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An Innovations-Based Data-Driven Kalman Predictor for Predictive Control

Mohamed Abdalmoaty, Roy S. Smith

TL;DR

This work addresses predictive control when the process disturbance is unmeasured by reframing the system in the innovations form, where disturbances are captured by the innovations $e(k)$. It develops an innovations-based data-driven Kalman predictor built on a parsimonious innovations-based SMM, enabling online state estimation and future-output prediction from IO data without direct disturbance measurements. Key contributions include: (i) an innovations-based SMM that uses $\bar{u}=[u;e]$ and measured $y$ to characterize IO trajectories, (ii) a recursive Kalman predictor for the innovations-based model with a predictor for $y_f$ from $u_f$, and (iii) a data-driven procedure to estimate the innovations $e^d$ from IO data and construct the innovations-based SMM, with guidance on parameter choices and consistency. Simulation on a Boeing 747 longitudinal model shows the proposed innov-SMM-KF outperforms disturbance-measured baselines, validating its practicality for realistic predictive control where disturbances are not accessible.

Abstract

A recently developed data-driven Kalman filter requires offline measurement of the process disturbance; a requirement that is often unmet for many practical applications. We propose a solution that parametrizes the Kalman filter exclusively using measured input and output data. The key idea is to use the innovations form which naturally accounts for the process disturbance and measurement noise into a single orthogonal stochastic process. Unlike process disturbances, the innovations process can be estimated directly from input-output data via a numerically efficient projection step. The performance of the method is demonstrated using a benchmark simulation.

An Innovations-Based Data-Driven Kalman Predictor for Predictive Control

TL;DR

This work addresses predictive control when the process disturbance is unmeasured by reframing the system in the innovations form, where disturbances are captured by the innovations . It develops an innovations-based data-driven Kalman predictor built on a parsimonious innovations-based SMM, enabling online state estimation and future-output prediction from IO data without direct disturbance measurements. Key contributions include: (i) an innovations-based SMM that uses and measured to characterize IO trajectories, (ii) a recursive Kalman predictor for the innovations-based model with a predictor for from , and (iii) a data-driven procedure to estimate the innovations from IO data and construct the innovations-based SMM, with guidance on parameter choices and consistency. Simulation on a Boeing 747 longitudinal model shows the proposed innov-SMM-KF outperforms disturbance-measured baselines, validating its practicality for realistic predictive control where disturbances are not accessible.

Abstract

A recently developed data-driven Kalman filter requires offline measurement of the process disturbance; a requirement that is often unmet for many practical applications. We propose a solution that parametrizes the Kalman filter exclusively using measured input and output data. The key idea is to use the innovations form which naturally accounts for the process disturbance and measurement noise into a single orthogonal stochastic process. Unlike process disturbances, the innovations process can be estimated directly from input-output data via a numerically efficient projection step. The performance of the method is demonstrated using a benchmark simulation.

Paper Structure

This paper contains 11 sections, 3 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem formulation and Background
  3. Problem Formulation
  4. Data-driven simulation
  5. Data-based state-space characterization
  6. Data-Driven Kalman Prediction
  7. Innovations Form Representation
  8. Innovations-based SMM
  9. An innovations-based data-driven KF
  10. "Measuring" the innovations $e^d$
  11. Simulation Example

Key Result

Proposition 1

Consider the system $G$ and assume that $w$ is measurable, and $v(k) = 0$$\forall k$ so that $y = \bar{y}$. Let $\{(\bar{u}^d(k), y^d(k))\}_{k=1}^N$ be a trajectory of $G$. Then, if the pair $(A,B_u)$ is controllable and the input is persistently exciting of order $T+n_x$, i.e., $\mathcal{H}_{T+n_x}

Figures (2)

  • Figure 1: Performance indices boxplots. The performance of innov-SMM-KF is a bit better than SMM-Kal that uses measured $w$ to construct the SMM.
  • Figure 2: True versus estimated innovations sequence with $L = 150$ samples. For clarity, only the tail of the 2500 samples trajectory is shown.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3