Finding Control Invariant Sets via Lipschitz Constants of Linear Programs

TL;DR

An approach to safely expand an initial set while always guaranteeing that the set is control invariant is presented, and the safe expansion law of the control invariant set is derived and can be interpreted as a second invariance problem in the space of possible boundaries.

Abstract

Control invariant sets play an important role in safety-critical control and find broad application in numerous fields such as obstacle avoidance for mobile robots. However, finding valid control invariant sets of dynamical systems under input limitations is notoriously difficult. We present an approach to safely expand an initial set while always guaranteeing that the set is control invariant. Specifically, we define an expansion law for the boundary of a set and check for control invariance using Linear Programs (LPs). To verify control invariance on a continuous domain, we leverage recently proposed Lipschitz constants of LPs to transform the problem of continuous verification into a finite number of LPs. Using concepts from differentiable optimization, we derive the safe expansion law of the control invariant set and show how it can be interpreted as a second invariance problem in the space of possible boundaries. Finally, we show how the obtained set can be used to obtain a minimally invasive safety filter in a Control Barrier Function (CBF) framework. Our work is supported by theoretical results as well as numerical examples.

Figures (5)

• Figure 1: Illustration of the proposed invariant set expansion applied to the double integrator. Starting from a small invariant set, we apply a virtual expansion control input to each control point and ensure that there exists a control input keeping the state inside the set for all states on the boundary. Here, we vary the number of control points on the boundary to illustrate the conservatism.
• Figure 2: Illustration of a closed Catmull-Rom curve consisting of four segments. The control and mid points are shown by red and blue dots, respectively. The grey rectangles show the bounding box of each curve segment.
• Figure 3: Illustration of the reference virtual control input $\bm{\eta}^{\mathrm{ref}}$ that balances expansion and distribution of control points along the boundary.
• Figure 4: Simulation results for the running example of a double integrator. Left: SDF constructed numerically from the boundary $\partial \Omega$. Blue dots indicate the position of control points in the phase space. Right: Illustration of the invariant set $\Omega$ with sampled state trajectories shown in blue. If we do not apply the proposed safety filter, the reference controller (red) drives the state into the unsafe area.
• Figure 5: Simulation results for the inverted pendulum system. The value function calculated using HJ reachability is shown by the blue contours and its zero levelset (maximum control invariant set) is shown by the dashed lines. Our obtained control invariant set is shown in grey. The colored trajectories illustrate forward simulations of the dynamical system.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Theorem 1
• Proposition 1
• proof
• Example 1
• Example 2
• Example 3
• Theorem 2
• proof