control system,

Hierarchical Control using Approximate Simulation Relations

Duc-Tri Duc-Tri Follow Mar 30, 2025 · 5 mins read

Curious about how complex systems can be controlled with simpler models? In this post, I will introduce you an interesting approach called hierarchical control using approximate simulation relations. You’ll find a clear, easy-to-follow summary of Girard’s 2006 paper, along with insights into how the method works, how it’s evolved in recent research, and where it’s being applied today–from robot motion control to multi-agent systems. Plus, I’ve included my own implementation code so you can try it out for yourself!

Contents:

Paper Summary

The paper (Girard & Pappas, 2006) introduces a hierarchical control framework for complex systems by approximating them with simpler models, facilitating the controller design. Central to this approach are the concepts of approximate simulation relations, which define how closely the outputs \(\mathbf{y}\) and \(\mathbf{y}'\) of the concrete system \(\Sigma\) and the abstract system \(\Sigma'\) align. These relations are constructed using simulation functions, which are positive functions bounding the differences between the outputs of the two systems and ensuring that these bounds are non-increasing during their parallel evolution.

The framework’s architecture consists of two main layers: the concrete system \(\Sigma\), which represents the detailed and complex model of the plant, and the abstract system \(\Sigma'\)1, a simplified model characterized using approximate simulation relations to simplify control design at the Abstract System Controller. These layers are connected via a control interface, which synthesizes the control inputs for the concrete system.

Architecture of the controller.
Figure: Architecture of the controller.

Consider a control problem for the concrete system with specific invariance and reachability properties. Section III proves that the proposed control architecture guarantees that the output \(\mathbf{y}\) of \(\Sigma\) satisfies these desired properties, provided the following conditions are met:

  • \(\Sigma'\) is a complete approximate sub-system of \(\Sigma\) with precision \(\delta\), meaning that for every initial state \(\mathbf{x}_0\) of \(\Sigma\), there exists a corresponding initial state \(\mathbf{z}_0\) of \(\Sigma'\) such that, starting from these initial conditions, the outputs are bounded, i.e., \(\|\mathbf{h}(\mathbf{x}) - \mathbf{k}(\mathbf{z})\| \leq \delta\) for all \(t \geq 0\);2
  • \(\Sigma'\) satisfies the invariance and reachability properties within a safety margin of \(\delta\).

This approach is demonstrated through an application in robot motion control. The authors used a first-order kinematic model of the robot as \(\Sigma'\) to approximate the second-order dynamical model of the robot, treated as \(\Sigma\). Using the simulation function proposed in Proposition 4.1, the conditions and precision for which \(\Sigma'\) becomes a complete approximate sub-system of \(\Sigma\) are computed as functions of the bounds on \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\), as detailed in Theorem 4.3. Finally, the application of this architecture to autonomous robots is explored in the latter part of the paper, showcasing its practicality and effectiveness.3

Reviews on Hierarchical Control based on Approximate Simulation Relations

In (Kurtz et al., 2020), the architecture shown in Fig. 1 was extended to account for disturbances in the concrete system by introducing the Robust Simulation Function and Robust Approximate Simulation Relation. Subsequently, in (Wooding et al., 2023), an interface function for disturbances was integrated into the same architecture, allowing the abstract system to handle significant disturbances in the concrete system. Building on this, (Firouzmand et al., 2024) incorporated an observer to estimate the state of the concrete system, enabling Extended Robust Approximate Simulation Relations for linear systems.

The study of hierarchical control systems has also evolved in other directions. In (Yang & Ji, 2017), vector simulation functions were used to analyze large-scale hierarchical control systems. Meanwhile, (Tang & Hong, 2012) explored hierarchical control for a class of nonlinear systems using approximate simulation relations. This work was extended to distributed multi-agent systems in (Tang & Wang, 2018) and further applied to the Nash equilibrium of distributed multi-agent systems in (Tang & Yi, 2023).

Comments

The hierarchical control architecture based on approximate simulation relations offers an exciting perspective, mainly because my background is in process control, where hierarchical control is typically viewed as a multi-layered structure operating at different time scales (see (Scattolini, 2009)). In this context, each layer generally focuses on controlling a sub-system rather than addressing control and abstraction of the concrete system as presented in (Girard & Pappas, 2006).

References

  1. Girard, Antoine and Pappas, George J., "Hierarchical Control Using Approximate Simulation Relations", , 2006. DOI.
  2. Kurtz, Vince and Wensing, Patrick M. and Lin, Hai, "Robust Approximate Simulation for Hierarchical Control of Linear Systems under Disturbances", , 2020. DOI.
  3. Wooding, Ben and Lavaei, Abolfazl and Vahidinasab, Vahid and Soudjani, Sadegh, "Robust Simulation Functions with Disturbance Refinement", , 2023. DOI.
  4. Firouzmand, Elnaz and Talebi, H.A. and Sharifi, Iman, "Hierarchical Control of Linear Systems Using Extended Robust Approximate Simulation", European Journal of Control, 2024. DOI.
  5. Yang, Kaihong and Ji, Haibo, "Hierarchical Analysis of Large-Scale Control Systems Via Vector Simulation Function", Systems and Control Letters, 2017. DOI.
  6. Tang, Yutao and Hong, Yiguang, "Hierarchical Control Design of Nonlinear Systems Based on Approximate Simulation", , 2012. DOI.
  7. Tang, Yutao and Wang, Xinghu, "Optimal Output Consensus for a Class of Uncertain Nonlinear Multi-Agent Systems", , 2018. DOI.
  8. Tang, Yutao and Yi, Peng, "Nash Equilibrium Seeking for High-Order Multiagent Systems with Unknown Dynamics", IEEE Transactions on Control of Network Systems, 2023. DOI.
  9. Scattolini, Riccardo, "Architectures for distributed and hierarchical Model Predictive Control – A review", Journal of Process Control, 2009. DOI. Link.

Footnotes

  1. The method for characterizing the abstract system \(\Sigma'\) in the context of linear systems \(\Sigma\) is detailed in a subsequent paper by the author (Girard & Pappas, 2009). ↩

  2. The precision \(\delta\) can be computed from the simulation function using equation (6) in (Girard & Pappas, 2006). ↩

  3. I tried to reproduce the simulation results, which are available at this link: Google Colab ↩

featured
Install Ignition on VPS
Kinematics of Planar Robots with Revolut Joints