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Model-Predictive Control with NUP Priors

Raphael Keusch, Hans-Andrea Loeliger

TL;DR

This paper elaborate on recently proposed NUP representations of half-space constraints, box constraints, and finite-level constraints, and demonstrates the use of such NUP representations for exemplary applications in model predictive control with a variety of constraints on the input, the output, or the internal state of the controlled system.

Abstract

Normals with unknown variance (NUV) and, more generally, normals with unknown parameters (NUP) can represent many useful priors including L_p norms and other sparsifying priors, and they blend well with linear-Gaussian models and Gaussian message passing algorithms. In this paper, we elaborate on recently proposed NUP representations of half-space constraints, box constraints, and finite-level constraints. We then demonstrate the use of such NUP representations for exemplary applications in model predictive control with a variety of constraints on the input, the output, or the internal state of the controlled system. In such applications, the computations boil down to iterations of Kalman-type forward-backward recursions, with a complexity (per iteration) that is linear in the planning horizon. In consequence, this approach can handle long planning horizons, which distinguishes it from the prior art. For nonconvex constraints, this approach has no claim to optimality, but it is empirically very effective.

Model-Predictive Control with NUP Priors

TL;DR

This paper elaborate on recently proposed NUP representations of half-space constraints, box constraints, and finite-level constraints, and demonstrates the use of such NUP representations for exemplary applications in model predictive control with a variety of constraints on the input, the output, or the internal state of the controlled system.

Abstract

Normals with unknown variance (NUV) and, more generally, normals with unknown parameters (NUP) can represent many useful priors including L_p norms and other sparsifying priors, and they blend well with linear-Gaussian models and Gaussian message passing algorithms. In this paper, we elaborate on recently proposed NUP representations of half-space constraints, box constraints, and finite-level constraints. We then demonstrate the use of such NUP representations for exemplary applications in model predictive control with a variety of constraints on the input, the output, or the internal state of the controlled system. In such applications, the computations boil down to iterations of Kalman-type forward-backward recursions, with a complexity (per iteration) that is linear in the planning horizon. In consequence, this approach can handle long planning horizons, which distinguishes it from the prior art. For nonconvex constraints, this approach has no claim to optimality, but it is empirically very effective.
Paper Structure (37 sections, 53 equations, 24 figures, 2 tables)

This paper contains 37 sections, 53 equations, 24 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Models with NUP Priors: A Brief Review
  3. Joint MAP Estimation with Alternating Maximization
  4. Type-II MAP Estimation with Expectation Maximization
  5. Mixed MAP Estimation
  6. Proposed Approach
  7. System Model and Examples
  8. The Statistical Model
  9. Multiple Inputs, Outputs, and State Constraints
  10. Iterative Augmented Kalman Estimation (IAKE)
  11. Tabulated Update Rules for Step 2 of IAKE
  12. Outer Loop for Constraint Satisfaction
  13. Why Deterministic Systems?
  14. NUP Priors for Half-Space Constraints and Box Constraints
  15. NUP Representation of Vapnik Loss
  16. ...and 22 more sections

Figures (24)

  • Figure 1: Factor graph of system model (\ref{['eqn:NUV:Sys']}) with NUP priors (\ref{['eqn:NUP']}) and fixed observation(s) $\breve Y = \breve y$.
  • Figure 2: Factor graph of the model (\ref{['eqn:GenStatModel']}) and (\ref{['eqn:GenStatModelwithFinal']}).
  • Figure 3: Multiple inputs and outputs as in Section \ref{['sec:MultInOut']}.
  • Figure 4: Cost function (\ref{['eqn:box:CostFunctionBoxPrior']}) for $a=-1, b=1$, and $\gamma = 1$.
  • Figure 5: Estimate (\ref{['eqn:box:Analysis:EstX']}) for $a=-1$, $b=1$, $\gamma=1$, and different values of $s^2$.
  • ...and 19 more figures