Exam Preparation Notes

Statistical Hypothesis Testing

• $H_0$: Null hypothesis (default, uninteresting outcome).
• $H<em>1 \sim \neg H</em>0$: Alternative hypothesis.
• Process:
  1. Assume $H_0$.
  2. Under $H_0$, compute statistic of interest (e.g., $Q$).
  3. Compare statistic with its distribution under $H_0$ (e.g., $X^2(\text{df})$).
  • $p$-value: Probability of observing $Q$ or something more extreme if $H_0$ was true.

P-Value and Significance Level

• Probability Density Function (PDF): $p = f<em>Q(Q</em>{\text{data}})$
• Cumulative Distribution Function (CDF): $F<em>Q(Q</em>{\text{data}}) = \int<em>{-\infty}^{Q</em>{\text{data}}} f_Q(Q) dQ$
• $p$-value $= 1 - F<em>Q(Q</em>{\text{data}})$
• Reject $H_0$ if p < \alpha (e.g., 0.05).
  • $\alpha$: Probability of False Positives that we accept.

State-Space Identification

• Motivation: Need "states" for control / fault detection / etc.
• System representation:
  • Input: $u(t)$
  • Output: $y(t)$
  • State: $x(t)$
  • Equation: $y(t) = G(q) u(t)$
• Controller design (e.g., LQR - Linear Quadratic Regulator) uses state feedback.

Approaches to State-Space Modeling

• App 1) Learn I/O model $\rightarrow$ transform it to state space.
• App 2) Learn state-space model directly.
• State-space models are identifiable up to a change of coordinates of the state.

State-Space Model Equations

• State equation: $x(t+1) = Ax(t) + Bu(t)$
• Output equation: $y(t) = Cx(t) + Du(t)$
• Transfer function: $G(z) = C(zI - A)^{-1}B + D$

Similarity Transformation

• Change of coordinates: $\tilde{x}(t) = Px(t)$
• Transformed state equation: $\tilde{x}(t+1) = PAP^{-1}\tilde{x}(t) + Bu(t)$
• Transformed output equation: $y(t) = CP^{-1}\tilde{x}(t) + Du(t)$
• Transfer function invariance: $G(z) = C(zI - A)^{-1}B + D = \tilde{C}(zI - \tilde{A})^{-1}\tilde{B} + \tilde{D}$

Extended Observability Matrix

• Under similarity transformation:
  • $\tilde{A}=PAP^{-1}$
  • $\tilde{B}=PB$
  • $\tilde{C}=CP^{-1}$
  • $\tilde{D}=D$
• Extended Observability Matrix changes under coordinates
  • $O = \begin{bmatrix} C \ CA \ CA^2 \ … \ CA^{r-1} \end{bmatrix}$

Realization Theory

• Goal: Given a learned I/O model, find a state-space representation.
• Learned I/O model: $y(t) = G(q; \theta)u(t)$
• $G(q; \theta) = \frac{b<em>0 + b</em>1q^{-1} + b<em>2q^{-2} + … + b</em>mq^{-m}}{1 + a<em>1q^{-1} + a</em>2q^{-2} + … + a_nq^{-n}}$
• Objective: Solve $G(q) = C(qI - A)^{-1}B + D$

Example: Controllable Canonical Form

• $G(q; \theta) = \frac{b<em>0 + b</em>1q^{-1} + b<em>2q^{-2} + b</em>3q^{-3}}{1 + a<em>1q^{-1} + a</em>2q^{-2} + a_3q^{-3}}$
• State equation:
  $x(t+1) = \begin{bmatrix} -a<em>1 & -a</em>2 &amp; -a_3 \ 1 &amp; 0 &amp; 0 \ 0 &amp; 1 &amp; 0 \end{bmatrix} x(t) + \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix} u(t)$
• Output equation:
  $y(t) = \begin{bmatrix} b<em>1 & b</em>2 &amp; b<em>3 \end{bmatrix} x(t) + [b</em>0]u(t)$

State Estimation: Kalman Filter

• Optimal state estimator is the Kalman Filter.
• State-space model:
  • State equation: $x(t+1) = Ax(t) + Bu(t) + w(t)$
  • Output equation: $y(t) = Cx(t) + Du(t) + v(t)$
  • $w(t)$: Process noise.
  • $v(t)$: Measurement noise.
• Rearrange output equation: $Cx(t) = y(t) - Du(t) - v(t)$

State Estimation Equations

• $y(t) = Cx(t) + Du(t) + v(t)$
• $y(t+1) = C[Ax(t) + Bu(t) + w(t)] + Du(t+1) + v(t+1) = CAx(t) + CBu(t) + Du(t+1) + Cw(t) + v(t+1)$
• Vector of outputs:
  $\begin{bmatrix} y(t) \ y(t+1) \ y(t+2) \ … \ y(t+r-1) \end{bmatrix} = \begin{bmatrix} C \ CA \ CA^2 \ … \ CA^{r-1} \end{bmatrix} x(t) + \begin{bmatrix} D &amp; 0 &amp; 0 &amp; … \ CB &amp; D &amp; 0 &amp; … \ CAB &amp; CB &amp; D &amp; … \ … &amp; … &amp; … &amp; … \ CA^{r-2}B &amp; CA^{r-3}B &amp; CA^{r-4}B &amp; … \end{bmatrix} \begin{bmatrix} u(t) \ u(t+1) \ u(t+2) \ … \end{bmatrix} + \begin{bmatrix} v(t) \ w(t) \ w(t+1) \ … \end{bmatrix}$

Re-arranging equations

• Define:
  $O<em>r = \begin{bmatrix} C \ CA \ CA^2 \ … \ CA^{r-1} \end{bmatrix}$$S</em>r = \begin{bmatrix} D &amp; 0 &amp; 0 &amp; … \ CB &amp; D &amp; 0 &amp; … \ CAB &amp; CB &amp; D &amp; … \ … &amp; … &amp; … &amp; … \ CA^{r-2}B &amp; CA^{r-3}B &amp; CA^{r-4}B &amp; … \end{bmatrix}$
• Rewrite: $O<em>r x(t) = y</em>r(t) - S<em>r u</em>r(t) - v_r(t)$
• In theory: $x(t) = O<em>r^{-1} (y</em>r(t) - S<em>r u</em>r(t) )$
• But use Kalman Filter (KF) instead.

Solving Linear Equations

• $Ax = b$: 3 possibilities
  1. Unique solution: $A^{-1}b$, A is full column rank.
  2. 0 solutions: $x$ solves $min ||Ax - b||^2$
  3. $\infty$ solutions: $x$ solves $min ||x||^2$ s.t. $Ax = b$

Kalman Filter Algorithm

• State equation: $x(t+1) = Ax(t) + Bu(t) + w(t)$
• Output equation: $y(t) = Cx(t) + Du(t) + v(t)$
• Initialization: $\hat{x}_{0|0}$
• Prediction:
  • $\hat{x}<em>{1|0} = A \hat{x}</em>{0|0} + Bu(0)$
• Update:
  • $\hat{x}<em>{1|1} = \hat{x}</em>{1|0} + K (y(1) - C \hat{x}_{1|0} - Du(0))$
  • $K$: Kalman Gain.
• Output prediction error: $y(1) - C \hat{x}_{1|0} - Du(0)$
• Next step prediction: $\hat{x}<em>{2|1} = A \hat{x}</em>{1|1} + Bu(1)$
• Continue with update step