Making Spatial Trade-Offs Using MultiobjectiveH2 Synthesis

In feedback control design, the Bode sensitivity integral dictates that trade-offs must be made frequency-wise. For multiple-in-multiple-out (MIMO) systems, trade-offs can also be made spatially (element-wise). To do this using ‘norm based’ control methods, such as H2 or H∞, one can only change weights on the inputs and the outputs of the systems. This way, it is not possible to exploit all spatial design freedom, e.g. it is not possible the put a lot of weight on one particular element of the transfer function matrix. To exploit all spatial design freedom, multiobjective methods can be employed. In the first part of this report, we discuss a multiobjective H2 synthesis method, which uses a vector valued performance criterion. This criterion is handled using the notion of Pareto optimality. Using this, the multiobjective problem is converted into a ‘simultaneous model matching problem’ by using the Youla parametrisation. Finally, a closed form expression is formulated to calculate one (out of many) Pareto optimal compensator. Second, we show that a H2 optimal control problem can be formulated by giving a frequency domain interpretation to the ‘classical’ LQG optimal control problem. Then, given a SISO system, it is shown that this method of multiobjective control synthesis is capable of trading off competing objectives. Then, this method is extended to MIMO systems, so spatial design freedom can be studied. First, we show that it is possible for a system with a non-minimum-phase zero (NMPhZ) to shift the effect of the NMPhZ from one output to another output. Then, the multiobjective problem is formulated for a 2 × 2 system, with one (real) NMPhZ. Four objectives can be distinguished: from each input to each output, which can be traded off against each other. Spatial design freedom is depicted by a curve of all Pareto optimal compensators. This curve shows that the output in which the zero is most present, is more restricted in bandwidth (resulting in a higher minimal 2-norm). Moreover, it can be seen that decoupling of the system decreases performance of the other loops mostly in the direction which the NMPhZ is present. Subsequently, a similar multiobjective problem is formulated for a 2×2 modal MIMO system, which is is not fundamentally restricted in bandwidth, but restricted due to robustness specifications. The curve of Pareto optimal compensators shows that both outputs are equally restricted and that the cost of decoupling is equal in both outputs. Conclusively, it can be said, that multiobjective control synthesis can be used to make spatial trade-offs. However, we were unable to apply this methodology to more realistic systems. Thus, the implementation need some improvements.