On the Determination of the Worst Allowable Persistent Bounded Disturbance for a System with Constraints

The problem of explicitly determining the worst persistent input disturbance that a closed loop system can tolerate under prespecified state and control constraints is studied. Verification of designs specifically aimed at maximizing the size of persistent bounded disturbances while satisfying system constraints, typically requires extensive simulations because the exact nature of the worst input is not known. In this paper the exact nature of the worst input is completely characterized for both SISO and MIMO cases. A finite number of specific impulse responses of the closed loop system determines the worst persistent input disturbance. In the case of a SISO system with n state constraints (|x
_i
| ≤ ß
_i
), a, control constraint (|u| ≤ ß
_u
) and an output constraint (|y| ≤ ß
_o
), n + 2 impulse responses are generaly needed. With this new result the large number of simulations that is typically needed for design verification can be significantly reduced. Three examples illustrate how the new characterization can be utilized.