17 May

#### Problem statement for my Ph.D.

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: $$B^{i}_{S}(k)u^{i}_{S}(k), \ u^{i}_{S}(k) \in \mathbb{R}^{p_i}$$

.

##### 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.

Use unscented kalman filter, if you know the distribution of each measurement, we can propagate the mean and variance to get updated mean and variance.

How can I integrate those given equations eq. $$(3,4,5)$$?

##### 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 for low communication burden compared to centralized fusion.**

Why?

In a distributed wireless sensor network, each sensor consumes much power to communicate with its neighbors for exchanging the estimates at each time. However, it is not necessary to send data to all the adjacent sensors at each time, particularly in the case that the sensor power or the channel bandwidth is constrained

Given

Discrete-time LTI system eq. $$(3)$$

Measurement in sensor network composed by $$r$$ sensors eq. $$(5)$$

Communication network modeld with directed graph by $$G=(V, E)$$ with the nodes $$V=\{1,2,...r\}$$ and edges $$E \subset V \times V$$ representing the communication links. The existence of edge $$(i,j)$$ means the $$i$$th sensor receives data from the $$j$$th sensor.

To design scheduling strategy for sensors to decide when to communicate with its adjacent senors, we use binary value $$\gamma_{ij}(k)$$. If $$\gamma_{ij}(k) = 1$$, the $$j$$th sensor sends data to the $$i$$th sensor at time step $$k$$. otherwise, it is vice versa.

Find set of $$\gamma_{ij}(k)$$.

Such that

Minimize objective function: $$tr(P(k)), \\ where \ P(k) = E[e(k)e(k)^T|y^i(k),\gamma_{ij}(k), j \in adj_i], e(k) = [e^1(k)^T, e^2(k)^T, \dots, e^r(k)^T]^T, \\ e^i(k) = E[x(k)-\hat{x}^i(k)|y^i(k),\gamma_{ij}(k), j \in adj_i]$$

Considering the following distributed state estimator at the $$i$$th sensor:

$$\hat{x}^i(k+1) = A\hat{x}^i(k)+K^i(k)[y^i(k)-C^i\hat{x}^i(k)]-A\sum_{j \in adj_i} \gamma_{ij}(k)[\hat{x}^i(k)-\hat{x}^j(k)]$$

Power constraints : $$\sum_{j \in adj_i} h_{ji}\gamma_{ij}(k) \leq \delta_i$$ weights of energy resource for sensors $$h_{ji}$$, resource limit $$\delta_i$$