Robust Control, Sensitivity & Loop Shaping

A nominal controller is designed for one plant model $P_0$. A robust claim quantifies what remains true for a set of plants, disturbances, sensor noise and implementation errors. “It worked in simulation” is nominal evidence unless the simulation varied the uncertainties that matter.

Sensitivity partitions the loop #

For loop transfer $L=PC$,

$$S=\frac{1}{1+L},\qquad T=\frac{L}{1+L},\qquad CS=\frac{C}{1+L}.$$

• $S$ shapes output response to many load disturbances and plant perturbations.
• $T$ shapes reference transmission and measurement-noise paths, depending on architecture.
• $CS$ exposes control effort and actuator demand.

At low frequency, large $|L|$ makes $|S|$ small and improves regulation. At high frequency, small $|L|$ keeps $|T|$ small and avoids chasing noise and unmodeled dynamics. Crossover is the negotiated boundary.

flowchart TD
  D["low-frequency disturbances"] --> S["small S wanted"]
  N["high-frequency noise and uncertainty"] --> T["small T wanted"]
  S --> L["shape loop L = PC"]
  T --> L
  L --> M["margins, effort and bandwidth"]

Because $S+T=1$, both cannot be small at the same frequency. Bode's sensitivity integral adds a waterbed constraint: for important plant classes, pushing sensitivity down in one region forces it up elsewhere.

Describe uncertainty before proving robustness #

Useful models include:

• parametric ranges for gains, time constants and delays;
• additive uncertainty $P=P_0+W_a\Delta$;
• multiplicative uncertainty $P=P_0(1+W_m\Delta)$;
• neglected high-frequency dynamics;
• nonlinear limits and mode changes tested outside linear robust analysis.

The weight $W_m(j\omega)$ should bound relative model error over frequency. If $|\Delta|\le1$, a small-gain robust-stability condition for output-multiplicative uncertainty is

$$\lVert W_m T\rVert_\infty<1.$$

This statement is only as credible as the uncertainty bound. Data from one operating point do not justify a global $W_m$.

Classical loop shaping #

Shape $L$ so it has:

• high enough low-frequency gain for tracking and disturbance rejection;
• crossover compatible with plant delay and uncertainty;
• adequate gain and phase margins;
• roll-off before noise and unmodeled resonances;
• tolerable peaks in $S$ and $T$;
• feasible control effort and rate.

The maximum sensitivity $M_s=\max_\omega|S(j\omega)|$ summarizes the worst disturbance/uncertainty amplification and relates to distance from the Nyquist critical point. It is more informative than tuning for a pretty step response alone.

H∞ formulation #

The $H_\infty$ norm of a stable transfer matrix is its maximum singular value over frequency. Weighted mixed-sensitivity design asks for a stabilizing controller that makes

$$\left\Vert \begin{bmatrix} W_S S\\W_U CS\\W_T T \end{bmatrix} \right\Vert_\infty<\gamma.$$

$W_S$ encodes desired error/disturbance attenuation, $W_U$ control limits, and $W_T$ high-frequency robustness/noise requirements. The optimization is powerful because it handles multivariable interaction and worst-case amplification. It does not choose physically meaningful weights or uncertainty sets for the engineer.

Robustness workflow #

1. Identify the decision-relevant operating region and uncertain mechanisms.
2. Validate nominal dynamics and bound error in time and frequency.
3. Write performance outputs and exogenous inputs explicitly.
4. Shape or synthesize the controller.
5. Verify nominal stability and performance.
6. Verify robust stability separately from robust performance.
7. Test nonlinear saturation, rate limits, sampling, quantization and mode changes.
8. Validate on representative plants and disturbances, then update uncertainty bounds.

Robust control is not conservatism for its own sake. It is a reproducible claim of what the closed loop tolerates, where the claim applies, and which uncertainty would invalidate it.