Mixed H2/H-Infinity Control
Mixed synthesis optimizes average performance under an H2 objective while enforcing a hard worst-case robustness bound in the H-infinity norm.
Two objectives, one controller
Real designs rarely want only average performance or only worst-case robustness. Mixed H2/H-infinity synthesis minimizes the H2 norm of one channel, capturing nominal disturbance rejection, subject to a constraint that the H-infinity norm of a second channel stays below a bound, capturing robustness. The result is a controller that performs well on typical inputs while never violating a guaranteed margin.
LMI formulation
The problem has no closed-form Riccati solution in general, but it is naturally posed as a convex program in linear matrix inequalities. One Lyapunov matrix certifies the H-infinity bound through a bounded-real lemma constraint; a second, coupled to an auxiliary matrix, bounds the H2 cost. Convexity is preserved when the two conditions share a single Lyapunov variable, at the price of some conservatism.
Because the constraints are convex, additional specifications, such as pole placement in a region of the complex plane or actuator-effort bounds, can be appended as further LMIs and solved together by interior-point methods.
Design trade-offs
Sharing one Lyapunov matrix couples the two objectives and can degrade the achievable H2 cost. Relaxations that use separate matrices restore performance but sacrifice the exact certificate, so practitioners choose based on how tight the robustness requirement is.
- Minimize H2 cost subject to an H-infinity constraint
- Convex when both channels share a Lyapunov matrix
- Extra regional or effort specs added as more LMIs
- Trade conservatism for exactness by decoupling matrices
For plant such as the plasma-shaping loops of a design-stage machine, mixed synthesis lets engineers demand a firm stability margin against modeling error while still minimizing mean-square shape deviation under stochastic disturbances. All such work for Kronos machines is model-based, since the hardware is not built.
The method connects directly to the broader robust-control toolkit and is often the practical middle ground between pure H2 and pure H-infinity design.