Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum

Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: We're looking at the paper: "Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum".

Dev: Here's the gist of the paper. The gist: This work combines a certified Lyapunov-based positively invariant set with a predecessor-based implicit representation to compute a significantly less conservative inner approximation of the Region of Attraction while preserving formal stability guarantees.

Rosa: System Modeling and Control Design The nonlinear mathematical model of the Two-Wheeled Inverted Pendulum (TWIP) is presented, comprising two mechanical degrees of freedom and assuming longitudinal motion only, neglecting yaw dynamics The control objective is to stabilize the robot about its upright equilibrium configuration, corresponding to the robot balancing configuration with zero translational and angular velocities The equations of motion are expressed as x¨w = 1/d1(aIOθ˙ 2sinθ−a 2gsinθcosθ+T(IO r+acosθ)), θ¨ = 1/d1(-a 2θ˙ 2sinθcosθ+amOgsinθ−T(mO + a r cos θ)) The closed-loop system is stabilized using a discrete-time LQR controller This LQR controller is obtained by solving the infinite-horizon quadratic optimization problem min(uk)∞∑k=0(xk⊤Qxk+u⊤kRuk) s.t.

Dev: xk+1 = Adxk +Bd uk The resulting saturated state-feedback control law is given by uk = sat(-KLQRxk), where KLQR =

-4.291, -8.142, -9.271, -0.574: Region of Attraction Estimation Methodology The methodology for computing a certified inner approximation of the RoA starts from a certified positively invariant set, which provides a formal but conservative approximation of the RoA Building upon this initial invariant set, a predecessor-based implicit set expansion is employed to iteratively enlarge the certified region The one-step predecessor of V0 is defined as V1 = f−1(V0) = x: f(x) ∈ V0 This construction can be applied recursively, yielding the sequence VN = f−N(V0), where f−N(·) denotes the N-step predecessor operator The resulting sets satisfy V0 ⊆ V1 ⊆ ··· ⊆ VN ⊆ R OA The predecessor expansion is represented implicitly, where membership is determined by evaluating the implicit function vN(x) = v0(fN(x)) Consequently, a state belongs to the predecessorexpanded approximation if vN(x) ≤ 1 Robust Positive Invariance Analysis The RPI analysis of the Lyapunov-certified invariant set extends the nominal closed-loop dynamics introduced in (6) by incorporating bounded additive disturbances acting at the control input The disturbed closed-loop system is therefore written as xk+1 = Aclxk +g(xk) +Bwk +h(xk,wk), where Bwk denotes the linear contribution of the disturbance and h(xk,wk) collects the higher-order nonlinear interaction terms introduced by the disturbance Using the quadratic Lyapunov function V(xk) = x⊤k Pxk, obtained from the solution of the discrete-time Lyapunov equation in Section 3.1, the Lyapunov difference is given by ∆V(xk) = V(xk+1)−V(xk) By combining the nonlinear remainder and the disturbance contribution into a single perturbation term, and applying norm inequalities together with the local Taylor bound on the nonlinear dynamics, a sufficient condition for ∆V(xk) < 0 is obtained as p(ρ,w¯) < 0 For a prescribed disturbance magnitude, the positive roots of p(ρ,w¯) define the interval over which the Lyapunov function is guaranteed to decrease Denoting these roots by ρin and ρout, with 0 < ρin < ρout, the decrease condition is certified for all states satisfying ρin < xk ≤ ρout Numerical Results and Validation The predecessor-based expansion was computed over a prediction horizon of N = 1500 sampling instants, corresponding to a simulation horizon of 15 s The approximation was obtained by evaluating the implicit predecessor function on a uniform grid over the state subspace x˙w ∈

-1,1: m/s, θ ∈

-1.5, 1.5: rad while fixing the remaining states to xw = 0 and θ˙ = 0 Monte Carlo validation was performed using the nonlinear closed-loop dynamics The resulting stable and non-converging samples were then superimposed on the contour of the predecessor-expanded approximation defined by VN(x) = 1 Experimental validation showed that for each disturbed state, the implicit predecessor function VN(x) = V0(fN(x)) is evaluated In the first experiment, the robot successfully recovers and returns to the desired balancing configuration In the second experiment, under the same disturbance conditions, the tracking error reaches e =

0-0.10-0.50-2.60: ⊤ This indicates that the tracking error lies outside the predecessor-expanded RoA approximation The proposed approach significantly enlarges the initial certified region, yielding an RoA approximation that closely matches the empirically observed closed-loop behavior In addition, the RPI analysis provides a formal robustness certificate for the initial Lyapunov-based invariant set under bounded additive input disturbances Future work will focus on deriving an explicit analytical representation of the predecessorexpanded region One direction consists of computing the largest analytically describable set contained within the implicit predecessor representation, thereby obtaining an enlarged explicit certified inner approximation of the RoA Alternatively, a certified branch-and-bound approach could be employed to verify the inclusion of candidate invariant sets, such as ellipsoids, within the implicit predecessor representation Finally, future research will investigate less conservative approaches for robust invariance analysis How it works The core of the method involves combining a certified Lyapunov-based positively invariant set with a predecessor-based implicit representation to compute a significantly less conservative inner approximation of the RoA while preserving formal stability guarantees The initial Lyapunov-based set is derived from the proof of Theorem 4.7 in Khalil, 2002 This set serves as the starting point for the predecessor-based expansion and as the basis for the RPI analysis The predecessor expansion enlarges this set by iteratively propagating invariant sets through the system dynamics This is achieved by defining V1 = f−1(V0) = x: f(x) ∈ V0, where f(·) denotes the nonlinear closed-loop dynamics The recursive application of this operator yields the sequence VN = f−N(V0), which satisfies V0 ⊆ V1 ⊆ ··· ⊆ VN ⊆ R OA This construction allows for the computation of progressively larger certified approximations when initialized from a positively invariant set Robustness Assessment The robust positive invariance (RPI) analysis extends the nominal closed-loop dynamics introduced in (6) by incorporating bounded additive disturbances acting at the control input This analysis uses the quadratic Lyapunov function V(xk) = x⊤k Pxk, obtained from the solution of the discrete-time Lyapunov equation in Section 3.

Paper summary: Rosa: Okay, so to recap this paper "Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum", their core idea is using a combination of tools to improve how we estimate the safe region for these nonlinear systems. They start with what they call a certified positively invariant set derived from Lyapunov analysis, which gives them an initial, but conservative, inner approximation of the Region of Attraction.

Dev: And then they build on that set by using a predecessor-based implicit representation to iteratively expand it. This allows them to compute a significantly less conservative inner approximation while still preserving the formal stability guarantees you get from the original Lyapunov method.

Taro: So, instead of just stopping at that first certified set, they are using this iterative propagation idea, where they define a one-step predecessor as V one being the inverse image of V zero under the system's nonlinear dynamics, meaning if you start in V zero you want to know where that state can go in the next step.

Rosa: That iterative sequence leads to sets like V N, and they claim these sets satisfy V zero is contained within all subsequent sets, all the way up to the actual Region of Attraction, which is a big deal for getting a better boundary.

Dev: Beyond just expanding the region, they also analyze the robust positive invariance of that initial set under bounded additive input disturbances. This provides formal robustness guarantees for how well the system behaves when things go slightly wrong with the control inputs.

Taro: That robustness analysis seems crucial because in real-world applications, we always have some kind of noise or disturbance acting on the system, and knowing how much that noise can push us outside the safe zone is important for a safety case.

Rosa: So, this paper tackles two major things: getting a tighter approximation of the safe operating region through predecessor expansion, and proving robustness against disturbances in that initial set. This methodology is shown on the nonlinear TWIP system using an LQR controller derived from a quadratic optimization problem.

Conclusion: Dev: Thinking about this whole paper, "Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum", the main thing they achieved is making the safe region estimate much closer to what happens in reality, without sacrificing those formal stability proofs.

Rosa: Yeah, it’s about taking that conservative bound from traditional Lyapunov methods and stretching it out using this implicit predecessor technique. It shows that you can get a more practical estimate of the Region of Attraction while keeping the mathematical safety guarantees sound.

Taro: For someone who just listens to the show, what does that actually mean for them? It means if they are designing a system, they don't have to rely on an overly cautious region just because it's mathematically derived from a standard Lyapunov function.

Dev: It means you can use this approach to define a safer operating envelope that is much more realistic, and you get the added benefit of knowing how resistant that envelope is to small disturbances, which the robust positive invariance analysis gives them.

Rosa: So, when we look at the authors Lorenzo Fici and Guillaume Ducard’s work here, it highlights a practical path forward in nonlinear control: using advanced set propagation ideas to bridge the gap between certified theory and real-world performance.

Dev: It shows that for systems like the TWIP robot, combining these techniques can lead to a region of attraction approximation that actually reflects how the system behaves when you run it in the field.