Recursively Feasible Stochastic Model Predictive Control for Time-Varying Linear Systems Subject to Unbounded Disturbances

Abstract

Model predictive control solves a constrained optimization problem online in order to compute an implicit closed-loop control policy. Recursive feasibility -- guaranteeing that the optimal control problem will have a solution at every time step -- is an important property to guarantee the success of any model predictive control approach. However, recursive feasibility is difficult to establish in a stochastic setting and, in particular, in the presence of disturbances having unbounded support (e.g., Gaussian noise). The problem is further exacerbated for time-varying systems, in which case recursive feasibility must be established also in a robust sense, over all possible future time-varying parameter values, as well as in a stochastic sense, over all potential disturbance realizations. This work presents a method for ensuring the recursive feasibility of a convex, affine-feedback stochastic model predictive control problem formulation for systems with time-varying system matrices and unbounded disturbances using ideas from covariance steering stochastic model predictive control. It is additionally shown that the proposed approach ensures the closed-loop operation of the system will satisfy the desired chance constraints in practice, and that the stochastic model predictive control problem may be formulated as a convex program so that it may be efficiently solved in real-time.

Figures (4)

• Figure 1: Kinematic Bicycle Model in Curvilinear Coordinates
• Figure 2: Computed robust terminal covariance $\Sigma_f$, compared with a nominal terminal covariance for a single set of system parameters $\Sigma_{f, \mathrm{nom}}$, the noise covariance $D D^\top$, and the initial state covariance $\Sigma_0$.
• Figure 3: Computed robust terminal set $\mathcal{X}_{f}^{\mu}$, compared with a nominal terminal set for a single set of system parameters $\mathcal{X}^{\mu}_{f, \mathrm{nom}}$, and the tightened state constraints $\mathcal{X}_{\mathrm{safe}}$.
• Figure 4: Monte Carlo closed-loop CS-SMPC trajectories in curvilinear coordinates with robust $\mathcal{X}_{f}^{\mu}$ and $\Sigma_f$, compared with CS-SMPC using nominal and no terminal constraints, respectively.