Geometry-Preserving Optimal Control of Implicitly Singularly Perturbed Systems with Sliding and Exit Constraints

Technical Summary: Geometry-Preserving Optimal Control of Implicitly Singularly Perturbed Systems

Problem Statement
The paper addresses the optimal regulation of a class of implicitly singularly perturbed nonlinear systems where a state-dependent coefficient loses rank on a dynamically accessible hypersurface. The system is modeled as:
$$\begin{cases} \dot{x} = f(x, y) + B(x, y)u \\ \phi(y) \dot{y} = g(x, y) \end{cases}$$
where $\phi(y^*) = 0$ and $\phi'(y^*) \neq 0$. The control input $u$ acts only on the "ordinary" equation ($\dot{x}$), leaving the singular equation ($\phi(y)\dot{y} = g$) unactuated. This structure creates a singular hypersurface $S = \{y = y^*\}$ where the vector field becomes unbounded ($g/\phi \to \infty$). The control objective involves steering the system while managing attraction to $S$ and avoiding "exit" from the sliding domain (where $g(x, y^*) \neq 0$), without displacing the singular geometry itself.

Methodology
The author proposes a reciprocal Cauchy regularization to handle the singularity. Instead of relying on classical Filippov set-valued constructions or asymptotic singular perturbation limits ($\epsilon \to 0$), they introduce a regularization parameter $\epsilon > 0$ and define a regularized field:
$$R_\epsilon(y) = \frac{\phi(y)}{\phi(y)^2 + \epsilon^2}$$
The regularized system becomes:
$$\begin{cases} \dot{x} = f(x, y) + B(x, y)u \\ \dot{y} = g(x, y)R_\epsilon(y) \end{cases}$$
Key methodological features include:

1. Exact Invariance: For any $\epsilon > 0$, the hyperplane $S = \{y = y^*\}$ is an exact invariant set because $R_\epsilon(y^*) = 0$. The dynamics restricted to $S$ are independent of $\epsilon$.
2. Normal Attraction: The regularization induces a normal attraction rate toward $S$ of order $O(\epsilon^{-2})$, ensuring rapid collapse of trajectories starting near $S$.
3. Separation of Concerns: The paper distinguishes between the on-manifold optimal control problem (restricted to $S$) and the off-manifold consistency analysis. It treats exit avoidance as a state-constrained optimal control problem (OCP) separate from real-time safety filtering using Control Barrier Functions (CBFs).

Key Contributions
The paper makes the following specific contributions:

• Regular-Channel Architecture: A control structure that modifies tangential motion without altering the location of the singular hypersurface or its exit boundary.
• Exact Invariant Reduction: Proof that the reciprocal regularization selects a unique, $\epsilon$-independent generalized sliding dynamics on $S$ for every $\epsilon > 0$. This is an exact finite-dimensional reduction, not an asymptotic approximation.
• Quantitative Estimates:
  • Normal Contraction: Trajectories contract toward $S$ at a rate proportional to $\epsilon^{-2}$.
  • Tangential Deviation: For initial normal offsets of size $O(\epsilon)$, the tangential deviation between the regularized and reduced trajectories is $O(\epsilon^3)$ on fixed finite horizons.
• Exact Equivalence Theorem: If the initial state, reference, and terminal target lie on $S$, the full regularized optimal control problem and the reduced lower-dimensional problem are identical for every $\epsilon > 0$. They share the same state maps, costs, optimal values, minimizers, and Pontryagin Minimum Principle (PMP) systems.
• Off-Manifold Consistency: For initial states with $O(\epsilon)$ offsets, the cost difference between the regularized and reduced problems is $O(\epsilon^3)$, provided the starting time is bounded away from the terminal time.
• Safety Certification: A nonlinear Lyapunov argument based on a Linear-Quadratic Regulator (LQR) baseline is used to certify an invariant safe ellipsoid that is strictly separated from the singular exit surface.
• Constraint Handling: The formulation separates path constraints (handled via state-constrained OCP theory) from real-time safety layers (CBF/HOCBF), clarifying that barrier filters are instantaneous modifications rather than global horizon optimizers.

Results and Validation
The theoretical framework is validated through several examples:

1. Duffing–Rayleigh System: A 3D nonlinear example where the author:
   • Designed an LQR controller for the selected sliding dynamics.
   • Computed a certified nonlinear safe ellipsoid ($\rho_{safe} \approx 1.50$) that guarantees actuator compatibility and separation from the exit boundary.
   • Solved a constrained finite-horizon OCP using direct transcription (SLSQP) and cross-verified with IPOPT. The constrained solution satisfied path constraints with lower cost than HOCBF-filtered LQR.
   • Demonstrated that full regularized simulations with $\epsilon \in \{0.2, 0.1, 0.05, 0.025\}$ collapse onto the selected dynamics, with tangential errors scaling as $O(\epsilon^4)$ for this specific example (due to quadratic tangential dependence) and $O(\epsilon^3)$ for a general tracking example.
2. Analytical Examples:
   • An exactly solvable minimum-energy transfer problem where the optimal trajectory is analytically proven to avoid the singular exit.
   • A higher-dimensional multi-input system demonstrating the framework's applicability beyond 2D sliding modes.
   • A periodic tracking problem showing that unconstrained LQ tracking can naturally maintain a safety margin without explicit constraint imposition.
3. Turbomachinery Surrogate: A dimensionless model of cavitating pump surging illustrates the application of regular-channel actuation in a mechanical context without claiming parameter identification from experimental data.

Significance and Claims
The paper claims that its primary significance lies in providing a geometry-preserving framework that avoids the ambiguities of classical Filippov solutions for unbounded vector fields. By using reciprocal regularization, the author establishes:

• Exactness: The equivalence between the regularized and reduced problems is exact for all $\epsilon > 0$, not just in the limit.
• Structural Clarity: The method explicitly separates the singular geometry (which is fixed) from the control action (which shapes tangential motion).
• Quantitative Rigor: It provides explicit bounds on off-manifold convergence and safety, moving beyond qualitative sliding mode arguments.

The author modestly notes limitations: the exact equivalence holds strictly for on-manifold data; off-manifold convergence of costates is not claimed; and the theory is "event-free," meaning it does not optimize trajectories that cross the singular boundary (ejection and re-entry), which would require hybrid optimal control techniques. The work is presented as a structural and analytical advancement for a specific class of implicitly singularly perturbed systems, offering a rigorous alternative to standard singular perturbation and DAE optimal control theories.