Immersion and Invariance Technique

• Immersion and Invariance is a control design framework for nonlinear systems that immerses plant behavior into a lower-dimensional target manifold to achieve desired stability or periodic orbits.
• It utilizes methodologies such as PDE solving, Lyapunov analysis, and contraction theory to enforce manifold invariance and guarantee exponential convergence.
• The approach is applied in adaptive control, observer design, disturbance estimation, and privacy-preserving learning, offering robust performance and precise parameter tuning.

Immersion and Invariance Technique

The Immersion and Invariance (I&I) technique is a general, constructive framework for controller, observer, and estimator design in nonlinear systems. It addresses both stabilization to equilibrium and orbital stabilization (periodic orbits), and is applicable in adaptive control, observer design, disturbance estimation, and privacy-preserving optimization. The fundamental strategy is to construct a lower-dimensional "target" system exhibiting desired behavior, then design mappings and control laws to immerse the plant into this target while enforcing invariance and attractivity of a corresponding manifold. The approach is systematic, modular, and unifies perspectives from passivity theory, Lyapunov analysis, and contraction theory (Wang et al., 2015, Cervantes-Pérez et al., 18 Jan 2026, Nayyer et al., 2022, Yi et al., 2020, Wang et al., 2023, Hayati et al., 2024, Ortega et al., 2018, Romero et al., 2021).

1. Theoretical Foundations and Problem Statement

The I&I framework begins with a general control-affine nonlinear system: $$\dot{x} = f(x) + g(x)u, \qquad x \in \mathbb{R}^n,\quad u \in \mathbb{R}^m$$ The objective is to imbue the system with prescribed closed-loop properties—typically stabilization, orbital stabilization, or estimation—by "immersing" its behavior into that of a simpler "target" (reduced-order) system: $\dot{\xi} = \alpha(\xi)$ where $\xi \in \mathbb{R}^p$, $p < n$, and $\alpha$ is designed to generate equilibrium or periodic orbits as required by the application (Ortega et al., 2018, Wang et al., 2015).

Key steps in the I&I paradigm:

• Target system selection: Choose reduced dynamics $\alpha(\xi)$ encapsulating the desired behavior.
• Immersion mapping: Define $\pi: \mathbb{R}^p \to \mathbb{R}^n$ such that $x = \pi(\xi)$ represents the embedded target within the plant.
• Invariance enforcement: Develop a feedback law ensuring the submanifold $\Gamma = \{x = \pi(\xi)\}$ is invariant under the closed-loop flow, and off-manifold trajectories are attracted to $\Gamma$.
• Matching condition: On $\Gamma$, the dynamics must match: $$f(\pi(\xi)) + g(\pi(\xi))c(\pi(\xi)) = \nabla \pi(\xi)\alpha(\xi)$$ where $c(\pi(\xi))$ is the on-manifold control.

2. Representative Methodologies: From PDEs to Contraction

The enforcement of immersion and invariance can be achieved through various analytical tools:

• Partial Differential Equation (PDE) Approach:

Solve invariance PDEs to find $u(x)$ such that, on the manifold, the above matching holds (Nayyer et al., 2022).

• Lyapunov-Based Attractivity:

Construct Lyapunov or storage functions $V(x, \xi)$ measuring distance to $\Gamma$, and design $u(x)$ for exponential or asymptotic decay of $V$ off the manifold. For systems with affine control extensions, this yields explicit feedback laws combining invariance and attractivity (Nayyer et al., 2022).

• Horizontal Contraction Theory:

Contractive metrics are constructed transversally to $\Gamma$, with a contraction inequality on the prolonged (variational) system:

$\dot{V}(x,\delta x) \leq -\rho(x) V(x,\delta x)$

where $V$ is a Finsler-Lyapunov function measuring deviation from $\Gamma$. Solving a matrix inequality involving the Jacobian of the closed-loop dynamics ensures exponential convergence (attractivity) to the manifold (Wang et al., 2015).

• Passivity-Based Synthesis:

The I&I invariance manifold can be interpreted as generating a passive output, and storage functions can be linked to the system's energy, blending passivity-based and immersion approaches (Nayyer et al., 2022).

Approach	Core Tool	Main Guarantee
PDE (Invariance)	PDE for mapping/control	Manifold invariance
Lyapunov (Attractivity)	Quadratic/energy function	Off-manifold decay
Contraction	Finsler metric, LMI	Exponential convergence
Passivity	Storage, output injection	Dissipativity & stability

3. Applications: Control, Observation, and Beyond

Adaptive Control and Friction Compensation

I&I is employed to design adaptive observers for mechanical systems with parametric uncertainties (e.g., friction). In a single-link pendulum with viscous and Coulomb friction, the approach involves mapping plant states and unknown parameters to observer coordinates, and defining adaptation laws to enforce invariance and drive estimation errors to zero. A Lyapunov function combining deviation in state and parameter estimates produces strict decay off the invariant manifold, ensuring global convergence:

• Velocity estimation error $| \hat{x}_2 - x_2 | \leq 0.01$ rad/s within $0.2$ s.
• Parameter estimates converge under sufficient excitation.
• Smooth, chatter-free closed-loop performance superior to sliding-mode control (Cervantes-Pérez et al., 18 Jan 2026).

Orbital Stabilization of Nonlinear Systems

I&I admits orbitally stable closed orbits as attractors, not just equilibrium points. For path following in underactuated mechanical systems (e.g., marine vessels), the design selects an autonomous target oscillator with desired limit cycles, then immerses this oscillator into the plant via a static, analytic mapping. The off-manifold coordinate is stabilized, and the constructed feedback achieves boundedness, invariance, and forward motion without time parameterization or guidance laws (Yi et al., 2020, Romero et al., 2021, Ortega et al., 2018).

Disturbance Observers

Immersion and invariance is extended to disturbance estimation for plants with nonlinear disturbance input matrices. The I&I observer dynamics ensure uniform ultimate boundedness (GUUB) of the disturbance estimation error. The technique is integrated into control barrier function quadratically-programmed (CBF-QP) safe control, yielding tight safety guarantees and improved nominal performance under uncertainty (Wang et al., 2023).

Privacy-Preserving Federated Learning

The I&I toolset is applied to encode optimization dynamics in federated learning by immersing the client's model-update trajectory in a higher-dimensional random affine embedding. This guarantees that, after suitable encoding/decoding and addition of calibrated noise, the federated scheme attains $(\epsilon, \delta)$-differential privacy at both local and global levels, with zero loss in learning accuracy and convergence (Hayati et al., 2024).

4. Practical Tuning and Implementation

• Selection of convergence and adaptation gains (e.g., $k_1,\, \gamma_1,\, \gamma_2$) affects rates of state and parameter convergence, as well as sensitivity to measurement noise and modeling uncertainties (Cervantes-Pérez et al., 18 Jan 2026).
• Relay-approximation parameters (e.g., $\vartheta$ for friction modeling) must balance sign-function representation fidelity against numerical stiffness.
• Lateral gain parameters are typically tuned via pole placement or desired orbital convergence speed in path-following applications (Yi et al., 2020).
• In privacy-preserving FL, embedding matrix design and noise variance are governed by the required sensitivity and privacy budget (Hayati et al., 2024).

5. Extensions, Generality, and Relationship to Other Design Paradigms

• I&I design subsumes and unifies methods such as backstepping, forwarding, passivity-based control, and contraction-based synthesis, as demonstrated in the Passivity and Immersion (P&I) construct (Nayyer et al., 2022).
• Nonlinear target dynamics, such as oscillators for orbital stabilization, are admissible; the I&I technique generalizes to limit-cycle and multi-periodic objectives (Ortega et al., 2018, Yi et al., 2020).
• The framework is applicable to both single-input and multi-input systems, as well as underactuated mechanical systems under mild structural assumptions (Romero et al., 2021).
• Adaptations include approximate solution construction when the core invariance PDE is analytically intractable, compensated by auxiliary adaptive or filtering dynamics (Wang et al., 2023).

6. Comparative Performance and Empirical Results

Direct experimental comparison with sliding-mode adaptive controllers demonstrates tangible benefits:

• Elimination of chattering, torque smoothness, absence of actuator heating.
• Steady-state tracking error less than $0.005$ rad; rapid (sub-second) parameter convergence.
• Superior parameter identifiability and estimation precision under sufficient excitation (Cervantes-Pérez et al., 18 Jan 2026).

In advanced domains such as federated learning, the I&I-based coding mechanism enables "privacy for free," simultaneously attaining rigorous differential privacy guarantees without accuracy or convergence compromise. In disturbance rejection and safety-critical control, the immersion-based observer achieves faster, tighter estimation and less conservative constraint enforcement than robust CBF approaches (Wang et al., 2023, Hayati et al., 2024).

7. Summary of Key Ingredients and Design Recipe

The systematic workflow for I&I-based controller or estimator design is as follows (Wang et al., 2015, Nayyer et al., 2022, Cervantes-Pérez et al., 18 Jan 2026):

1. Pick target reduced dynamics $(\dot{\xi} = \alpha(\xi))$ encapsulating desired closed-loop behavior.
2. Construct immersion mapping $\pi(\xi)$ embedding the target in the plant's state space.
3. Establish manifold invariance through the invariance PDE or matching condition.
4. Design off-manifold coordinates $\phi(x)$ and derive a feedback $u(x)$ (or composite estimator/observer law) rendering $\phi(x)\to 0$ by Lyapunov, contraction, or passivity arguments.
5. Tune design parameters for desired convergence rates, robustness, and computational tractability.
6. Validate through stability proofs, boundedness arguments, and, as applicable, experiments or simulations.

This canonical sequence yields explicit (often analytic) feedback or observer/controller architectures ensuring global attractivity to equilibrium, periodic orbits, or encoded state trajectories, with demonstrably superior performance to classical robust or adaptive schemes in nonlinear and uncertain settings (Cervantes-Pérez et al., 18 Jan 2026, Wang et al., 2015, Hayati et al., 2024, Yi et al., 2020, Wang et al., 2023, Ortega et al., 2018, Romero et al., 2021, Nayyer et al., 2022).