H∞ Control as a Differential Game

The Stability Problem under Unknown Disturbances

In real control systems—aircraft, power systems, chemical reactors—there are always unknown disturbances: turbulence, load changes, parametric uncertainty. Classical control (LQR) is optimal for a known model, but can "fail" in the presence of disturbances. H∞ control builds a regulator that is guaranteed to be stable under the worst-case disturbances—this is a "minimax" optimization problem, naturally formulated as a differential game.

The H∞ Problem: Formulation

System: ẋ = Ax + Bu + Dw (w is an unknown disturbance), output z = Cx + Eu.

H∞ Problem: find control u = K(x) minimizing the "gain coefficient" from w to z in the worst case:

‖T_{zw}‖∞ = sup{w≠0, w∈L₂} ‖z‖{L₂} / ‖w‖{L₂} < γ

‖T_{zw}‖_∞ is the H∞ norm of the transfer function from w to z. This is the "worst-case signal gain."

As a differential game: control u is the minimizer, disturbance w is the maximizer:

J = ∫₀^∞ [‖z(t)‖² − γ²‖w(t)‖²] dt

min_u max_w J: find u such that, for the worst w, the total "energy" of z is less than γ² times the energy of w. If min max J < 0—the goal is achieved.

Game-Theoretic Formulation

J = ∫₀^∞ [‖Cx+Eu‖² − γ²‖w‖²] dt + boundary condition.

This is an LQ-differential game with: Q = CᵀC, R = EᵀE, S = γ²I (for D, use the disturbance matrix instead of C).

Hamiltonian: H = ‖Cx+Eu‖² − γ²‖w‖² + pᵀ(Ax+Bu+Dw).

Optimality condition:

• min over u: 2EᵀCx + 2EᵀEu + BᵀP = 0 → u* = −(EᵀE)⁻¹(EᵀCx + (1/2)BᵀPx)
• max over w: −2γ²w + DᵀPx = 0 → w* = DᵀPx/(2γ²)

LMI Solution to H∞

The condition ‖T_{zw}‖_∞ < γ is equivalent to the existence of P = Pᵀ ≻ 0 such that the matrix inequality holds:

[AᵀP+PA+CᵀC PB PD ] [ BᵀP −EᵀE 0 ] ≺ 0 [ DᵀP 0 −γ²I ]

This is an LMI (Linear Matrix Inequality)! The task: minimize γ under this LMI constraint → SDP problem. Solved by standard software packages in seconds.

Physical Interpretation

What does H∞ norm < γ mean?

Imagine the system as a "black box" that takes input w and outputs z. If ‖T‖_∞ = 2 and γ = 2, this means: for any input w with "energy" 1 (∫w² dt = 1), the output z has "energy" ≤ 4. The system "amplifies" by no more than γ times.

Connection with stability: if the H∞ norm is finite—the system is stable. The minimal γ for which the problem is solvable is the system's "robustness margin."

Full Example: H∞ Synthesis for a Car Suspension

Model: ẋ₁ = x₂ (body velocity), mẋ₂ = −k(x₁−x₃) − c(x₂−x₄) + u (damper), Mẋ₃ = x₄, Mẋ₄ = k(x₁−x₃) + c(x₂−x₄) − kₜ(x₃−w). Here w is the road roughness, u is active suspension.

Goal: minimize ‖body acceleration ẍ₁‖ under limited ‖u‖ and worst-case ‖w‖.

This is an H∞ problem: z = [ẍ₁, u]ᵀ, disturbance w = road irregularities. We minimize γ = ‖T_{zw}‖_∞.

Result: An H∞ regulator reduces body acceleration by 30-40% compared to passive suspension under the same constraints on suspension travel.

Applications

H∞ control is a standard in aerospace and missile guidance (Boeing, Airbus), power systems (generator control under random loads), automotive industry (active suspensions, ABS, ESP), robotics (robustness to unknown loads). Packages: MATLAB Control Toolbox hinfsyn, Python-Control h_inf_synthesize.

H∞ Norm and Interpretation

For a linear system with input w and output z, the H∞ norm of the transfer function: ‖T_{zw}‖_∞ = sup_w ‖z‖₂ / ‖w‖₂

This is the "worst-case gain coefficient." H∞ control minimizes this norm, guaranteeing z is bounded for any w of unit norm. Parameter γ sets the target level: ‖T_{zw}‖_∞ < γ.

Connection with Differential Game

The H∞ synthesis problem is equivalent to a differential game with the functional: J = ∫₀^T (zᵀz − γ²wᵀw) dt

The regulator u "plays for itself" (minimizes), disturbance w is the "opponent" (maximizes). Under Isaacs' condition, the optimal pair yields the H∞ regulator.

Synthesis Algorithms

• Riccati equations with γ-iteration: binary search over γ + solve the Riccati equation for each γ
• LMI (Linear Matrix Inequalities): formulation via convex constraints, solution by SDP solvers (SeDuMi, MOSEK)
• μ-synthesis: accounts for structured uncertainty (D-K iteration)
• Loop-shaping: classical H∞ approach via shaping the frequency response

Applications

H∞ control is standard in civil aviation (closed-loop autopilots for Boeing 777, Airbus A380), hydropower plant control (turbine oscillation damping), magnetic levitation (Maglev), active car suspensions. Robustness to parametric uncertainty makes it indispensable where system models are inaccurate.