01
Generalized Plant P(s)
Full representation of the design problem.
Generalised plant combining the nominal model, performance weights W_p and uncertainty weights W_u.
H∞ Control
Minimization of the H∞ gain (supremum norm of the transfer function in the frequency domain) as a robust controller design criterion against worst-case disturbance signals.
Category
Abstraction level
Passive FTC — fault-tolerant baseline controllerActive vibration isolationFlight control (fly-by-wire)Stabilization of optical platformsPower grid controlRobust loop shaping for manipulators
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.
02
Weighting Functions
Specifies the robustness–performance tradeoff.
Frequency-domain filters W_1(s) (perf.), W_2(s) (control effort), W_3(s) (robustness) shaping the open loops.
03
H∞ Optimizer (Riccati / LMI)
Synteza regulatora.
Solver minimising ||T_{zw}||_∞ by solving Riccati equations or an LMI formulation.
Common pitfalls
High controller orderMEDIUM
H∞ synthesises controllers of order equal to the generalised plant (can be very high). High-order controllers are hard to implement in embedded systems.
Model order reduction (Schur balanced truncation, Hankel norm approximation).
Weight selection for W — iterative and non-intuitiveMEDIUM
Selecting shaping weights is iterative and requires expertise; bad weights produce a conservative or unstable controller.
Loop shaping (McFarlane–Glover), systematic H∞ loop shaping iteration.
1981
Zames — H∞ optimal control
breakthroughG. Zames formulates the H∞ problem in "Feedback and optimal sensitivity" IEEE TAC.
1989
Doyle, Glover, Khargonekar, Francis — Riccati algorithm
breakthroughState-space solution to H∞ via Riccati equations — practical synthesis algorithm.
1994
Gahinet & Apkarian — LMI-based H∞
H∞ formulation via LMI opens unified synthesis for broader system classes.
1996
μ-synthesis (DK iteration)
μ-analysis and DK iteration tools for robust stability with structured uncertainty.
CPU AVXPRIMARY
H∞ controller after order reduction implemented as LTI state-space on RT CPU.
Hardware agnosticGOOD
The synthesised controller (state-space) is hardware-agnostic once implemented.
ALTERNATIVE TO
SMC
Sliding Mode Control (SMC) belongs to the class of Variable Structure Systems (VSS), developed primarily by Utkin in the USSR in the 1960s–1970s. An SMC controller consists of two parts: the switching law, which drives the state trajectory towards the sliding surface s(x)=0, and the sliding dynamics, which describe motion along the surface once the sliding regime is reached. The key property of SMC is invariance to disturbances and uncertainties satisfying the matching condition: disturbances entering through the same channels as the control input are completely rejected during sliding. This makes SMC a natural choice for FTC — actuator faults or plant parameter changes can be treated as matched disturbances. The main drawback of classical SMC is chattering — rapid control oscillations around the sliding surface caused by the discrete nature of implementation. Chattering mitigation methods: boundary layer (smooth control inside a thin layer), Higher-Order SMC (super-twisting for 2nd-order SMC), continuous approximation (signum → tanh). Applications: manipulators, electric vehicles, aerial robots, wind energy, power grids.
GO TO CONCEPT| Title | Publisher | Type |
|---|---|---|
| State-space solutions to standard H2 and H∞ control problems Doyle, Glover, Khargonekar, Francis, 1989. H∞ synthesis algorithm. | IEEE Transactions on Automatic Control | scientific article |
| Robust and Optimal Control (Zhou, Doyle, Glover) Standard reference text on H∞ and μ-synthesis. | Prentice Hall | documentation |
State-space solutions to standard H2 and H∞ control problems
Doyle, Glover, Khargonekar, Francis, 1989. H∞ synthesis algorithm.
scientific articleIEEE Transactions on Automatic Control
Robust and Optimal Control (Zhou, Doyle, Glover)
Standard reference text on H∞ and μ-synthesis.
documentationPrentice Hall