Stabilisation of time-delay systems using nonsmooth optimisation techniques

status: publishe

Using spectral discretization for the optimal H₂ design of time-delay systems

The stabilization and robustification of a time-delay system is the topic of this paper. More precisely, we want to minimize the H₂ norm of the transfer function corresponding to this class of linear time-invariant input-output systems with fixed time delays in the states. Due to the presence of the delays, the transfer function is a nonrational, nonlinear function, and the classical procedure which involves solving Lyapunov equations is no longer applicable. We therefore propose an approach based on a spectral discretization applied to a reformulation of the time-delay system as an infinite-dimensional standard linear system. In this way, we obtain a large delay-free system, which serves as an approximation to the original time-delay system, and which allows the application of standard H₂ norm optimization techniques. We give an interpretation of this approach in the frequency domain and relate it to the approximation of the nonlinear terms in the time-delay transfer function by means of a rational function. Using this property, we can provide some insight in the convergence behaviour of the approximation, justifying its use for the purpose of H₂ norm computation. Along with this, the easy availibility of derivatives with respect to the original matrices allows for an efficient integration into any standard optimization framework. A numerical example finally illustrates how the presented method can be employed to perform optimal H₂ norm design using smooth optimization techniques.nrpages: 24status: publishe

Stabilisation of time-delay systems using nonsmooth optimisation techniques

status: publishe

Characterizing and computing the ℋ₂ norm of time-delay systems by solving the delay Lyapunov equation

It is widely known that the solutions of Lyapunov equations can be used to compute the H2 norm of linear time-invariant (LTI) dynamical systems. In this paper, we show how this theory extends to dynamical systems with delays. The first result is that the two-norm can be computed from the solution of a generalization of the Lyapunov equation, which is known as the delay Lyapunov equation. From the relation with the delay Lyapunov equation we can prove an explicit formula for the H² norm if the system has commensurate delays, here meaning that the delays are all integer multiples of a basic delay. The formula is explicit and contains only elementary linear algebra operations applied to matrices of finite dimension. The delay Lyapunov equations are matrix boundary value problems. We show how to apply a spectral discretization scheme to these equations for the general, not necessarily commensurate, case. The convergence of spectral methods typically depend on the smoothness of the solution. To this end we describe the smoothness of the solution to the delay Lyapunov equations, for the commensurate as well as for the non-commensurate case. The smoothness properties allow us to completely predict the convergence order of the spectral method.nrpages: 25status: publishe

Smooth stabilization and optimal H2 design

In this paper we propose two smooth optimization methods, one that can stabilize a system, and the other that can perform a stabilization as well as solve the optimal $H_2$-norm design problem. For both methods, we make use of the smoothed spectral abscissa, a stabilization measure which originates from the inversion of an $H_2$-norm type function, and that behaves as a smooth approximation of the spectral abscissa. In this way, we can set up an optimization framework in which a stabilizing point can efficiently be found. Taking advantage of its computation via Lyapunov equations, we derive computationally attractive formulae for the first-order and second-order derivatives of this smooth objective, which allows for the use of standard gradient- or Hessian-based optimization techniques. A second optimization framework, also involving the smoothed spectral abscissa, can be designed to deal with the $H_2$-norm synthesis. This method has the advantage that it is not necessary to find a stable point for the system on beforehand, as the stabilization is done simultaneously with the actual minimization of the $H_2$-norm. We apply the discussed methods to the class of systems with low-order, or fixed-order, feedback laws, where the number of controller parameters is smaller than the dimension of the plant.status: publishe

Optimising the robust stability of time-delay systems via nonsmooth eigenvalue optimisation

status: publishe

A nonsmooth optimisation approach for the stabilisation of time-delay systems

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.status: publishe