11.6 Sensitivity Improvement of the Optimal Smoother
The improvement that smoothing represents upon filtering is a well-known fact in the filtering literature. Indeed, it has been proven for the case of op- timal smoothers that the smoothing error variance is always lower than the filtering error variance (Anderson and Chirarattananon, 1971).
As we have discussed in §11.5.1, there is also an improvement in sen- sitivity associated with smoothing, in the sense that design restrictions imposed by ORHP poles and zeros of the plant relax with the smoothing lag. However, even for a large smoothing lag, these results only show less stringent design constraints, and do not guarantee the reduction of peaks in the values of|P|and|M|on thejω-axis.
In this section, we give a more precise result concerning improvement in sensitivity by smoothing. We will see that for the class of smoothers based on the Kalman filter, i.e., smoothers that are optimal in the least- squares sense, the frequency responses of both smoothing sensitivitiesP andMhave no peak values greater than one provided that the smoothing lag is sufficiently large. In fact, we will see that a smoothing lag of several times the dominant time constant of the Kalman filter, essentially achieves all the possible improvement in sensitivity reduction. We then illustrate these results with two numerical examples.
Consider then the smoother based on least squares presented in Exam- ple 11.1.1, where the plant is given by (11.1) withD1 = D2 = 0. We will assume that the solution,Φ, of the Riccati equation (11.6) is positive defi- nite. Further, we assume that the system evolution matrixAin (11.1) has no eigenvalues on the imaginary axis. Also, we take the process noise in- cremental covarianceR=1for simplicity of notation.
Letτ = τ¯ τmax( ^A), whereτmax( ^A)is the dominant time constant2 of the Kalman filter. It is shown in Lemma D.0.3 in Appendix D that, for τ=τ, the scalar smoothing sensitivities in (11.21) are approximately given¯ by
M[τ]¯(s) =QHιv(−s)Hιv(s),
P[τ]¯(s) =1−QHιv(−s)Hιv(s), (11.27) whereHιvis the transfer function from the process input,v, to the innova- tions processι=y−C2^xcorresponding to the Kalman filter (see (D.2) in Appendix D), and given by
Hιv=H(sI− ^A)−1B . (11.28)
2See the footnote on page 220.
238 11. Extensions to SISO Smoothing The following result shows that the modulus ofM[¯τ]and P[τ]¯ on the imaginary axis is bounded above by one.
Theorem 11.6.1. Consider the expressions (11.27) of the smoothing sensi- tivities for a smoothing lagτ= τ¯ τmax( ^A). Assume that the evolution matrixAof the system to be estimated has no eigenvalues on the imagi- nary axis. Then
kM[τ]¯k < 1 ,
kP[τ]¯k =1 . (11.29)
Proof. We first show thatkHιv√
Qk < 1, whereHιvis given in (11.28).
SinceA^ has no imaginary eigenvalues, we can use the result that kHιv
pQk < 1 if and only if the Hamiltonian matrix
AH,
A^ BQB0
−C20C2 − ^A0
(11.30) has no eigenvalues on the imaginary axis (Willems, 1971b).
We thus compute
|sI−AH|=
sI− ^A −BQB0 C20C2 sI+ ^A0
=
I 0 0 Φ
sI− ^A −BQB0 C20C2 sI+ ^A0
I 0 0 Φ−1
=
sI− ^A −BQB0Φ−1 ΦC20C2 sI+ΦA^0Φ−1
=
sI−A+ΦC20C2 −BQB0Φ−1 ΦC20C2 sI−A−BQB0Φ−1
=
sI−A 0
ΦC20C2 sI−A−BQB0Φ−1+ΦC20C2
=
sI−A 0
ΦC20C2 sI+ΦA0Φ−1 ,
where we have used the Riccati equation (11.6) in the formΦA^0Φ−1 = Φ(A0−C20C2Φ)Φ−1 = −(A+BQB0Φ−1), and applied some elementary algebra of determinants.
11.6 Sensitivity Improvement of the Optimal Smoother 239 It follows thatAHhas no eigenvalues on the imaginary axis since the evolution matrix of the Kalman filter,A, is a stability matrix, and^ Ahas no eigenvalues on the imaginary axis by assumption. Hence,kHιv√
Qk < 1 and thus, from (11.27),
kM[τ]¯k ≤ kHιv
pQk2 < 1.
Now note that0 ≤ Q|Hιv(jω)|2 < 1 implies thatP[τ]¯ in (11.27) satisfies 0 < P[τ]¯(jω)≤1. Thus
kP[¯τ]k =1
sinceHιvis strictly proper.
Theorem 11.6.1 shows that the frequency responses of the smoothing sensitivities will experience no peaks above one if the smoothing lag is chosen several times greater than the dominant time constant of the filter.
We emphasize that the result holds for unstable and nonminimum phase systems, which were shown in Chapter 8 to produce filtering sensitivities with large peaks above one.
The following examples illustrate Theorem 11.6.1 for the cases of unsta- ble and stable plants.
Example 11.6.1 (Unstable Plant). We consider again the Kalman filter given in Example 8.4.1 of Chapter 8. Here we construct a smoother as in (11.7), derived from the Kalman filter obtained for the weightsQ = 100 andR=1. We take the parameterain (8.23) asa=1.
The frequency responses of|P|and|M|are shown in Figures 11.2 and 11.3, respectively, for three different values of the smoothing lagτ. Note that the filtering sensitivities correspond toτ=0.
10−4 10−2 100 102 104
0 1 2 3 4 5 6
ω
|P(jω)|
τ=0 τ=0.5
τ=3
FIGURE 11.2. achieved by the Kalman filter-based smoother for the unstable plant of Example 11.6.1.
10−4 10−2 100 102 104
0 1 2 3 4 5 6
ω
|M(jω)|
τ=0 τ=0.5
τ=3
FIGURE 11.3. achieved by the Kalman filter-based smoother for the unstable plant of Example 11.6.1.
These plots clearly indicate that the process of smoothing reduces the constraints on the frequency responses of both sensitivities, which tend to
“ideal” shapes as we increase the smoothing lag. ◦
240 11. Extensions to SISO Smoothing Example 11.6.2 (Stable Plant). Consider again the stable plant used in Example 8.4.2 of Chapter 8 and revisited in §10.6.2 of Chapter 10. For this plant, we construct a smoother as in (11.7). The generating Kalman filter is again designed for an aggregated plant that includes the disturbance model (8.23) with the parametera=1. The weights are chosen asQ=100 andR=0.1.
The plots of Figures 11.4 and 11.5 show the frequency responses ofP andMfor three values ofτ. These figures show essentially the same effect of an improved frequency response as for the unstable system in Exam- ple 11.6.1.
Note that the poles of the generating Kalman filter for this example are −2.1035±j2.5701 and −1.8335, which implies that τ = 0.8 is ap- proximately 0.9 times the filter dominant time constant, taken as τf ≈ 1/1.8335 ≈ 0.55, whilst τ = 2 is approximately 3.6 times τf. As seen in §11.6, taking a smoothing lag of several times (approximately5times according to Anderson (1969)) the dominant time constant of the filter, gives all the improvement possible using smoothing. We can see that this phenomenon of “saturation” of improvement is also present in the fre- quency responses of the error sensitivities.
10−4 10−2 100 102 104
0 0.5 1 1.5 2
ω
|P(jω)|
τ=0 τ=0.5
τ=3
FIGURE 11.4. achieved by the Kalman filter-based smoother for the stable plant of Example 11.6.2.
10−4 10−2 100 102 104
0 0.5 1 1.5 2
ω
|M(jω)|
τ=0 τ=0.5
τ=3
FIGURE 11.5. achieved by the Kalman filter-based smoother for the stable plant of Example 11.6.2.
◦