Robust relative stability of time-invariant and time-varying lattice filters

We consider the relative stability of time invariant and time varying unnormalized lattice filters. First, we consider a set of lattice filters whose reflection parameters alpha(i), obey \alpha(i)\ less than or equal to delta(i), and provide necessary and sufficient conditions on the Si that guarantee that each time invariant Lattice in the set has poles inside a circle of prescribed radius 1/rho < 1, i.e. they are relatively stable with degree of stability In rho. We also show that the relative stability of the whole family is equivalent to the relative stability of a single filter obtained by fixing each alpha(i) to delta(i), and can be checked with only the real poles of this filter. Counterexamples are given to show that a number of properties that hold for stability of LTI Lattices do not apply to relative stability verification. Second, we give a diagonal Lyapunov matrix that is useful in checking the above pole condition. Finally, we consider the time varying problem where the reflection coefficients vary in a region where the frozen transfer functions have poles with magnitude less than 1/rho, and provide bounds on their rate of variations that ensure that the zero input state solution of the time varying Lattice decays exponentially at a rate faster than 1/rho' > 1/rho.