Overview of State Space Models
Standard State Space Model
• Standard state space model
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
• Straight-forward extensions
• Properties
n≥0
where
• Solving the normal equations
• Covariance matrices
• Wide-sense Markov processes
Fn ∈ C×
Gn ∈ C×m
Hn ∈ Cp×
xn ∈ C×1
un ∈ Cm×1
yn ∈ Cp×1
vn ∈ Cp×1
• un and vn are multivariate white noise processes
• Designing state space models
– Common choices for the state dynamics
• In many cases vn is interpreted as an output disturbance or
measurement noise
• Continuous-time to discrete-time conversion
– Coarse estimates
– Exact conversions
• un is usually called process noise
• This is a more general form than is used in many texts
• Nonlinear state space models
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Standard State Space Model Matrices
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
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State Space Model With Known Input
n≥0
xn+1 = Fn xn + Gn (un + un )
n≥0
yn = Hn xn + vn
• Conceptually, yn usually contains all the signals that are
observable
– May come from multiple sensors
• 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
• xn contains all of the parameters of the process that you are
interested in estimating
• This changes little in the development of the Kalman filter
algorithm
• Note that the process is nonstationary in general since all of the
matrices are allowed to change with time
• 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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State Space Model With Known Input Continued
xn+1 = Fn xn + Gn (un + un )
State Space Model With Known Input Continued
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
n≥0
yn = Hn xn + vn
• An additional known input signal could be added to the output
equation in a similar fashion
• 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
yn = Hn xn + wn + vn
• Suppose we wish for the scaled state noise Gn un to have a
covariance matrix Tn
• This is most easily handled by defining a different observed signal
as
−∗
Qn G−1
n T n Gn
zn yn − wn
un , un = Qn
zn = Hn xn + vn
Gn un , Gn un = Gn Qn G∗n = Tn
• Thus, it easily reduces to the form of our standard state space
model output equation
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Covariance Matrices
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
⎡ ⎤
⎡ x ⎤ x0 ⎡Π
0
0
0
0
⎢ ⎥
⎣un ⎦ , ⎢uk ⎥ = ⎣ 0 Qn δnk Sn δnk
⎣ vk ⎦
vn
0 Sn∗ δnk Rn δnk
1
⎤
0
0⎦
0
State Space Models
State Space Models
0
Qn δnk
Sn∗ δnk
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0
Sn δnk
Rn δnk
6
⎤
0
0⎦
0
• 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
≥0
S∗ R
• This characterization ensures the process is wide-sense Markov (to
be discussed later)
• Clever use of the inner product notation has been used here to
indicate these RVs have zero mean
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• The matrices Π0 , Qn , Rn , and Sn are assumed to be known
• The model is causal in that xn and yn are completely determined
by x0 and past and present values of vn and un
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Covariance Matrices
⎡ ⎤
⎡ x ⎤ x0 ⎡
Π0
0
xn+1 = Fn xn + Gn un
⎢ ⎥
⎣un ⎦ , ⎢uk ⎥ = ⎣ 0
⎣ vk ⎦
yn = Hn xn + vn
vn
0
1
• 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
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Correlation of Process and Measurement Noise
⎡ ⎤
⎡ ⎤ x0 ⎡
x0 ⎢ ⎥
0
0
Π0
xn+1 = Fn xn + Gn un
⎣un ⎦ , ⎢uk ⎥ = ⎣ 0 Qn δnk Sn δnk
⎣ vk ⎦
yn = Hn xn + vn
vn
0 Sn∗ δnk Rn δnk
1
Orthogonality Properties
⎤
0
0⎦
0
You should be able to show that the following properties are true:
• It is common to assume the process and measurement noise
processes are completely uncorrelated, Sn = 0 for all n
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
• Due to domain knowledge that they have different origins
iff un , xn = 0
for n ≥ 0
• The key idea is to leverage the orthogonality of the initial state
and the noise covariance matrices
• 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
Example 1: Orthogonality Properties
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
x̂n|n = xn , yy−2 y
y col{y0 , . . . , yn }
• We know that the optimal causal estimator of xn is given by the
normal equations
Prove some of the orthogonality properties on the previous slide.
• Requires knowledge of two terms
• The first is easy to solve for
xn , y = xn , Hn xn + vn = xn , xn Hn∗
• 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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Solving for the State Covariance Matrix
Solving for the State Covariance Matrix Continued
Given the state space model,
Now let us define the state transition matrix
Fn−1 Fn−2 . . . Fk n > k
Φ(n, k) I
n=k
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
Then
we can easily show that
xn = Φ(n, k)xk + A col{uk , uk+1 , . . . , un−1 }, n ≥ k
Πn+1 xn , xn = Fn xn + Gn un , Fn xn + Gn un 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
= Fn xn , xn Fn∗ + Gn un , un G∗n
= Fn Πn Fn∗ + Gn Qn G∗n
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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xn , xk = Φ(n, k)Πk for n ≥ k
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Given the state space model,
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
For since x, y = y, x∗ for any random vectors x and y, it follows
immediately that
xn , xk = xk , xn ∗
The output covariance matrix is given by
∗
yn , yk = Hn xn + vn , Hk xk + vk = (Φ(k, n)Πn )
= Hn xn , xk Hk∗ + Hn xn , vk + vn , xk Hk∗ + vn , vk = Πn Φ∗ (k, n) for n ≤ k
The cross-terms xn , vk and vn , xk don’t necessarily cancel in this
case sense un and vn are correlated.
These results can be summarized as
⎧
⎪
n≥k
⎨Φ(n, k)Πk
xn , xk = Πn
n=k
⎪
⎩
Πn Φ∗ (k, n) n ≤ k
However they do if n = k since then xn is only a function of past
values of un
If
then
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State Space Models
Solving for the Output Covariance Matrix Ry
Solving for the State Covariance Matrix Continued
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vn , xk = 0
yn , yn =
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Hn Πn Hn∗
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xn , vk = 0
+ Rn
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Solving for the Output Covariance Matrix Ry Continued
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
⎧
⎪
n>k
⎨Hn Φ(n, k + 1)Nk
∗
yn , yk = Hn Πn Hn + Rn
n=k
⎪
⎩ ∗ ∗
∗
Nn Φ (k, n + 1)Hk n < k
yn , yk = Hn xn , xk Hk∗ + Hn xn , vk + vn , xk Hk∗ + vn , vk Now if n > k,
vn , xk = 0
vn , vk = 0
For n > k,
yn , yk = Hn xn , xk Hk∗ + Hn xn , vk where
= Hn Φ(n, k)Πk Hk∗ + Hn Φ(n, k + 1) (Fk xk + Gk uk ) , vk =
Hn Φ(n, k)Πk Hk∗
Nn Fn Πn Hn∗ + Gn Sn
+ Hn Φ(n, k + 1)Gk uk , vk It may not be apparent, but this has introduced exactly the type of
symmetry in Ry that we needed
= Hn Φ(n, k + 1)Fk Πk Hk∗ + Hn Φ(n, k + 1)Gk Sk
= Hn Φ(n, k + 1) (Fk Πk Hk∗ + Gk Sk )
= Hn Φ(n, k + 1)Nk
where
Nk Fk Πk Hk∗ + Gk Sk
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Symmetry in Ry
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Wide Sense Markov (WSM) Processes
Consider three observation vectors {y0 , y1 , y2 } produced by the
standard state space model
⎡
R0 + H0 Πo H0∗
H1 N 0
Ry = ⎣
H2 F1 N0
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N0∗ H1∗
R1 + H1 Π1 H1∗
H2 N 1
• Markov stochastic processes have the following relationship for
their conditional probability density functions (pdfs)
⎤
N0∗ F1∗ H2∗
⎦
N1∗ H2∗
R2 + H2 Π2 H2∗
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
• However, it is not yet clear how we might be able to invert Ry
efficiently
• 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 }.
• 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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State Space Models are WSM Processes
WSM Processes have State Space Models
Suppose we have a wide-sense Markov process xn . Consider its
innovations process
xn+1 = Fn xn + Gn uN
x0
x
Π0
0
, 0
=
un
uk
0 Qn δnk
e0 = x0
Now define
Since xn ∈ L{x0 , u0 , . . . , un−1 } we know that
un ⊥xk ,
un en+1
n≥k≥0
un , uk = en+1 , ek+1 = 0 for n = k
un 2 = en+1 2 = Πn+1 − Fn Πn Fn∗ Qn
= Fn xn + 0
= Fn x̂n|xn
Since en+1 ⊥e0 , un = en+1 , and e0 = x0 , we have
= x̂n+1|xn
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xn 2 Πn
Now since the innovations are orthogonal
x̂n+1|{x0 ,...,xn } = Fn x̂n|{x0 ,...,xn } + Gn ûn|{x0 ,...,xn }
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Fn Ko,n
xn+1 = Fn xn + un
Thus the orthogonal projection of xn+1 onto L{x0 , . . . , xn } is given
by
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en+1 = xn+1 − Ko,n xn
un ⊥x0
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WSM Processes have State Space Model Representations
Continued
Thus, given a wide-sense Markov process xn , we can define a state
space model
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Examples of State Space Models
• Autoregressive (AR) processes
• ARMA processes
• AR processes revisited
xn+1 = Fn xn + Gn uN
x0
x
Π0
0
, 0
=
un
uk
0 Qn δnk
• Physical models
• Nonlinear models
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 = Hn xn + 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
Autoregressive Processes
yn+1 = un +
−1
AR Process State Space Model
Let us define
xn col{yn , yn−1 , . . . , yn−+1 }
ak yn−k
k=0
⎡
Note that this differs from the usual model of AR processes
x(n) = w(n) +
p
yn+1
yn
yn−1
..
.
⎤
⎡
a0
⎥ ⎢1
⎥ ⎢
⎥ ⎢0
⎥=⎢
⎥ ⎢ ..
⎦ ⎣.
⎢
⎢
⎢
⎢
⎢
⎣
yn−+2
xn+1
ak x(n − k)
k=1
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)
a1
0
1
..
.
. . . a−2
...
0
...
0
..
..
.
.
0 0 ...
1
= F xn + Gun
⎤ ⎡ ⎤
⎤⎡
a−1
1
yn
⎢ yn−1 ⎥ ⎢0⎥
0 ⎥
⎥ ⎢ ⎥
⎥⎢
⎥ ⎢ ⎥
⎢
0 ⎥
⎥ ⎢ yn−2 ⎥ + ⎢0⎥ u(n)
.. ⎥ ⎢ .. ⎥ ⎢ .. ⎥
. ⎦ ⎣ . ⎦ ⎣.⎦
yn−+1
0
0
Note that in the vector case, yn ∈ Cp×1 , F and G are block matrices.
• The order of the model is denoted as • 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
⎡
yn = 1
0
yn
yn−1
yn−2
..
.
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Autoregressive Processes Continued
⎤
Thus with the definition of a state vector
⎢
⎥
⎥
⎢
⎢
⎥
0 ... 0 ⎢
⎥
⎢
⎥
⎣
⎦
yn−+1
xn col{yn , yn−1 , . . . , yn−+1 }
we can write an autoregressive process in state space form
xn+1 = F xn + Gun
yn = Hxn
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) z − a0 z −1 − · · · − a−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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ARMA Processes
ARMA In State Space Form
⎡
a0
⎢1
⎢
⎢
=⎢0
⎢ ..
⎣.
yn+1 = a0 yn + · · · + a−1 yn−+1 + b0 un + · · · + b−1 un−+1
. . . a−2
...
0
.
.
.
0
xn+1
..
..
.
.
0 0 ...
1
yn = b0 . . . b−1 xn
xn+1 a0 xn + · · · + a−1 xn−+1 + un
yn+1 = b0 xn + · · · + b−1 xn−+1
xn+1 = F xn + 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
a1
0
1
..
.
xn+1 = F xn + Gun
• We already have an expression for the output of the AP system
yn = Hxn
• The output of the AZ system is then just a linear combination of
outputs of the AP system
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⎡ ⎤
⎤
1
a−1
⎢0⎥
0 ⎥
⎢ ⎥
⎥
⎢ ⎥
0 ⎥
⎥ xn + ⎢0⎥ u(n)
⎢ .. ⎥
.. ⎥
⎣.⎦
. ⎦
0
0
29
AR Processes Revisited
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AR Process Alternative Formulation
Let us define the state as the parameters,
T
xn a1 a2 . . . a
xn col{yn , yn−1 , . . . , yn−+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
Then the output of the AR process is given by
• The state of the system is trivial to estimate once observations
of the output process have been made
yn =
ak yn−k + vn = Hn xn + vn
k=1
• Recall that the state of the system is usually consists of the
parameters that you want (or have) to estimate
where
Hn yn−1
• For AR processes, the parameters we’re interested in estimating
are the model coefficients {a0 , . . . , a−1 }
yn−2
. . . yn−
• 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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Designing State Space Models
Constant Parameter Models
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
xn+1 = xn
yn = Hn xn + vn
• Fn = I, Gn = 0 or Qn = 0
• In many signal processing applications you can write a
mathematical expression that relates the observed signal to the
parameters of interest
• 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
• However, there is often no obvious state space model that can be
obtained from first-principles or domain knowledge of the problem
• Example: Suppose we are trying to estimate the mean of a WN
process
• The Kalman filter approach to linear estimation is still often useful
in these circumstances
• Note that there will be a transition from the a priori estimate
x0|−1 (provided by user) to the unbiased estimate x as n → ∞
• There are several models for the state space equations that are
frequently used in these cases
• 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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Random Walk Models
xn+1 = Fn xn + Gn un
yn = Hn xn + vn
xn+1 = xn + un
yn = Hn xn + vn
xn+1 = Fn xn + Gn un
34
• Often, Qn = αI where the scalar α is chosen by the user
• The scalar measurement noise in this case is set Rn = 1
• The model is unstable, but the estimates may (usually) still
converge!
• It turns out that if Rn = αI for any scalar, positive parameter α,
the same solution is obtained! (prove this)
• Perhaps the most widely used model
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yn = Hn xn + vn
• 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 degree of the drift is controlled by the process noise
covariance Qn
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• Oddly, the adaptive filter algorithm recursive least squares is
equivalent to using the Kalman filter with an unstable state space
model
• This is a good model of processes in which the model parameters
slowly drift over time
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State Space Models
xn+1 = λ−1/2 xn
yn = Hn xn + vn
• The state at time (n + 1) is the same as xn plus a random
perturbation
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Unstable Deterministic Models
• Fn = I, as for constant parameter models
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White Noise Models
xn+1 = Fn xn + Gn un
Continuous-Time State Space Models
The continuous time state space model is given by
xn+1 = Gn un
yn = Hn xn + vn
ẋ(t) = F (t)x(t) + G(t)u(t)
yn = Hn xn + vn
y(t) = H(t)x(t) + v(t)
⎡
⎤
⎤
⎡
⎤
x(0) ⎡
x(0)
0
0
0
Π0
⎢u(τ )⎥
⎥
⎣ u(t) ⎦ , ⎢
⎣
Q(t)δ(t − τ ) S(t)δ(t − τ ) 0⎦
⎣ v(τ ) ⎦ = 0
0 S(t)∗ δ(t − τ ) R(t)δ(t − τ ) 0
v(t)
1
• 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 = Hn xn + vn
• ẋ(t) is the derivative with respect to time of each element of x(t)
• As before, all RVs have zero mean
Find the best linear estimate of xn :
x̂n = xn , yn yn −2 yn
• Treatment of the noise processes is tricky (they have infinite
power and bandwidth)
xn , yn = Πn Hn∗
x̂n = Πn Hn∗ (Hn Πn Hn∗ + Rn )−1 yn
yn 2 = Hn Πn Hn∗ + Rn
• Will not discuss here
• Our only interest is to determine how the model terms relate to
the discrete-time model
• Not used in practice
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Continuous-Time to Discrete-Time
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x(t + Ts ) = x(t) +
ẋ(t) = F (t)x(t) + G(t)u(t)
t+Ts
ẋ(τ ) dτ
x(t + Ts ) = x(t) +
t
= x(t) +
t
= x(t) +
t+Ts
t+Ts
t
F (τ )x(τ ) dτ +
t
t+Ts
t+Ts
t
x(τ ) ≈ xn
G(τ )u(τ ) dτ
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t+Ts
t
G(τ )u(τ ) dτ
G(τ ) ≈ Gn
for
t = nTs ≤ τ < t + Ts = (n + 1)Ts
un Portland State University
F (τ )x(τ ) dτ +
F (τ ) ≈ Fn
Let us also define
J. McNames
Ver. 1.03
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
F (τ )x(τ ) + G(τ )u(τ ) dτ
State Space Models
Continuous-Time to Discrete-Time Coarse Approximations
Suppose our sample interval is given by Ts = 1/fs . Then
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J. McNames
Portland State University
t
t+Ts
u(τ ) dτ
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CT to DT Coarse Approximations: State Updates
CT to DT Coarse Approximations: Measurement Equation
Our approximations for nTs ≤ t < (n + 1)Ts are
F (t) ≈ F (nTs )
x(t) ≈ x(nTs )
y(t) = H(t)x(t) + v(t)
G(t) ≈ G(nTs )
yn = Hn xn + vn
Then we have
Let us define
xn+1 x ((n + 1)Ts )
(n+1)Ts
F (τ )x(τ ) dτ +
= x(nTs ) +
nTs
Hn H(nTs )
(n+1)Ts
nTs
vn v(nTs )
G(τ )u(τ ) dτ
≈ x(nTs ) + F (nTs )x(nTs ) + G(nTs )un
= (I + F (nTs )) x(nTs ) + G(nTs )un
= Fn xn + Gn un
where we have defined
Fn I + F (nTs )
J. McNames
Portland State University
Gn G(nTs )
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CT to DT Coarse Approximations: Measurement Noise
J. McNames
x0 = x(0)
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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= Π0
The process noise is a little more tricky
(n+1)Ts
u(t) dt,
un , uk =
such that
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State Space Models
vn , vk = v(nTs ), v(kTs ) = Rn δnk
v(t)2 = R(t) < ∞
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CT to DT Coarse Approximations: Noise Covariance Matrices
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(nTs ), v(kTs ) = R(nTs )δnk = Rn δnk
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nTs
(n+1)Ts
nTs
(n+1)Ts
nTs
= δn,j
43
J. McNames
kTs
(k+1)Ts
kTs
(j+1)Ts
jTs
(n+1)Ts
nTs
Portland State University
(k+1)Ts
u(τ ) dτ
u(t), u(τ ) dτ dt
Q(t)δ(t − τ ) dτ dt
Q(t) dt
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CT to DT Coarse Approximations: Process Noise
CT to DT Coarse Approximations: Measurement Equation
Now if we make the approximation
Q(τ ) = Q(t)
ẋ(t) = F (t)x(t) + G(t)u(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
xn+1 = Fn xn + Gn un
y(t) = H(t)x(t) + v(t)
yn = Hn xn + vn
xn x(nTs )
yn y(nTs )
Fn I + F (nTs )
Hn H(nTs )
Gn G(nTs )
t+Ts
un u(τ ) dτ
vn v(nTs )
Qn Ts Q(nTs )
Rn R(nTs )
t
where we defined
Qn Ts Q(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
Notice that the covariance of Qn scales linearly with the sampling
interval Ts
• May require large sample rates and excessive computation
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J. McNames
Portland State University
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State Space Models
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CT to DT Better Approximations
Matrix Exponentials
ẋ(t) = F (t)x(t) + G(t)u(t)
Consider the power series approximation of an exponential, generalized
to a matrix exponential
y(t) = H(t)x(t) + v(t)
eAt I + At +
• A better approximation can be obtained using a more exact model
of how the state evolves over a sampling interval Ts
=
• Requires knowledge of matrix exponentials
∞
Ak t k
k=0
• This material is from [1]
46
1 2 2
1
A t + · · · + Ak t k + . . .
2!
k!
k!
If the series converges, we can differentiate it term by term to give
1
1
d At
e = A + A2 t + A3 t2 + · · · +
Ak tk−1 + . . .
dt
2!
(k − 1)!
1 k k
1 2 2
= AeAt
= A I + At + A t + · · · + A t + . . .
2!
k!
1
1
= I + At + A2 t2 + · · · + Ak tk + . . . A
= eAt A
2!
k!
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Portland State University
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Matrix Exponential Properties
CT State Space Approximations
Let us assume F (t) and G(t) are constant over the interval
nTs ≤ t < (n + 1)Ts , as before
d At
e = AeAt = eAt A
dt
Now consider
eAt eAτ =
=
∞
Ak t k
k=0
∞
Ak
k=0
k!
∞
Ak τ k
k=0
k!
(t + τ )k
k!
=
∞
k=0
G(t) ≈ G(nTs )
F (t) ≈ F (nTs )
k
ti τ k−1
k
A
i!(k − i)!
i=0
ẋ(t) = F (t)x(t) + G(t)u(t)
ẋ(t) − F (t)x(t) = G(t)u(t)
= eA(t+τ )
−F (t)t
e
[ẋ(t) − F (t)x(t)] = e−F (t)t G(t)u(t)
Now over the interval nTs ≤ t < (n + 1)Ts we have the approximation
Thus, if τ = −t we have
e−F (nTs )t [ẋ(t) − F (nTs )x(t)] ≈ e−F (nTs )t G(nTs )u(t)
eAt e−At = e−At eAt = eA(t−t) = I
Note, however, that
e(A+B)t = eAt eBt if and only if AB = BA
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J. McNames
CT State Space Approximations Continued
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If we evaluate
F (nTs )Ts
x(t+Ts ) = e
F (nTs )(t+Ts )
50
t+Ts
t
e−F (nTs )τ G(nTs )u(τ ) dτ
at t = nTs and use the definition xn x(nTs ), we get
(n+1)Ts
xn+1 = eF (nTs )Ts xn +eF (nTs )(n+1)Ts
e−F (nTs )τ G(nTs )u(τ ) dτ
nTs
Suppose for a moment that we define
Solving for x(t + Ts ) then gives us
e−F (nTs )(t+Ts ) x(t + Ts ) = e−F (nTs )t x(t) +
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x(t)+e
t
J. McNames
Ver. 1.03
CT State Space Approximations Continued
e−F (nTs )t [ẋ(t) − F (nTs )x(t)] ≈ e−F (nTs )t G(nTs )u(t)
d −F (nTs )t
e
x(t) = e−F (nTs )t G(nTs )u(t)
dt
t+Ts
t+Ts
d −F (nTs )τ
e
x(τ ) dτ =
e−F (nTs )τ G(nTs )u(τ ) dτ
dτ
t
t
t+Ts
e−F (nTs )τ G(nTs )u(τ ) dτ
e−F (nTs )(t+Ts ) x(t + Ts ) − e−F (nTs )t x(t) =
x(t + Ts ) = eF (nTs )Ts x(t) + eF (nTs )(t+Ts )
State Space Models
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t
t+Ts
t
t+Ts
Gn I
e−F (nTs )τ G(nTs )u(τ ) dτ
e−F (nTs )τ G(nTs )u(τ ) dτ
State Space Models
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un eF (nTs )(n+1)Ts
(n+1)Ts
nTs
e−F (nTs )τ G(nTs )u(τ ) dτ
What is the covariance of un ?
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CT State Space Approximations: Process Noise Covariance
un , uk =
eF (nTs )(n+1)Ts
e
=e
F (nTs )(n+1)Ts
nTs
F (kTs )(k+1)Ts
(n+1)Ts
nTs
(n+1)Ts
kTs
(n+1)Ts
e−F (nTs )s G(nTs )
S(τ )δ(t − s)G(kTs )∗ e−F (nTs )s
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∗
ds dτ
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eF (nTs )(n+1)Ts
State Space Models
∗
• 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 Gn un
∗
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J. McNames
Portland State University
CT State Space Approximations Continued
Then we have
xn+1 ≈ eF (nTs )Ts xn + eF (nTs )(n+1)Ts
=e
F (nTs )Ts
∗
× eF (nTs )(n+1)Ts Qn δnk
• Unlike the measurement noise, we cannot filter the process noise
∗
nTs
nTs
e−F (kTs )s G(kTs )u(s) ds u(τ ), u(s)G(kTs )∗ e−F (nTs )s ds dτ eF (nTs )(n+1)Ts
(n+1)Ts (n+1)Ts
F (nTs )(n+1)Ts
=e
e−F (nTs )s G(nTs )
nTs
un , uk = δnk eF (nTs )(n+1)Ts
(n+1)Ts
∗
e−F (nTs )s G(nTs )S(τ )G(kTs )∗ e−F (nTs )τ
dτ
e−F (nTs )τ G(nTs )u(τ ) dτ,
(k+1)Ts
nTs
CT State Space Approximations: Process Noise Covariance
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nTs
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CT State Space Approximations Continued
ẋ(t) = F (t)x(t) + G(t)u(t)
(n+1)Ts
State Space Models
xn+1 = Fn xn + Gn un
y(t) = H(t)x(t) + v(t)
e−F (nTs )τ G(nTs )u(nTs ) dτ
yn = Hn xn + vn
xn x(nTs )
xn + u n
yn y(nTs )
= Fn xn + Gn un
F (nTs )Ts
Fn e
Hn H(nTs )
where we have defined
Gn I
Fn eF (nTs )Ts
vn v(nTs )
Gn I
F (nTs )(n+1)Ts
un e
(n+1)Ts
nTs
F (nTs )(n+1)Ts
Qn δnk e
F (nTs )(n+1)Ts
× e
∗
−F (nTs )τ
e
(n+1)Ts
nTs
G(nTs )u(τ ) dτ
−F (nTs )s
e
G(nTs )S(τ )G(kTs )
∗
−F (nTs )τ
e
∗
dτ
Rn R(nTs )
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Nonlinear State Space Models
• This approximation is a more accurate representation of the CT
model
xn+1 = fn (xn ) + gn (xn )un
yn = hn (xn ) + vn
• Both approximations can be generalized to the nonlinear case
easily
where
⎡
⎤H
um
⎡Q
un
n δn,m
⎢ vm ⎥
⎥
⎣ 0
⎣ vn ⎦ ⎢
=
⎣x0 − x̄0 ⎦
x0 − x̄0
0
1
⎡
⎤
0
Rn δn,m
0
0
0
Π0
⎤
0
0⎦
0
• 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 = δnk Sn
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The Nonlinear Functions
The functions fn (xn ), gn (xn ) and hn (xn ) are multivariate, nonlinear,
time-varying functions
⎤
⎡
fn,1 (xn )
⎢fn,2 (xn )⎥
⎥
⎢
fn (xn ) = ⎢
⎥
..
⎦
⎣
×1
.
fn, (xn )
⎤
⎡
gn,11 (xn ) . . . gn,1m (xn )
⎢ gn,21 (xn ) . . . gn,2m (xn )⎥
⎥
⎢
gn (xn ) = ⎢
⎥
..
..
.
.
⎦
⎣
.
×m
.
.
gn,m (xn ) . . . gn,m (xn )
⎤
⎡
hn,1 (xn )
⎢hn,2 (xn )⎥
⎥
⎢
hn (xn ) = ⎢
⎥
..
⎦
⎣
p×1
.
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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
hn,p (xn )
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Example 2: Random Walk Model of Position
Summary
• State space models can often be obtained directly from the
physical system (by Newton’s laws, Kirchoff’s laws, etc.)
Suppose we have an accelerometer.
• To be completed
• 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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• Give examples of processes modeled in MATLAB
• Work out what the process noise power is in the exact CT to DT
conversion
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[1] Katsuhiko Ogata. Discrete-time Control Systems. Prentice-Hall,
Inc., 1995.
• Give complete summary of the more exact CT to DT conversion
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References
To Do
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