Approximate-Simulation-Based Hierarchical Control (ASHC)

• ASHC is a hierarchical control framework that employs approximate simulation relations to bridge high-dimensional concrete models with lower-dimensional abstractions.
• It designs interfaces using simulation functions and LMIs to guarantee robust output tracking and worst-case error bounds even under bounded disturbances.
• ASHC unifies abstraction-based control, formal synthesis, and moment matching to enable scalable, compositional controller synthesis for complex and interconnected systems.

Approximate-Simulation-Based Hierarchical Control (ASHC) is a formal methodology for hierarchical control of complex dynamical systems, rooted in approximate simulation relations between high-dimensional "concrete" systems and lower-dimensional "abstract" models. ASHC constructs and certifies interfaces such that the output of a concrete (often nonlinear, hybrid, or piecewise affine) system tracks the output of a simpler abstraction within a provable worst-case error bound, even under bounded disturbances. The framework enables scalable controller synthesis, compositional reasoning for large-scale and interconnected systems, and quantitative refinement of abstract-level plans or specifications, including temporal logic. Recent developments position ASHC as the mathematical bridge connecting abstraction-based control, formal synthesis, model reduction, and moment matching.

1. Formal Problem Setting and Abstraction Construction

ASHC begins with a full-order system, typically a piecewise affine (PWA) or linear time-invariant (LTI) system:

• For PWA: Each mode $i \in \mathcal{I}$ is defined on a polyhedral cell $$X_1^i = \{ x_1 \in \mathbb{R}^n \mid E_i x_1 \geq f_i \}$$ with dynamics

$$\dot x_1 = A_i x_1 + B_i u_1 + c_i, \qquad y_1 = C_i x_1,$$

subject to $\|c_i\|_\infty \le \bar c_i$.

• The abstraction is typically a lower-dimensional linear system:

$\dot x_2 = F x_2 + G u_2, \qquad y_2 = H x_2,$

or, for greater expressiveness, a lower-dimensional PWA system.

Abstraction construction seeks matrices $P_i, Q_i$ (injective $P_i \in \mathbb{R}^{n \times m}$) for each concrete mode $i$ solving:

• $H = C_i P_i$
• $P_i F = A_i P_i + B_i Q_i$

If solutions exist, the abstraction approximately simulates the concrete system in mode $i$. For PWA abstraction, this is extended over joint mode pairs $(i,j)$, with $H_j = C_i P_i$, $P_i F_j = A_i P_i + B_i Q_i$ (Song et al., 2022).

2. Interface and Simulation Function Design

The core of ASHC is construction of an interface enforcing the simulation relation:

• Transform the abstraction input: $u_2 = L x_2 + \bar u_2$ with $F + G L$ Hurwitz.
• State-error: $\tilde x = x_1 - P_i x_2$
• Interface (linear abstraction):

$$u_v = R_i \bar u_2 + (Q_i + R_i L) x_2 + K_i \tilde x$$

Parameters $R_i, K_i$ are chosen (with $K_i$ tunable for stability).

• Substitution into the full dynamics yields joint error dynamics $(\tilde x, x_2)$, and output error $e = C_i \tilde x$.

Simulation functions (Lyapunov-like, quadratic in $\omega = [\tilde x; x_2]$) are synthesized via LMI conditions on each cell:

• $M - C^T C \succeq 0$ (output error bounding)
• $M - E^T U E \succ 0$ (positivity within cell)
• $A^T M + M A + E^T W E + \lambda M \preceq 0$ (decay under joint drift) with $U, W \geq 0$, $\lambda > 0$ (Song et al., 2022).

Let $$V(\omega) = \frac{1}{\kappa} \sqrt{ \omega^T M \omega }$$. Along the closed-loop, $$\dot V \leq -\alpha V + \beta_1\|\bar u_2\| + \beta_2\|c\| + \beta_3\|x_2\|$$. Thus, outside a compact tube, $V$ decays, and eventually $V \le b$ yields a worst-case error $\|e\| \le \kappa b$.

3. Theoretical Guarantees and Algorithmic Workflow

The main ASHC result establishes robust satisfaction of the output error bound $\delta = \kappa b$:

• For all modes $i$, given $P_i, Q_i, L, R_i, K_i$, the relation

$$R = \{ (x_1, x_2) \mid V(x_1 - P_i x_2, x_2) \le b \}$$

is a robust approximate-simulation relation of precision $\delta$.

Controller synthesis proceeds by:

1. Choosing abstraction dimension $m$ and model $(F, G, H)$ or $(F_j, G_j, H_j)$ for PWA abstraction.
2. Solving $H = C_i P_i$, $P_i F = A_i P_i + B_i Q_i$ for each $i$ (and optionally $(i, j)$).
3. Selecting $L$ so $F + GL$ is Hurwitz.
4. Selecting $R_i$, $K_i$.
5. Formulating and solving cellwise LMIs for $M, U, W, \lambda$.
6. Computing $b$ from error gains, setting $\delta = \kappa b$ (Song et al., 2022).

Computational complexity is polynomial in the number of modes ($s$ for concrete, $r$ for abstraction), with simultaneous solving of $s$ or $sr$ LMIs of size $(n+m)$.

4. Extensions: Compositionality, Formal Synthesis, and Robustness

Compositional construction is addressed by small-gain theorems for interconnections:

• Individual subsystem abstractions with simulation functions $V_i$ are aggregated using nonlinear gain operators ($\Gamma$, $\Lambda$, etc). A composite simulation function is $$V(\hat x, x) = \max_{i=1..N} [ \sigma_i^{-1} \circ \lambda_i \circ V_i(\hat x_i, x_i) ]$$, under small-gain conditions (Rungger et al., 2015).
• Linear systems admit geometric characterization of abstraction via controlled-invariant subspaces and explicit interface construction.
• These compositional results guarantee global output tube invariance and permit controller synthesis for complex symbolic specifications (e.g., LTL) at the abstract level, then refining to concrete systems with quantified satisfaction margins.

ASHC is robust to bounded disturbances and impulse disturbances:

• Simulation function invariance and incremental stability ensure the output tubes are preserved under bounded additive and impulsive inputs, with modified error bounds reflecting disturbance magnitude and dwell time (Kurtz et al., 2020).
• Formally, for bounded disturbances $d$, output error is bounded by $$\|y_c - y_a\| \le \max \{ V(0), \gamma_1(\|d\|_\infty) + \gamma_2(\|u_a\|_\infty) \}$$, with explicit gain computations. Similar results hold for impulsive disturbances with additional additive terms.

5. Connections to Moment Matching and Model Reduction

ASHC's two key requirements—bounded output discrepancy and the $M$-relation—are shown to be moment-matching conditions for system interconnections:

• The bounded output condition ($P F = A P + B L$, $H = C P$) is interpreted as matching system moments via the Sylvester equation.
• The $M$-relation ($M A = F M + G N$, $G \Gamma = M B$, $C = H M$) corresponds to output-trajectory recovery, with moments determined by direct or swapped interconnection structures.
• These findings establish a deep conceptual bridge between ASHC and classical moment matching, enabling new directions in nonlinear, time-delay, or data-driven hierarchical control, and synergistic development with compositional/symbolic abstraction techniques (Niu et al., 7 Dec 2025).

6. Applications and Implementation in Complex Systems

ASHC has been validated in diverse contexts:

• Piecewise affine systems under bounded disturbances, with both linear and PWA abstractions, demonstrate output tube invariance and rigorous worst-case tracking error, even at partition switches (Song et al., 2022).
• Large-scale interconnected linear systems admit compositional abstraction construction and controller synthesis for symbolic/temporal logic objectives (Rungger et al., 2015).
• Hierarchical MPC for building temperature control employs reduced-order models for high-level tube-based robust MPC, with local fast-rate regulators guaranteeing recursive feasibility and robust convergence (Farina et al., 2017).
• Template-based whole-body control for humanoid robots exploits Hamiltonian structure and passivity, yielding controllers with robust tracking and disturbance rejection (Kurtz et al., 2020).
• Reinforcement learning architectures separate high- and low-level policy synthesis, provide PAC guarantees on abstraction quality and end-to-end performance, and leverage ASHC for scalable controller composition in complex RL-driven environments (Delgrange et al., 2024).

Table: Instantiations of ASHC in Recent Literature

System Type	Abstraction	Robustness Target
PWA (multi-segment, hybrid)	Linear/PWA	Bounded disturbances
Interconnected LTI blocks	Linear (compositional)	Symbolic/LTL specs
Multi-zone building MPC	Reduced linear	Tube-based invariance
Legged robot (Valkyrie)	Template/Hamiltonian	Push/model error
RL-composed rooms (MDP)	Latent DRL policies	PAC value guarantees

ASHC thus unifies abstraction-based, compositional, and robust hierarchical control across a range of complex systems.

7. Current Research Directions and Limitations

• The unification of ASHC with moment matching suggests extensions to nonlinear, time-delay, and stochastic systems via invariance equations and nonlinear moment matching (Niu et al., 7 Dec 2025).
• Data-driven estimation of simulation manifolds facilitates scalable ASHC in RL settings and high-dimensional data environments (Delgrange et al., 2024).
• Computational limits arise from the explosion in the number of modes (for hybrid/PWA) and complexity in solving large-scale LMIs. Scalability, compositionality, and efficient solver development are active areas of investigation (Song et al., 2022, Rungger et al., 2015).
• Current proofs and guarantees rely on spectrum separation and simple eigenvalues; generalizations to non-minimal, non-smooth, or time-varying systems remain open problems. Real-time optimal control in high DOF manipulators is constrained by sample complexity and hardware acceleration demands (Patil et al., 28 Mar 2025).

In summary, Approximate-Simulation-Based Hierarchical Control (ASHC) is a mathematically rigorous and algorithmically constructive framework for robust, compositional hierarchical control, synthesizing abstraction-based design, formal verification, and moment matching into a unified methodology for modern complex systems.