LOCALIZED LINEAR QUADRATIC REGULATOR
5.6 Localized Distributed Kalman Filter
5.6.3 LDKF Formulation
We now use the system response to analyze the estimation error dynamics of the Kalman filter.
Delayed Form LDKF
Consider first the delayed form (5.42) of the Kalman filter. Taking the z-transform of equation (5.42), we get
(zI −A)ˆx = B2u+H(y−C2ˆx). (5.44) In the following, we assume that the estimator structure (5.44) is fixed, but the gain matrixH is unknown and needs to be designed. Although the Kalman filter can be implemented via a static gainH, this is not necessary. We relax the static gainHto be a proper transfer matrixHand rewrite (5.44) as
(zI− A)ˆx = B2u+H(y−C2ˆx) (5.45) in the sequel. This extra freedom will be key in allowing us to incorporate spa- tiotemporal constraints on the transfer matrices that define the estimator. Combining (5.45) and (5.35), we have the estimation error dynamics
(zI − A+HC2)˜x= δx−Hδy. (5.46) DefineR:= (zI − A+HC2)−1to be the system response from process noiseδx to estimation error ˜x, and likewise letN:= −RHbe the system response from sensor noiseδy to estimation error˜x. Equation (5.46) can then be rewritten as
˜x= Rδx+Nδy. (5.47)
Rather than finding a suitable gain matrixHto optimize the system response(R,N) indirectly, we instead characterize the set of stable achievable system response(R,N) and optimizedirectlyover those sets. Note that the characterization of the set(R,N) can be obtained by applying the dual of Theorem 1 in Section 3.1, which gives the following lemma.
Lemma 12. The system response(R,N)with finite mean square error can be induced by an estimator with structure (5.45) if and only if the following two affine constraints hold:
h R N
i
"
zI −A
−C2
#
= I (5.48)
h R N
i
∈ 1
zRH∞. (5.49)
Proof. To prove the necessary direction, we show that (5.48) and (5.49) must hold for an estimator with structure (5.45) and suitable gain matrix H. Equation (5.48) can be verified using the identityR(zI−A+HC2)= Idirectly. For (5.49), note that the transfer matrices R andN must be stable so that the mean square error of the estimator is finite. Besides, as R = (zI − A+HC2)−1and N = −RH, the transfer matricesRand Nmust be strictly proper, i.e., R[0] = 0 and N[0] = 0. Therefore, (5.49) must holds.
To prove the sufficient direction, we show that the desired system response (R,N) can be induced by an estimator with structure (5.45) if (5.48) and (5.49) hold. For any solution of (5.48) - (5.49), we construct a gain matrixH0 =−R−1Nfor (5.45).
In this case, the estimation error dynamics (5.46) become
(zI− A−R−1NC2)˜x= δx+R−1Nδy. (5.50) MultiplyingRto both sides of (5.50) and substituting (5.48) into the equation, we can show that the desired system response ˜x= Rδx +Nδyis achieved.
Lemma 12 suggests that we can implement the estimator using the gain matrix H=−R−1Nto achieve the desired system response. Substituting this identity back to (5.45) and multiplyingRto both sides of the equation, we get
ˆx =RB2u−Ny. (5.51)
This gives a simpler estimator implementation.
Given this characterization of valid system response, we now aim to find an expres- sion for the Kalman filter objective function in terms ofR andN. Using the error dynamics (5.47) and the AWGN assumptions on the noise dynamics, it is straight- forward to show that the Kalman filter is the optimal solution to the following SLS problem
minimize
{R,N}
k h R N
i
"
B1 D21
# kH2
2
subject to (5.48) − (5.49) (5.52)
with "
B1 D21
#
=
"
W12 0 0 V12
# .
We can view optimization problem (5.52) as an alternative formulation for the Delayed Form of the Kalman filter problem.
In order to enhance the scalability of estimator design and implementation, we impose an additional d-localized SLC Ld, a FIR SLC FT, and a QI information sharing SLC C on the system response. This leads to the Delayed Form LDKF given by
minimize
R,N k h
R N i
"
B1 D21
# kH2
2
subject to (5.48) − (5.49) h
R N i
∈ Ld∩ FT ∩ C. (5.53) Problem (5.53) is just the transpose of a LLQR problem. Therefore, we can solve (5.53) using the method described in the previous sections in a localized and scalable way. Specifically, we first perform a row-wise separation to decompose (5.53) into row-wise LDKF subproblems. For each LDKF subproblem, the d-localized con- straintLdfurther allows us to reduce the dimension of each LDKF subproblem from globalscale tolocalscale. Each LDKF subproblem in the reduced dimension can then be solved by a local optimization problem using local plant model information only. In particular, the complexity of solving a LDKF subproblem is independent with the size of the global system. In other words, our method can scale to systems of arbitrary size if parallel computation is available.
Remark 16. In (5.53), we try to find a localized system response for the state estimation error. It should be noted that we arenotlocalizing the effect of the pro- cess/sensor noise on the state vector. Rather, we are localizing thestate estimation errordue to the noise.
Remark 17. It should be noted that the localized synthesis method is valid for arbitrary noise covariance matrices (W,V), i.e., even when the noise is globally correlated. In addition, when the system matrices (A,B2,C2) change locally, we only need to resolve some of the row-wise LDKF subproblems and update the estimator locally. It follows that this allows for the incremental addition of new subsystems to the global system without the need for a complete redesign of the estimator.
Current Form LDKF
Consider the current form of Kalman filter in (5.43). Taking the z-transform of (5.43) and rearranging some terms, we have
(zI −A)ˆx =−KC2Aˆx+(I−KC2)B2u+zKy.
We relax the static Kalman gainK to a proper transfer matrixKin the sequel. We then have
(zI − A)ˆx = −KC2Aˆx+(I−KC2)B2u+zKy
= −KC2Aˆx+(I−KC2)B2u+zK(C2x+δy)
= −KC2Aˆx+(I−KC2)B2u+zKδy +KC2(zx)
= −KC2Aˆx+(I−KC2)B2u+zKδy +KC2(Ax+ B2u+δx)
= KC2A˜x+B2u+zKδy+KC2δx. (5.54) Combining (5.54) with (5.35), we have
(zI −A)˜x=−KC2A˜x+(I −KC2)δx−zKδy.
The system response from noise to the estimation error are therefore given by R = (zI − A+KC2A)−1(I −KC2)
N = −(I− 1
z(I−KC2)A)−1K. (5.55) For the characterization of all valid system response, we still have the identity (5.48).
The constraint in (5.49) changes slightly however, as the transfer matrix from sensor
noise to estimation errorNis only restricted to be proper in the Current Form of the Kalman filter. The estimator equation in (5.51) remains unchanged. The Current Form LDKF is then given by
minimize
R,N k h
R N i
"
B1 D21
# k
subject to (5.48) R∈ 1
zRH∞, N∈ RH∞ h
R N i
∈ Ld∩ FT ∩ C. (5.56) Similar to the Delayed Form LDKF, the Current Form LDKF (5.56) can also be solved in a localized way using the LLQR algorithm.