Parametric Modeling and System Identification

Parametric Modeling

• ARX Model Structure
  • System identification with noise.
  • Data: ${(u(t), y(t))}$
  • ARX model:
    • $A(q^{-1})y(t) = B(q^{-1})u(t) + e(t)$
    • $[1 + a<em>1q^{-1} + … + a</em>{na}q^{-na}]y(t) = [b<em>0 + b</em>1q^{-1} + … + b_{nb}q^{-nb}]u(t) + e(t)$
    • ${q^{-1}}$ is the delay operator.
    • $y(t)$ is the output at time $t$.
    • $u(t)$ is the input at time $t$.
    • $e(t)$ is the noise at time $t$.
    • $a_i$ are the coefficients of the polynomial $A$.
    • $b_i$ are the coefficients of the polynomial $B$.
  • Regression form:
    • $y(t) = \phi^T(t) \theta + e(t)$
    • $\phi(t)$ is the regressor vector (past inputs and outputs).
    • $\theta$ is the parameter vector.
  • Regressors: $\phi(t)$
  • Parameters: $\theta$
  • $A, B \implies \theta$

ARX Representation

• Equivalent Form
  • $y(t; \theta) = \phi^T(t) \theta$
  • $\theta = \begin{bmatrix} a<em>1 \ … \ a</em>{na} \ b<em>0 \ … \ b</em>{nb} \end{bmatrix}$
  • $\phi(t) = \begin{bmatrix} -y(t-1) \ … \ -y(t-na) \ u(t) \ … \ u(t-nb) \end{bmatrix}$
• Multiple Linear Regression
  • $J(\theta) = \frac{1}{N} \sum_{t=0}^{N-1} \epsilon^2(t; \theta)$
  • One-step ahead prediction error: $\epsilon(t; \theta) = y(t) - y(t; \theta)$
  • Prediction error.
  • Mean-squared error (MSE) cost function.

Cost Function and Linear System

• Expressing the cost function:
  • $J(\theta) = \frac{1}{N} \sum_{t=0}^{N-1} \epsilon^2(t; \theta)$
• Taking the derivative to find the minimum:
  • $\hat{\theta} = argmin J(\theta) \implies \sum_{t=0}^{N-1} \phi(t) \epsilon(t; \theta) = 0$
• This leads to a linear system of equations.
• Solution:
  • $\hat{\theta}_N = R^{-1}(N) f(N)$
  • $R(N) = \sum_{t=0}^{N-1} \phi(t) \phi^T(t)$
  • $f(N) = \sum_{t=0}^{N-1} \phi(t) y(t)$
• If the data is "rich enough", $R(N)$ will be full rank (invertible).
  • $R(N)$ is a $p \times p$ matrix.
  • $f(N)$ is a $p \times 1$ matrix.

Ordinary Least Squares (OLS) Estimation

• LS Estimate
  • $\hat{\theta}_N = R^{-1}(N) f(N)$
• Consistency: Is OLS a "good" estimate?
• Assumption: The data-generating process is an ARX model.
  • $y(t) = \phi^T(t) \theta_0 + e(t)$
  • $\theta_0$ represents the ground-truth parameters.
• Ideally, we want $\hat{\theta}<em>N = \theta</em>0$

OLS and Estimation Error

• Substituting the Data Generating Process
  • $\hat{\theta}<em>{LS} = R^{-1}(N) \sum</em>{t=0}^{N-1} \phi(t) y(t)$
  • $\hat{\theta}<em>{LS} = R^{-1}(N) \sum</em>{t=0}^{N-1} \phi(t) [\phi^T(t) \theta_0 + e(t)]$
  • $\hat{\theta}<em>{LS} = \theta</em>0 + R^{-1}(N) \sum_{t=0}^{N-1} \phi(t) e(t)$
• Estimation Error
  • $\tilde{\theta}<em>N = \hat{\theta}</em>N - \theta<em>0 = R^{-1}(N) f</em>e(N)$
  • $f<em>e(N) = \sum</em>{t=0}^{N-1} \phi(t) e(t)$

Consistency of OLS

• Definition of Consistency
  • An estimate $\hat{\theta}$ is consistent if $\lim<em>{N \to \infty} \hat{\theta}</em>N = \theta_0$
• Question: Is OLS consistent?
• Analysis
  • $\lim<em>{N \to \infty} R^{-1}(N) f</em>e(N) = 0$
  • $\lim<em>{N \to \infty} R(N) = \lim</em>{N \to \infty} \frac{1}{N} \sum_{t=0}^{N-1} \phi(t) \phi^T(t)$
  • $\lim<em>{N \to \infty} \frac{1}{N} \sum</em>{t=0}^{N-1} \begin{bmatrix} -y(t-1) \ … \ -y(t-na) \ u(t) \ … \ u(t-nb) \end{bmatrix} \begin{bmatrix} -y(t-1) &amp; … &amp; -y(t-na) &amp; u(t) &amp; … &amp; u(t-nb) \end{bmatrix}$
• Covariance Matrices
  • $R<em>y(\tau) = \lim</em>{N \to \infty} \frac{1}{N} \sum_{t=0}^{N-1} y(t+\tau) y(t)$

Covariance Matrix and Open-Loop Experiment

• $\lim<em>{N \to \infty} \frac{R(N)}{N} = \begin{bmatrix} R</em>y(0) &amp; R<em>y(1) & … & R</em>y(na) &amp; R<em>{yu}(0) & … & R</em>{yu}(nb) \ R<em>y(1) & R</em>y(0) &amp; … &amp; R<em>y(na-1) & R</em>{yu}(1) &amp; … &amp; R<em>{yu}(nb) \ … & … & … & … & … & … & … \ R</em>y(na) &amp; R<em>y(na-1) & … & R</em>y(0) &amp; R<em>{yu}(na) & … & R</em>{yu}(nb) \ R<em>{uy}(0) & R</em>{uy}(1) &amp; … &amp; R<em>{uy}(na) & R</em>u(0) &amp; … &amp; R<em>u(nb) \ … & … & … & … & … & … & … \ R</em>{uy}(nb) &amp; R<em>{uy}(nb-1) & … & R</em>{uy}(na) &amp; R<em>u(nb) & … & R</em>u(0) \end{bmatrix}$
• $\lim<em>{N \to \infty} \frac{f</em>e(N)}{N} = \lim<em>{N \to \infty} \frac{1}{N} \sum</em>{t=0}^{N-1} \begin{bmatrix} -y(t-1) \ … \ -y(t-na) \ u(t) \ … \ u(t-nb) \end{bmatrix} e(t) = \begin{bmatrix} -R<em>{ye}(1) \ … \ -R</em>{ye}(na) \ R<em>{ue}(0) \ … \ R</em>{ue}(nb) \end{bmatrix} = 0$
• Since $e(t)$ is assumed to be white noise.
• Any open-loop experiment.

Regularization and Prediction Error

• Regularization
  • $J(\theta) = MSE(\theta) + \lambda ||\theta||^2$
  • Adds bias but controls variance when $N \approx p$
• Prediction
  • $\hat{y}(t|t-1) = E[y(t) | \Omega(t-1)]$
  • $\epsilon(t) = y(t) - \hat{y}(t|t-1)$
  • $\Omega(t-1)$ is the information set up to time $t-1$.
  • $\epsilon(t)$ is the "new" information in $y(t)$ that did not exist in $\Omega(t-1)$.
  • If the model learned well, $\epsilon(t)$ should be uncorrelated with $\Omega(t-1)$.

Prediction Error Framework

• General Principle
  • Good Model = Good Prediction
  • Minimize prediction error.
  • $V<em>N(\theta) = \frac{1}{N} \sum</em>{t=1}^{N} \epsilon^2(t, \theta)$
• For ARX Models
  • Prediction error should be uncorrelated with past data.
  • $\sum_{t=1}^{N} \epsilon(t) \phi(t) = 0$

Model Mismatch

• Question: What happens if we fit an ARX model to data coming from a non-ARX system?
• Scenario: True system is not ARX.
  • $y(t) = \phi^T(t) \theta_0 + v(t)$
  • $v(t)$ is colored noise.
• Model: ARX model.
  • $y(t; \theta) = \phi^T(t) \theta + e(t)$
• Model Mismatch
  • $A<em>0 y(t) = B</em>0 u(t) + C_0 e(t)$
  • $A y(t) = B u(t) + e(t)$
  • We can always re-arrange into an ARX-looking form, except that the noise won't be white.

Consequences of Model Mismatch

• LS Estimate
  • $\hat{\theta}_{LS} = R^{-1}(N) f(N)$
  • $\hat{\theta}<em>{LS} = \theta</em>0 + R^{-1}(N) f_v(N)$
• $\lim<em>{N \to \infty} R^{-1}(N) f</em>v(N) \neq 0$
  • Not generally zero.
  • Bias if $v$ is colored.
  • $\hat{\theta}$ won't be consistent if the experiment is open-loop.

Instrumental Variable (IV) Method

• System and Model
  • Assume System: $y(t) = \phi^T(t) \theta_0 + v(t)$
  • Model: $\hat{y}(t; \theta) = \phi^T(t) \theta + e(t)$
• Instead of minimizing MSE
  • $\sum_{t=0}^{N-1} \zeta(t) \epsilon(t; \theta) = 0$
  • $\zeta(t)$ is the instrumental variable.
  • $\zeta(t) e(t; \theta) = 0$
• For now, assume $\zeta(t)$ is given
• I.V.

IV Estimation

• Cost Function
  • $J(\theta) = \frac{1}{N} \sum_{t=0}^{N-1} \zeta(t) [y(t) - \phi^T(t) \theta]$
• $\hat{\theta}<em>{IV} = \Bigg[\sum</em>{t=0}^{N-1} \zeta(t) \phi(t)^T \Bigg]^{-1} \Bigg[\sum_{t=0}^{N-1} \zeta(t) y(t) \Bigg]$
• Asymptotic Analysis
  • $\lim<em>{N \to \infty} \frac{1}{N} \sum</em>{t=0}^{N-1} \zeta(t) v(t) = 0$
  • $\lim<em>{N \to \infty} \hat{\theta}</em>{IV} = \theta<em>0 + \Bigg[ R</em>{\zeta \phi} \Bigg]^{-1} \Bigg[ R_{\zeta v} \Bigg]$

Designing the Instrumental Variable

• $\lim<em>{N \to \infty} \hat{\theta}</em>{IV} = \theta<em>0 + \Bigg[ R</em>{\zeta \phi} \Bigg]^{-1} \Bigg[ R_{\zeta v} \Bigg]$
• We want:
  • $R_{\zeta v} = 0$
  • $\zeta$ should be uncorrelated with $v$.
  • $R_{\zeta \phi}$
  • Should be invertible and even well-conditioned.
  • $\zeta$ should be correlated with $\phi$.