Discussion on: “Design of Low Order Robust Controllers for a VSC HVDC Power Plant Terminal”

This note aims at providing some background andcomments to the paper by M. Durrant, H. Wernerand K. Abbott published in this issue. This paperproposes a comparison between the so-called Glover-McFarlane loop shaping design procedure (LSDP)and the Lyapunov-based design procedure (calledellisoidal set formulation) proposed in [25]. Thesynthesis procedures are evaluated via the design ofrobust multi-performance reduced-order controllersfor a voltage source converter high voltage directcurrent (VSC HVDC) power plant terminal that isoperatingoverdifferentpoints.Thepresentdiscussionwill focus on the theoretical problem of the synthesisof robust multi-performance reduced-order control-lers for linear uncertain models rather than on thepractical control problem. It is well-known that thisparticular control problem is extremely difficult [1].Even the simplified problem of reduced-order con-troller synthesis for a given plant is still an openproblem. The present problem has been extensivelystudied and the format of a discussion section inEuropean Journal of Control is clearly too stringentto include all the possible references. Therefore, theobjective of this note is not to give a complete and fairevaluation of the existing methods in a very rich lit-erature but rather to give some brief comments andadditionalreferencestotheinterestedreader.First,wewould like to comment two issues with respect to theellipsoidal set formulation.When comparing with other formulations, thisparameterization has the advantage to allow multi-objective or multi-operating points design without theconservative constraints introduced by the Lyapunovshaping paradigm (LSP) [23]. Yet, this property maybeusedinaricherwaythanitisherein[8].Indeed,themulti-objective problems may also have inhomoge-neous dimensions. In particular, the specificationsmay be defined for models of different orders. This isoften the case when including weighting functions inthe design process.Another debatable point may be raised about theapplication of the convexifying approach to thisparticular formulation. Indeed, it seems that the con-verging point of the algorithm is located on theboundaries of the nonlinear nonconvex inequality con-straint leading to the loss of the main characteristic ofthe proposed parameterization: convex sets (ellipsoids)of controllers. This property has been used in [25] tochoose the best controllers in the set with respect toresilience properties. We have some doubts about theeffectiveness of this resilience procedure when usingthe convexifying approach. This last remark naturallyleads to the discussion about numerical issues.The control problem encountered in the presentpaper may be recast as the following nonconvexnonlinear optimization problem:min