Abstract
Consider the linear stochastic system with the following statespace description.
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Exercises
Exercises

5.1.
Let \(\{\underline{\beta }_{k}\}\) be a sequence of zeromean Gaussian white noise and \(\{\mathbf {v}_{k}\}\) a sequence of observation data as in the system (5.1). Set
$$\begin{aligned} \underline{\tilde{\beta }}_{k}=\left[ \begin{array}{c} 0\\ \underline{\beta }_{k} \end{array}\right] ,\qquad \mathbf {v}^{k}=\left[ \begin{array}{c} \mathbf {v}_{0}\\ \vdots \\ \mathbf {v}_{k} \end{array}\right] , \end{aligned}$$and define \(L(\mathbf {x},\ \mathbf {v})\) as in (4.6). Show that
$$\begin{aligned} L\left( \underline{\tilde{\beta }}_{k},\ \mathbf {v}^{k}\right) =0. \end{aligned}$$ 
5.2.
Let \(\{\underline{\gamma }_{k}\}\) be a sequence of zeromean Gaussian white noise and \(\mathbf {v}^{{k}}\) and \(L(\mathbf {x},\ \mathbf {v})\) be defined as above. Show that
$$\begin{aligned} L\left( \mathbf {v}_{k1},\ \mathbf {v}^{k1}\right) =\mathbf {v}_{k1} \end{aligned}$$and
$$\begin{aligned} L\left( \underline{\gamma }_{k},\mathbf {v}^{k1}\right) =0. \end{aligned}$$ 
5.3.
Let \(\{\underline{\gamma }_{k}\}\) be a sequence of zeromean Gaussian white noise and \(\mathbf {v}^{{k}}\) and \(L(\mathbf {x},\ \mathbf {v})\) be defined as in Exercise 5.1. Furthermore, set
$$\begin{aligned} \hat{\mathbf {z}}_{k1}=L\left( \mathbf {z}_{k1},\ \mathbf {v}^{k1}\right) \qquad and \qquad \mathbf {z}_{k1}=\left[ \begin{array}{c} \mathbf {x}_{k1}\\ \underline{\xi }_{k1} \end{array}\right] . \end{aligned}$$Show that
$$\begin{aligned} \langle \mathbf {z}_{k1}\hat{\mathbf {z}}_{k1},\ \underline{\gamma }_{k}\rangle =0. \end{aligned}$$ 
5.4.
Let \(\{\underline{\beta }_{k}\}\) be a sequence of zeromean Gaussian white noise and set
$$\begin{aligned} \underline{\tilde{\beta }}_{k}=\left[ \begin{array}{c} 0\\ \underline{\beta }_{k} \end{array}\right] . \end{aligned}$$Furthermore, define \(\hat{\mathbf {z}}_{k1}\) as in Exercise 5.3. Show that
$$\begin{aligned} \langle \mathbf {z}_{k1}\hat{\mathbf {z}}_{k1},\underline{\tilde{\beta }}_{k}\rangle =0. \end{aligned}$$ 
5.5.
Let \(L(\mathbf {x},\ \mathbf {v})\) be defined as in (4.6) and set \(\hat{\mathbf {z}}_{0}=L(\mathbf {z}_{0},\ \mathbf {v}_{0})\) with
$$\begin{aligned} \mathbf {z}_{0}=\left[ \begin{array}{c} \mathbf {x}_{0}\\ \underline{\xi }_{0}\end{array}\right] . \end{aligned}$$Show that
$$\begin{aligned}&\quad Var(\mathbf {z}_{0}\hat{\mathbf {z}}_{0})\\&=\left[ \begin{array}{cc} Var(\mathbf {x}_{0}) &{}\quad 0\\ [Var(\mathbf {x}_{0})]C_{0}^{\top }[C_{0}Var(\mathbf {x}_{0})C_{0}^{\top }+R_{0}]^{1}C_{0}[Var(\mathbf {x}_{0})] &{}\quad \\ 0 &{}\quad Q_0 \end{array}\right] . \end{aligned}$$ 
5.6.
Verify that if the matrices \(M_{k}\) and \(N_{k}\) defined in (5.1) are identically zero for all k, then the Kalman filtering algorithm given by (5.18–5.20) reduces to the one derived in Chaps. 2 and 3 for the linear stochastic system with uncorrelated system and measurement white noise processes.

5.7.
Simplify the Kalman filtering algorithm for the system (5.1) where \(M_{k}=0\) but \(N_{k}\ne 0\).

5.8.
Consider the tracking system (5.22) with colored input (5.23).

(a)
Reformulate this system with colored input as a new augmented system with Gaussian white input by setting
$$\begin{aligned}\begin{gathered} \underline{X}_{k}=\left[ \begin{array}{c} \mathbf {x}_{k}\\ \underline{\xi }_{k}\\ \eta _{k} \end{array}\right] ,\quad \underline{\zeta }_{k}=\left[ \begin{array}{c} 0\\ \underline{\beta }_{k+1}\\ \gamma _{k+1} \end{array}\right] ,\\ A_{c}=\left[ \begin{array}{ccc} A &{}\quad I &{}\quad 0\\ 0 &{}\quad F &{}\quad 0\\ 0 &{}\quad 0 &{}\quad g \end{array}\right] \quad and \quad C_{c}=[C \ 0\ 0\ 0\ 1]. \end{gathered}\end{aligned}$$ 
(b)
By formally applying formulas (3.25) to this augmented system, give the Kalman filtering algorithm to the tracking system (5.22) with colored input (5.23).

(c)
What are the major disadvantages of this approach?

(a)
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Chui, C.K., Chen, G. (2017). Colored Noise Setting. In: Kalman Filtering. Springer, Cham. https://doi.org/10.1007/9783319476124_5
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DOI: https://doi.org/10.1007/9783319476124_5
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