Immersion & Invariance Adaptive Observers

• Immersion and Invariance (I&I) Adaptive Observers are a framework for nonlinear state estimation that uses immersion mappings to form invariant manifolds ensuring robust error dynamics.
• They generalize and unify prior designs like the Kazantzis–Kravaris–Luenberger Observer and the Parameter Estimation-Based Observer through systematic PDE solutions.
• The methodology integrates adaptive estimation and ISS principles to deliver global asymptotic convergence and enhanced robustness in nonlinearly parameterized systems.

The Immersion and Invariance (I&I) Adaptive Observer is a systematic framework for designing nonlinear state observers and adaptive estimators that unify and generalize various prior strategies, notably the Kazantzis-Kravaris-Luenberger Observer (KKLO) and Parameter Estimation-Based Observer (PEBO) methods. I&I observers employ immersion mappings to create an invariant manifold in the extended observer state space, ensuring that the estimation error dynamics remain attractive and robust to modeling uncertainties, disturbance, and parameter variations. Central elements include a well-posed immersion, a parameterized output injection mechanism, and the solution of an invariance partial differential equation (PDE) to ensure the stability of the estimation process. The framework supports both standard state-observer synthesis and adaptive estimation for nonlinearly parameterized systems (Yi et al., 2017, Wang et al., 2021).

1. General Structure of I&I Adaptive Observers

Consider a general nonlinear plant: $$\dot x = f(x, u), \quad y = h(x), \quad x \in \mathbb{R}^n, \; u \in \mathbb{R}^m, \; y \in \mathbb{R}^p$$ Assuming $x(t)$ is bounded, the I&I approach requires:

• An immersion $\phi:\mathbb{R}^n \to \mathbb{R}^{n_z}$, injective with $n_z \geq n$.
• A left-inverse $$L: \mathbb{R}^{n_z} \times \mathbb{R}^p \to \mathbb{R}^n$$ satisfying $L(\phi(x), h(x)) = x$.
• An output injection $$\beta: \mathbb{R}^p \times \mathbb{R}^{n_\chi} \to \mathbb{R}^{n_z}$$ and a “free” injection $Q(y,\chi,u)$ into the kernel of $\nabla_\chi \beta^\top$.
• An observer state $\chi \in \mathbb{R}^{n_\chi}$, with $n_\chi \geq n_z$.

The observer estimates the state as: $\hat x = L(\beta(y,\chi), y)$ with dynamics: $$\dot \chi = -[\nabla_\chi \beta(y,\chi)]^\dagger \left(\nabla_y \beta(y,\chi)^\top \nabla h(\hat x) f(\hat x, u) - \nabla \phi(\hat x)^\top f(\hat x, u)\right) + [I - [\nabla_\chi \beta]^\dagger \nabla_\chi \beta^\top ] Q(y, \chi, u)$$ where $[\cdot]^\dagger$ denotes the Moore–Penrose pseudoinverse (Yi et al., 2017).

2. Immersion Mapping, Invariant Manifold, and the Invariance PDE

The immersion mapping $\phi$ constructs an invariant manifold in the augmented observer space. Define the off-manifold error: $$M(t) = \beta(y(t), \chi(t)) - \phi(x(t)) \in \mathbb{R}^{n_z}$$

Differentiating $M$ along system trajectories yields: $$\dot M = \nabla_y \beta(y, \chi)^\top \nabla h(x) f(x, u) + \nabla_\chi \beta(y, \chi)^\top \dot \chi - \nabla \phi(x)^\top f(x, u)$$

To ensure manifold invariance and local attractivity, the observer's dynamics are designed so that $M=0$ is an attractive, invariant set. This leads to the invariance PDE: $$\nabla \phi(x)^\top f(x, u) = \nabla_y \beta(y, \chi)^\top \nabla h(x) f(x, u) + \nabla_\chi \beta(y, \chi)^\top \alpha(y, \chi, u)$$ where $\alpha(y,\chi,u)$ is the principal part of the observer dynamics (Yi et al., 2017). The free mapping $Q$ shapes the kernel dynamics.

For parameter- and state-unknown systems (e.g., with $f(x)$ dependent on unknown $\theta$), I&I can embed standard adaptation in the observer structure (Wang et al., 2021).

3. Convergence and Stability Analysis

Under conditions:

• Full rank $\nabla_\chi \beta(y, \chi)$ for all $(y, \chi)$.
• Asymptotic stability of the off-manifold error $M=0$ in the closed-loop system,

the I&I observer ensures global asymptotic convergence of $\hat x$ to $x$: $\lim_{t \to \infty} |\hat x(t) - x(t)| = 0$ Proofs employ Lyapunov analysis with $V(M)=\frac{1}{2}|M|^2$ and ensure strict decrease of $V(M)$ along trajectories: $\dot V = M^\top \Upsilon_0(x, M, u) \leq -W(M)$ for some positive definite $W(M)$ (Yi et al., 2017). For adaptive estimation, the framework connects the estimation error subsystem and the tracking error subsystem via Input-to-State Stability (ISS) arguments, leading to explicit small-gain conditions and sum-type strict Lyapunov functions (Wang et al., 2021).

4. Design Methodology

The I&I observer design follows a systematic sequence:

1. Immersion: Select injective $\phi:\mathbb{R}^n \to \mathbb{R}^{n_z}$ and construct $L(\cdot, y)$.
2. Output Injection: Choose $\beta(y,\chi)$ with rank of $\nabla_\chi \beta$ equal to $n_z$.
3. Invariance PDE: Derive and solve for $\alpha(y, \chi, u)$ ensuring manifold invariance.
4. Free Injection $Q$: Assign $Q(y,\chi,u)$ within $\ker \nabla_\chi \beta^\top$ as a degree of freedom to influence off-manifold dynamics.
5. Adaptive Element: For parameter identification, incorporate adaptation laws for unknown $\theta$ within the observer state $\chi$.
6. Stability Verification: Check the attractivity of $M=0$ for the closed-loop off-manifold system, typically by Lyapunov methods (Yi et al., 2017, Wang et al., 2021).

For nonlinearly parameterized systems, invariance manifold shaping may be conducted via either explicit PDE solutions for $\beta$ or via filter-based approximations, as in the ISS-based designs (Wang et al., 2021).

5. Special Cases: KKLO and PEBO

The I&I framework encompasses prior major observer paradigms as special cases.

• Kazantzis–Kravaris–Luenberger Observer (KKLO): Setting $n_z=q=n_\chi=n_\xi$, $\beta(y,\chi) = \chi$, and arbitrary $Q$, the PDE reduces to $$\nabla \phi(x)^\top f(x, u) = \Lambda \phi(x) + B(y, u)$$ (with $\Lambda$ Hurwitz). The observer takes the standard KKLO form: $$\dot \xi = \Lambda \xi + B(y, u), \quad \hat x = L(\xi, y)$$
• Parameter-Estimation-Based Observer (PEBO): For partially measured states ($h(x)=y$), with $\Lambda = 0$, $\chi = (\xi, \zeta)$, $\beta = \xi + N(\zeta, \xi, y)$, and appropriately designed adaptation, the structure matches PEBO with parameter adaptation in $N$.

A table summarizing these specializations is provided below:

Observer	$\beta(y,\chi)$	Special Structure
KKLO	$\chi$	$\nabla_\chi \beta = I$, $\nabla_y \beta = 0$
PEBO	$\xi + N(\zeta,\xi,y)$	$h(x)=y$ (“partial-state measured”)

(Yi et al., 2017)

6. ISS-based Adaptive Observers and Robustness

Building on the I&I foundation, Input-to-State Stability (ISS) based approaches extend the methodology to address robustness with respect to perturbations and to structure adaptive observers for nonlinearly parameterized systems. For the system

$\dot x = f_1(x) + \phi(\theta, x) + g_1(x) u$

the estimation error coordinate is defined as $$\tilde \theta = \hat \theta - \theta + \beta(e,x_r)$$, introducing an “immersion manifold” that is enforced invariant via an estimator $\dot{\hat \theta}$ engineered to guarantee exponential stability of the estimation error subsystem. Interconnection of the ISS estimation-error subsystem with the ISS tracking-error subsystem yields uniform global asymptotic stability under a small-gain condition.

A salient feature is the explicit construction of a sum-type strict Lyapunov function: $$V_{cl}(t, e, \tilde \theta) = \int_0^{V_{err}(t, e)} \lambda_{err}(s) ds + \int_0^{V_{est}(t, \tilde \theta)} \lambda_{est}(s) ds$$ making the ISS gains explicit and facilitating robustness analysis against input perturbations (Wang et al., 2021).

7. Practical Performance and Simulation Benchmarks

Simulation results on an average-model Cúk converter compare I&I observers, KKLO, PEBO, a hybrid [KKL+PEB]O, and two high-gain observers:

• In the presence of measurement noise, KKLO achieved the lowest RMS estimation error owing to its linear LTI filtering.
• I&I, [KKL+PEB]O, and PEBO demonstrated slightly higher noise sensitivity due to their nonlinear estimation dynamics.
• High-gain observers exhibited pronounced chattering and significant transient oscillations at switching events.

The I&I framework enables systematic trade-offs between convergence speed and noise robustness via selection of $\beta$, $Q$, and adaptation dynamics within a Lyapunov-based design process (Yi et al., 2017).

• “On State Observers for Nonlinear Systems: A New Design and a Unifying Framework”, Yi, Ortega & Zhang (2018) (Yi et al., 2017)
• “Robust I&I Adaptive Tracking Control of Systems with Nonlinear Parameterization: An ISS Perspective”, Karagiannis et al. (2021) (Wang et al., 2021)