[PPT] - Overview of State Space Models Standard State Space Model Standard PowerPoint Presentation

SLIDE 1

Standard State Space Model Matrices xn+1 = Fnxn + Gnun n ≥ 0 yn = Hnxn + vn

Conceptually, yn usually contains all the signals that are
bservable

– May come from multiple sensors

xn contains all of the parameters of the process that you are

interested in estimating

Note that the process is nonstationary in general since all of the

matrices are allowed to change with time

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Overview of State Space Models

Standard state space model
Straight-forward extensions
Properties
Solving the normal equations
Covariance matrices
Wide-sense Markov processes
Designing state space models

– Common choices for the state dynamics

Continuous-time to discrete-time conversion

– Coarse estimates – Exact conversions

Nonlinear state space models
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State Space Model With Known Input xn+1 = Fnxn + Gn(un + un) n ≥ 0 yn = Hnxn + vn

The state space model can be extended when there are known

input signals (collected in the vector un ∈ Cm×1) that affect the state of the system

This changes little in the development of the Kalman filter

algorithm

The apparent constraint that both un and un are multiplied by

Gn does not really impair the flexibility of the model

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Standard State Space Model xn+1 = Fnxn + Gnun n ≥ 0 yn = Hnxn + vn where Fn ∈ Cℓ×ℓ Gn ∈ Cℓ×m Hn ∈ Cp×ℓ xn ∈ Cℓ×1 un ∈ Cm×1 yn ∈ Cp×1 vn ∈ Cp×1

un and vn are multivariate white noise processes
In many cases vn is interpreted as an output disturbance or

measurement noise

un is usually called process noise
This is a more general form than is used in many texts
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SLIDE 2

Covariance Matrices xn+1 = Fnxn + Gnun yn = Hnxn + vn ⎡ ⎣ x0 un vn ⎤ ⎦ , ⎡ ⎢ ⎢ ⎣ x0 uk vk 1 ⎤ ⎥ ⎥ ⎦

=

⎡ ⎣ Π0 Qnδnk Snδnk S∗

nδnk

Rnδnk ⎤ ⎦

The random variable x0 and the two white noise processes drive

the state space model – This means that un, uk = 0 for n = k – Does not mean Qn or Rn are diagonal matrices

The model is causal in that xn and yn are completely determined

by x0 and past and present values of vn and un

Clever use of the inner product notation has been used here to

indicate these RVs have zero mean

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State Space Model With Known Input Continued xn+1 = Fnxn + Gn(un + un) n ≥ 0 yn = Hnxn + vn

We can specify the covariance matrix of un to be anything we like

such that the additive noise term has whatever covariance matrix we like

Suppose we wish for the scaled state noise Gnun to have a

covariance matrix Tn Qn G−1

n TnG−∗ n

un, un = Qn Gnun, Gnun = GnQnG∗

n = Tn

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Covariance Matrices xn+1 = Fnxn + Gnun yn = Hnxn + vn ⎡ ⎣ x0 un vn ⎤ ⎦ , ⎡ ⎢ ⎢ ⎣ x0 uk vk 1 ⎤ ⎥ ⎥ ⎦

=

⎡ ⎣ Π0 Qnδnk Snδnk S∗

nδnk

Rnδnk ⎤ ⎦

The matrices Π0, Qn, Rn, and Sn are assumed to be known
By definition, the matrices Qn and Rn must be Hermitian and

non-negative definite

The matrix Sn doesn’t have to be Hermitian, but it must be such

that the joint covariance matrix is non-negative definite

Q

S S∗ R

≥ 0
This characterization ensures the process is wide-sense Markov (to

be discussed later)

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State Space Model With Known Input Continued xn+1 = Fnxn + Gnun yn = Hnxn + vn

An additional known input signal could be added to the output

equation in a similar fashion yn = Hnxn + wn + vn

This is most easily handled by defining a different observed signal

as zn yn − wn zn = Hnxn + vn

Thus, it easily reduces to the form of our standard state space

model output equation

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SLIDE 3

Example 1: Orthogonality Properties xn+1 = Fnxn + Gnun yn = Hnxn + vn Prove some of the orthogonality properties on the previous slide.

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Correlation of Process and Measurement Noise xn+1 = Fnxn + Gnun yn = Hnxn + vn ⎡ ⎣ x0 un vn ⎤ ⎦ , ⎡ ⎢ ⎢ ⎣ x0 uk vk 1 ⎤ ⎥ ⎥ ⎦

=

⎡ ⎣ Π0 Qnδnk Snδnk S∗

nδnk

Rnδnk ⎤ ⎦

It is common to assume the process and measurement noise

processes are completely uncorrelated, Sn = 0 for all n

Due to domain knowledge that they have different origins
When feedback is used and the output modifies the state

equation, the noises may become correlated, Sn = 0 for some n

There are many forms, but the most useful is to assume that they

are correlated only at the same time instant

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Relationship to Ry ˆ xn|n = xn, yy−2y y col{y0, . . . , yn}

We know that the optimal causal estimator of xn is given by the

normal equations

Requires knowledge of two terms
The first is easy to solve for

xn, y = xn, Hnxn + vn = xn, xnH∗

n

In order to solve the normal equations, we need to know how

Ry = y2 is related to all of the state space model parameters and covariance matrices

However, we need to solve for xn, xk in order to obtain

expressions for both xn, y and Ry

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Orthogonality Properties You should be able to show that the following properties are true: un, xk = 0 vn, xk = 0 for n ≥ k un, yk = 0 vn, yk = 0 for n > k un, yk = Sn vn, yk = Rn for n = k un, x0 = 0 iff un, xn = 0 for n ≥ 0

The key idea is to leverage the orthogonality of the initial state

and the noise covariance matrices

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SLIDE 4

Solving for the State Covariance Matrix Continued xn, xk = Φ(n, k)Πk for n ≥ k For since x, y = y, x∗ for any random vectors x and y, it follows immediately that xn, xk = xk, xn∗ = (Φ(k, n)Πn)∗ = ΠnΦ∗(k, n) for n ≤ k These results can be summarized as xn, xk = ⎧ ⎪ ⎨ ⎪ ⎩ Φ(n, k)Πk n ≥ k Πn n = k ΠnΦ∗(k, n) n ≤ k

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Solving for the State Covariance Matrix Given the state space model, xn+1 = Fnxn + Gnun yn = Hnxn + vn we can easily show that Πn+1 xn, xn = Fnxn + Gnun, Fnxn + Gnun = Fnxn, xnF ∗

n + Gnun, unG∗ n

= FnΠnF ∗

n + GnQnG∗ n

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Solving for the Output Covariance Matrix Ry Given the state space model, xn+1 = Fnxn + Gnun yn = Hnxn + vn The output covariance matrix is given by yn, yk = Hnxn + vn, Hkxk + vk = Hnxn, xkH∗

k + Hnxn, vk + vn, xkH∗ k + vn, vk

The cross-terms xn, vk and vn, xk don’t necessarily cancel in this case sense un and vn are correlated. However they do if n = k since then xn is only a function of past values of un If vn, xk = 0 xn, vk = 0 then yn, yn = HnΠnH∗

n + Rn

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Solving for the State Covariance Matrix Continued Now let us define the state transition matrix Φ(n, k)

Fn−1Fn−2 . . . Fk

n > k I n = k Then xn = Φ(n, k)xk + A col{uk, uk+1, . . . , un−1}, n ≥ k for some matrix A. In other words, xn consists of Φ(n, k)xk plus some linear combination of {uk, uk+1, . . . , un−1}, which is uncorrelated with xk. Thus xn, xk = Φ(n, k)xk + A col{uk, uk+1, . . . , un−1}, xk = Φ(n, k)xk, xk + 0 = Φ(n, k)Πk for n ≥ k

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SLIDE 5

Symmetry in Ry Consider three observation vectors {y0, y1, y2} produced by the standard state space model Ry = ⎡ ⎣ R0 + H0ΠoH∗ N ∗

0 H∗ 1

N ∗

0 F ∗ 1 H∗ 2

H1N0 R1 + H1Π1H∗

1

N ∗

1 H∗ 2

H2F1N0 H2N1 R2 + H2Π2H∗

2

⎤ ⎦

However, it is not yet clear how we might be able to invert Ry

efficiently

Would be much easier to invert if was block diagonal
The correlation matrix of the innovations is block diagonal
But then we need a way of obtaining the innovations and it’s

covariance matrix

Stay tuned . . .
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Solving for the Output Covariance Matrix Ry Continued yn, yk = Hnxn, xkH∗

k + Hnxn, vk + vn, xkH∗ k + vn, vk

Now if n > k, vn, xk = 0 vn, vk = 0 For n > k, yn, yk = Hnxn, xkH∗

k + Hnxn, vk

= HnΦ(n, k)ΠkH∗

k + HnΦ(n, k + 1) (Fkxk + Gkuk) , vk

= HnΦ(n, k)ΠkH∗

k + HnΦ(n, k + 1)Gkuk, vk

= HnΦ(n, k + 1)FkΠkH∗

k + HnΦ(n, k + 1)GkSk

= HnΦ(n, k + 1) (FkΠkH∗

k + GkSk)

= HnΦ(n, k + 1)Nk where Nk FkΠkH∗

k + GkSk

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Wide Sense Markov (WSM) Processes

Markov stochastic processes have the following relationship for

their conditional probability density functions (pdfs) fyn|yj,yk(yn|yj, yk) = fyn|yj(yn|yj) if n > j > k

For linear MMSE estimation we only use first- and second-order

statistics

A random process is Wide-sense Markov (WSM) if the linear

MMSE estimator of xn given L{xn−1, . . . , x0} equals the linear MMSE estimator of xn given L{xn−1}.

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Solving for the Output Covariance Matrix Ry Finished Again we x, y = y, x∗ for any random vectors x and y, to complete the solution yn, yk = ⎧ ⎪ ⎨ ⎪ ⎩ HnΦ(n, k + 1)Nk n > k HnΠnH∗

n + Rn

n = k N ∗

nΦ∗(k, n + 1)H∗ k

n < k where Nn FnΠnH∗

n + GnSn

It may not be apparent, but this has introduced exactly the type of symmetry in Ry that we needed

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SLIDE 6

WSM Processes have State Space Model Representations Continued Thus, given a wide-sense Markov process xn, we can define a state space model xn+1 = Fnxn + GnuN

x0

un

,
x0

uk

=
Π0

Qnδnk

Thus a process {xn, n ≥ 0} is WSM if and only if it has a forwards

Markovian state space representation of the form above.

However, it turns out that yn is not WSM
In general a sum of WSM processes is generally not WSM
Processes such as yn = Hnxn + vn are called projections of

Markov processes

However the aggregate process obtained by considering

col{xn, yn} is WSM

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State Space Models are WSM Processes xn+1 = Fnxn + GnuN

x0

un

,
x0

uk

=
Π0

Qnδnk

Since xn ∈ L{x0, u0, . . . , un−1} we know that

un⊥xk, n ≥ k ≥ 0 Thus the orthogonal projection of xn+1 onto L{x0, . . . , xn} is given by ˆ xn+1|{x0,...,xn} = Fn ˆ xn|{x0,...,xn} + Gn ˆ un|{x0,...,xn} = Fnxn + 0 = Fn ˆ xn|xn = ˆ xn+1|xn

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Examples of State Space Models

Autoregressive (AR) processes
ARMA processes
AR processes revisited
Physical models
Nonlinear models
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WSM Processes have State Space Models Suppose we have a wide-sense Markov process xn. Consider its innovations process e0 = x0 en+1 = xn+1 − Ko,nxn Now define un en+1 Fn Ko,n xn2 Πn xn+1 = Fnxn + un Now since the innovations are orthogonal un, uk = en+1, ek+1 = 0 for n = k un2 = en+12 = Πn+1 − FnΠnF ∗

n Qn

Since en+1⊥e0, un = en+1, and e0 = x0, we have un⊥x0

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SLIDE 7

Autoregressive Processes Continued yn =

1

. . .

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎣ yn yn−1 yn−2 . . . yn−ℓ+1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ yn = Hxn

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Autoregressive Processes yn+1 = un +

ℓ−1

k=0

akyn−k Note that this differs from the usual model of AR processes x(n) = w(n) +

p

k=1

akx(n − k) in several ways

The white noise process is denoted as un
The AR process is denoted as yn
yn+1 is a function of un, instead of un+1 (non-minimum phase)
The order of the model is denoted as ℓ
yn can be a vector-valued process, though I will still denote the

coefficients with lower-case letters ak

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Autoregressive Processes Continued Thus with the definition of a state vector xn col{yn, yn−1, . . . , yn−ℓ+1} we can write an autoregressive process in state space form xn+1 = Fxn + Gun yn = Hxn

The companion matrix F is stable if and only if all its eigenvalues

are less than one in magnitude

This is equivalent to requiring that

a(z) zℓ − a0zℓ−1 − · · · − aℓ−1 has all of its roots inside the unit circle

Note that this could easily be generalized to the case where any of

the model parameters are time-varying

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AR Process State Space Model Let us define xn col{yn, yn−1, . . . , yn−ℓ+1} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ yn+1 yn yn−1 . . . yn−ℓ+2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ a0 a1 . . . aℓ−2 aℓ−1 1 . . . 1 . . . . . . . . . ... . . . . . . . . . 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ yn yn−1 yn−2 . . . yn−ℓ+1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 . . . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ u(n) xn+1 = Fxn + Gun Note that in the vector case, yn ∈ Cp×1, F and G are block matrices.

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SLIDE 8

AR Processes Revisited xn col{yn, yn−1, . . . , yn−ℓ+1}

The AR model we set up earlier is of little practical utility
Just like the earlier case, all of the model parameters must be

known

The state of the system is trivial to estimate once ℓ observations
f the output process have been made
Recall that the state of the system is usually consists of the

parameters that you want (or have) to estimate

For AR processes, the parameters we’re interested in estimating

are the model coefficients {a0, . . . , aℓ−1}

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ARMA Processes yn+1 = a0yn + · · · + aℓ−1yn−ℓ+1 + b0un + · · · + bℓ−1un−ℓ+1 xn+1 a0xn + · · · + aℓ−1xn−ℓ+1 + un yn+1 = b0xn + · · · + bℓ−1xn−ℓ+1 xn+1 = Fxn + Gun yn = Hxn

An ARMA process can be expressed as white noise filtered by a

cascade of an AP system followed by an AZ system

We already have an expression for the output of the AP system
The output of the AZ system is then just a linear combination of
utputs of the AP system
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AR Process Alternative Formulation Let us define the state as the parameters, xn a1 a2 . . . aℓ T Then the output of the AR process is given by yn =

ℓ

k=1

akyn−k + vn = Hnxn + vn where Hn yn−1 yn−2 . . . yn−ℓ

Now the state represents the parameters we’re interested in

estimating

This is a time-varying model
But how do we write an expression for the state update equation?
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ARMA In State Space Form xn+1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ a0 a1 . . . aℓ−2 aℓ−1 1 . . . 1 . . . . . . . . . ... . . . . . . . . . 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ xn + ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 . . . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ u(n) yn = b0 . . . bℓ−1

xn

xn+1 = Fxn + Gun yn = Hxn

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SLIDE 9

Random Walk Models xn+1 = Fnxn + Gnun xn+1 = xn + un yn = Hnxn + vn yn = Hnxn + vn

Fn = I, as for constant parameter models
The state at time (n + 1) is the same as xn plus a random

perturbation

This is a good model of processes in which the model parameters

slowly drift over time

The degree of the drift is controlled by the process noise

covariance Qn

Often, Qn = αI where the scalar α is chosen by the user
The model is unstable, but the estimates may (usually) still

converge!

Perhaps the most widely used model
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Designing State Space Models xn+1 = Fnxn + Gnun yn = Hnxn + vn

In many signal processing applications you can write a

mathematical expression that relates the observed signal to the parameters of interest

However, there is often no obvious state space model that can be
btained from first-principles or domain knowledge of the problem
The Kalman filter approach to linear estimation is still often useful

in these circumstances

There are several models for the state space equations that are

frequently used in these cases

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Unstable Deterministic Models xn+1 = Fnxn + Gnun xn+1 = λ−1/2xn yn = Hnxn + vn yn = Hnxn + vn

Oddly, the adaptive filter algorithm recursive least squares is

equivalent to using the Kalman filter with an unstable state space model

The user-specified scalar parameter λ controls the bias-variance

tradeoff – λ = 1: Equivalent to regularized least squares – 0 < λ < 1: Equivalent to recursive least squares – λ = 0: ?

The scalar measurement noise in this case is set Rn = 1
It turns out that if Rn = αI for any scalar, positive parameter α,

the same solution is obtained! (prove this)

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Constant Parameter Models xn+1 = Fnxn + Gnun xn+1 = xn yn = Hnxn + vn yn = Hnxn + vn

Fn = I, Gn = 0 or Qn = 0
The state at time xn+1 is the same as xn
The estimate will be time varying because we can improve the

estimate of x as we gain more observations yn

Example: Suppose we are trying to estimate the mean of a WN

process

Note that there will be a transition from the a priori estimate

x0|−1 (provided by user) to the unbiased estimate x as n → ∞

In this sense, the KF can be expressed as a type of linear Bayesean

estimation

Could be used for a quasi-stationary AR parameter estimate
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SLIDE 10

Continuous-Time to Discrete-Time Suppose our sample interval is given by Ts = 1/fs. Then ˙ x(t) = F(t)x(t) + G(t)u(t) x(t + Ts) = x(t) + t+Ts

t

˙ x(τ) dτ = x(t) + t+Ts

t

F(τ)x(τ) + G(τ)u(τ) dτ = x(t) + t+Ts

t

F(τ)x(τ) dτ + t+Ts

t

G(τ)u(τ) dτ

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White Noise Models xn+1 = Fnxn + Gnun xn+1 = Gnun yn = Hnxn + vn yn = Hnxn + vn

The state is just a white noise process
In this case the state space model contains no dynamics or

memory that can be used to help estimate xn

No advantage over the more general linear estimation problem:

Given yn = Hnxn + vn Find the best linear estimate of xn: ˆ xn = xn, ynyn−2yn xn, yn = ΠnH∗

n

ˆ xn = ΠnH∗

n(HnΠnH∗ n + Rn)−1yn

yn2 = HnΠnH∗

n + Rn

Not used in practice
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Continuous-Time to Discrete-Time Coarse Approximations x(t + Ts) = x(t) + t+Ts

t

F(τ)x(τ) dτ + t+Ts

t

G(τ)u(τ) dτ Now if Ts is small enough, a coarse approximation of the CT state space equations can be obtained by assuming x(τ), F(τ), and G(τ) are constant over short intervals of t ≤ τ < t + Ts x(τ) ≈ xn F(τ) ≈ Fn G(τ) ≈ Gn for t = nTs ≤ τ < t + Ts = (n + 1)Ts Let us also define un t+Ts

t

u(τ) dτ

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Continuous-Time State Space Models The continuous time state space model is given by ˙ x(t) = F(t)x(t) + G(t)u(t) y(t) = H(t)x(t) + v(t) ⎡ ⎣ x(0) u(t) v(t) ⎤ ⎦ , ⎡ ⎢ ⎢ ⎣ x(0) u(τ) v(τ) 1 ⎤ ⎥ ⎥ ⎦

=

⎡ ⎣ Π0 Q(t)δ(t − τ) S(t)δ(t − τ) S(t)∗δ(t − τ) R(t)δ(t − τ) ⎤ ⎦

˙

x(t) is the derivative with respect to time of each element of x(t)

As before, all RVs have zero mean
Treatment of the noise processes is tricky (they have infinite

power and bandwidth)

Will not discuss here
Our only interest is to determine how the model terms relate to

the discrete-time model

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SLIDE 11

CT to DT Coarse Approximations: Measurement Noise Let us relax the requirement that v(t), v(τ) = R(t)δ(t − τ) so that v(t) is bandlimited white noise with finite power, v(t)2 = R(t) < ∞ such that v(nTs), v(kTs) = R(nTs)δnk = Rnδnk for any integers n, k This is a reasonable model if

1. The signal component (H(t)x(t)) of y(t) is bandlimited
2. An anti-aliasing filter (AAF) is applied to y(t) prior to sampling

As usual, the AAF should eliminate high-frequency components of v(t) without causing aliasing of the signal component.

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CT to DT Coarse Approximations: State Updates Our approximations for nTs ≤ t < (n + 1)Ts are x(t) ≈ x(nTs) F(t) ≈ F(nTs) G(t) ≈ G(nTs) Then we have xn+1 x ((n + 1)Ts) = x(nTs) + (n+1)Ts

nTs

F(τ)x(τ) dτ + (n+1)Ts

nTs

G(τ)u(τ) dτ ≈ x(nTs) + F(nTs)x(nTs) + G(nTs)un = (I + F(nTs)) x(nTs) + G(nTs)un = Fnxn + Gnun where we have defined Fn I + F(nTs) Gn G(nTs)

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CT to DT Coarse Approximations: Noise Covariance Matrices x0 = x(0) = Π0 vn, vk = v(nTs), v(kTs) = Rnδnk The process noise is a little more tricky un, uk = (n+1)Ts

nTs

u(t) dt, (k+1)Ts

kTs

u(τ) dτ

=

(n+1)Ts

nTs

(k+1)Ts

kTs

u(t), u(τ) dτdt = (n+1)Ts

nTs

(j+1)Ts

jTs

Q(t)δ(t − τ) dτdt = δn,j (n+1)Ts

nTs

Q(t) dt

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CT to DT Coarse Approximations: Measurement Equation y(t) = H(t)x(t) + v(t) yn = Hnxn + vn Let us define Hn H(nTs) vn v(nTs)

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SLIDE 12

CT to DT Better Approximations ˙ x(t) = F(t)x(t) + G(t)u(t) y(t) = H(t)x(t) + v(t)

A better approximation can be obtained using a more exact model
f how the state evolves over a sampling interval Ts
Requires knowledge of matrix exponentials
This material is from [1]
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CT to DT Coarse Approximations: Process Noise Now if we make the approximation Q(τ) = Q(t) for t ≤ τ < t + Ts we have un, uk = δn,j (n+1)Ts

nTs

Q(t) dt ≈ Ts Q(nTs) δn,j = Qn δn,j where we defined Qn Ts Q(nTs) Notice that the covariance of Qn scales linearly with the sampling interval Ts

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Matrix Exponentials Consider the power series approximation of an exponential, generalized to a matrix exponential eAt I + At + 1 2!A2t2 + · · · + 1 k!Aktk + . . . =

∞

k=0

Aktk k! If the series converges, we can differentiate it term by term to give d dteAt = A + A2t + 1 2!A3t2 + · · · + 1 (k − 1)!Aktk−1 + . . . = A

I + At + 1

2!A2t2 + · · · + 1 k!Aktk + . . .

= AeAt

=

I + At + 1

2!A2t2 + · · · + 1 k!Aktk + . . .

A

= eAtA

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CT to DT Coarse Approximations: Measurement Equation ˙ x(t) = F(t)x(t) + G(t)u(t) xn+1 = Fnxn + Gnun y(t) = H(t)x(t) + v(t) yn = Hnxn + vn xn x(nTs) yn y(nTs) Fn I + F(nTs) Hn H(nTs) Gn G(nTs) un t+Ts

t

u(τ) dτ vn v(nTs) Qn Ts Q(nTs) Rn R(nTs)

This DT approximation of CT state space systems is only accurate

if the noise covariance matrices, model matrices, and state change very little over the period of a sampling interval Ts

May require large sample rates and excessive computation
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SLIDE 13

CT State Space Approximations Continued e−F (nTs)t [ ˙ x(t) − F(nTs)x(t)] ≈ e−F (nTs)tG(nTs)u(t) d dt

e−F (nTs)tx(t)
= e−F (nTs)tG(nTs)u(t)

t+Ts

t

d dτ

e−F (nTs)τx(τ)
dτ =

t+Ts

t

e−F (nTs)τG(nTs)u(τ) dτ e−F (nTs)(t+Ts)x(t + Ts) − e−F (nTs)tx(t) = t+Ts

t

e−F (nTs)τG(nTs)u(τ) dτ Solving for x(t + Ts) then gives us e−F (nTs)(t+Ts)x(t + Ts) = e−F (nTs)tx(t) + t+Ts

t

e−F (nTs)τG(nTs)u(τ) dτ x(t + Ts) = eF (nTs)Tsx(t) + eF (nTs)(t+Ts) t+Ts

t

e−F (nTs)τG(nTs)u(τ) dτ

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Matrix Exponential Properties d dteAt = AeAt = eAtA Now consider eAteAτ = ∞

k=0

Aktk k! ∞

k=0

Akτ k k!

=

∞

k=0

Ak k

i=0

tiτ k−1 i!(k − i)!

=

∞

k=0

Ak (t + τ)k k! = eA(t+τ) Thus, if τ = −t we have eAte−At = e−AteAt = eA(t−t) = I Note, however, that e(A+B)t = eAteBt if and only if AB = BA

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CT State Space Approximations Continued If we evaluate x(t+Ts) = eF (nTs)Tsx(t)+eF (nTs)(t+Ts) t+Ts

t

e−F (nTs)τG(nTs)u(τ) dτ at t = nTs and use the definition xn x(nTs), we get xn+1 = eF (nTs)Tsxn+eF (nTs)(n+1)Ts (n+1)Ts

nTs

e−F (nTs)τG(nTs)u(τ) dτ Suppose for a moment that we define Gn I un eF (nTs)(n+1)Ts (n+1)Ts

nTs

e−F (nTs)τG(nTs)u(τ) dτ What is the covariance of un?

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CT State Space Approximations Let us assume F(t) and G(t) are constant over the interval nTs ≤ t < (n + 1)Ts, as before F(t) ≈ F(nTs) G(t) ≈ G(nTs) ˙ x(t) = F(t)x(t) + G(t)u(t) ˙ x(t) − F(t)x(t) = G(t)u(t) e−F (t)t [ ˙ x(t) − F(t)x(t)] = e−F (t)tG(t)u(t) Now over the interval nTs ≤ t < (n + 1)Ts we have the approximation e−F (nTs)t [ ˙ x(t) − F(nTs)x(t)] ≈ e−F (nTs)tG(nTs)u(t)

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SLIDE 14

CT State Space Approximations Continued Then we have xn+1 ≈ eF (nTs)Tsxn + eF (nTs)(n+1)Ts (n+1)Ts

nTs

e−F (nTs)τG(nTs)u(nTs) dτ = eF (nTs)Tsxn + un = Fnxn + Gnun where we have defined Fn eF (nTs)Ts Gn I

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CT State Space Approximations: Process Noise Covariance un, uk =

eF (nTs)(n+1)Ts

(n+1)Ts

nTs

e−F (nTs)τG(nTs)u(τ) dτ, eF (kTs)(k+1)Ts (k+1)Ts

kTs

e−F (kTs)sG(kTs)u(s) ds = eF (nTs)(n+1)Ts (n+1)Ts

nTs

(n+1)Ts

nTs

e−F (nTs)sG(nTs) u(τ), u(s)G(kTs)∗ e−F (nTs)s∗ ds dτ eF (nTs)(n+1)Ts∗ = eF (nTs)(n+1)Ts (n+1)Ts

nTs

(n+1)Ts

nTs

e−F (nTs)sG(nTs) S(τ)δ(t − s)G(kTs)∗ e−F (nTs)s∗ ds dτ eF (nTs)(n+1)Ts∗

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CT State Space Approximations Continued ˙ x(t) = F(t)x(t) + G(t)u(t) xn+1 = Fnxn + Gnun y(t) = H(t)x(t) + v(t) yn = Hnxn + vn

xn x(nTs) yn y(nTs) Fn eF (nTs)Ts Hn H(nTs) Gn I vn v(nTs) un eF (nTs)(n+1)Ts

(n+1)Ts

nTs

e−F (nTs)τ G(nTs)u(τ) dτ Qn δnkeF (nTs)(n+1)Ts

(n+1)Ts

nTs

e−F (nTs)sG(nTs)S(τ)G(kTs)∗

e−F (nTs)τ

∗ dτ

×
eF (nTs)(n+1)Ts

∗

Rn R(nTs)

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CT State Space Approximations: Process Noise Covariance un, uk = δnkeF (nTs)(n+1)Ts (n+1)Ts

nTs

e−F (nTs)sG(nTs)S(τ)G(kTs)∗ e−F (nTs)τ∗ dτ

×
eF (nTs)(n+1)Ts∗

Qnδnk

Unlike the measurement noise, we cannot filter the process noise
However, even if the process noise is continuous-time white noise,

the discrete-time noise has finite variance!

Not clear how to partition G(t)u(t) into Gnun
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SLIDE 15

The Nonlinear Functions The functions fn(xn), gn(xn) and hn(xn) are multivariate, nonlinear, time-varying functions fn(xn)

ℓ×1

= ⎡ ⎢ ⎢ ⎢ ⎣ fn,1(xn) fn,2(xn) . . . fn,ℓ(xn) ⎤ ⎥ ⎥ ⎥ ⎦ gn(xn)

ℓ×m

= ⎡ ⎢ ⎢ ⎢ ⎣ gn,11(xn) . . . gn,1m(xn) gn,21(xn) . . . gn,2m(xn) . . . ... . . . gn,ℓm(xn) . . . gn,ℓm(xn) ⎤ ⎥ ⎥ ⎥ ⎦ hn(xn)

p×1

= ⎡ ⎢ ⎢ ⎢ ⎣ hn,1(xn) hn,2(xn) . . . hn,p(xn) ⎤ ⎥ ⎥ ⎥ ⎦

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This approximation is a more accurate representation of the CT

model

Both approximations can be generalized to the nonlinear case

easily

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Nonlinear State Space Models Comments xn+1 = fn(xn) + gn(xn)un yn = hn(xn) + vn

There are other types of nonlinear state space models
This is one of the most general
Note that the noise is still additive for both the state update and

measurement equations

We will see that the optimal estimator uses Taylor series

approximations

This only works if the nonlinear functions are smooth
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Nonlinear State Space Models xn+1 = fn(xn) + gn(xn)un yn = hn(xn) + vn where ⎡ ⎣ un vn x0 − ¯ x0 ⎤ ⎦ ⎡ ⎢ ⎢ ⎣ um vm x0 − ¯ x0 1 ⎤ ⎥ ⎥ ⎦

H

= ⎡ ⎣ Qnδn,m Rnδn,m Π0 ⎤ ⎦

Here the matrices Fn, Gn, and Hn have been replaced with

nonlinear functions of the state

Does not assume zero mean
Does assume no interaction between process and measurement

noise, un, vk = 0, but could probably be generalized to the earlier case of un, vk = δnkSn

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SLIDE 16

To Do

Give examples of processes modeled in MATLAB
Give complete summary of the more exact CT to DT conversion
Work out what the process noise power is in the exact CT to DT

conversion

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Example 2: Random Walk Model of Position Suppose we have an accelerometer.

To be completed
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References

[1] Katsuhiko Ogata. Discrete-time Control Systems. Prentice-Hall, Inc., 1995.

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Summary

State space models can often be obtained directly from the

physical system (by Newton’s laws, Kirchoff’s laws, etc.)

State space models are wide sense Markov (WSM)
The nonstationary case is nearly as easy to handle as the

stationary case: a straightforward generalization

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