Fractional-Order Extended State Observers

• Fractional-Order ESO (FESO) is a state observer that uses fractional calculus to estimate both the system state and total disturbance in fractional-order dynamics.
• Its improved variant, IFESO, explicitly separates known plant dynamics from residual disturbances, resulting in enhanced disturbance rejection, faster transient response, and better robustness.
• Empirical and simulation studies demonstrate that FESO-based ADRC transforms fractional-order plant behavior into near-ideal integrator performance while maintaining stability under parameter variations.

Fractional-Order Extended State Observers (FESO), and their improved variants, represent a significant advancement in active disturbance rejection control (ADRC) for fractional-order dynamical systems. FESOs are designed to accurately estimate both the plant state and total disturbance, explicitly incorporating fractional calculus into the observer architecture, thereby exploiting the real system's dynamics with greater fidelity than integer-order counterparts. Recent developments, notably the Improved Fractional-Order @@@@1@@@@ (IFESO), offer enhanced performance on the key metrics of disturbance rejection, transient response, and robustness, as extensively analyzed in the literature (Li et al., 2021, Li et al., 2021).

1. Mathematical Model and Observer Structure

Consider a fractional-order single-input/single-output system modeled as

$$G_p(s) = \frac{b_0}{s^\mu + a_0}, \qquad 0<\mu<1,\; a_0 > 0,\; b_0 > 0.$$

The time-domain representation is

$D^\mu y(t) + a_0 y(t) = b_0 u(t) + d_{\rm ext}(t),$

where $D^\mu$ denotes a fractional derivative (Caputo or Riemann–Liouville), $y(t)$ is the output, $u(t)$ is the control, and $d_{\rm ext}(t)$ is an external disturbance. The total disturbance is defined as

$$f_0(t) = -a_0 y(t) + (b_0 - b) u(t) + d_{\rm ext}(t),$$

with $b$ a nominal gain.

The state-augmented plant dynamics become

$$\begin{cases} D^\mu x_1 = x_2 + b u, \ D^\mu x_2 = D^\mu f_0, \end{cases}$$

where $x_1 = y$, $x_2 = f_0$. The ADRC approach uses an observer to estimate $(x_1, x_2)$ in real time, facilitating real-time disturbance compensation.

The canonical IFESO is

$$\begin{aligned} D^\mu z_1(t) &= z_2(t) + b u(t) + \beta_1 (y(t) - z_1(t)), \ D^\mu z_2(t) &= \beta_2 (y(t) - z_1(t)), \end{aligned}$$

where $(z_1, z_2)$ are the observer states and $(\beta_1, \beta_2)$ the observer gains; typically, $\beta_1 = 2\omega_o$, $\beta_2 = \omega_o^2$ under bandwidth parameterization.

2. Improved Fractional-Order ESO (IFESO): Formulation and Structural Advances

Traditional fractional-order ESOs (FO-ESO) estimate all unmodeled dynamics and external inputs as a generic, fully unknown disturbance. The IFESO leverages explicit structural information from the plant, separating the known component $q$ from the unknown remainder $f_{ifo}$: $$y^{(n\gamma)} = q(y^{(n\gamma)}, y^{(m)}, t) + f_{ifo} + b_0 u,$$ where

$q := y^{(n\gamma)} - y^{(m)},$

$$f_{ifo} = -\sum_{i=1}^{m-1} a_i y^{(i)} - a_0 y + (b-b_0)u + d.$$

The IFESO observer structure is then given by

$$z^{(\gamma)} = A z + B u + F \hat{q} + L (y - C z),$$

with $\hat{f}_{ifo}=z_{n+1}$, and block-diagram architecture capturing both plant states and disturbance estimates. This explicit use of known plant structure (see (Li et al., 2021), eqs. 41–44) results in lower observer dimension and simplified gain design vs. conventional FO-ESO.

3. Stability Conditions and Controller Synthesis

The IFESO and its closed-loop integration in ADRC admit bounded-input bounded-output (BIBO) stability under explicit and easily verifiable parameter choices. For the base-case IFESO, stability is ensured if

$$\beta_1 > 0, \quad \beta_2 > 0, \quad \mu \in (0,1).$$

In the closed loop, a proportional control law $v(t) = K(r(t)-z_1(t))$ and the IFESO-based control input

$u(t) = \frac{v(t) - z_2(t)}{b},$

yield the closed-loop characteristic equation

$$b \omega_o^2 s^\kappa + (\cdots) s^{\kappa-1} + \cdots + (b \beta_2 K) = 0,$$

where $\kappa = 1 + (\mu^{-1} - 1)$. The roots must lie in the sector $|\arg(s)| < \frac{\pi}{2q}$ to guarantee BIBO stability (Li et al., 2021). In the higher-order IFO-ESO case, specifically constructed observer gains (binomial scaling with a bandwidth parameter $\omega_0$) guarantee the observer estimation error remains BIBO stable for any bounded input, via the shape of the characteristic polynomial in the Laplace domain (Li et al., 2021).

4. Frequency-Domain and Time-Domain Performance Comparison

Applying Laplace analysis to the closed-loop system, one obtains compensated plant transfer functions $G_{\rm ifo}(s)$ (fractional ESO) and $G_{i}(s)$ (integer-order ESO). The degree of approximation to the ideal integrator ($1/s$ or fractional integrator $1/s^{n\gamma}$) is quantified by the mean-square error over frequency: $$e_o(\omega) = |\Delta_o(\omega)|^2, \quad \Delta_o(\omega) = 1 - (j\omega)^{n\gamma} \frac{Y(j\omega)}{U_0(j\omega)}.$$ Empirical and analytic results show that IFESO/IFO-ADRC yields

• Lower mean-square error $e_{ifo}$ than both FO-ESO and integer-ESO variants
• Reduced sensitivity to variation in plant parameters ($a_0$), observer rate ($\omega_0$), and fractional order ($\gamma$)
• Better tracking of the ideal integrator behavior across control bandwidth

Bode and MSE plots, as in (Li et al., 2021, Li et al., 2021), confirm these advantages empirically and in simulation.

Controller	MSE Tracking Error	Robustness to Gain Changes	Bandwidth Fidelity
IO-ADRC	High	Poor	Limited
FO-ADRC	Moderate	Improved	Better
IFADRC/IFO-ADRC	Minimal	Superior	Highest

5. Simulation and Experimental Evaluation

Benchmarks on representative fractional plants (e.g., $G_p(s) = \frac{50.8}{s^{0.8}+10}$ (Li et al., 2021) and $G_p(s) = \frac{2380.9}{s^2+138.1 s+3819.7}$ for PMSM systems (Li et al., 2021)) demonstrate that IFADRC/IFO-ADRC achieves:

• Fastest rise and settling times
• Minimal percent overshoot
• Invariance of transient and disturbance rejection features against large gain variations
• Robust performance under both step inputs and parameter drift

For instance, on a PMSM speed-servo, IFO-ADRC produced an overshoot $M_p=3.3\%$ vs. $4.8\%$ (FO-ADRC) and $5.4\%$ (IO-ADRC), with the shortest settling time and lowest overshoot fluctuation under ±40% controller gain perturbation.

6. Practical Implementation and Application Scope

Practical implementation of IFESO-based ADRC has been validated both in real-time simulation (MATLAB/Simulink) and in laboratory PMSM servo systems, with the discrete fractional operators realized via impulse-response invariant discretization at high sampling frequencies (e.g., 8 kHz). The architecture preserves simplicity in controller tuning—often requiring only a bandwidth parameter and proportional gain—while delivering superior performance over a greater frequency range.

A notable property is the capability of the IFESO+ADRC structure to convert a fractional-order plant to an integrator in closed-loop, effectively "cancelling" the plant’s fractional dynamics and yielding a fully predictable, robust, integer-order response to disturbances.

7. Context, Robustness Analysis, and Impact

FESO and especially its improved forms (IFESO, IFO-ESO) address the limitations of conventional integer-order and fractional-order observers, primarily by integrating known plant structure explicitly and reducing reliance on generic disturbance lumping. The result is a marked improvement in both estimation accuracy and system robustness—notably, improved disturbance rejection, less-peaked step responses, and minimal performance degradation under plant and controller variations.

This paradigm establishes FESO-based approaches as state-of-the-art for applications where plant fractional dynamics are significant, such as electrical drives, viscoelastic actuators, and diffusion systems. Furthermore, the theoretical stability assurances and ease of parameter selection (via observer bandwidth and gain scaling) make these controllers tractable for both simulation and real-time embedded deployment (Li et al., 2021, Li et al., 2021).

A plausible implication is that, as the theoretical and numerical tools for fractional calculus mature, further refinements and automated gain tuning methodologies will further increase the applicability of FESO-based ADRC to increasingly complex, high-performance control scenarios.