Gaussian sum approximation filter for nonlinear dynamic time-delay system

Abstract

This paper focuses on designing Gaussian sum approximation filter for both accurately and rapidly estimating the state of a class of nonlinear dynamic time-delay systems. Firstly, a novel nonaugmented Gaussian filter (GF) is derived, whose superiority in computation efficiency is theoretically analyzed as compared to the standard augmented GF. Secondly, a nonaugmented Gaussian sum filter (GSF) is proposed to accurately capture the state estimates by a weight sum of the above-proposed GF. In GSF, each GF component is independent from the others and can be performed in a parallel manner so that GSF is conducive to high-performance computing across many compute nodes. Finally, the performance of the proposed GSF is demonstrated by a vehicle suspension system with time delay, where the GSF achieves higher accuracy than the single GF and is computationally much more efficient than the particle filter with the almost same accuracy.

Appendices

Appendix 1

For symmetric and invertible matrices ${{\mathbf {A}}}\in \mathbf{R}^{n}$, ${{\mathbf {B}}}\in \mathbf{R}^{n\times m}$ and ${{\mathbf {C}}}\in \mathbf{R}^{m}$, if ${{\mathbf {A}}}-{{\mathbf {B}}}{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}}$ is invertible, then

• (i) ${{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}}$ is invertible

• (ii) ${{\mathbf {D}}}=\left[ {{\begin{array}{c@{\quad }c} {{\mathbf {A}}}&{} {{\mathbf {B}}} \\ {{{\mathbf {B}}}^{\mathrm{T}}}&{} {{\mathbf {C}}} \\ \end{array} }} \right]$ are invertible

• (iii)
  $$\begin{aligned}&{{\mathbf {D}}}^{-1}=\left[ \begin{array}{cc} ({{\mathbf {A}}}-{{\mathbf {B}}}{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}})^{-1}&{} -{{\mathbf {A}}}^{-1} {{\mathbf {B}}}({{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1} {{\mathbf {B}}})^{-1} \\ -{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}} ({{\mathbf {A}}}-{{\mathbf {B}}}{{\mathbf {C}}}^{-1} {{\mathbf {B}}}^{\mathrm{T}})^{-1}&{} ({{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}} {{\mathbf {A}}}^{-1}{{\mathbf {B}}})^{-1}\\ \end{array} \right] ,\nonumber \\ \end{aligned}$$

  (61)

or

$$\begin{aligned} {{\mathbf {D}}}^{-1}= & {} \left[ {{\begin{array}{c@{\quad }c} 0&{} 0 \\ 0&{} {{{\mathbf {C}}}^{-1}} \end{array} }} \right] +\left[ {{{\mathbf {I}}}_n -{{\mathbf {B}}}{{\mathbf {C}}}^{-1}} \right] ^{\mathrm{T}}\nonumber \\&\times \left( {{{\mathbf {A}}}-{{\mathbf {B}}}{{\mathbf {C}}}^{-1} {{\mathbf {B}}}^{\mathrm{T}}} \right) ^{-1} \left[ {{{\mathbf {I}}}_n -{{\mathbf {B}}}{{\mathbf {C}}}^{-1}} \right] . \end{aligned}$$

(62)

Proof

Note that

$$\begin{aligned} \left| {{\mathbf {D}}} \right|= & {} \left| {{\mathbf {A}}} \right| \left| {{{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}}} \right| =\left| {{\mathbf {C}}} \right| \left| {{{\mathbf {A}}}-{{\mathbf {B}}}{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}}} \right| \ne 0,\nonumber \\ \end{aligned}$$

(63)

where $\left| {{{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}}} \right| \ne 0$ is obvious such that ${{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}}$ and ${{\mathbf {D}}}$ are invertible.

From (61), it can be obtained that

$$\begin{aligned} {{\mathbf {DD}}}^{-1}={{\mathbf {D}}}^{-1}{{\mathbf {D}}}={{\mathbf {I}}}_{n+m}. \end{aligned}$$

(64)

Therefore the inverse of ${{\mathbf {D}}}$ can be described by (61) in result (iii). $\square$

In addition, the following two equations are available

$$\begin{aligned}&{{\mathbf {A}}}^{-1}{{\mathbf {B}}}({{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}})^{-1}=({{\mathbf {A}}}-{{\mathbf {B}}}{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}})^{-1} \bullet \nonumber \\&\quad ({{\mathbf {A}}}-{{\mathbf {B}}}{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}}) {{\mathbf {A}}}^{-1}{{\mathbf {B}}}({{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}})^{-1}\nonumber \\&\qquad =({{\mathbf {A}}}-{{\mathbf {B}}}{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}})^{-1}{{\mathbf {B}}}{{\mathbf {C}}}^{-1}, \end{aligned}$$

(65)

$$\begin{aligned}&{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}}({{\mathbf {A}}}-{{\mathbf {B}}}{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}})^{-1}{{\mathbf {B}}}{{\mathbf {C}}}^{-1}\nonumber \\&\quad ={{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}} ({{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}})^{-1}\nonumber \\&\quad =({{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}})^{-1}-{{\mathbf {C}}}^{-1}{{\mathbf {C}}}({{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}})^{-1}\nonumber \\&\qquad +\,{{\mathbf {C}}}^{-1}{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}}({{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}})^{-1}\nonumber \\&\quad =({{\mathbf {C}}}-{{\mathbf {B}}}^{\mathrm{T}}{{\mathbf {A}}}^{-1}{{\mathbf {B}}})^{-1}-{{\mathbf {C}}}^{-1}. \end{aligned}$$

(66)

From (65) and (66), rearranging (62) yield the result (iii).

Appendix 2

Considering the augmented state ${{\mathbf {X}}}_k$, the nonlinear dynamic model in (1) is transformed to the following equivalent model, i.e.,

$$\begin{aligned} {{\mathbf {X}}}_{k +1} ={{\mathbf {G}}}({{\mathbf {X}}}_k )+{\varvec{\varGamma }} _k {{\mathbf {w}}}_k , \end{aligned}$$

(67)

with ${\varvec{\varGamma }} _k =[1,0,\ldots ,0]^{\mathrm{T}}$ and

$$\begin{aligned} {{\mathbf {G}}}({{\mathbf {X}}}_k )=\left[ {{\begin{array}{c} {{{\mathbf {f}}}({\mathbf {x}}_k )+\sum _{j=1}^d {{{\mathbf {g}}}_j ({\mathbf {x}}_{k-j} )} } \\ {{{\mathbf {x}}}_k } \\ \vdots \\ {{{\mathbf {x}}}_{k-d+1} } \\ \end{array} }} \right] . \end{aligned}$$

According to [18], the augmented GF for the nonlinear time delay system can be given as follows,

Step 1: Prediction. Given the Gaussian approximation density $p_{\mathrm{GF}}^i ({{\mathbf {X}}}_k |{{\mathbf {Z}}}_k )$ with the estimates ${\hat{{\mathbf {X}}}}_k^i$ and ${{\mathbf {P}}}_k^{X,i}$ at time k, the predicted estimates of the augmented state can be computed as follows

$$\begin{aligned} {\hat{{\mathbf {X}}}}_{k +1| k}^i= & {} \hbox {E}^{i}[{{\mathbf {G}}}({{\mathbf {X}}}_k )|{{\mathbf {Z}}}_k ]=\int {{{\mathbf {G}}}({{\mathbf {X}}}_k )p_{\mathrm{GF}}^i ({{\mathbf {X}}}_k |{{\mathbf {Z}}}_k )} \mathrm{d}{{\mathbf {X}}}_k , \nonumber \\\end{aligned}$$

(68)

$$\begin{aligned} {{\mathbf {P}}}_{k+1 | k}^{X,i}= & {} \hbox {E}^{i}[{\tilde{{\mathbf {X}}}}_{k\hbox {+1|}k}^i ({\tilde{{\mathbf {{X}}}}}_{k +1| k}^i )^{\mathrm{T}}|{{\mathbf {Z}}}_k ]\nonumber \\= & {} \int {{{\mathbf {G}}}({{\mathbf {X}}}_k ){{\mathbf {G}}}^{\mathrm{T}}({{\mathbf {X}}}_k )p_{\mathrm{GF}}^i ({{\mathbf {X}}}_k |{{\mathbf {Z}}}_k )} \mathrm{d}{{\mathbf {X}}}_k.\nonumber \\&-{\hat{{\mathbf {X}}}}_{k + 1|k}^i ({\hat{{\mathbf {X}}}}_{k + 1|k}^i )^{\mathrm{T}}+{\varvec{\varGamma }} _k Q_k {\varvec{\varGamma }} _k^{\mathrm{T}} \end{aligned}$$

(69)

Step 2 : Correction. Given the Gaussian approximation density $p_{\mathrm{GF}}^i ({{\mathbf {X}}}_{k+1} |{{\mathbf {Z}}}_k )$with the estimates ${\hat{{\mathbf {X}}}}_{k+1|k}^i$ and ${{\mathbf {P}}}_{k+1|k}^{X,i}$ at time k+1, the filtering density at time k+1 can be updated as the Gaussian distribution, i.e.,

$$\begin{aligned} p_{\mathrm{GF}}^i ({{\mathbf {X}}}_{k + 1} |{{\mathbf {Z}}}_{k + 1} )={\mathcal {N}}({{\mathbf {X}}}_{k+1} ;{\hat{{\mathbf {X}}}}_{k+1}^i ,{{\mathbf {P}}}_{k+1}^{X,i} ), \end{aligned}$$

(70)

with the analytical computation

$$\begin{aligned} \left\{ {\begin{array}{c} {\hat{{\mathbf {X}}}}_{k+1}^i ={\hat{{\mathbf {X}}}}_{k+1|k}^i +{{\mathbf {K}}}_{k+1}^i ({{{\mathbf {z}}}}_{k+1} -{\hat{{\mathbf {z}}}}_{k+1|k}^i ), \\ {{\mathbf {P}}}_{k+1}^{X,i} ={{\mathbf {P}}}_{k+1 |k}^{X,i} -{{\mathbf {K}}}_{k+1}^i {{\mathbf {P}}}_{k+1|k}^{z,i} ({{\mathbf {K}}}_{k+1}^i )^{\mathrm{T}}, \\ {{\mathbf {K}}}_{k+1}^i ={{\mathbf {P}}}_{k+1 |k}^{Xz,i} ({{\mathbf {P}}}_{k+1|k}^{z,i} )^{-1}, \\ \end{array}} \right. \end{aligned}$$

(71)

and the nonlinear integrals

$$\begin{aligned} {\hat{{\mathbf {z}}}}_{k + 1|k}^i= & {} \int {{{\mathbf {h}}}({\mathbf {x}}_{k + 1} )p_{\mathrm{GF}}^i ({{\mathbf {X}}}_{k+1} |{{\mathbf {Z}}}_k )} \mathrm{d}{{\mathbf {X}}}_{k + 1}, \end{aligned}$$

(72)

$$\begin{aligned} {{\mathbf {P}}}_{k + 1|k}^{z,i}= & {} \int {{{\mathbf {h}}}({\mathbf {x}}_{k + 1} ){{\mathbf {h}}}^{\mathrm{T}} ({\mathbf {x}}_{k + 1}) p_{\mathrm{GF}}^i ({{\mathbf {X}}}_{k+1} |{{\mathbf {Z}}}_k )\mathrm{d}{{\mathbf {X}}}_{k + 1}}\nonumber \\&-{\hat{{\mathbf {z}}}}_{k + 1|k}^i ({\hat{{\mathbf {z}}}}_{k + 1|k}^i )^{\mathrm{T}}+{{\mathbf {R}}}_{k + 1} , \end{aligned}$$

(73)

$$\begin{aligned} {{\mathbf {P}}}_{k+1 |k}^{Xz,i}= & {} \int {{{\mathbf {X}}}_{k+1} {{\mathbf {h}}}^{\mathrm{T}} ({\mathbf {x}}_{k+1} )p_{\mathrm{GF}}^i ({{\mathbf {X}}}_{k+1} |{{\mathbf {Z}}}_k )\mathrm{d}{{\mathbf {X}}}_{k+1} }\nonumber \\&-{\hat{{\mathbf {X}}}}_{k+1|k}^i ({\hat{{\mathbf {z}}}}_{k + 1|k}^i )^{\mathrm{T}}. \end{aligned}$$

(74)