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

This note aims at providing some background and comments to the paper by M. Durrant, H. Werner and K. Abbott published in this issue. This paper proposes a comparison between the so-called GloverMcFarlane loop shaping design procedure (LSDP) and the Lyapunov-based design procedure (called ellisoidal set formulation) proposed in [25]. The synthesis procedures are evaluated via the design of robust multi-performance reduced-order controllers for a voltage source converter high voltage direct current (VSC HVDC) power plant terminal that is operating over different points. The present discussion will focus on the theoretical problem of the synthesis of robust multi-performance reduced-order controllers for linear uncertain models rather than on the practical control problem. It is well-known that this particular control problem is extremely difficult [1]. Even the simplified problem of reduced-order controller synthesis for a given plant is still an open problem. The present problem has been extensively studied and the format of a discussion section in European Journal of Control is clearly too stringent to include all the possible references. Therefore, the objective of this note is not to give a complete and fair evaluation of the existing methods in a very rich literature but rather to give some brief comments and additional references to the interested reader. First, we would like to comment two issues with respect to the ellipsoidal set formulation. When comparing with other formulations, this parameterization has the advantage to allow multiobjective or multi-operating points design without the conservative constraints introduced by the Lyapunov shaping paradigm (LSP) [23]. Yet, this property may be used in a richer way than it is here in [8]. Indeed, the multi-objective problems may also have inhomogeneous dimensions. In particular, the specifications may be defined for models of different orders. This is often the case when including weighting functions in the design process. Another debatable point may be raised about the application of the convexifying approach to this particular formulation. Indeed, it seems that the converging point of the algorithm is located on the boundaries of the nonlinear nonconvex inequality constraint leading to the loss of the main characteristic of the proposed parameterization: convex sets (ellipsoids) of controllers. This property has been used in [25] to choose the best controllers in the set with respect to resilience properties. We have some doubts about the effectiveness of this resilience procedure when using the convexifying approach. This last remark naturally leads to the discussion about numerical issues. The control problem encountered in the present paper may be recast as the following nonconvex nonlinear optimization problem: