Reliability of Control and Robust Control

LQR/LQG assume that we know the exact model of the system. But real parameters (mass, stiffness, resistance) are known only approximately: manufacturing tolerances, wear, temperature fluctuations. Moreover, real systems have unmodeled dynamics (for example, the flexibility of metal, which is ignored in the model as a “rigid body”). Robust control—a field developed in the 1980s–90s (Doyle, Glover, Khargonekar, Stein)—provides tools for designing controllers that are guaranteed to work under permissible uncertainty. Today it is a standard in aviation, automation, and autonomous systems.

Sources of Uncertainty

Parametric. Specific model parameters are inaccurate: mass $m \in [0.9, 1.1]$ kg, spring stiffness $k \in [k_0(1 − 0.2), k_0(1 + 0.2)]$. Described by intervals or probabilistically.

Structural (unmodeled dynamics). Actual transfer function $G_\text{real}(s) = G_\text{nominal}(s) \cdot (1 + \Delta(s))$ where $\Delta$ is an unknown “perturbation”, usually bounded by the norm $||\Delta||_\infty < \gamma$. For example, flexible modes of the structure, which manifest at high frequencies.

Disturbances and noises. External forces (wind gusts, road bumps), sensor noises. Described as stochastic processes or bounded signals ($||d||_2 \leq D$).

Measures of Robustness

Gain Margin (GM). By what factor can the open-loop gain be increased before the closed-loop becomes unstable. Standard: GM $\geq$ 6 dB (factor of 2).

Phase Margin (PM). By how many degrees can the phase be shifted before losing stability. Standard: PM $\geq$ 45°.

Low gain and low phase shift—large margin. These measures are easily calculated from the Bode diagram: GM—at frequency where phase = −180°, PM—at frequency where $|G| = 1$.

Sensitivity function S and complementary T. $S = \frac{1}{1 + L},;; T = \frac{L}{1 + L}$ where $L$ is the open-loop system. $||S||\infty$—how much the loop amplifies disturbances. $||T||\infty$—sensitivity to measurement noise. Conservation law: $S + T = 1$, i.e. it is impossible to make both small throughout the whole frequency range.

$H_\infty$ Control

$H_\infty$-norm of transfer matrix: $||G||\infty = \sup{\omega} \sigma_\text{max}(G(j\omega))$—maximum singular value over all frequencies. This is the “worst-case” transmission coefficient of disturbance to output.

$H_\infty$ problem statement: Find controller $K$ minimizing $||T_{zw}||\infty$, where $T{zw}$ is the transfer function from disturbance $w$ to evaluated output $z$ (usually includes tracking error, control force, state).

Mixed-sensitivity: Standard formulation with weighting functions $W_S$, $W_T$, $W_{KS}$: Minimize $|| [W_S \cdot S;; W_{KS} \cdot K \cdot S;; W_T \cdot T] ||\infty$. $W_S$—desired sensitivity (want low at low frequencies to suppress disturbances). $W_T$—desired $T$ (want low at high frequencies against noise). $W{KS}$—constraint on control effort.

Solution via two Riccati equations (Doyle-Glover-Khargonekar-Francis, 1989). Implemented in MATLAB Robust Control Toolbox (hinfsyn), Python (python-control).

$\mu$-Synthesis

For structured uncertainties (several independent blocks $\Delta_1$, ..., $\Delta_k$): $\mu$—structured singular value.

$\mu$-synthesis (D-K iteration): Alternating optimization of $K$ ($H_\infty$) and scaling matrices $D$. In practice, converges to a local optimum. Used for complex systems with many sources of uncertainty.

Adaptive Control

Idea. Controller parameters are adjusted online to actual plant parameters.

Self-Tuning Regulator (STR, Åström). Online identification of parameters (RLS—recursive least squares) $\rightarrow$ recalculation of $K$. Suitable for slowly changing parameters.

Model Reference Adaptive Control (MRAC, Landau). Reference model sets desired behavior $y_m(t)$. Parameter adaptation law minimizes $e_m = y - y_m$. Stability guarantees through Lyapunov functions.

Gain scheduling. Ready-made set of controllers $K_1$, ..., $K_M$ for different modes; switching based on measured parameters. Aircraft: different $K$ at low/high speeds. Simple, but requires careful justification of transitions.

Numerical Example: Pendulum with Unknown Length

Pendulum: $\ddot{\theta} = \frac{g}{l} \cdot \sin \theta + \frac{u}{m l^2}$. Length $l \in [0.5, 1.5]$ m, nominal $l_0 = 1$ m.

Linearization around $\theta = 0$: $\ddot{\theta} = \frac{g}{l} \cdot \theta + \frac{u}{m l^2}$—unstable. PD controller: $u = -K_P \cdot \theta - K_D \cdot \dot{\theta}$.

Closed-loop: $\ddot{\theta} = \left( \frac{g}{l} - \frac{K_P}{m l^2} \right) \theta - \frac{K_D}{m l^2} \dot{\theta}$.

Stability condition: $\frac{g}{l} < \frac{K_P}{m l^2}\ \rightarrow K_P > m l g$.

With $K_P$ calculated for $l = 1$ ($m = 1$, $g = 9.81$): $K_P > 9.81$. Let’s take $K_P = 30$, $K_D = 5$.

Check for $l = 1.5$: $\frac{g}{l} = 6.54$, $\frac{K_P}{m l^2} = 30/2.25 = 13.33 > 6.54$ ✓ stable. For $l = 0.5$: $\frac{g}{l} = 19.62$, $\frac{K_P}{m l^2} = 30/0.25 = 120 \gg 19.62$, hyper-stable (possible overshoot).

With $l = 3$: $\frac{g}{l} = 3.27$, $\frac{K_P}{m l^2} = 30/9 = 3.33$—at the stability margin. If $l > 3.06$—stability is lost.

Solution with adaptive element. Estimate $l$ online via RLS from $\ddot{\theta}_\text{observed} = \frac{g}{l} \cdot \theta + \frac{u}{m l^2}$. Adaptive $K_P(\hat{l}) = m \hat{l} g$ (safety factor 3)—guaranteed stable for any $l \in [0.5, 3]$.

Real Applications

Aviation. All passenger aircraft Airbus A320–A380 are designed on principles of $H_\infty$/$\mu$-synthesis, including “control law switching” system in failures. F-16, F-22—gain-scheduled controllers for different flight regimes.
Hard disks and drives. Head positioning control onto track—a high-speed $H_\infty$ regulator considering flexibility of suspension. Precision 1–10 nm.
Nuclear reactors. Power control with stability guarantees under temperature changes, fuel burnup—$H_\infty$ synthesis.
Semiconductor manufacturing. ASML EUV lithography machines—positioning control with accuracy < 1 nm at 10g accelerations. Robust synthesis—a critical component.
Automotive cruise control. Adaptive controllers (ACC) account for changes in mass (cargo, passengers) and road slope.

Assignment. Pendulum with unknown length $l \in [0.5, 1.5]$ m, $m = 1$ kg, $g = 9.81$. Linearized model near $\theta = 0$: $\ddot{\theta} = \frac{g}{l} \cdot \theta + \frac{u}{m l^2}$.
(a) For nominal model $l_0 = 1$, design a PD controller $u = -K_P \cdot \theta - K_D \cdot \dot{\theta}$ with: damping $\zeta = 0.7$, $\omega_n = 5$ rad/s.
(b) Check stability for $l = 0.5,\ 1.0,\ 1.5$: compute eigenvalues of closed system.
(c) Simulate response for $\theta(0) = 0.1$ rad for all three values of $l$.
(d) Implement simple adaptation: estimate $l$ via least squares from last 20 observations $(\theta, \dot{\theta}, u)$, recalculate $K_P, K_D$ in real time.
(e) What is robustness at $l = 2$ (outside nominal range)?