Stochastic Model Predictive Control with Online Risk Allocation and Feedback Gain Selection

Abstract

Stochastic Model Predictive Control addresses uncertainties by incorporating chance constraints that provide probabilistic guarantees of constraint satisfaction. However, simultaneously optimizing over the risk allocation and the feedback policies leads to intractable nonconvex problems. This is due to (i) products of functions involving the feedback law and risk allocation in the deterministic counterpart of the chance constraints, and (ii) the presence of the nonconvex Gaussian quantile (probit) function. Existing methods rely on two-stage optimization, which is nonconvex. To address this, we derive disjunctive convex chance constraints and select the feedback law from a set of precomputed candidates. The inherited compositions of the probit function are replaced with power- and exponential-cone representable approximations. The main advantage is that the problem can be formulated as a mixed-integer conic optimization problem and efficiently solved with off-the-shelf software. Moreover, the proposed formulations apply to general chance constraints with products of exclusive disjunctive and Gaussian variables. The proposed approaches are validated with a path-planning application.

Figures (7)

• Figure 1: The curves of the nonconvex function compositions resulting from the formulations in \ref{['sec:convex-reformulations']} (solid) and their corresponding exponential cone-representable approximations (dashed).
• Figure 2: A polyhedron representing a stay-in region $\mathcal{I}$ encoded as a conjunction of linear equality constraints.
• Figure 3: A polyhedron representing a stay-out region $\mathcal{O}$ encoded as a disjunction of linear equality constraints.
• Figure 4: 1000 Monte Carlo simulations of the state predictions at the first sampling instant, obtained using the best and worst feasible integer solutions. The plots show the prediction envelopes of the best integer solution $\mathbf{X}^{\mathrm{min}}$ (gray) and the worst integer solution $\mathbf{X}^{\mathrm{max}}$ (magenta). The results in each subplot correspond to different parameter ranges used to construct $\bm{\mathcal{M}}(\delta)$.
• Figure 5: 100 Monte Carlo simulations of the performed trajectories in Case 1 (without obstacles). The vehicle navigates inside $\mathcal{I}$ (light green), from the initial state $\bm x_{0} = \left[0,-1.18,0,0.16\right]^{\top}$, to $\mathcal{T}$ (dark green). As the vehicle's initial position is close to the boundaries of $\mathcal{I}$, a safer trajectory toward $\mathcal{T}$ involves moving away from these limits.

Theorems & Definitions (1)

• Remark 1