ActiveDisturbanceRejection.md

Active Disturbance Rejection Control (ADRC + ESO)

Overview & Motivation

Active Disturbance Rejection Control addresses a fundamental tension in feedback design: high-performance control normally requires an accurate plant model, yet accurate models are expensive to identify and degrade with temperature, load, and wear. ADRC resolves this by treating everything beyond a known input gain — unmodeled dynamics, parameter variation, and external disturbances — as a single lumped signal called the total disturbance. An Extended State Observer (ESO) estimates this signal in real time, and the control law subtracts the estimate before issuing the command. What remains behaves like a clean chain of integrators that a simple PD law can regulate with textbook bandwidth.

The practical payoff on embedded hardware is significant: you need only one plant number ($b_0$, the rough input gain) and two tuning dials. The controller then survives a bad model because any mismatch is absorbed into the disturbance estimate.

Mathematical Theory

Plant Representation

An $n$-th order SISO plant is written as the canonical integrator chain plus a total-disturbance term $f$:

$$y^{(n)} = f(t, y, \dot{y}, \ldots, d) + b_0 u$$

where $f$ captures unmodeled dynamics, nonlinearities, and external loads; $b_0$ is a nominal input-gain estimate; and $u$ is the control input.

Extended State Observer

Augmenting the $n$ plant states with $x_{n+1} = f$ yields an $(n+1)$-dimensional system. The continuous ESO is a Luenberger-type observer driven by the output error:

$$\dot{\hat{x}}_i = \hat{x}_{i+1} + \beta_i (y - \hat{x}_1), \quad i = 1, \ldots, n$$ $$\dot{\hat{x}}_{n+1} = \beta_{n+1} (y - \hat{x}_1)$$

with the convention $\hat{x}_{n+1} = \hat{f}$ and $\hat{x}_2$ through $\hat{x}_n$ as derivative estimates.

The forward-Euler discretization used here is:

$$\hat{x}_i[k+1] = \hat{x}_i[k] + T_s \bigl(\beta_i , e[k] + \hat{x}_{i+1}[k]\bigr), \quad e[k] = y[k] - \hat{x}_1[k]$$

with $b_0 u[k-1]$ injected into the $(n)$-th state to drive the highest derivative.

Bandwidth Parameterization (Gao)

All observer poles are placed at $-\omega_o$ (Gao's bandwidth parameterization). The resulting gains follow the binomial expansion of $(\lambda + \omega_o)^{n+1}$:

$$\beta_i = \binom{n+1}{i} \omega_o^i, \quad i = 1, \ldots, n+1$$

All control poles are placed at $-\omega_c$ via the expansion of $(\lambda + \omega_c)^n$:

$$k_i = \binom{n}{i} \omega_c^i, \quad i = 1, \ldots, n$$

For a second-order plant ($n = 2$):

$$\beta = [3\omega_o,; 3\omega_o^2,; \omega_o^3], \quad k = [\omega_c^2,; 2\omega_c]$$

Control Law

After disturbance estimation the control is:

$$u = \frac{u_0 - \hat{f}}{b_0}, \qquad u_0 = k_1(r - \hat{x}_1) - \sum_{i=2}^{n} k_i \hat{x}_i$$

Substituting into the plant equation and using $\hat{f} \approx f$ gives the closed-loop residual $y^{(n)} \approx u_0$, a pure integrator chain under a PD law — independent of the original plant dynamics.

Complexity Analysis

Case	Time	Space	Notes
Best	$O(n)$	$O(n)$	Linear sweep over $n+1$ ESO states
Average	$O(n)$	$O(n)$	Same; gains precomputed at construction
Worst	$O(n)$	$O(n)$	No branching in the hot path

Gains are computed once at construction from closed-form binomial formulas in $O(n)$ time. The Compute hot path is a pair of $O(n)$ loops with no dynamic allocation.

Step-by-Step Walkthrough

Second-order plant ($n=2$), $\omega_o = 30$, $\omega_c = 6$, $b_0 = 1$, $T_s = 0.001$ s.

Observer gains: $\beta_1 = 90$, $\beta_2 = 2700$, $\beta_3 = 27000$.
Control gains: $k_p = 36$, $k_d = 12$.

At sample $k$ with state $\hat{x} = [\hat{y}, \hat{\dot{y}}, \hat{f}]$, measurement $y[k]$, reference $r$:

1. Output error: $e = y[k] - \hat{y}$.
2. Inject correction into all three states: $\hat{x}_i \mathrel{+}= T_s \beta_i e$.
3. Chain integration: $\hat{y} \mathrel{+}= T_s \hat{\dot{y}}$; then $\hat{\dot{y}} \mathrel{+}= T_s b_0 u_\text{applied}[k-1]$, where $u_\text{applied}$ is the value actually delivered by the actuator. When the actuator does not saturate this equals the commanded $u[k-1]$; when it does, pass the clipped value via the three-argument Compute(reference, measuredOutput, actualApplied) overload.
4. PD law on integrator chain: $u_0 = k_p(r - \hat{y}) - k_d \hat{\dot{y}}$.
5. Disturbance cancellation: $u = (u_0 - \hat{f}) / b_0$.

After a transient of roughly $5/\omega_o \approx 0.17$ s the observer converges; the output tracks $r$ with bandwidth $\omega_c$.

Pitfalls & Edge Cases

Actuator saturation (anti-windup). When the downstream actuator clips the commanded value, the ESO integrates the wrong input and its disturbance estimate drifts, causing significant overshoot. Close the anti-windup loop by feeding the actually-applied signal back: u_applied = clamp(Compute(r, y, u_applied_prev), lo, hi), then pass u_applied as the third argument on the next call. The two-argument Compute(reference, measuredOutput) is a convenience overload that assumes no saturation (commanded = applied) and is equivalent to passing the last commanded value.

ESO peaking. Large initial estimation errors drive high-magnitude corrections, temporarily saturating the actuator. Mitigation: initialize the observer near the first measurement, or schedule $\omega_o$ upward from a low value during the first few samples.

Observer bandwidth vs. noise. Increasing $\omega_o$ speeds convergence but amplifies measurement noise because $\beta_3 = \omega_o^3$ grows cubically. A practical rule of thumb is $\omega_o \in [3\omega_c, 10\omega_c]$.

$b_0$ mismatch. The ESO is robust to moderate mismatch (factor of 2–3), but large errors shrink the stability margin. If $b_0 \gg b_\text{true}$ the effective loop gain drops and response slows; if $b_0 \ll b_\text{true}$ the loop gain rises and may oscillate.

Euler discretization stability and accuracy. The forward-Euler ESO is unconditionally unstable once $\omega_o T_s$ exceeds the stability boundary of the $(n+1)$-th order observer polynomial. For a first-order plant ($n=1$) this boundary is $\omega_o T_s \approx 0.83$; higher orders have progressively tighter boundaries (approximately $0.54$ for $n=2$, $0.40$ for $n=3$). The constructor enforces $\omega_o T_s < 0.5/n$ as a conservative order-aware hard precondition (aborts if violated). Keeping $\omega_o T_s \leq 0.1$ is further recommended for accuracy; beyond that a ZOH or bilinear discretization is preferred.

Integer overflow in gain computation. Binomial coefficients are computed with integer arithmetic at compile time. For large orders or very high bandwidths the intermediate product may exceed std::size_t before the division; keep $n \leq 5$ in practice.

Variants & Generalizations

Nonlinear ESO (NESO). Replace the linear correction $\beta_i e$ with Han's fal function to reduce peaking while preserving fast convergence.

Discrete ESO. Exact discretization of the observer (ZOH or pole-matched) improves accuracy when $\omega_o T_s$ is not small.

Higher-order plants. The template parameter Order generalizes the same bandwidth-parameterized structure to $n > 2$ — gains grow binomially and the Compute loop extends automatically.

Multi-input / multi-output (MIMO). Each output channel runs an independent ADRC; cross-coupling is absorbed into the respective disturbance estimates.

Applications

• Electric motor drives (rejects friction, load torque, and back-EMF variation with a single $b_0$ estimate).
• Attitude and position control of UAVs and satellites (absorbs aerodynamic and thruster uncertainty).
• Industrial process control where the plant model is poorly known or time-varying.
• Hard-disk drive servo (high-bandwidth disturbance rejection without a detailed head-media model).

Connections to Other Algorithms

• Luenberger Observer — the ESO is a Luenberger observer augmented with one extra disturbance state.
• Disturbance Observer (DOB) — the transfer-function sibling; DOB works in the frequency domain while ESO works in the state-space domain.
• PID — ADRC generalizes PID: a first-order ADRC with proportional-plus-integral action recovers a PI with disturbance feed-forward.
• LQR / LQI — state-feedback alternatives that require a full model; ADRC trades optimality for model-independence.

References & Further Reading

• J. Han, "From PID to Active Disturbance Rejection Control," IEEE Transactions on Industrial Electronics, vol. 56, no. 3, pp. 900–906, 2009.
• Z. Gao, "Scaling and Bandwidth-Parameterization Based Controller Tuning," Proceedings of the American Control Conference, 2003, pp. 4989–4996.
• R. Miklosovic, A. Radke, Z. Gao, "Discrete implementation and generalization of the extended state observer," ACC, 2006.