(188h) Strategies for Minimum Variance ALS Estimation of Noise Covariance Matrices

In process systems, engineers generally have a deterministic model describing the dynamics of the plant. Uncertainty manifests as noises and disturbances that are not included in this model, so it is common to augment the deterministic model with a stochastic model. Disturbances may occur within the process itself, affecting the dynamic response of the system, or they may occur at the output (i.e., sensor noise). Motivated by the central limit theorem, a typical choice is to model the process and measurement noises as independent zero mean Gaussian random variables. In this case the stochastic model is fully defined by the process and measurement noise covariance matrices, denoted Q and R respectively. Determining good values for Q and R is necessary for applications such as state estimation and controller performance monitoring.

The variances are generally not known a priori, so they must be estimated from plant data. The autocovariance least squares (ALS) algorithm casts the estimation as a linear regression problem (Odelson et al., 2006; Rajamani and Rawlings, 2009). To get the minimum variance ALS estimates of Q and R, the appropriate weighting matrix W* is required. However, W* is itself a function of the unknown Q and R. We will discuss the following strategies for obtaining minimum variance ALS estimates:

1. Pick an initial Q₀ and R₀ , calculate W*(Q₀, R₀), and solve the ALS problem with this weight to obtain Q₁ and R₁. Repeat until convergence.
2. Estimate W* itself directly from the plant data.

Strategy 1 was originally suggested by Rajamani and Rawlings (2009), but has not been widely adopted because calculation of W* from Q and R was thought to be intractable in most practical cases. We will present a new method of calculation that reduces the computational burden to more manageable levels.

Strategy 2 was originally suggested by Zagrobelny and Rawlings (2015). We will present improvements and new insights related to this strategy.

B. J. Odelson, M. R. Rajamani, and J. B. Rawlings. A new autocovariance least-squares method for estimating noise covariances. Automatica, 42(2):303–308, February 2006.

M. R. Rajamani and J. B. Rawlings. Estimation of the disturbance structure from data using semidefinite programming and optimal weighting. Automatica, 45(1):142–148, 2009.

M. A. Zagrobelny and J. B. Rawlings. Practical improvements to autocovariance least-squares. AIChE J., 61:1840–1855, 2015.