Derivation of Transfer Functions for Kalman Filter

Question

Hi All: I'm somewhat familiar with the kalman filter from a statistical point of view. But lately I've been trying to familarize myself with the linear systems-EE way of looking at it. So, I've been trying to read the Hannan and Deistler 1988 text slowly. I find it difficult and there is material in there that I don't follow so this is what my question is about.

I will write down verbatim what the book has and then explain the parts of it that I don't understand. I don't expect the actual detailed answer but rather a book or notes that explain the derivation. I'm totally new to this notation and terminology so maybe my questions can be easily answered but I don't expect it ? In fact, notes or a reference would still be appreciated even if the answers are straightforward.

Quote from Hannan and Deistler: Page 8

As will be shown, equations ... can be transformed into a state space system

1.1.22)$ ~~ x(t+1) = Fx(t) + Lz(t) + K \epsilon(t) $

1.1.23) $ ~~ y(t) = H x(t) + \epsilon(t) $

where $F \in R^{n \times n}$, $L \in R^{n \times m}$, $K \in R^{n \times s}$ and $H \in R^{s \times n}$.

Then there about five pages before the authors comes back to these equations above. I won't quote these five pages but they are mostly about how to use the lag operator to and the z-generating function. Although I find these pages confusing, I do understand the concept of a lag operator. I have a feeling that, if someone can answer my question or point to a useful reference or notes, then they won't need these pages. If someone does need them, I can copy them and send them in a pdf.

So, now I continue to where the authors comes back to above and I quote:

Hannan and Deistler, page 14

For a state space system (1.1.22), (1.1.23) we have det$(I - Fz) = 1$ for $z = 0$ and thus there always exists a causal solution

1.2.21)$ ~~~ x(t) = (I - Fz)^{-1}z[Lz(t) + K e(t)]$

where $(I - Fz)^{-1} = \sum_{j = 0}^\infty F^{j}z^{j}$, provided that the sum in 1.2.21) exists. In this sense, causality is built into the definition of state-space systems.

The transfer functions corresponding to (1.1.22), (1.1.23) are

$k(z) = H(I_{n} - Fz)^{-1}zK + I_{s} = I_{s} + \sum_{j=1}^{\infty} HF^{j-1}Kz^{j}$

$l(z) = H(I_{n} - Fz)^{-1} zL = \sum_{j=1}^{\infty} HF^{j-1}Lz^{j}$

Starting the system with given initial values $x(0)$, $z(0)$, $\epsilon(0)$ at t = 1, we have

$ x(t) = \sum_{j=1}^{t} F^{j-1}Lz(t-j) + \sum_{j=1}^{t} F^{j-1}K\epsilon(t-1) + F^{t}x(0)$

$y(t) = \sum_{j=1}^{t} HF^{j-1}Lz(t-j) + \sum_{j=1}^{t} HF^{j-1}K\epsilon(t-1) + HF^{t}x(0) + \epsilon_{t}$

The three parts that I don't understand regarding above are

1) the statement about the determinant of $(I-Fz)$ being 1 when $z$ = zero. I mean I understand why that's true but what does it have to do with the price of tea in china ? or causality ?

2) the derivation of the transfer functions, $k(z)$ and $l(z)$. I do understand 1.2.21) so I imagine that 1.2.21) is used to derive them ????? I also understand the second equalities in $k(z)$ and $l(z)$. But not the first equality signs.

3) the equation for $x(t)$ at the very end. I do see how $y(t)$ is obtained once one has $x(t)$.

I imagine that the material that I'm confused about above ( which is most of it ) is hidden deeply in the 5 pages that I didn't quote but, like I said, I found those pages to be quite "opaque", to say the least.

My guess is that I need is nice reference or lectures notes that explain why, when it comes to the KF,

A) the determinant of $(I-Fz)$ is important and

B) how the associated transfer response functions are obtained from a state space representation?

C) How the $x(t)$ equation is obtained given the transfer function ?

Note that I'm definitely opened to fighting with this material on my own so even just a good link is fine.

Finally, if anyone knows of a book that has similar objectives to Hannan and Deistler but is less concise and shows more steps to things (i.e. doesn't assume as much mathematical maturity ), it's appreciated.

Answer 1

This is an attempt to answer. It's not complete, and a little fuzzy, but feel free to comment and I can edit / update.

1) the statement about the determinant of $(I-Fz)$ being 1 when $z$ = zero. I mean I understand why that's true but what does it have to do with the price of tea in china ? or causality ?

So let's have a look: $$ x(t+1) = Fx(t) + Lz(t) + K \epsilon(t)\\ zx(t) - Fx(t) = Lz(t) + K \epsilon(t)\\ (I-Fz^{-1})x(t) = z^{-1}[Lz(t) + K \epsilon(t)]\\ x(t) = (I-Fz^{-1})^{-1}z^{-1}[Lz(t) + K \epsilon(t)]\\ $$

One thing to note about this is that $z$ is used twice: once as the lag operator and once as the "reference" signal. That confused me for a while. Another thing is that they appear to use $z$ as the lag operator, whereas $z^{-1}$ is usually used. Above I've assumed the conventional approach to $z$ as the advance operator.

My understanding of causality comes from the ROC associated with the $z$ transform. A $z$-transform is causal if its ROC extends outwards from the outermost pole. A $z$-transform is BIBO stable if its ROC includes the unit circle.

I believe what they really should have said is:

Because we assume that $\displaystyle(I - Fz)^{-1} = \sum_{j = 0}^\infty F^{j}z^{j}$, we assume $j\not < 0$ and so assume causality.

which is also sort of implied in their use of state space equations, as they say In this sense, causality is built into the definition of state-space systems.

2) the derivation of the transfer functions, $k(z)$ and $l(z)$. I do understand 1.2.21) so I imagine that 1.2.21) is used to derive them ????? I also understand the second equalities in $k(z)$ and $l(z)$. But not the first equality signs.

Assuming for now that $L=0$, then the state space form of the $z$ transform of (1.1.22) and (1.1.23) is just: $$ H(I - z^{-1})K + I $$ see these notes, particularly equation (10.20):

Though they use $A, B, C,$ and $D$ instead of $F, K, H,$ and $I$.

This is just the standard way to represent transfer functions in state space.

3) the equation for $x(t)$ at the very end. I do see how $y(t)$ is obtained once one has $x(t)$.

I believe the $x(t)$ equation is just had by the usual solution to a linear, constant coefficient difference equation with non-zero initial conditions (but using their notation). It's not the same notation, but try this.

My guess is that I need is nice reference or lectures notes that explain why, when it comes to the KF,

A) the determinant of $(I-Fz)$ is important and

See answer to 2) above.

B) how the associated transfer response functions are obtained from a state space representation?

See references above.

C) How the $x(t)$ equation is obtained given the transfer function ?

See answer to 3) above.