Formulate the problem as a generalized plant P(s) with outputs z (to be minimized) and y (measurements), and inputs w (disturbances) and u (control). Find K(s) minimizing ||F_l(P,K)||_∞ < γ by solving two Riccati equations (or LMIs). The resulting controller K is optimal for the worst-case w.
How to design a controller that guarantees bounded performance degradation across an entire class of disturbances — rather than just their nominal values — without explicit knowledge of the disturbance shape.
Generalised plant combining the nominal model, performance weights W_p and uncertainty weights W_u.
Frequency-domain filters W_1(s) (perf.), W_2(s) (control effort), W_3(s) (robustness) shaping the open loops.
Solver minimising ||T_{zw}||_∞ by solving Riccati equations or an LMI formulation.
H∞ synthesises controllers of order equal to the generalised plant (can be very high). High-order controllers are hard to implement in embedded systems.
Selecting shaping weights is iterative and requires expertise; bad weights produce a conservative or unstable controller.
G. Zames formulates the H∞ problem in "Feedback and optimal sensitivity" IEEE TAC.
State-space solution to H∞ via Riccati equations — practical synthesis algorithm.
H∞ formulation via LMI opens unified synthesis for broader system classes.
μ-analysis and DK iteration tools for robust stability with structured uncertainty.
H∞ controller after order reduction implemented as LTI state-space on RT CPU.
The synthesised controller (state-space) is hardware-agnostic once implemented.
