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 α_i, obey |α_i|⩽δ_i, and provide necessary and sufficient conditions on the δ_i that guarantee that each time invariant Lattice in the set has poles inside a circle of prescribed radius 1/ρ<1, i.e. they are relatively stable with degree of stability in ρ. 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 α_i to δ_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/ρ, 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/ρ´>1/ρ