Correlated System and Measurement Noise Processes

Abstract

In the previous two chapters, Kalman filtering for the model involving uncorrelated system and measurement noise processes was studied.

Exercises

• 4.1 Let $\mathbf {v}$ be a random vector and define

  $$\begin{aligned} L(\mathbf {x},\ \mathbf {v})=E(\mathbf {x})+\langle \mathbf {x},\ \mathbf {v}\rangle [\Vert \mathbf {v}\Vert ^{2}]^{-1}(\mathbf {v}-E(\mathbf {v})). \end{aligned}$$

  Show that $L(\cdot ,\ \mathbf {v})$ is a linear operator in the sense that

  $$\begin{aligned} L(A\mathbf {x}+B\mathbf {y},\ \mathbf {v})=AL(\mathbf {x},\ \mathbf {v})+BL(\mathbf {y},\ \mathbf {v}) \end{aligned}$$

  for all constant matrices A and B and random vectors $\mathbf {x}$ and $\mathbf {y}$.

• 4.2 Let $\mathbf {v}$ be a random vector and $L(\cdot ,\ \mathbf {v})$ be defined as above. Show that if a is a constant vector then $L(\mathbf {a},\ \mathbf {v})=\mathbf {a}$.

• 4.3 For a real-valued function f and a matrix $A=[a_{ij}]$, define

  $$\begin{aligned} \frac{df}{dA}=\left[ \frac{\partial f}{\partial a_{ij}}\right] ^{\top }. \end{aligned}$$

  By taking

  $$\begin{aligned} \frac{\partial }{\partial H}(tr\Vert \mathbf {x}-\mathbf {y}\Vert ^{2})=0, \end{aligned}$$

  show that the solution $\mathbf {x}^{*}$ of the minimization problem

  $$\begin{aligned} tr\Vert \mathbf {x}^{*}-\mathbf {y}\Vert ^{2}=\mathop {min}\limits _{H}tr\Vert \mathbf {x}-\mathbf {y}\Vert ^{2}, \end{aligned}$$

  where $\mathbf {y}=E(\mathbf {x})+H(\mathbf {v}-E(\mathbf {v}))$, can be obtained by setting

  $$\begin{aligned} \mathbf {x}^{*}=E(\mathbf {x})-\langle \mathbf {x},\ \mathbf {v}\rangle [\Vert \mathbf {v}\Vert ^{2}]^{-1}(E(\mathbf {v})-\mathbf {v}), \end{aligned}$$

  where

  $$\begin{aligned} H^{*}=\langle \mathbf {x},\ \mathbf {v}\rangle [\Vert \mathbf {v}\Vert ^{2}]^{-1}. \end{aligned}$$

• 4.4 Consider the linear stochastic system (4.16). Let

  $$\begin{aligned} \mathbf {v}^{k-1}=\left[ \begin{array}{c} \mathbf {v}_{0}\\ \vdots \\ \mathbf {v}_{k-1} \end{array}\right] \qquad and\qquad \mathbf {v}^{k}=\left[ \begin{array}{c} \mathbf {v}^{k-1}\\ \mathbf {v}_{k} \end{array}\right] . \end{aligned}$$

  Define $L(\mathbf {x},\ \mathbf {v})$ as in Exercise 4.1 and let $\mathbf {x}_{k}^{\#}=\mathbf {x}_{k}-\hat{\mathbf {x}}_{k|k-1}$ with $\hat{\mathbf {x}}_{k|k-1} :=L(\mathbf {x}_{k},\mathbf {v}^{k-1})$. Prove that

  $$\begin{aligned}&\langle \underline{\xi }_{k-1},\ \mathbf {v}^{k-2}\rangle =0,\qquad \langle \underline{\eta }_{k-1},\ \mathbf {v}^{k-2}\rangle =0,\\&\langle \underline{\xi }_{k-1},\ \mathbf {x}_{k-1}\rangle =0,\qquad \langle \underline{\eta }_{k-1},\ \mathbf {x}_{k-1}\rangle =0,\\&\langle \mathbf {x}_{k-1}^{\#},\ \underline{\xi }_{k-1}\rangle =0,\qquad \langle \mathbf {x}_{k-1}^{\#},\ \underline{\eta }_{k-1}\rangle =0,\\&\langle \hat{\mathbf {x}}_{k-1|k-2},\ \underline{\xi }_{k-1}\rangle =0,\qquad \langle \hat{\mathbf {x}}_{k-1|k-2},\ \underline{\eta }_{k-1}\rangle =0. \end{aligned}$$

• 4.5 Verify that

  $$\begin{aligned} (I-G_{k}C_{k})P_{k, k-1}C_{k}=G_{k}R_{k} \end{aligned}$$

  and

  $$\begin{aligned} \langle \mathbf {x}_{k-1}-\hat{\mathbf {x}}_{k-1|k-1},\ \mathrm {\Gamma }_{k-1}\underline{\xi }_{k-1}-K_{k-1}\underline{\eta }_{k-1}\rangle =O_{n\times n}. \end{aligned}$$

• 4.6 Consider the linear deterministic/stochastic system

  $$\begin{aligned} \left[ \begin{array}{rcl} \mathbf {x}_{k+1}&{}=&{}A_{k}\mathbf {x}_k+B_{k}\mathbf {u}_{k}+\mathrm {\Gamma }_{k}\underline{\xi }_{k}\\ \mathbf {v}_{k}&{}=&{}C_{k}\mathbf {x}_{k}+D_{k}\mathbf {u}_{k}+\underline{\eta }_{k}, \end{array}\right. \end{aligned}$$

  where $\{\mathbf {u}_{k}\}$ is a given sequence of deterministic control inputs. Suppose that the same assumption for (4.16) is satisfied. Derive the Kalman filtering algorithm for this model.

• 4.7 Consider the following so-called ARMAX (auto-regressive moving-average model with exogeneous inputs) model in signal processing:

  $$\begin{aligned} v_{k}=-a_{1}v_{k-1}-a_{2}v_{k-2}-a_{3}v_{k-3}+b_{0}u_{k}+b_{1}u_{k-1}+b_{2}u_{k-2}+c_{0}e_{k}+c_{1}e_{k-1}, \end{aligned}$$

  where $\{v_{j}\}$ and $\{u_{j}\}$ are output and input signals, respectively, $\{e_{j}\}$ is a zero-mean Gaussian white noise sequence with $Var(e_{j})=s_{j}>0$, and $a_{j}, b_{j}, c_{j}$ are constants.

  • (a) Derive a state-space description for this ARMAX model.

  • (b) Specify the Kalman filtering algorithm for this state-space description.

• 4.8 More generally, consider the following ARMAX model in signal processing:

  $$\begin{aligned} v_{k}=-\sum _{j=1}^{n}a_{j}v_{k-j}+\sum _{j=0}^{m}b_{j}u_{k-j}+\sum _{j=0}^{\ell }c_{j}e_{k-j}, \end{aligned}$$

  where $0\le m,\ell \le n$, $\{v_{j}\}$ and $\{u_{j}\}$ are output and input signals, respectively, $\{e_{j}\}$ is a zero-mean Gaussian white noise sequence with $Var(e_{j})=s_{j}>0$, and $a_{j}, b_{j}, c_{j}$ are constants.

  • (a) Derive a state-space description for this ARMAX model.

  • (b) Specify the Kalman filtering algorithm for this state-space description.