Abstract
Monte Carlo simulations have long been a widely used method in the
industry for control system validation. They provide an accurate
probability measure for sufficiently frequent phenomena, but are often
time-consuming and may fail to detect very rare events. Conversely,
deterministic techniques such as µ or IQC-based analysis allow
fast calculation of worst-case stability margins and performance levels,
but in the absence of a probabilistic framework, a control system may be
invalidated on the basis of extremely rare events. Probabilistic
µ-analysis has therefore been studied since the 1990s to bridge
this analysis gap by focusing on rare but nonetheless possible
situations that may threaten system integrity. The solution adopted in
this paper implements a branch-and-bound algorithm to explore the whole
uncertainty domain by dividing it into smaller and smaller subsets. At
each step, sufficient conditions involving µ upper bound
computations are used to check whether a given requirement – related to
the delay margin in the present case – is satisfied or violated on the
whole considered subset. Guaranteed bounds on the exact probability of
delay margin satisfaction or violation are then obtained, based on the
probability distributions of the uncertain parameters. The difficulty
here arises from the exponential term classically used to represent a
delay, which must be replaced by a rational expression to fit into the
Linear Fractional Representation (LFR) framework imposed by
µ-analysis. Two different approaches are proposed and compared in
this paper. First, an equivalent representation using a rational
function of degree 2 with the same gain and phase as the real delay,
which results into an LFR with frequency-dependent uncertainty bounds.
Then, a Padé approximation, whose order should be chosen carefully to
handle the trade-off between conservatism and complexity. A constructive
way to derive minimal LFR from Padé approximations of any order is also
provided as an additional contribution. The whole method is first
assessed on a simple satellite benchmark, and its applicability to
realistic problems involving a larger number of states and uncertainties
is then demonstrated.