Adaptive Observers: Methods & Applications

• Adaptive observers are dynamic systems that estimate both the state and unknown parameters to cope with time-varying and uncertain dynamics.
• They employ methods like dynamic regressor extension and mixing (DREM) and parameter augmentation to decouple estimation in switched and nonlinear settings.
• Robust design guarantees exponential error convergence even with disturbances, offering practical tools for real-time monitoring, fault detection, and adaptive control.

Adaptive observers are dynamic systems designed for the joint estimation of states and unknown parameters in dynamical systems, particularly when some parameters are time-varying or subject to structural changes. Their development is fundamental for advanced control, identification, and monitoring of systems where partial knowledge, switching, nonlinearity, or uncertainties preclude classical observer solutions.

1. Definitions and Theoretical Foundations

An adaptive observer estimates both the (possibly unmeasured) state vector $x(t)$ and a vector of unknown system parameters $\theta$. The canonical continuous-time adaptive observer for a system with known structure but unknown constant parameters is

$$\begin{aligned} \dot x &= f(x, u, \theta),\ y &= h(x),\ \dot{\hat x} &= f(\hat x, u, \hat\theta) + K(y - h(\hat x)), \ \dot{\hat\theta} &= \Gamma \, \Phi(\cdot) (y - h(\hat x)), \end{aligned}$$

where $K$ is a user-designed observer gain and $\Gamma$ sets the adaptation rate. The function $\Phi(\cdot)$ depends on the system regressor structure, and $u$ is a known input. Key requirements are system observability, a persistency of excitation (PE) condition on the regressor, and boundedness of all necessary signals.

For systems with nonlinearly parameterized dependencies, switching, or time-varying modes, the basic structure is extended using tools such as parameter augmentation, regressor extension, or dynamic regressor mixing (DREM).

2. Adaptive Observers for Switched and Hybrid Systems

Systems with unknown switched parameters exhibit mode-dependent dynamics governed by a piecewise-constant switching signal $\sigma(t)$. Standard regression designs are obstructed by zero-input responses (SASI) at switching instants, which act as additive disturbances and prevent parameter convergence.

The DREM-based approach for such systems, as formalized in "Adaptive Observer for a Class of Systems with Switched Unknown Parameters Using DREM" (Liu et al., 2022), leverages parameter augmentation. Defining an augmented parameter vector that combines the current subsystem parameters and the state at switching, the observer employs dynamic regression mixing to decouple estimation of true parameters from SASI. The resulting adaptive observer ensures that both parameter and state errors converge to zero provided a non-square-integrability (NSI) condition on the scalar mixing signal is satisfied: $$\int_0^\infty \chi_i(t) \Delta^2(t) \, dt = \infty \quad (i = 1, ..., s)$$ where $\chi_i(t)$ is the indicator for mode $i$, and $\Delta(t)$ is the DREM-generated scalar regressor. This condition is strictly weaker than classical PE.

3. Dynamic Regressor Extension and Mixing (DREM)

DREM is a regressor transformation methodology that enables adaptive estimation in multidimensional or weakly excited settings by:

• Forming multiple filtered regressions (dynamic extension) to build a high-rank stacked regressor matrix $N^T$;
• Performing algebraic mixing using adjugate and determinant operations, ultimately yielding scalar decoupled regression equations for each parameter:

$$\bar{\mathcal{Z}}_i(t) = \Delta(t) \bar{\theta}^*_i(t)$$

DREM elementwise adaptation allows each parameter to evolve independently, mitigating the effect of unknown persistent disturbances (such as SASI after switches) and regularizing excitation requirements. This framework supports reduced-order observer architectures and is robust to disturbances and measurement noise, provided a basic PE or NSI condition on the scalar regressor is met (Tavan, 26 Jan 2025).

4. Robustness and Performance Guarantees

Adaptive observers are subject to robustness properties determined by disturbance bounds, regressor excitation, and observer gain selection. The Lyapunov-stability-based analyses show that under persistent excitation (PE) of the DREM regressor, both state and parameter errors converge exponentially to an invariant set whose size is set by the disturbance amplitude and observer gain: $$\|\tilde{x}\| + \|\tilde{\theta}_i\| \le \mu \sup_t |\Delta D| + c$$ where $D$ collects disturbance and noise terms, and $\mu$, $c$ are computable from observer and system properties (Liu et al., 2022).

Adaptive observers can also be made resilient to both bounded process and measurement noise. This is critical for applications in chaotic systems, such as Chua circuits, where switched parameters or strong nonlinearities occur and simulation demonstrates robust convergence even in the presence of significant perturbations.

5. Observer Design for Discrete-Time and Nonlinear Systems

For discrete-time and nonlinear systems, adaptive observer design often relies on transforming the original plant dynamics into quasi-LPV or Takagi-Sugeno (T-S) polytopic forms. This embedding converts nonlinearities and parameter dependencies into a weighted sum over affine or linear-in-parameter submodels, parametrized by the premise variables.

Joint state and parameter estimation is then addressed using a parallel structure: $$\begin{aligned} \hat{x}_{k+1} &= \sum_{i=1}^s h_i(\hat{z}_k, \hat{\theta}_k)\left[A_i \hat{x}_k + B_i u_k + L_i (y_k - C\hat{x}_k)\right], \ \hat{\theta}_{k+1} &= \hat{\theta}_k + \sum_{i=1}^s h_i(\hat{z}_k, \hat{\theta}_k) K_{y,i}(y_k - C\hat{x}_k) \end{aligned}$$ The observer gains are chosen to satisfy Lyapunov matrix inequalities (LMIs) that guarantee exponential error convergence and $\mathbb L_2$-disturbance attenuation, even when premise variables are unmeasured and induce "uncertain-like" system-parameter mismatches (Srinivasarengan et al., 2017).

6. Applications, Numerical Studies, and Practical Guidelines

Switched Parameter Systems:

The DREM-based observer has been validated on a piecewise-linear Chua-type oscillator, where the switching law is induced by a state-dependent region partition. Numerical studies confirm that:

• The DREM determinant $\Delta(t)$ satisfies NSI, validating excitation.
• Parameter tracking is strictly monotonic within the active mode and remains frozen otherwise, ensuring stability across all switches.
• Both parameter and state estimation errors exhibit monotonic decay and vanish asymptotically.
• Robust versions remain accurate under additive process and measurement disturbances (Liu et al., 2022).

Nonlinear/LPV Discrete-Time Systems:

Discrete-time observers employing T-S polytopic embeddings and sector-nonlinearity transformations have been applied to chemical reaction models (e.g., wastewater treatment) and are shown to converge under practical finite-state switching and bounded parametric uncertainty. Observer gains are efficiently computed via LMI solvers. Guarantees extend to cases with only partial premise measurement (Srinivasarengan et al., 2017).

Practical Design Considerations:

• Gains should be selected to ensure system (A-KC or analog) is Hurwitz, balancing speed against noise sensitivity.
• Excitation requirements are relaxed in DREM-based approaches (NSI instead of classical PE).
• Disturbances and noise necessitate stronger excitation or adaptation gain, but the invariant set bound scales with their amplitude.
• For discrete and T-S models, block-diagonal or structured Lyapunov matrices enable efficient LMI-based observer synthesis.

7. Significance and Broader Implications

Adaptive observers constitute a central methodology in nonlinear, hybrid, and switched-systems theory, bridging model-based estimation with robust real-time adaptation to unmodeled changes and system uncertainties. The DREM principle has introduced a paradigm shift by enabling decoupled parameter adaptation, relaxing excitation conditions, and providing strong robustness guarantees in switched systems and nonlinear affine cases (Liu et al., 2022, Tavan, 26 Jan 2025).

Current directions include robustification with projection operators and dead-zones in adaptation laws, extension to MIMO, distributed, or descriptor systems, and integration within output regulation, adaptive control, and cooperative estimation architectures.

Adaptive observer theory thus provides foundational and practical tools for advanced monitoring, fault detection, and real-time control in applications ranging from nonlinear electronics (as in chaotic circuits) to large-scale networked systems and multi-agent consensus (Liu et al., 2022, Cai et al., 2020).