Title :
Exponential condition number of solutions of the discrete Lyapunov equation
Author :
Mullhaupt, Andrew P. ; Riedel, Kurt S.
Author_Institution :
S.A.C. Capital Manage., New York, NY, USA
fDate :
5/1/2004 12:00:00 AM
Abstract :
The condition number of the n×n matrix P is examined, where P solves P-APA*=BB*, and B is a n×d matrix. Lower bounds on the condition number κ of P are given when A is normal, a single Jordan block, or in Frobenius form. The bounds show that the ill-conditioning of P grows as exp(n/d)≫1. These bounds are related to the condition number of the transformation that takes A to input normal (IN) form. A simulation shows that P is typically ill-conditioned in the case of n≫1 and d=1. When Aij has an independent Gaussian distribution (subject to restrictions), we observe that κ(P)1n/∼3.3. The effect of autocorrelated forcing on the conditioning on state space systems is examined.
Keywords :
Gaussian distribution; Lyapunov matrix equations; filtering theory; Frobenius form; discrete Lyapunov equation; exponential condition number; independent Gaussian distribution; input normal; single Jordan block; state space system; Autocorrelation; Controllability; Equations; Filters; Gaussian distribution; Numerical simulation; Signal processing; State-space methods; Symmetric matrices; System identification;
Journal_Title :
Signal Processing, IEEE Transactions on
DOI :
10.1109/TSP.2004.826177