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Stability optimization and robust control design for the class of time-delay systems
August 17, 2010
Advisor(s): Stefan Vandewalle and Wim Michiels
In the field of feedback control, measured information about the state of a system is put available as output, processed in a controller, and used to influence the system's behaviour (by actuation via the inputs) in a desirable way. The problem of finding a controller with a prescribed structure or dimension such that it is optimal with respect to a certain objective is called optimal fixed-structure/fixed-order controller design.
This thesis concerns the development of methods to solve optimal fixed-structure, fixed-order controller design problems for the class of time-delay systems. Time delays are widely used to describe physical phenomena and appear in many industrial applications that have a dependence on the past. This typically reflects that some part of the process needs a non-negligible amount of time to take effect. The explicit insertion of such time delays in mathematical models, leading to a description by delay differential equations, complicates matters a great deal.
In the approaches taken in this thesis, optimization plays a crucial role. Firstly, we discuss the stabilization problem, for which we combine the direct application of non-smooth eigenvalue optimization together with efficient methods to compute the characteristic roots of a time-delay system. In contrast to existing approaches, our approach has the advantage of not introducing any conservatism, meaning that a solution can always be found if it exists.
Next, we consider the H2-norm of an appropriately defined transfer function in the context of time-delay systems. We generalize the ideas involving a relation with Lyapunov equations, and also develop an alternative computational method for the H2-norm that is both efficient and particularly well suited for the purpose of optimization. It is based on the discretization of the time-delay system, seen as a linear first-order system over an infinite-dimensional space.
Finally, we introduce a novel stability measure, called the smoothed spectral abscissa. It offers the representation of a range of trade-offs between optimality in terms of stability and optimality in terms of the H2-norm, thereby compromising between response time and robustness. Moreover, we can employ this smooth measure to develop an algorithm that computes the trade-off solely using standard, derivative-based methods and without the need to perform a preliminary stabilization step.Doctadmin 3E051131 / lirias 270816 / mailto: twr team