# : Lyapunov Equation for backward continuous-time Kalman Filter

## Main Question or Discussion Point

URGENT: Lyapunov Equation for backward continuous-time Kalman Filter

Hi,

Consider a continuous Kalman filter running backward in time as desired in a "two-filter" smoother. What would be the form of Lyapunov equation for this backward-time filter?

Given a system: dx/dt = Fx + Gv, and, say, a forward Kalman filter: dX/dt = CX + Dx + Ew (where X is the filtered esimate of the state x, and v, w are white noises), the augmented system would be: d/dt [x X]' = A [x X]' + B [v w]'

Then, the (steady-state) Lyapunov equation for the augmented (forward-time) system is: AP + PA' + BB' = 0.

In backward-time, the system modifies to: - dx/dt = Fx + Gv, and accordingly the (backward) Kalman filter may be obtained using standard Riccati equation approach. How would the augmented system be obtained to be used for Lyapunov equation? What is the form of the Lyapunov equation in this case, if different from the forward-case mentioned earlier? I need to be able to use Lyapunov equation to derive the state-covariance matrix and the error-covariance from that.

Shibdas.

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berkeman
Mentor

Hi,

Consider a continuous Kalman filter running backward in time as desired in a "two-filter" smoother. What would be the form of Lyapunov equation for this backward-time filter?

Given a system: dx/dt = Fx + Gv, and, say, a forward Kalman filter: dX/dt = CX + Dx + Ew (where X is the filtered esimate of the state x, and v, w are white noises), the augmented system would be: d/dt [x X]' = A [x X]' + B [v w]'

Then, the (steady-state) Lyapunov equation for the augmented (forward-time) system is: AP + PA' + BB' = 0.

In backward-time, the system modifies to: - dx/dt = Fx + Gv, and accordingly the (backward) Kalman filter may be obtained using standard Riccati equation approach. How would the augmented system be obtained to be used for Lyapunov equation? What is the form of the Lyapunov equation in this case, if different from the forward-case mentioned earlier? I need to be able to use Lyapunov equation to derive the state-covariance matrix and the error-covariance from that.

Shibdas.
Welcome to the PF.

Can you give us the context of your question? Why is it urgent? Is it for schoolwork?

Yes, this is for my research work in my school. It is urgent because I need to present my work soon.

Note that I know that the error-covariance for the backward filter is simply the solution of the steady-state Riccati equation used to obtain the backward Kalman filter. But, I need to use Lyapunov equation (for the augmented system) to evaluate it and demonstrate that the two errors are the same.

It appears to me that the correct answer may be obtained ONLY if the augmented system considers, alongwith the backward filter equation, the forward system equation: dx/dt = Fx + Gv, rather than the backward system equation: - dx/dt = Fx + Gv (which is strange).

Thanks,
Shibdas.