In this blog post, I described several problem formulations I'd like to solve. I started with easy formulation(Discrete linear time-invariant system) that would be extended to more complicated formulations such as non-linear examples.

  • Step 0. Preliminary knowledge

    \(x_{k+1} = Ax_{k} + w_{k}, \ w_k \sim N(0 ,Q), Q \in \mathbb{S}^{n}_{++}, \tag{1}\)

    \(y_{k} = Cx_{k} + v_{k}, \ v_k \sim N(0 ,R), R \in \mathbb{S}^{m}_{++}, \tag{2}\)

These equations form the basis for many state estimation, where the goal is to estimate the true state \(x_k \in \mathbb{R}^{n}\) of the system from noisy measurements \(y_k \in \mathbb{R}^{m}\).

My research seeks to address several critical challenges in the field of state estimation to enhance the reliability, accuracy, and efficiency of monitoring and control systems. Specifically, the research application will focus on estimating the state of Earth from different perspectives:

  • Step 1. Scratch problem formulation for Earth observing system, incorporating other state information for more optimal state estimation

Why?

Understanding

Given

With eq. \((3,4,5)\).

Find \(\hat{x}_E(k)\)

Such that Minimize state error covariance \(P_E(k)\)

Linear discrete-time system dynamics of earth

\(x_{E}(k+1) = A_{E}(k)x_{E}(k)+w_{E}(k), \tag{3}\)

Dynamics of i-th satellites \(x^{i}_{S}(k+1) = A^{i}_{S}(k)x^{i}_{S}(k)+w^{i}_{S}(k), \ where \ i = 1\dots r \tag{4}\)

Measurement

\(y^{i}(k) = C^{i} \begin{pmatrix} x^{i}_S(k) \\ x_E(k) \end{pmatrix}+ v^{i}(k) \tag{5}\)

\(w_E(k) \sim N(0,Q_E), \ Q_E \in \mathbb{S}^{n}_{++} \ w^i_S(k) \sim N(0,Q^i_S), \ Q^i_S \in \mathbb{S}^{n_i}_{++}, \ v^i(k) \sim N(0,R^i), \ R^i \in \mathbb{S}^{m_i}_{++},\)


1-1) Resilient optimal state estimation from unknown input or sensor attack

Given

In eq.\((3)\), adding unknown input term: \(G_{E}(k)d_{E}(k), \ d_E(k) \in \mathbb{R}^{l}\)

1-2) Control input

Given In eq.\((4)\), adding control input term: \(B^{i}_{S}(k)u^{i}_{S}(k), \ u^{i}_{S}(k) \in \mathbb{R}^{p_i}\)

2) Integrating heterogeneous sensor information

.

Motion model

\(q(k) = \theta_1+(k-k_{l-1})\theta_2 + \frac{(k-k_{l-1})^2}{2}\theta_3\), (\(q(k), \theta_1,\theta_2, \theta_3 \in \mathbb{R}^{2}\))

Camera measurements

\(I(k) = \frac{M(k)(q(k)-p(k))}{m(k)(q(k)-p(k))}\), \(\in \mathbb{R}^{2}\)

where

\(M(k) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}AR_c(k)\), \(M(k) \in \mathbb{R}^{2 \times 3}\) and

\(m(k) = \begin{pmatrix} 0 & 0 & 1 \end{pmatrix}AR_c(k)\), \(m(k) \in \mathbb{R}^{1 \times 3}\)

\(p(k)\): (known) Optical center at the agent's position. \(R_c(k)\in \mathbb{R}^{3}\): (known) Camera orientation. \(A \in \mathbb{R}^{3 \times 3}\): (known) Camera intrinsic matrix.

\(P(y^i;\Sigma_{I}) = \frac{1}{2\pi\sqrt{det\Sigma_{I}}}e^{\frac{1}{2}(I(k)-y^i)^T\Sigma_{I}(I(k)-y^i)}\)

Likelihood of a measurement \(y^i\) and covariance matrix (\(\Sigma_{I}\)) independent of the unknown parameters.

RF(Doppler) measurements

\(\lambda F(k) = \theta_{\lambda}-\dot{\rho}(k)\)

where

\(\theta_{\lambda} = \lambda\Delta f\), \(\lambda = c/f_c\),

\(f_c\) is the carrier frequency of the transmitter. \(\Delta f\) is the difference between the carrier frequencies of the transmitter and receiver.

\(\dot{\rho}(k) = \frac{(q(k)-p(k))^T(\dot{q}(k)-\dot{p}(k))}{||q(k)-p(k)||}\)

Frequency shift: \(F(k) = \Delta f(k) - \frac{\dot{\rho}(k)}{\lambda}\)

\(P(y^i;\sigma_{\lambda F},\theta_{\lambda}) = \frac{1}{\sqrt{2\pi}\sigma_{\lambda F}}e^{-\frac{(\lambda F(k)-y^i)^2}{2\sigma_{\lambda F}^2}}\)

Likelihood of a measurement \(y^i\) and covariance matrix (\(\sigma_{\lambda F}\)) independent of the unknown parameters.

How do we define the heterogeneous sensors in linear dynamical system?

3) Selective(selection/scheduling) Measurement under Resource Constraints

Why?

Understanding which measurements to include or exclude, dynamically, can significantly improve the efficiency of state estimation, ensuring optimal resource utilization within a certain level of compromising accuracy.

Given

With eq. \((1,2)\) and resource constraints. Resource constraints depend on how we define the resource e.g) the limited number of sensors (\(n_s\)), and energy resource (\(H \in \mathbb{R}^{h \times mT}\), \(b \in \mathbb{R}^h\)).

Find \(\gamma_i\).

\(\gamma_i\) takes binary values in \(\{0, 1\}\). \(\gamma_{m(k-1)+j} = 1\) means that sensor \(j\) where \(j = 1, . . . ,m\) , is active at the \(k^{th}\) time instance for \(k = 1, . . . ,T\). This collection of binary indicators, \(\gamma = [\gamma_1, \ldots, \gamma_{mT}]\) forming a vector in \(\{0, 1\}^{mT}\), is defined as the sensor scheduling vector.

Such that

To find the true state \(x_k\), minize the trace of \(P_k\) under the resource constraints (e.g. \(H\gamma \leq b\)).

\(P_k = Cov(x_k|\Gamma_kY_k)\)

\(Y_k = [y_1^T, y_2^T, \dots, y_k^T]^T \in \mathbb{R}^{mk}\), \(\Gamma_k = diag([\gamma_1, \ldots, \gamma_{mk}]), \in \mathbb{R}^{mk \times mk}\)


4) Distributed, Decentralized (reinforcement learning) for putting more autonomy and agility to adapt different environment**
  • Step 2. Literature review (How much developed and solved already so far)

  • Step 3. Speciality and strength of my research

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