Necessary and sufficient conditions for robust identification of uncertain LTI systems☆
Introduction
Robust identification typically arises whenever the real system does not belong to the model class chosen for the purpose of identification. Such problems arise in a number of both stationary and non-stationary situations. Acoustic echo-cancellation systems designed for cabins must deal with time-varying and complex acoustic dynamics. These result from a combination of large-scale multi-path reflections coupled with time-variation at high frequencies [17]. The time variation limits the complexity of model classes that can be chosen leading to undermodeling. A similar situation is encountered in the problem of channel identification in air-to-ground communication scenarios. Finally, these situations can also arise when lumped parameter models are utilized for convenience to model distributed systems that are governed by partial differential equations.
In all of the above cases it is reasonable to seek limited complexity models for characterizing the underlying complex system. The objective of undermodeling a complex system has been suggested in [4], [5], [19], [22] and has been extensively dealt with by the author in [19]. A consequence of undermodeling is the presence of unmodeled dynamics in addition to the inherent measurement noise present in any data. The question of how to “model” unmodeled dynamics has been longstanding and fundamentally impacts our ability to extract reasonable limited complexity models for complex systems. The arbitrary but bounded noise model (see [10]) for “modeling” unmodeled errors have been shown to have an identification error that is bounded from below by twice the amplitude of disturbance [14], [15], thus resulting in inconsistent estimates. Furthermore, for ℓ1 identification it has also been shown that to achieve any measure of accuracy an exponential length of data [3], [6], [11] is required. In contrast structured noise models such as the deterministic white noise models [20] have been shown to be too stringent in being able to account for unmodeled errors. Based on these considerations in [19] structural constraints on the unmodeled errors were imposed. These structural constraints arose from the objective of seeking best approximations among limited complexity models resulting in a “unique” decomposition of a system into a limited complexity component and an “orthogonal” unmodeled component. These structural assumptions on the unmodeled dynamics was utilized in establishing convergence of limited complexity estimates to their corresponding best approximations in the model class.
This paper deals with the problem of characterizing necessary and sufficient conditions for robust consistency of uncertain systems. These conditions imply that robust consistency holds if and only if a specific instrumental variable method exists that annihilates both unmodeled error as well as noise. These conditions imply that establishing consistency amounts to inputs that have infinite informativity, as opposed to the well-known persistency of excitation equal to the model order required in traditional system identification theory [8], [12]. Another implication of these results is that, although, consistency does not hold in the bounded but worst-case noise setting, it does hold almost surely except on a set of Lebesgue measure zero. This shows that the well-known worst-case error bound of twice the amplitude of the disturbance is pathological.
Section snippets
Robust consistency
In this section we define the notion of robust consistency and provide the motivation for introducing this concept. Mathematically, we are given a sequence of real-valued observations, y(k), which are assumed to be generated according to an underlying model. The model is uncertain in that it has both random and non-random aspects that provide information on how the sequence of observations are generated. More precisely, letwhere Fk is a time-dependent real-valued regressor; θ is
Robust consistency for FIR model classes
Suppose, the input output data is given bywith w(0),w(1),… is a sequence of jointly Gaussian i.i.d. random variables. The inputs are bounded, i.e., . The input–output data is observed over a time interval starting from t=0 to t=N.
Now, it is impossible to identify H with finite data since it is an arbitrary element in ℓ1. Indeed, even if H were bounded, say ||H||1⩽γ, with known bound, γ, and data length N it is still not possible to estimate, H, to a resolution
Discussion
In this section we point to several generalizations to the FIR result derived in the previous section. A comprehensive discussion of these topics appear in our forthcoming papers [16], [18].
Conclusions
We have presented conditions for robust identifiability of uncertain systems. This work complements existing body of work on system identification with stochastic noise or bounded noise type disturbances. Our paper focuses on identification of uncertain systems which naturally arise in situations where the model class is inadequate to characterize the actual complex system. We show that the instrumental variables approach provides a complete characterization for achieving robust consistency. In
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This research was supported by ONR Young Investigator Award N00014-02-100362.