Region of Attraction Estimation for Linear Quadratic Regulator, Linear and Robust Model Predictive Control on a Two-Wheeled Inverted Pendulum

The gist The work investigates the estimation of the Region of Attraction (RoA) for a two-wheeled inverted pendulum stabilized under three model-based control strategies: saturated LQR, linear MPC, and constraint tightening MPC.

Methodology Overview

The paper proposes a methodology that combines an analytical inner approximation with data-driven Monte Carlo estimation to characterize the RoA of the system <ref:2604.04455#pg2>. This approach provides both a formally certified inner bound and an empirical characterization of the RoA, offering a practical way to evaluate controller performance <ref:2604.04455#pg4>. The analysis begins with deriving a Lyapunov-based invariant set that serves as an inner approximation of the RoA under the LQR control law <ref:2604.04455#pg2>. This analytical bound is then refined using a Monte Carlo–based numerical estimation procedure to obtain a more representative approximation <ref:2604.04455#pg4>.

Analytical Inner Approximation

The analytical inner approximation of the RoA is derived from the nonlinear closed-loop dynamics under the saturated LQR control law, which defines an invariant set <ref:2604.04455#pg2>. This set is obtained by selecting the largest Lyapunov sublevel set fully contained in the region where a quadratic Lyapunov function decreases for all states satisfying a specific condition <ref:2604.04455#pg4>. The invariant set is defined as X = [x: x T P x ≤ λmin(P) ρ 2] <ref:2604.04455#pg4>. This region represents a mathematically justified invariant set under the LQR closed-loop dynamics, which serves as a stopping condition for the Monte Carlo–based RoA estimation because convergence to the equilibrium is guaranteed once a trajectory enters this set <ref:2604.04455#pg4>.

Validity of Inner Set for Other Controllers

The validity of this analytical inner approximation is verified for Linear MPC and Constraint Tightening MPC by showing that the same condition holds when the terminal cost QN is chosen as the solution of the Discrete-time Algebraic Riccati Equation (DARE) <ref:2604.04455#pg4>. Furthermore, it is confirmed that both controllers match the LQR law inside this terminal set when no constraints are active, and they also satisfy their respective state, input, and terminal constraints within this region <ref:2604.04455#pg4>. The scaling factor α⋆ is computed to ensure that the ellipsoid defined by the invariant set remains fully contained within these constraint sets for both controllers <ref:2604.04455#pg4>.

Monte Carlo Estimation and Empirical Validation

The RoA is estimated using a Monte Carlo-based numerical procedure where a large number of initial states are randomly sampled from a bounded region of the state space <ref:2604.04455#pg4>. The stopping condition for these simulations is the analytically derived invariant set introduced in Section III-A, which provides a certified stopping condition <ref:2604.04455#pg4>. A total of 5000 initial conditions were sampled from the 3D grid defined by x˙ w ∈ [-1, 1] m/s, θ ∈ [-1.5, 1.5] rad, and ˙θ ∈ [-1.5, 1.5] rad/s <ref:2604.04455#pg4>. The results show that all three controllers yielded the same RoA estimates: 58.48% of the sampled states entered the invariant set, and thus were guaranteed to converge <ref:2604.04455#pg4>. This indicates that LQR, MPC, and CTMPC stabilize essentially the same set of initial conditions on the TWIP <ref:2604.04455#pg4>.

Hardware Validation

To provide practical insight, hardware experiments were conducted using the saturated LQR under reference tracking conditions <ref:2604.04455#pg4>. Two representative experiments were performed using the LQR controller under reference tracking conditions <ref:2604.04455#pg4>. In one experiment, external input disturbances drove the system state toward the boundary of the estimated RoA <ref:2604.04455#pg4>. In a second experiment, the system was driven beyond the estimated RoA boundary, leading to loss of stability <ref:2604.04455#pg4>. These results are consistent with the estimated RoA and demonstrate the practical applicability of the proposed analysis <ref:2604.04455#pg4>.

Conclusion

The proposed framework successfully combines analytical guarantees with empirical characterization, enabling RoA analysis for nonlinear systems when analytical methods are overly conservative and data-driven approaches lack formal guarantees <ref:2604.04455#pg4>. The results show that LQR, linear MPC, and CTMPC achieve the same estimated region of attraction on the TWIP, indicating that their true RoAs are probably very similar <ref:2604.04455#pg4>. This methodology provides a general but model-dependent framework for RoA estimation <ref:2604.04455#pg4>. The inner approximation is inherently local and becomes more conservative in higher-dimensional systems <ref:2604.04455#pg4>. The invariant set is obtained in a single step, which represents a promising direction for future work <ref:2604.04455#pg4>.

How it works

1. A mathematically-certified invariant set is derived from the nonlinear closed-loop dynamics of the TWIP under the LQR control law, providing an inner approximation of the region of attraction that guarantees asymptotic stability within its domain <ref:2604.04455#pg4>.

2. A Monte Carlo–based numerical simulation is developed to empirically characterize the region of attraction by identifying initial conditions that reach the invariant set without violating input constraints <ref:2604.04455#pg4>.

3. The methodology combines analytical guarantees with data-driven estimation, providing both a formally certified inner bound and an empirical characterization of the RoA <ref:2604.04455#pg4>.

Contributions

The main contributions include:

1. A mathematically-certified invariant set derived from the nonlinear closed-loop dynamics of the TWIP under the LQR control law, providing an inner approximation of the region of attraction that guarantees asymptotic stability within its domain <ref:2604.04455#pg4>.

2. A Monte Carlo–based numerical simulation developed to empirically characterize the region of attraction of the three controllers considered in this study: saturated LQR, linear MPC, and CTMPC, by identifying initial conditions that reach the invariant set without violating input constraints <ref:2604.04455#pg4>.

Controller Synthesis

The nonlinear dynamics are linearized around the upright equilibrium point x = 0, u = 0 <ref:2604.04455#pg2>. The resulting discrete-time linear time-invariant (LTI) model is used for controller synthesis <ref:2604.04455#pg2>.

Saturated Linear Quadratic Regulator (LQR)

The LQR controller is obtained by solving the infinite-horizon quadratic optimization problem <ref:2604.04455#pg2>. The resulting saturated state-feedback control law is uk = sat(-KLQRxk) <ref:2604.04455#pg2>.

Linear Model Predictive Control (MPC)

Linear MPC solves a finite-horizon constrained optimal control problem at each sampling instant based on the linearized model of the system <ref:2604.04455#pg2>. The constraints X and U are defined, and slack variables epsilonk are penalized by ρ = 10 5 <ref:2604.04455#pg2>.

Constraint Tightening Model Predictive Control (CTMPC)

CTMPC optimizes a nominal trajectory and constrains it to lie inside tightened state and input sets, ensuring constraint satisfaction in the presence of bounded disturbances on the input <ref:2604.04455#pg2>. The tube tightening follows the recursion δi+1 = α δi + δ1, where α = 0.815 <ref:2604.04455#pg4>.

Model Parameters

The model parameters of the TWIP robot are listed in Table I <ref:2604.04455#pg2>. The equations of motion are expressed as (1) <ref:2604.04455#pg2>. The linearized dynamics are given by (2) <ref:2604.04455#pg2>.

References

[1] H. K. Khalil, Nonlinear systems, 3rd ed. Prentice Hall, 2002 <ref:2604.04455#pg2>.

[3] J. Tellez and J.

1. Saturated LQR Controller Improvement: The saturated LQR controller can be improved by using a larger state-feedback gain KLQR, which would increase the size of the certified invariant set X = x: x T Px <= lambda min(P) rho squared, potentially allowing for a larger region of attraction.

2. Linear MPC Improvement: Linear MPC can be improved by increasing the prediction horizon N to capture longer-term dynamics more effectively, or by reducing the penalty parameter rho to allow for larger state and terminal constraints in the optimization problem.

3. Constraint Tightening MPC Improvement: Constraint tightening MPC can be improved by adjusting the contraction rate alpha, which is a chosen design parameter that specifies the desired contraction rate of the tube, potentially leading to a faster convergence of the error dynamics ek+1 = (A + BKtube)ek + wk.