Passivity and Immersion (P&I) Construct

• Passivity and Immersion (P&I) construct is a unified framework that merges control theory’s passivity and immersion-invariance methods with immersive narrative measurement techniques.
• It employs rigorous methods like Lyapunov analysis and invariant manifold feedback to ensure system stability and adaptive control in complex dynamical environments.
• In immersive studies, P&I quantifies subjective presence and agency by mapping navigation approaches against metrics such as the Presence Questionnaire and NASA-TLX scores.

The Passivity and Immersion (P&I) construct represents a rigorous theoretical and applied framework intersecting two domains: (1) the analysis and design of control systems through the fusion of passivity theory and immersion–invariance methodologies, and (2) the quantitative study of agency and experienced immersion in human–computer interfaces, especially in immersive visual storytelling. Across these domains, P&I encompasses both a mathematically-precise design paradigm for dynamical systems stabilization (notably via Lyapunov/energy methods and manifold-based feedback) (Nayyer et al., 2022, Nayyer et al., 2022, Alkrunz et al., 2024, He et al., 2017) and an operational framework for psychophysical measurement of user experience in highly immersive virtual environments (Lu et al., 6 Feb 2025, Naz et al., 2017). Definitions, metrics, and implementation details differ in each field but share an underlying concern with the structure and consequences of “passivity” versus “active” engagement, and the nature of “immersion” either as an engineering target (dissipation into a manifold) or as a subjective presence in virtual space.

1. Formal Definitions and Conceptual Foundations

Control Theory

In nonlinear control, the P&I approach unifies passivity-based design with immersion and invariance (I&I) (Nayyer et al., 2022, Nayyer et al., 2022, Alkrunz et al., 2024, He et al., 2017). Passivity refers to systems whose storage function (often energy-like) does not increase faster than the supplied input power. Immersion denotes embedding the system into a lower-dimensional manifold with target dynamics guaranteeing stability or tracking. The P&I method: - Selects an invariant manifold (typically $\Psi(x,\lambda)=0$ for plant state $x$ and extended coordinate $\lambda$), chosen so that restriction to the manifold yields desired closed-loop (stable) behavior. - Identifies a passive output $y$ (typically $y=\lambda+q(x)$) and designs a storage function $V=\tfrac12y^2$. - Derives feedback laws enforcing $\dot V\leq -\alpha V$, ensuring off-manifold states dissipate.

Immersive Storytelling and VR

In spatial computing and immersive journalism, P&I denotes the controlled trade-off between navigation agency (“passivity” vs. “active interaction”) and the resulting level of “immersion” as a subjective user state (Lu et al., 6 Feb 2025, Naz et al., 2017). - Passivity is operationalized as system-driven navigation—e.g., the system controls all camera/viewpoint changes, minimizing user agency. - Immersion refers to the user’s felt sense of presence, measured via established psychometric questionnaires along subdimensions such as Realism, Possibility to Act, Interface Quality, and others. - The design space is structured as a 2x2 matrix: Viewpoint (egocentric vs. exocentric) × Navigation (active vs. passive), forming the experimental framework for analysis.

2. Mathematical and Experimental Frameworks

Nonlinear Control: P&I Algorithm Structure

The P&I design methodology in control can be condensed into the following steps (Nayyer et al., 2022, Nayyer et al., 2022): 1. Select invariant manifold: $\Psi(x,\lambda)=0$ to embed target dynamics. 2. Compute passive output: $y=\lambda+q(x)$ where $\nabla q(x)$ solves a metric-splitting condition. 3. Define storage function: $V = \tfrac12 y^2$. 4. Feedback law: $$u = -\frac\alpha2\,\lambda - \frac\alpha2\,q(x) - \nabla q(x) f(x,\lambda)$$. 5. Lyapunov analysis: Show $\dot V \leq -\alpha V$ and exponential convergence.

For example, in port-controlled Hamiltonian systems with parameter uncertainty, P&I combined with I&I yields adaptive laws with composite Lyapunov functions

$V_k = H_a(x_k) + \gamma z_k^\top P z_k$

guaranteeing local asymptotic convergence of both states and parameter estimates (Alkrunz et al., 2024).

Immersive Studies: P&I Measurement Protocols

Lu et al. (Lu et al., 6 Feb 2025) and Naz et al. (Naz et al., 2017) operationalize P&I through factorial experimental design, with P&I implemented as: - 2 × 2 within-subjects (or between-subjects) structure: Viewpoint × Navigation or Task (active vs. passive tasking). - Standardized presence/immersion questionnaires: e.g., I-Group Presence Questionnaire (PQ, with subscores computed as $$PQ_{\text{Realism}} = \frac1n \sum_{\rm relevant\ items} \mathrm{Likert}_i$$), NASA-TLX for workload. - Statistical analyses: GLMMs, mixed-design ANOVA, effect size $\eta_p^2$, and post-hoc pairwise contrasts.

3. Principal Empirical Findings Across Domains

Immersive Storytelling/Vr

Empirical results demonstrate: - Active navigation consistently yields higher immersion/presence scores than passive navigation, particularly in Realism, Possibility to Act, Examine, and Interface Quality. E.g., $PQ_{\text{Realism}}$ mean: Active ≈ 5.4 ± 0.8, Passive ≈ 4.8 ± 0.9, $\Delta\approx0.6$, $p<0.05$ (Lu et al., 6 Feb 2025). - Strong Viewpoint × Navigation interactions: Egocentric+Active maximizes presence/engagement, while Exocentric+Passive yields minimal presence but higher content focus. - Passive navigation reduces agency and novelty, attenuating spatial immersion; in exocentric view, active navigation increases physical demand and effort substantially (NASA-TLX: Exo+Active ≈ 15.2 vs. Exo+Passive ≈ 7.3, $p<0.001$).

Emotional Qualities of VR Space

Naz et al. (Naz et al., 2017) report: - No main effects of passivity/activity on emotional dimension ratings, though activity modulates the impact of color and brightness on feelings of intimacy and warmth. - Strong main effects for color/brightness; interaction Color × Activity ($F(1,30)=5.221$, $p<0.05$) whereby inactive users find orange environments more intimate, but no such distinction among active users. - Design heuristics derived from these findings for architectural/VR deployment.

Control Systems

Simulations and analysis in adaptive control illustrate: - P&I-based controllers with I&I estimators ensure boundedness and local asymptotic stability, outperforming classical PD controllers in nonminimum-phase settings (He et al., 2017, Alkrunz et al., 2024). - Storage-dissipation via quadratic Lyapunov functions yields explicit convergence rates for both tracking error and parameter estimation (Nayyer et al., 2022).

4. Technical Construction and Theoretical Underpinnings

The P&I approach brings together several strands: - Target manifold selection and immersion: Embedding into low-order GES (globally exponentially stable) dynamics via proper manifold design. - Passivity theory: Passive outputs and storage functions constructed by metric splitting on the tangent bundle; ensures dissipation and convergence. - Lyapunov analysis: Quadratic/composite Lyapunov functions for closed-loop stability and convergence of augmented error states. - Unification of nonlinear control methods: Backstepping, forwarding, incremental backstepping, control contraction metrics (CCM), and I&I are recoverable as specializations within the P&I meta-framework (Nayyer et al., 2022).

Key Structural Elements	Control Theory Domain	Immersive Storytelling Domain
Passivity	Dissipativity, storage function	Lack of agency, system-driven progress
Immersion	Convergence into invariant manifold	Subjective presence, measured by PQ
Key Metric	Lyapunov storage $V$	PQ, NASA-TLX, semantic differentials
Primary Feedback Law	$u = -\frac\alpha2\,\lambda - \cdots$	User input or passive
Analytical Goal	Global asymptotic stability	Maximize user presence/engagement

5. Practical Implications and Design Guidelines

Immersive Media & VR

• P&I trade-offs are central to immersive narrative design: maximizing agency (active navigation) increases spatial immersion but may elevate cognitive load; passivity reduces workload but sacrifices engagement (Lu et al., 6 Feb 2025).
• Specific user types must be considered: “information-oriented” users may prefer Exo+Passive; “experience-oriented” users benefit most from Ego+Active.
• Design taxonomies based on P&I are nascent; future work should extend to branching, multi-scene stories and multimodal guidance schemes.

Control Systems

• P&I controllers can be systematically synthesized for wide classes of nonlinear and port-Hamiltonian systems, including those with nonminimum-phase properties or unknown parameters (Alkrunz et al., 2024, He et al., 2017).
• Robustness to parameter uncertainty is achieved via I&I estimators and energy-shaping feedback.

6. Limitations and Research Directions

• VR studies: P&I operationalizations are limited by ecological validity (restricted narrative scopes, single-scene stories) and by fixed immersion profiles (fixed hardware, no variation in multisensory feedback) (Lu et al., 6 Feb 2025, Naz et al., 2017).
• Control theory: P&I methodology requires well-posedness of manifold immersion and passivity, which may be technically limiting for some classes of underactuated or high-dimensional systems.
• In both fields, adaptive or user-segmented P&I frameworks, more general classes of Lyapunov/storage functions, and integration with learning-based controllers or affective computing remain relatively unexplored.

7. Cross-Domain Synthesis and Significance

The P&I construct, though arising from distinct disciplinary contexts, is unified by its dual deployment of passivity—as a theoretical property of dissipative dynamics or as a dimension of user engagement—and immersion—as either geometric convergence or a phenomenological target. In control theory, the P&I methodology yields rigorous, constructive algorithms for stabilization and parameter adaptation, with a strong geometric and Lyapunov-theoretic foundation (Nayyer et al., 2022, Nayyer et al., 2022, Alkrunz et al., 2024, He et al., 2017). In immersive media studies, it provides a typology and empirical framework for optimizing user experience by systematically manipulating agency and presence (Lu et al., 6 Feb 2025, Naz et al., 2017). The ongoing research agendas in both areas focus on expanding the operational envelope of P&I, elaborating the trade-offs and synergies implicit in its axes, and unifying insights into a generalized theory of engagement—whether for cybernetic systems or human–machine interaction.