Computing the structured distance to instability

We analyze robust stability and performance of dynamical systems with real uncertain parameters. We compute criteria like the distance to instability, the worst-case spectral abscissa, or the worst-case H∞-norm, which quantify the degree of robustness of such a system when parameters vary in a given set ∆. As computing these indices is NP-hard, we present a heuristic which finds good lower bounds fast and reliably. Posterior certification is then obtained by an intelligent global strategy. A test bench of 87 systems with up to 70 states 39 uncertain parameters with up to 11 repetitions demonstrates the potential of our approach. 1 Problem Specification. Robustness specifications limit the loss of performance and stability in a system where differences between the mathematical model and reality crop up. Robustness against real parametric uncertainties is among the most challenging calls in this regard. Already deciding whether a given system with uncertain real parameters δ is robustly stable over a given parameter range δ ∈ ∆ is NP-hard, and this is aggravated when it comes to deciding whether a given level of H2or H∞-performance is guaranteed over that range. In this work we address this type of uncertainty by computing three key criteria, which quantify the degree of parametric robustness of a system. These are (a) the worst-case H∞-norm, and (b) the worst-case spectral abscissa over a given parameter range, and (c) the distance to instability, or stability margin, of a system with uncertain parameters. Consider a Linear Fractional Transform [23] with real parametric uncertainties as in Figure 1, where P (s) :    ẋ = Ax + Bpp + Bww q = Cqx + Dqpp + Dqww z = Czx + Dzpp + Dzww (1.1) and x ∈ R is the state, w ∈ R1 a vector of exogenous inputs, and z ∈ R1 a vector of regulated outputs. The uncertainty channel is defined as p = ∆q, where the time-invariant uncertain matrix ∆ has the blockdiagonal form ∆ = diag [δ1Ir1 , . . . , δmIrm ] , (1.2) ONERA, Control System Department, Toulouse, France Universite de Toulouse, Institut de Mathematiques with δ1, . . . , δm representing real uncertain parameters, and ri giving the number of repetitions of δi. Here we assume without loss that δ = 0 ∈ ∆ represents the nominal parameter value, and we consider δ ∈ ∆ in one-to-one correspondence with the matrix ∆ in (1.2). For practical applications it is generally sufficient to consider the case ∆ = [−1, 1]. To analyze the performance of (1.1) in the presence of the uncertain δ ∈ R we compute the worst-case H∞-performance h = max{‖Twz(δ)‖∞ : δ ∈ ∆}, (1.3) where ‖·‖∞ is the H∞-norm, and where Twz(s, δ) is the transfer function z(s) = Fu(P (s), ∆)w(s), obtained by closing the loop between (1.1) and p = ∆q with (1.2) in Figure 1. The solution δ ∈ ∆ of (1.3) represents a worst possible choice of the parameters δ ∈ ∆, which may be an important element in analyzing performance and robustness of the system, see e.g. [1]. Our second criterion is similar in nature, as it allows to verify whether the uncertain system (1.1) is robustly stable over a given parameter range ∆. This can be tested by maximizing the spectral abscissa of the system A-matrix over the parameter range α = max{α(A(δ)) : δ ∈ ∆}, (1.4) where A(δ) = A + Bp∆(I − Dpq∆) Cq, and where the spectral abscissa of a square matrix A is defined as α(A) = max{Reλ : λ eigenvalue of A}. Since A is stable if and only if α(A) < 0, robust stability of (1.1) over ∆ is certified as soon as α < 0, while a destabilizing δ ∈ ∆ is found as soon as α ≥ 0. Note however that a decision in favor of robust stability over ∆ based on α < 0 is only valid when the global maximum over∆ is computed. This renders (1.4) a difficult problem, and it is in fact known that solving (1.4) globally is NP-hard. In [18] Poljak and Rohn have shown that for a given set of matrices A0, . . . , Ak, deciding whether A0 + r1A1 + . . . + rkAk is stable for all ri ∈ [0, 1] is NP-hard, and Braatz et al. [6] have shown that deciding whether a system with real (or mixed or complex) uncertainties is robustly stable over a range ∆ = [−1, 1] is harder than globally solving a nonconvex quadratic programming problem, hence is NP-hard. 423 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.