ON μ-ANALYSIS AND SYNTHESIS FOR SYSTEMS SUBJECT TO REAL UNCERTAINTY P. Iordanov * , M. J. Hayes † , and M. Halton ‡ Department of Electronic and Computer Engineering, University of Limerick, Ireland, * e-mail: petar.iordanov@ul.ie † e-mail: martin.j.hayes@ul.ie ‡ e-mail: mark.halton@ul.ie, Keywords: structured singular value, real-valued uncertainty, robustness analysis, µ-based controller design Abstract The paper introduces a new approach for the computation of a lower bound on the structured singular value (SSV), µ in the presence of purely real and mixed/complex uncertainties. The approach utilises a frequency sweeping technique based on a linear fractional transformation representation of structured un- certainty. The technique is applied to a well-known civil trans- port aircraft example. A fixed structure controller synthesis strategy is developed, which addresses potential stability prob- lems that can occur using standard design methodologies. Notation μ(M) structured singular value (SSV) of M σ(M) largest singular value of M F l (M, Δ) lower Linear Fractional Transformation of M, Δ ℜ real part of a complex number ℑ imaginary part of a complex number R field of real numbers C field of complex numbers λ(M) eigenvalues of M 1 Introduction The concept of robustness analysis for systems with structured uncertainties first appeared in 1980 when the so called excess stability margin, which later became known as the multiloop stability margin was introduced, [14]: k m = min Δ∈D {k ∈ [0, ∞): det(I − kΔM )=0} (1) where M represents the value of the transfer function matrix M (s) at s = ω, D is the set of all admissible perturbations and Δ is a structured perturbation defined as Δ = diag(δ r 1 I 1 ,...,δ r mr I mr ,δ c 1 I mr +1 ,...,δ c mc I mr +mc , Δ C 1 ,..., Δ C m C ), δ r i ∈R,δ c j ∈C , Δ C k ∈C p k ×q k i =1,...,m r ; j =1,...,m c ; k =1,...,m C (2) In [5], the term structured singular value for the reciprocal of k m was coined and denoted by µ: µ(M )=  min Δ∈D σ(Δ) : det(I − M Δ) = 0  -1 (3) The exact computation of µ is an NP-hard problem, there- fore lower and upper bounds are considered in the literature. The upper bounds are defined as convex optimisation problems [11]. MATLAB software, which uses a suitably defined fre- quency grid to determine an upper bound on µ [1, 8] is known to work quite well, with the possible exception of when the uncertainty consists of purely real blocks, [13]. This is a par- ticular problem for the practical interacting systems where the µ plot exhibits multiple narrow peaks. These peaks can neces- sitate a prohibitively narrow grid using standard µ-techniques. An upper bound solution to the frequency gridding problem has been presented by Feron [6], whereby stability guarantees, (albeit with fairly mild restrictions), are possible within a pre- specified frequency interval. However, the algorithm can po- tentially yield a conservative upper bound if the frequency in- terval is quite wide. All available lower bound computation techniques consist of finding a perturbation which corresponds to the limit of stabil- ity. A fixed point power algorithm is presented in [11]. Un- fortunately, when uncertainties are modelled as real parame- ter variations this approach does not converge well enough. An improvement can be achieved by adding a small amount of complex uncertainty [12] but this amount needs to be fixed by trial and induces approximation in the results that is diffi- cult to evaluate. Moreover, the solution is suboptimal in the real parameters. An approach for purely real uncertainties is presented in [3], but the method is of exponential time and its practical use is only for small uncertainty sets. An optimisation based approach presented in [9] provides a satisfactory lower bound for large Δ’s, but the algorithm is quite sensitive to the choice of initial starting point and recalculations are required at some frequencies. A different lower bound approach, which features a frequency independent µ computation is considered here. It is has been found that this approach works quite well on a wide variety of practically motivated problems, reducing the gap between the lower and upper bounds on µ to a very small level. A key feature of the approach is that an accurate combination of pa-