Continuous-time Errors-in-variables Filtering

Continuous-time Errors-in-variables Filtering

Ivan Markovsky, Jan C. Willems, and Bart De Moor

ESAT-SCD (SISTA), K.U.Leuven, Belgium

Outline

• Errors-in-variables model

• EIV smoothing and filtering

• Misfit and latency

• Estimation in the pure misfit case

• Estimation in the pure latency case

• Conclusion

Errors-in-variables model

consider the continuous-time linear state space system

d

dt

x = Ax + Bu

y = Cx + Du

(1)

let (u

, y

) be a

measured input/output trajectory

on the interval [0, t

]

in general, it is

not a trajectory of the model

, i.e.,

there is no x(0), such that the response of (1) to the signal u

is y

account for

modeling errors

by adding to the system an auxiliary input d

d

dt

x = Ax + Bu + Gd

y = Cx + Du + Hd

(2)

Errors-in-variables model (cont.)

measurement errors are modeled by appending to the system the eqns.

u

= u + ˜

u,

y

= y + ˜

y

(3)

the

measurement errors ˜

u, ˜

y

and the

disturbance d

are free variables that

allow us to “explain” the measured trajectory (u

, y

)

the system (2,3) is called an

errors-in-variables model

PSfrag replacements

u

˜

u

y

˜

y

u

y

x

(0)

d

EIV smoothing and filtering

problem:

estimate the

“true” I/S/O trajectory

from the measured traj.

assumptions:

1. the true I/S/O trajectory is generated by (1)

2. the measurements are generated by the EIV model

3. the initial condition, disturbance, and measurement errors are “small”

estimation principle:

find the “smallest” estimated ˆ

x(0), ˆ

d, ˆ

u, and ˆ

˜

y,

˜

that “explain” the measurements by the EIV model, i.e.,

d

dt

x = Aˆ

ˆ

x + B ˆ

u + G ˆ

d,

y = C ˆ

ˆ

x + D ˆ

u + H ˆ

d

u

= ˆ

u + ˆ

u,

˜

y

= ˆ

y + ˆ

y

˜

Misfit versus latency interpretation

ˆ

x(0) and ˆ

d are

latent variables

, i.e., variables that modify the system

equations to explain the model-data mismatch

ˆ

˜

u and ˆ

y are

˜

misfit variables

, i.e., variables that modify the data to make

it match the system equations

pure latency case

— consider the data as being error free and blame the

model as being imperfect description of the reality

pure misfit case

— consider the model as being perfect and blame the

measurements as being error corrupted

in general, both misfit and latency can be considered simultaneously, i.e.,

the uncertainty is in both the data and the model

Estimation cost functional

misfit and latency cost functionals:

J

:=

Z

||ˆ

u(t) − u

(t)||

+ ||ˆ

y(t) − y

(t)||

dt

J

:=

Z

|| ˆ

d(t)||

dt + ||ˆ

x(0)||

R > 0, Q ≥ 0, P > 0, Γ ≥ 0 — weighting matrices, reflecting the

relative importance of the terms

total cost functionals:

J := ρJ

+ (1 − ρ)J

varying ρ from 0 to 1, allows smooth transition from a purely misfit

contribution to a purely latency contribution

Smoothing versus filtering

the general EIV estimation problem, we will consider, is

min

ρJ

+ (1 − ρ)J

s.t.

x = Aˆ

ˆ

x + B ˆ

u + G ˆ

d

ˆ

y = C ˆ

x + D ˆ

u + H ˆ

d

(4)

as defined this is a

smoothing problem

, i.e., the estimates are wanted for

the whole interval [0, t

]

but we will derive also the solution of the corresponding filtering problem

filtering problem

— design a non-anticipating dynamical system that

assumes as input the measurements and produces as output at each

moment of time the optimal estimate

Common problem

we solve the following optimal control problem

min u,x Z t_f 0   u(t) x(t) 1   T M (t)   u(t) x(t) 1  dt + x T (0)Γx(0) s.t. _dtd x = Ax + Bu (5) where M (t) :=  _Q u Qux qu(t) ∗ Q_x q_x(t) ∗ ∗ q(t)  , Qu > 0, h Q_u Q_ux ∗ Q_x i ≥ 0, and Γ ≥ 0

completion of squares approach

Lemma 1. Let K : [0, tf] → Rn×n, K = KT, and s : [0, tf] → Rn be differentiable.

Then, for x and u related by _dtd x = Ax + Bu, the following identity holds

0 = − xTKx + 2sTx
|t₀f + Z t_f 0   u x 1   T   0 BTK BTs ∗ _dtd K + ATK + KA _dtd s + ATs ∗ ∗ 0     u x 1  dt.

Solution of the common problem

≥ 0, and Γ ≥ 0. Define K as the unique solution of the Riccati differential equation

0 = _dtd K + ATK + KA + Qx − (BTK + Qux)TQ−1_u (BTK + Qux), K(0) = −Γ,

and assume that K(tf) < 0.Then the unique solution of problem (5) is

u = −Q−1_u (BTK + Qux)x + BTs + qu
, and d dtx = A − BQ −1 u (B T K + Qux)
x − BQ−1_u BTs − BQ−1_u qu,

with final condition x(tf) = −K−1(tf)s(tf), where s is generated by d dts = − A − BQ −1 u (B T K + Qux)
T s − qx + (BTK + Qux)TQ−1_u qu, s(0) = 0.

Estimation in the pure misfit and pure latency cases

assume d = 0

⇒

only the misfit contributes to the cost functional

min

Z

||ˆ

u(t) − u

(t)||

+ ||ˆ

y(t) − y

(t)||

dt s.t.

x = Aˆ

ˆ

x + B ˆ

u

ˆ

y = C ˆ

x + D ˆ

u

assume

u = 0

˜

˜

y = 0

⇒ only the latency contributes to the cost functional

min

Z

|| ˆ

d(t)||

dt + x

(0)Γx(0) s.t.

x = Aˆ

ˆ

x + Bu

+ G ˆ

d

y

= C ˆ

x + Du

+ H ˆ

d.

this corresponds to the

deterministic Kalman filter

Theorem 1. [Pure misfit smoothing] Let R > 0, Q ≥ 0, and (A, CTQC) be observable. The unique solution of the pure misfit estimation problem is

ˆ u = −N−1 (BTK + DTQC)ˆx + BTs − DTQyd − Rud
d dtx = A − BNˆ −1 (BTK + DTQC)
Tx − BNˆ −1BTs + BN−1DTQyd + BN−1Rud, ˆx(tf) = −K−1(tf)s(tf) where N := R + DTQD, s is generated by d dts = − A − BN −1 (BTK + DTQC)
Ts + CTQyd − (BTK + DTQC)TN−1(DTQyd + Rud), s(0) = 0, and K is generated by (BTK +DTQC)TN−1(BTK +DTQC) = _dtd K +ATK +KA+CTQC, K(0) = 0. (6)

Corollary 1. [Pure misfit finite-horizon filtering] Under the assumptions of Theorem 1, the optimal misfit filter is ˆx = −K−1s, ˆy = C ˆx + D ˆu, and

ˆ u = −N−1 BT + (BTK + DTQC)K
s − DTQyd − Rud  , d dts = − A − BN −1 (BTK + DTQC)
Ts + CTQyd − (BTK + DTQC)TN−1(DTQyd + ud), s(0) = 0, N := R + DTQD and K is generated by (6).

In the infinite-horizon case, assuming in addition that (A, B) is controllable, the differential Riccati equation reduces to the algebraic Riccati equation

0 = ATK + KA + CTQC − (BTK + DTQC)TN−1(BTK + DTQC).

Its unique negative definite solution K− should be used (in place of K) in the optimal

filter.

Theorem 2. [Pure latency smoothing_{] Let H ∈ R}ny×nd be of full rank with n y ≤

nd, P > 0, and Γ > 0. The unique solution of the pure latency estimation problem is

ˆ u = −Q−1_u ( ¯BTK + H₂TF C)ˆx + ¯BTs − H₂TF (yd − Dud)
, d dtx =ˆ A − ¯¯ BQ −1 u ( ¯B T K + H₂TF C)
ˆx − ¯BQ−1_u B¯Ts + (B − ¯BQ−1_u H₂TF D − G1H₁−1D)ud + (G1H₁−1 + ¯BQ−1_u H₂TF )yd,

with final condition x(tf) = −K−1(tf)s(tf),where s and K are generated by d dts = − ¯A − ¯BQ −1 u ( ¯B T K + H₂TF C)
Ts + CT − ( ¯BTK + H₂TF C)TQ−1_u H₂T
F (yd − Dud), s(0) = 0, ( ¯BTK + H₂TF C)TQ−1_u ( ¯BTK + H₂TF C) = _dtd K + ¯ATK + K ¯A + CTF C, K(0) = −Γ. (7) ¯ A := A − G1H₁−1C, ¯B := G2 − G1H₁−1H2, Qu := P + H₂TF H2, F := H₁−TP H₁−1

Corollary 2. [Pure latency filtering] Under the assumptions of Theorem 2, the optimal latency filter is x = −K−1s, y = C ˆx + D ˆu

u = −Q_u−1 B¯T( ¯BTK + H₂TF C)K
s − H₂TF (yd − Dud)
, d dts = − ¯A − ¯BQ −1 u ( ¯B T K + H₂TF C)
Ts + CT − ( ¯BTK + H₂TF C)Q−1_u H₂T
F (yd − Dud), s(0) = 0,

where K is generated by (7), and ¯A, ¯B, Qu, and F are defined above.

In the infinite-horizon case, assuming in addition that ( ¯A, ¯B) is controllable and ( ¯A, C) is observable, K should be replaced by the unique negative definite solution of the algebraic Riccati equation

¯

ATK + K ¯A + CTF C = ( ¯BTK + H₂TF C)TQ−1_u ( ¯BTK + H₂TF C).

Misfit+latency case

Theorem 3. [Misfit and latency smoothing] Let R > 0, Q ≥ 0, P > 0, and Γ > 0. The unique solution of (4) for ρ ∈ (0, 1) is

ˆ

u = −Q−1_u (Lx + ¯BTs + qu), _dtd x = (A − ¯BQ_u−1L)x − ¯BQ−1_u B¯Ts − ¯BQ−1_u qu,

with final condition x(t_f) = −K−1(t_f)s(t_f),where s and K are generated by

d dts = −(A − ¯BQ −1 u L) T s + ρCTQyd + LQ−1_u qu, s(0) = 0, 0 = _dtd K + ATK + KA + ρCTQC − LTQ−1_u L, K(0) = −(1 − ρ)Γ (8) ¯ B := [B G], L := B T_{K + ρD}T_QC GTK + ρHTQC  , qu := −ρ Rud + DTQyd HTQy_d  Qu := ρ " R + DTQD DTQH ∗ (1−ρ)_ρ P + HTQH # .

Corollary 3. [Misfit and latency filtering] Under the assumptions of Theorem 3, the optimal misfit and latency filter is x = −K−1s,

ˆ u = −Q−1_u ( ¯BT − LK−1)s + qu
, y = − CKˆ −1+ DQ−1_u ( ¯BT − LK−1)
s − DQ−1_u qu, d dts = −(A − ¯BQ −1 u L) T s + ρCTQyd + LQ−1_u qu, s(0) = 0,

where K is generated by (8) and ¯B, Qu, L, qu are defined above.

The infinite horizon case, assuming in addition that (A, B) is controllable and (A, CTQC) is observable, the optimal time-invariant filter is obtained by replacing K with K−, the unique negative definite solution of the algebraic Riccati equation

0 = ATK + KA + ρCTQC − LTQ−1_u L.

Conclusion

• we posed an estimation problem for continuous-time LTI EIV model

(similar results are available for the discrete-time case)

• the disturbance is interpreted as a latent variable and the measurement errors as

misfit variables

• the estimation problem compensates for disturbance and measurement errors by minimizing an appropriately defined cost functional over all trajectories of the system • the solution of the general estimation problem and its extremes, pure misfit and pure latency, lead to one and the same problem—minimization of a quadratic function of the state and the input subject to the state equation