Paper Z

References

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→ overall

モデル予測制御
Minimax approaches to robust model predictive control

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→ Kalman filter

Understanding and Applying Kalman Filtering
Kalman Filtering - Theory and Practice Using MATLAB
Digital Control of Dynamic Systems
Tutorial - The Kalman Filter
the Kalman filter
MIT- Kalman Filter

Issues

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→ Kalman filter

 1. staffs

  1.1 materials

• book page 278, or pdf page 148 of モデル予測制御, a well interpreted version of Kalman filter in Japanese briefly
• Understanding and Applying Kalman Filtering, some easiest concept
• Kalman Filtering - Theory and Practice Using MATLAB, Matlab realization
• book page 389, or pdf page 205 of Digital Control of Dynamic Systems, original algorithm
• page 5~6 of Tutorial - The Kalman Filter, algorithm
• MIT- Kalman Filter
• Matlab introduction of Kalman filter
• Kalman filter design by Matlab

  1.2 codes

• Learning the Kalman Filter in Simulink v2.1
• An Intuitive Introduction to Kalman Filter
• Learning the Kalman Filter
• Learning the Kalman Filter: A Feedback Perspective
• Learning the Kalman-Bucy Filter in Simulink
• Linear Kalman Filter in Simulink

  1.3 GitHub assembly

https://github.com/Trancemania/Kalman-filter/tree/origin

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 2. pre-knowledge

  2.1 objective

Theoretically, the Kalman filter is an estimator for what is called the linear-quadratic problem. ^[kf1]

EXAMPLE 1: continuous-time plant model Kalman filter

$\left\{\begin{array}{l}\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}\mathit{u}+\mathit{\omega }\\ \mathit{y}=\mathit{C}\mathit{x}+\mathit{\nu }\end{array}\right.$ (kf.1)

with $\mathit{\omega }~\mathcal{N}(0,W^2),$ $\mathit{\nu }~\mathcal{N}(0,V^2),$ $\mathrm{E}\left\{{\mathit{\omega }}^T\mathit{\omega }\right\}=W^2\ge 0,$ $\mathrm{E}\left\{{\mathit{\nu }}^T\mathit{\nu }\right\}=V^2\ge 0,$ $\mathrm{E}\left\{{\mathit{\omega }}^T\mathit{\nu }\right\}=0$ , $W, V$ is the standard deviation, or square root of variance of white noise normal distribution $\mathit{\omega }, \mathit{\nu }$ , and $\mathrm{E}$ is the expectation operator. ^{[kf2] [kf3] [kf4] [kf5]}
A typical Kalman filter design can be demonstrated as:

where Kalman filter gain ^[kf6] $L$ is the variable needs to be worked out. The system in Fig. kf1 can be described as:

$\dot{\hat{\mathit{x}}}=\mathit{A}\hat{\mathit{x}}+\mathit{B}\mathit{u}+\mathit{L}(\mathit{y}-\mathit{C}\hat{\mathit{x}})$ (kf.2)

where $\hat{\mathit{x}}$ is the estimate of system state $\mathit{x}$ .
The estimation error may be defined as $\mathit{e}=\mathit{x}-\hat{\mathit{x}}$ , thus, ^[kf7]

$$\begin{aligned} \dot{\boldsymbol{e}}&=(\boldsymbol{A}\boldsymbol{x}+\boldsymbol{B}\boldsymbol{u}+\boldsymbol{\omega})-[\boldsymbol{A}\widehat{\boldsymbol{x}}+\boldsymbol{B}\boldsymbol{u}+\boldsymbol{L}(\boldsymbol{y}-\boldsymbol{C}\widehat{\boldsymbol{x}})]\\ &= (\boldsymbol{A}-\boldsymbol{L}\boldsymbol{C})\boldsymbol{e}+(\boldsymbol{\omega}-\boldsymbol{L}\boldsymbol{\nu}) \end{aligned}$$ (kf.3)

Kalman filter design provides $\mathit{L}$ that minimizes the scalar cost function ^[kf8]

$J=\mathrm{E}\left({\mathit{e}}^T\mathit{H}\mathit{e}\right)$ (kf.4)

where $\mathit{H}$ is an unspecified symmetric, positive definite weighting matrix, which penalize specified state(in this case) ^{[kf9] [kf10]} . A related matrix, the symmetric error covariance, is defined as

$\mathbf{\Sigma }=\mathrm{E}\left(\mathit{e}{\mathit{e}}^T\right)$ (kf.5)

Notice that $\mathbf{\Sigma }$ is a matrix.
Further reading about matrix, vector, derivative, calculus, quadratic forms, ${\mathit{x}}^T\mathit{P}\mathit{x}$ , ${\mathit{A}}^T\mathit{P}+\mathit{P}\mathit{A}$ .
A general solution of (kf.3) may be written as ^{[kf11] [kf12] [kf13]} :

$\mathit{e}\left(t\right)={\mathrm{e}}^{(\mathit{A}-\mathit{L}\mathit{C})t}\mathit{e}\left(0\right)+{\int }_0^t{\mathrm{e}}^{(\mathit{A}-\mathit{L}\mathit{C})(t-\tau )}\left[\right(\mathit{\omega }\left(\tau \right)-\mathit{L}\mathit{\nu }\left(\tau \right)]\mathrm{d}\tau$ (kf.6)

Thus,

$$\begin{aligned} \dot{\boldsymbol\Sigma}&=\mathrm E(\dot{\boldsymbol e}\boldsymbol e^T+\boldsymbol e\dot{\boldsymbol e}^T)\\&=\mathrm E[(\boldsymbol{A}-\boldsymbol{L}\boldsymbol{C})\boldsymbol{e}\boldsymbol e^T+(\boldsymbol{\omega}-\boldsymbol{L}\boldsymbol{\nu})\boldsymbol e^T+\boldsymbol{e}\boldsymbol e^T(\boldsymbol{A}^T-\boldsymbol{C}^T\boldsymbol{L}^T)+\boldsymbol e(\boldsymbol{\omega}^T-\boldsymbol{\nu}^T\boldsymbol{L}^T)] \end{aligned}$$ (kf.7)

by substitution of (kf.6) into (kf.7) , the final expression of which may be rewritten into

$$\begin{aligned} \mathrm E [\boldsymbol{e}(\boldsymbol{\omega}^T-\boldsymbol{\nu}^T\boldsymbol{L}^T)]&=\mathrm{e}^{(\boldsymbol{A}-\boldsymbol{LC})t}\mathrm E [\boldsymbol{e}(0)(\boldsymbol{\omega}^T-\boldsymbol{\nu}^T\boldsymbol{L}^T)]\\&+\int_0^t\mathrm e^{(\boldsymbol A-\boldsymbol L\boldsymbol C)(t-\tau)}\mathrm E\lbrack(\boldsymbol{\omega}^T(t)-\boldsymbol{\nu}^T(t)\boldsymbol{L}^T)(\boldsymbol\omega(\tau)-\boldsymbol L\boldsymbol\nu(\tau)\rbrack\mathrm d\tau \end{aligned}$$ (kf.8)

Notice that the initial condition $\mathit{e}\left(0\right)$ is uncorrelated with ${\mathit{\omega }}^T-{\mathit{\nu }}^T{\mathit{L}}^T$ , therefore,

$\mathrm{E}\left[\mathit{e}\right(0\left)\right({\mathit{\omega }}^T-{\mathit{\nu }}^T{\mathit{L}}^T\left)\right]=\mathrm{E}\left[\mathit{e}\right(0\left)\right]\mathrm{E}({\mathit{\omega }}^T-{\mathit{\nu }}^T{\mathit{L}}^T)=0$ (kf.9)

By reasoning process in book page 20 or pdf page 3 of MIT- Kalman Filter , (kf.8) may be reformed into

$\mathrm{E}\left[\mathit{e}\right({\mathit{\omega }}^T-{\mathit{\nu }}^T{\mathit{L}}^T\left)\right]=\frac{1}{2}(W^2+\mathit{L}{V}^2{\mathit{L}}^T)$ (kf.10)

According to page 9 theorem 1.5 of [kf14] , and [kf15] [kf16] [kf17] , (kf.4) may be rewritten as

$J=\mathrm{E}\left({\mathit{e}}^T\mathit{H}\mathit{e}\right)=tr\left(\mathbf{\Sigma }\mathit{H}\right)$ (kf.11)

An auxiliary cost function may be introduced and defined as

$J^{\text{'}}=tr(\mathbf{\Sigma }\mathit{H}+\mathbf{\Lambda }\mathit{F})$ (kf.12)

where $\mathit{F}$ is an $n\times n$ matrix of zeros, and $\mathbf{\Lambda }$ is an $n\times n$ matrix of unknown Lagrange multiplier ^[kf18] , which provide an ingenious mechanism for drawing constraints into the optimization. In this case, $\mathit{F}$ is designed as

$\mathit{F}=-\dot{\mathbf{\Sigma }}+\mathit{A}\mathbf{\Sigma }-\mathit{L}\mathit{C}\mathbf{\Sigma }+\mathbf{\Sigma }{\mathit{A}}^{\mathbf{T}}-\mathbf{\Sigma }{\mathit{C}}^{\mathbf{T}}{\mathit{L}}^{\mathbf{T}}+{\mathit{W}}^2+\mathit{L}{\mathit{V}}^{\mathbf{2}}{\mathit{L}}^{\mathbf{T}}$ (kf.13)

Riccati average model ref3