Region of attraction
- Also called: ROA, basin of attraction, domain of attraction
- Category: Control › Feedback control
- Tags: stability, nonlinear-control, lyapunov-analysis, verification, control-theory
The set of initial states from which a closed-loop system converges to a given equilibrium or setpoint; it measures how far a controller can be disturbed and still recover.
Definition
For a dynamical system under a fixed controller, the region of attraction (ROA) of an equilibrium is the set of all initial states whose trajectories converge to that equilibrium over time. Local stability results, such as linearizing around an upright pose and finding stable eigenvalues, only say that the equilibrium attracts states in some unspecified neighborhood; the region of attraction says how large that neighborhood actually is. For nonlinear systems with actuator limits it is usually bounded and often irregular in shape, and it can be much smaller than a linear analysis suggests. The same idea applies to periodic motions such as a walking gait, where the basin of attraction is the set of states that converge to the limit cycle.
The exact region of attraction can rarely be computed, so engineers work with inner approximations that are guaranteed to lie inside it. The standard tool is a Lyapunov function V(x): if V decreases along trajectories everywhere in a sublevel set {x : V(x) ≤ ρ}, then that set lies inside the region of attraction. For polynomial dynamics, sums-of-squares programming can search for the largest certified ρ, or for V itself. Simulation and sampling give quicker but uncertified estimates, and learning-based methods train neural Lyapunov functions. For LQR controllers, the quadratic cost-to-go is a natural Lyapunov candidate, and the analysis checks how far its level sets extend before nonlinearities or torque limits break the decrease condition.
Regions of attraction make controllers composable. In sequential composition, each controller or motion primitive is used only when the state is inside its region of attraction, and the goal set of one controller is placed inside the region of attraction of the next, so a planner can chain them with guarantees. LQR-Trees build a tree of stabilized trajectories whose verified funnels together cover a range of initial states. The legged locomotion paper on our site applies the same principle to a graph of motion primitives, where each primitive's region of attraction determines which transitions are safe.
Why it matters
A controller is only useful if the robot starts, and stays, within the states it can recover from. The region of attraction quantifies that robustness, guides when to switch controllers, and turns local stability claims into guarantees that a planner can use.
Key applications
- Verifying stabilizing controllers for underactuated systems
- Push recovery and switching between balance strategies on legged robots
- Sequential composition of controllers and motion primitives
- Feedback motion planning with verified funnels
- Safety analysis of learned controllers
In practice
A team designing a standing balance controller for a quadruped wants to know how hard a push it can survive. They simulate many perturbed initial states, varying base velocity and tilt, and record which ones return to the standing pose; the boundary between converging and failing samples gives an empirical region of attraction. For a reduced model they also compute a certified inner approximation from a Lyapunov function. On the robot, a supervisor compares the estimated state against this region and switches to a stepping controller whenever the state leaves it, instead of letting the standing controller try to recover from a push it cannot handle.
Common misconceptions
- A stable linearization means the robot is stable everywhere — linear analysis only guarantees convergence in some neighborhood, which can be very small once torque limits and nonlinearities act
- A Lyapunov sublevel set is the region of attraction — it is an inner approximation; the true region is usually larger
- The region of attraction is a property of the robot alone — it depends on the controller and its limits as well as the dynamics, and changes when gains or saturation limits change
Who should know this
- Control engineers designing stabilizing controllers
- Legged and humanoid robot developers working on balance
- Researchers in safe and verified control
- Planning engineers composing controllers or motion primitives
Related technologies
- Lyapunov Functions
- Sums-of-Squares Programming
- LQR-Trees
- Neural Lyapunov Functions
- Drake