturbation reduction technique with sufficient conditions to guarantee the closed loop structured singular value to remain less than unity. Reduction of unstable controllers is achieved through a new coprime factor reduction techniq...

Increasing performance demands in control applications necessitate accurate modeling of complex systems for control. The aim of this chapter is to develop a new system identification algorithm that delivers models that are suitable for subsequent robust control and can be reliably applied to complex systems. To achieve this, an identification algorithm is developed that delivers system model in...

the asymptotic behaviour of the sequence with general term $p_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $p_n-6/pi^2$ is found.

This paper discusses ltering and parametrization issues involved in the usage of fractional representations in multivariable approximate and feedback relevant identi cation of a possibly unstable plant operating under closed loop conditions The knowledge of the controller is used to access any stable right coprime factorization of the plant by measuring and ltering the signals present in the cl...