SoftwareforAuxiliarySignalDesign S. L. Campbell 1 R. Nikoukhah 2 Abstract—Recently an approach for multi-model identifica- tion and failure detection in the presence of model uncertainty and bounded energy noise over finite time intervals has been introduced. This approach involved offline computation of an auxiliary signal and online application of a hyperplane test. Thispaperdiscussesprogressindevelopingasoftwarepackage to carry out this procedure. I. I NTRODUCTION Failure detection has been the subject of many studies. Most of this work concerned passive failure detection. In the passive approach, for material or security reasons, the detector monitors the system but has no way of acting upon it. A major drawback with the passive approach is that failures can be masked by the operation of the system. This is true, in particular, for controlled systems where the desirable robustness of control systems tends to mask abnormal behaviors of the systems. In contrast, active detection consists in acting upon the system using a test signal in order to detect abnormal behaviors which would otherwise remain undetected during normal operation. The use of extra input signals specifically in the context of failure detection has been introduced by Zhang [11] and later developed by [5], [6]. We consider robustness in a deterministic setting. Here the normal and failed behaviors of a process are modeled by two or more linear uncertain systems. In this paper we restrict ourselves to two models. Failure detectability in linear systems thus becomes a linear multi- model identification problem. In most cases, there is no guarantee that one of the models can be ruled out by simply observing the inputs and outputs of the system. For this reason in some cases a test signal, usually referred to as auxiliary signal, is injected into the system to expose its behavior and facilitate the detection (identification) of the failure. Let v be the inputs taken over by the failure detector mechanism, u the rest of the inputs, if any, and y the outputs of the system. An auxiliary signal v guarantees failure detection if and only if A 0 (v) ∩A 1 (v)= ∅ where A i (v) is the set of input-outputs {u,y} consistent with Model i, i =0, 1, for a given input v. We call such a v a proper auxiliary signal. Unreasonably “large” signals are often proper, but cannot be applied in practice. There are many 1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205. USA. e-mail: slc@math.ncsu.edu Research sup- ported in part by the National Science Foundation under DMS-0101802, DMS-020695, and ECS-0114095. 2 INRIA, Rocquencourt BP 105, 78153 Le Chesnay Cedex, France. requirements on a test signal during the test period including the desire that the system should continue to operate in a reasonable manner, the test period [0 T ] should be short, and the effect of the auxiliary signal on the system minimal. In a series of papers [7], [8], [3] we have developed such an approach. The full mathematical description can be found in the monograph [2] which will appear in 2004. Here we discuss the numerical implementation of this approach. All material in Section 3 and beyond has not been published before. II. SUMMARY OF APPROACH:CONTINUOUS CASE Both continuous and discrete systems are of importance. Space limitations force us to restrict the summary to the continuous case. We first summarize the general procedure. The linear model (1) represents normal and failed systems. It can be considered as a generalization of the model used in Chapter 4 of [10]. ˙ x i = A i x i + B i v + M i ν i , (1a) E i y = C i x i + D i v + N i ν i . (1b) Here i =0, 1 correspond to the normal and failed system models respectively, y is the measured output, and ν i and x i are model noises and states. We assume here for simplicity that the systems have no measured inputs besides the auxiliary signal v. System matrices have arbitrary but consistent dimensions; the only conditions are that N i ’s have full row rank and E i ’s have full column rank. The constraint (or noise measure) on the initial condition and uncertainties is S i (v,s)= x i (0) T P -1 i0 x i (0) +  s 0 ν T i J i ν i dt< 1, ∀s ∈ [0,T ], (2) where J i ’s are signature matrices. The bound (2) allows for both additive and model uncertainty. With only additive uncertainty we have J i = I and need only consider s = T . The assumption is that for failure detection, we have access to y, given a v, consistent with one of the models. The problem is to find an optimal v for which observation of y provides enough information to decide from which model y has been generated. That is, there exist no solution to (1a), (1b) and (2) for i =0 and 1 simultaneously. We consider cost functions on v of the form: δ(v)= ξ (T ) T Wξ (T )+  T 0 |v| 2 + ξ T Uξdt, (3a) Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004 0-7803-8335-4/04/$17.00 ©2004 AACC FrA15.4 4414