Some results on dynamics of special double-loop \(\Sigma\Delta\)-modulators (Q1362197)

In this paper, the authors study the two-dimensional difference equation: \(x(t)= Ax(t-1)+ bu(t-1) +w\text{sgn} (x_2(t-1))\), where \(A\) is a \(2\times 2\) matrix, \(x(t)= (x_1(t), x_2 (t))\), \(t\in N\), is the two-dimensional state vector, \(u(t)\), \(t\in N\), the input sequence, and \(b= (b_1,b_2)\), \(w= (w_1,w_2)\) are \(2\times 1\) real vectors. The only nonlinearity is supplied by the signum function sgn(.). In this paper, let \(A\) be a rotation matrix, specifically, \(A= \left( \begin{smallmatrix} \cos \varphi & -\sin \varphi \\ \sin\varphi & \cos \varphi\end{smallmatrix} \right)\), and \(\varphi= 2\pi h\), \(h\) rational. This system is a generalised version of the standard model of the double-loop \(\Sigma\Delta\)-modulator. The authors proved a general result that these systems will have either unstable or eventually periodic orbits. Some criteria for the identification of initial conditions which lead to instability are provided. These criteria are based on elementary geometrical methods which can be applied in each particular case. Finally the authors present some simulations with time-varying input of the form \(u=\sin (\omega t)\). In all the cases the authors observed again either instability or periodicity.