A Novel Iterative Method To Approximate Structured Singular Values

Abstract

Abstract. A novel method for approximating structured singular values (also known as µ-values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in structured eigenvalue perturbation theory. Our approach consists of an inner-outer iteration. In the outer iteration, a Newton method is used to adjust the perturbation level. The inner iteration solves a gradient system associated with an optimization problem on the manifold induced by the structure. Numerical results and comparison with the well-known Matlab function mussv, implemented in the Matlab Control Toolbox, illustrate the behavior of the method.Key words. Structured singular value, µ-value, spectral value set, block diagonal perturbations, stability radius, differential equation, low-rank matrix manifold. AMS subject classifications. 15A18, 65K051. Introduction. The structured singular value (SSV) [14] is an important and versatile tool in control, as it allows to address a central problem in the analysis and synthesis of control systems: To quantify the stability of a closed-loop linear time-invariant systems subject to structured perturbations. The class of structures addressed by the SSV is very general and allows to cover all types of parametric uncertainties that can be incorporated into the control system via real or complex linear fractional transformations. We refer to [1,3,4,8,9,10,14,17,20] and the references therein for examples and applications of the SSV.The versatility of the SSV comes at the expense of being notoriously hard, in fact NP hard [2], to compute. Algorithms used in practice thus aim at providing upper and lower bounds, often resulting in a coarse estimate of the exact value. An upper bound of the SSV provides sufficient conditions to guarantee robust stability, while a lower bound provides sufficient conditions for instability and often also allows to determine structured perturbations that destabilize the closed loop linear system.The widely used function mussv in the Matlab Control Toolbox computes an upper bound of the SSV using diagonal balancing / LMI techniques [19,5]. The lower bound is computed by a generalization of the power method developed in [18,15]. This algorithm resembles a mixture of the power methods for computing the spectral radius and the largest singular value, which is not surprising, since the SSV can be viewed as a generalization of both. When the algorithm converges, a lower bound of the SSV results and this is always an equilibrium point of the iteration. However, in contrast to the standard power method, there are, in general, several stable equilibrium points and not all of them correspond to the SSV. In turn, one cannot guarantee convergence to the exact value but only to a lower bound. We remark that, despite this drawback, mussv is a very reliable and powerful routine, which reflects the state of the art in the approximation of the SSV.

Computing Structured Singular Values for Delay and Polynomial Eigenvalue Problems

“…Lemma 3.6. [13]. With the notation introduced above and , v u partitioned as in Equation (11) Proof.…”

“…This section provides the comparison of the lower bounds of SSV computed by well-known Matlab function mussv and the algorithm [13]. We consider the following delay eigenvalue problem of the form: Example 1.…”

“…Consider the following two dimensional matrix 1 A taken from above mentioned delay eigenvalue problem. By using the algorithm [13], one can obtain the perturbation * * ∆  with 1.0000 0.0000 0 , 0 1.0000 0.0000…”

“…In the following Table 1, it is presented the comparison of the bounds of SSV computed by MUSSV and the algorithm [13] for the matrix 1 A given bellow. In the very first column, it is presented the dimension of the matrix 1 A .…”

“…In the second column, it is presented the set of block diagonal matrices denoted by BLK. In the third, fourth and fifth columns, it is presented the upper and lower bounds By using the algorithm [13], one can obtain the perturbation * * ∆  with In Table 5, it is presented the comparison of the bounds of SSV computed by MUSSV and the algorithm [13] for the matrix 5 A given bellow. In the very first column, it is presented the dimension of the matrix 5 A .…”