Families of regimes for discrete control systems are studied possessing a
special quasi-controllability property that is similar to the Kalman
controllability property. A new approach is proposed to estimate the amplitudes
of transient regimes in quasi-controllable systems. Its essence is in obtaining
of constructive a priori bounds for degree of overshooting in terms of the
quasi-controllability measure. The results are applicable for analysis of
transients, classical absolute stability problem and, especially, for stability
problem for desynchronized systems.