Iterative Maximum Likelihood FIR Filter for State-Space Models with Time-Stamped Delayed and Missing Data

Abstract

Time delay and missing data are the primary reasons for instability and weak showing of control systems. This paper proposes a new finite impulse response (FIR) filter for the state-space models with time-stamped delayed and missing data. By reconstructing the original state-space model, the delayed and missing measurement model is transformed into an ordinary one with new parameters. By processing and calculating in both slow batch and quick iterative forms, a maximum likelihood FIR state estimator is developed. Simulation results based on the harmonic model and the drifting stochastic resonator are provided to demonstrate that the proposed method can estimate the control systems with time delay and missing data well and show better robustness than the existing methods when the statistics of the measurement noise are unavailable.

Appendix A Iterative ML FIR Algorithms

The derivations are obtained by following [34]. To be clarity, we do not present all the details and only give some essential formulas. Accordingly, we introduce

$$\begin{aligned} {\acute{\mathbf{C}}}_{i}&= \mathbf{C}_{i}^\mathrm{T}{\Pi }_{i}^{-1}{} \mathbf{C}_{i}, \end{aligned}$$

(A1)

$$\begin{aligned} {\varvec{\Pi }}_{i}^{-1}&= {\hat{\varvec{\Pi }}}_{i}^{-1}{\Lambda }_{i}^{-1},\end{aligned}$$

(A2)

$$\begin{aligned} {} \mathbf{P}_{i}&= {\bar{\mathcal {G}}}_{i}-\mathbf{A}\,{\mathcal {G}}_{i-1}{\varvec{\Pi }}_{i-1}^{-1}\mathbf{\mathcal {G}}_{i-1}^\mathrm{T}{} \mathbf{A}^\mathrm{T},\end{aligned}$$

(A3)

$$\begin{aligned} {} \mathbf{K}_{i}&= \begin{bmatrix} \mathbf{L}_{i}&\mathbf{A}{} \mathbf{K}_{i-1}-\mathbf{L}_{i}{\bar{\mathbf{C}}}_{\bar{m}+i}{} \mathbf{A}{} \mathbf{K}_{i-1} \end{bmatrix}, \end{aligned}$$

(A4)

where $\mathbf{L}_{i}={\bar{\mathbf{L}}}_{i}+{\tilde{\mathbf{L}}}_{i}$ and

$$\begin{aligned} {\bar{\mathbf{L}}}_{i}&= \mathbf{P}_{i}{\bar{\mathbf{C}}}_{\bar{m}+i}^\mathrm{T}({\bar{\mathbf{C}}}_{\bar{m}+i}{} \mathbf{P}_{i}{\bar{\mathbf{C}}}_{\bar{m}+i}^\mathrm{T}+{\bar{\mathbf{R}}}_{\bar{m}+i})^{-1}, \end{aligned}$$

(A5)

$$\begin{aligned} {\tilde{\mathbf{L}}}_{i}&= (\mathbf{I}-{\bar{\mathbf{L}}}_{i}{\bar{\mathbf{C}}}_{\bar{m}+i}){\bar{\mathbf{P}}}_{i}{\acute{\mathbf{C}}}_{i}^{-1}{\bar{\mathbf{P}}}_{i}^\mathrm{T}{\bar{\mathbf{C}}}_{\bar{m}+i}^\mathrm{T}({\bar{\mathbf{C}}}_{\bar{m}+i}{} \mathbf{P}_{i}{\bar{\mathbf{C}}}_{\bar{m}+i}^\mathrm{T}+{\bar{\mathbf{R}}}_{\bar{m}+i})^{-1}, \end{aligned}$$

(A6)

$$\begin{aligned} {\bar{\mathbf{P}}}_{i}&= \mathbf{A}^{i}-\mathbf{A}{\mathcal {G}}_{i-1}{\varvec{\Pi }}_{i-1}^{-1}{} \mathbf{C}_{i-1}, \end{aligned}$$

(A7)

where improvements are aiming at time delay and missing data having shown in the formulas above.

Next, for computing (34) iteratively, ${\hat{\mathbf{x}}}_{i}$ can be rewritten by (34), (35) and (36) in an iterative index i as

$$\begin{aligned} {\hat{\mathbf{x}}}_{i} = {\tilde{\mathbf{x}}}_{i}+{\bar{\mathbf{x}}}_{i}+{\check{\mathbf{x}}}_{i}, \end{aligned}$$

(A8)

where i is a positive integer ranging from $\phi +1$ to N and

$$\begin{aligned} {\tilde{\mathbf{x}}}_{i}= & {} \mathbf{K}_{i}{} \mathbf{Y}_{\bar{m}+i,m}, \end{aligned}$$

(A9)

$$\begin{aligned} {\bar{\mathbf{x}}}_{i}= & {} ({\bar{\mathbf{B}}}_{i}-\mathbf{K}_{i}{} \mathbf{D}_{i})\mathbf{U}_{\bar{m}+i-1,\bar{m}}, \end{aligned}$$

(A10)

$$\begin{aligned} {\check{\mathbf{x}}}_{i}= & {} -\mathbf{K}_{i}{\check{\mathbf{D}}}_{i}{} \mathbf{U}_{p_i,p_m}. \end{aligned}$$

(A11)

Formula (A9) and (A10) in iterative form can be obtained from [34] as follows:

$$\begin{aligned} {\tilde{\mathbf{x}}}_{i} = \mathbf{A}{\tilde{\mathbf{x}}}_{i-1}+\mathbf{L}_{i}(\mathbf{y}_{\bar{m}+i}-{ \bar{\mathbf{C}}}_{\bar{m}+i}{} \mathbf{A}{{\tilde{\mathbf{x}}}_{i-1}}), \end{aligned}$$

(A12)

$$\begin{aligned} {\bar{\mathbf{x}}}_{i} = (\mathbf{I}-\mathbf{L}_{i} \bar{\mathbf{C}}_{\bar{m}+i})(\mathbf{A}{\bar{\mathbf{x}}}_{i}+\mathbf{B}\mathbf{u}_{\bar{m}+i-1}). \end{aligned}$$

(A13)

To find a recursion for ${\check{\mathbf{x}}}_{i}$ given by (A11), by referring (A4) and invoking matrices

$$\begin{aligned} {\check{\mathbf{D}}}_{i}&= \begin{bmatrix} {\bar{\mathbf{D}}}_{\bar{m}+i} &{}\quad \mathbf{0} \\ \mathbf{0} &{}\quad {\check{\mathbf{D}}}_{i-1} \end{bmatrix}, \end{aligned}$$

(A11) can be transformed to

$$\begin{aligned} {\check{\mathbf{x}}}_{i}= & {} \mathbf{A}{\check{\mathbf{x}}}_{i-1}-\mathbf{L}_{i}({\bar{\mathbf{D}}}_{\bar{m}+i}{} \mathbf{U}_{p_i,\bar{m}+i-1}+{\bar{\mathbf{C}}}_{\bar{m}+i}{} \mathbf{A}{\check{\mathbf{x}}}_{i-1}). \end{aligned}$$

(A14)

Finally, by combining (A12), (A13) and (A14), the recursion for ${\hat{\mathbf{x}}}_{i}$ can be rewritten by ${\hat{\mathbf{x}}}^{-}_{i} = \mathbf{A}{\hat{\mathbf{x}}}_{i-1}+\mathbf{B}{} \mathbf{u}_{\bar{m}+i-1}$ as

$$\begin{aligned} {\hat{\mathbf{x}}}_{i}= & {} {\hat{\mathbf{x}}}_{i}^{-}+\mathbf{L}_{i}(\mathbf{y}_{\bar{m}+i}-{\bar{\mathbf{C}}}_{\bar{m}+i}{\hat{\mathbf{x}}}_{i}^{-}-{\bar{\mathbf{D}}}_{\bar{m}+i}{} \mathbf{U}_{p_i,\bar{m}+i-1}). \end{aligned}$$

(A15)