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Abstract. This paper is concerned with the stabilisation of linear time-delay systems by tuning a finite number of parameters. Such problems typically arise in the design of fixed-order controllers. As time-delay systems exhibit an infinite amount of characteristic roots, a full assignment of the spectrum is impossible. However, if the system is stabilisable for the given parameter set, stability can in principle always be achieved through minimising the real part of the rightmost characteristic root, or spectral abscissa, in function of the parameters to be tuned. In general, the spectral abscissa is a nonsmooth and nonconvex function, precluding the use of standard optimisation methods. Instead, we use a recently developed bundle gradient optimisation algorithm which has already been successfully applied to fixed-order controller design problems for systems of ordinary differential equations. In dealing with systems of time-delay type, we extend the use of this algorithm to infinite-dimensional systems. This is realised by combining the optimisation method with advanced numerical algorithms to efficiently and accurately compute the rightmost characteristic roots of such time-delay systems. Furthermore, the optimisation procedure is adapted, enabling it to perform a local stabilisation of a nonlinear time-delay system along a branch of steady state solutions. We illustrate the use of the algorithm by presenting results for some numerical examples.Mathematics Subject Classification. 65Q05, 65K10, 90C26

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Abstract. This paper concerns the stability optimization of (parameterized) matrices A(x), a problem typically arising in the design of fixed-order or fixed-structured feedback controllers. It is well known that the minimization of the spectral abscissa function α(A) gives rise to very difficult optimization problems, since α(A) is not everywhere differentiable, and even not everywhere Lipschitz. We therefore propose a new stability measure, namely the smoothed spectral abscissaαǫ(A), which is based on the inversion of a relaxed H 2 -type cost function. The regularization parameter ǫ allows tuning the degree of smoothness. For ǫ approaching zero, the smoothed spectral abscissa converges towards the non-smooth spectral abscissa from above, so thatαǫ(A) ≤ 0 guarantees asymptotic stability. Evaluation of the smoothed spectral abscissa and its derivatives w.r.t. the matrix parameters x can be performed at the cost of solving a primal-dual Lyapunov equation pair, allowing for an efficient integration into a derivative based optimization framework. Two optimization problems are considered: on the one hand the minimization of the smoothed spectral abscissaαǫ(A(x)) as a function of the matrix parameters for a fixed value of ǫ, and on the other hand the maximization of ǫ such that the stability requirement,αǫ(A(x)) ≤ 0, is still satisfied. The latter problem can be interpreted as an H 2 -norm minimization problem, and its solution additionally implies an upper bound on the corresponding H∞-norm, or a lower bound on the distance to instability. In both cases additional equality and inequality constraints on the variables can be naturally taken into account in the optimization problem.

Complications involving the patellofemoral joint, caused by malrotation of the femoral component during total knee replacement, are an important cause of persistent pain and failure leading to revision surgery. The aim of this study was to determine and quantify the influence of femoral component malrotation on patellofemoral wear, and to determine whether or not there is a difference in the rate of wear of the patellar component when articulated against oxidised zirconium (OxZr) and cobalt-chrome (CoCr) components. An in vitro method was used to simulate patellar maltracking for both materials. Both rates of wear and changes in height on the patellar articular surface were measured. The mean rates of wear measured were very small compared to standard tibiofemoral wear rates. When data for each femoral component material were pooled, the mean rate of wear was 0.19 mm3/Mcycle (sd 0.21) for OxZr and 0.34 mm3/Mcycle (sd 0.335) for CoCr. The largest change in height on each patella varied from -0.05 mm to -0.33 mm over the different configurations. The results suggest that patellar maltracking due to an internally rotated femoral component leads to an increased mean patellar wear. Although not statistically significant, the mean wear production may be lower for OxZr than for CoCr components.