A Sparse State Kalman Filter Algorithm Based on Kalman Gain

Abstract

In order to improve tracking accuracy of time-varying sparse signals, a sparse state Kalman filter algorithm based on Kalman gain matrix is proposed in this paper. Under the constraint of sparse state, minimizing the mean square error and solving the optimization problem by using the symmetrical alternating direction multiplier method (symmetrical ADMM), while keeping the update expression of state estimation unchanged, a Kalman gain matrix is obtained which can make the state estimation sparse. In addition, this paper also discusses two different sparse constraints which are L1 norm constraint and cardinality function constraint. The proposed algorithm can be implemented under the framework of conventional Kalman filter algorithm, no need to introduce additional frameworks. Two groups of dynamic signal models, a slow change of signal and a random walk of nonzero elements, are simulated. Simulation results show that the proposed algorithm can improve the tracking accuracy of conventional Kalman filter algorithm for time-varying sparse signal.

Appendix

The solution of $G_{t}$ subproblem:

1. When $g\left( {G_{t} } \right) = \left\| {{\hat{x}}_{t|t - 1} + G_{t} \cdot e_{t} } \right\|_{1}$:

$$ \begin{gathered} \varphi \left( {G_{t,i} } \right) = \beta \left| {\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} } \right| + \frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} \hfill \\ { = }\left\{ \begin{gathered} \frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} + \beta \left( {\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} } \right){,}\quad \hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} \ge 0\quad {\text{a}} \hfill \\ \frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} - \beta \left( {\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} } \right){,}\quad \hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} < 0\quad {\text{b}} \hfill \\ \end{gathered} \right.\quad 1 \le i \le n \hfill \\ \end{gathered} $$

(25)

The optimal solution of piecewise function can be classified and discussed in different regions. Writing a as an optimization problem:

$$ \begin{aligned} & \mathop {\min }\limits_{{G_{t,i} }} f_{1} \left( {G_{t,i} } \right) \triangleq \frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} + \beta \left( {{\hat{x}}_{t|t - 1,i} + G_{t,i} \cdot e_{t} } \right) \\ & {\text{s.t.}}\quad \hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} \ge 0 \\ \end{aligned} $$

(26)

Firstly, solving unconstrained optimization. By optimality conditions:

$$ \nabla f_{1} (G_{t,i} ) = 0 \Rightarrow G_{t,i} = - \frac{\beta }{\rho }e_{t}^{T} + V_{i}^{k} $$

(27)

When $\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} = V_{i}^{k} \cdot e_{t} + \hat{x}_{t|t - 1,i} - \frac{\beta }{\rho }e_{t}^{T} \cdot e_{t} \ge 0$, namely: when $V_{i}^{k} \cdot e_{t} + \hat{x}_{t|t - 1,i} \ge \frac{\beta }{\rho }e_{t}^{T} \cdot e_{t}$:

$$ {\text{ G}}_{t,i} = - \frac{\beta }{\rho }e_{t}^{T} + V_{i}^{k} $$

(28)

When $V_{i}^{k} \cdot e_{t} + \hat{x}_{t|t - 1,i} < \frac{\beta }{\rho }e_{t}^{T} \cdot e_{t}$, The optimal solution appears at boundary $\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} = 0$, This is equivalent to solve a constrained optimization problem. Let its Lagrange function be:

$$ L_{1} \left( {G_{t,i} ,\lambda_{1} } \right) = f_{1} \left( {G_{t,i} } \right) + \lambda_{1} \left( {\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} } \right) $$

(29)

According to the optimality conditions:

$$ \left\{ \begin{gathered} \rho \left( {G_{t,i} - V_{i}^{k} } \right) + \left( {\beta + \lambda_{1} } \right)e_{t}^{T} = 0 \hfill \\ \hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} = 0 \hfill \\ \end{gathered} \right. $$

(30)

Hence:

$$ {\text{ G}}_{t,i} = V_{i}^{k} - \frac{{\left( {\hat{x}_{t|t - 1,i} + V_{i}^{k} \cdot e_{t} } \right) \cdot e_{t}^{T} }}{{e_{t}^{T} \cdot e_{t} }} $$

(31)

In conclusion:

$$ {\text{ G}}_{t,i} = \left\{ \begin{gathered} - \frac{\beta }{\rho }e_{t}^{T} + V_{i}^{k} {,}\quad V_{i}^{k} \cdot e_{t} + \hat{x}_{t|t - 1,i} \ge \frac{\beta }{\rho }e_{t}^{T} \cdot e_{t} \hfill \\ V_{i}^{k} - \frac{{\left( {\hat{x}_{t|t - 1,i} + V_{i}^{k} \cdot e_{t} } \right) \cdot e_{t}^{T} }}{{e_{t}^{T} \cdot e_{t} }}{,}\quad V_{i}^{k} \cdot e_{t} + \hat{x}_{t|t - 1,i} < \frac{\beta }{\rho }e_{t}^{T} \cdot e_{t} \hfill \\ \end{gathered} \right. $$

(32)

Then write b in the form of optimization problem:

$$ \begin{aligned} & \mathop {\min }\limits_{{G_{t,i} }} f_{2} \left( {G_{t,i} } \right) \triangleq \frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} - \beta \left( {\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} } \right) \\ & {\text{s.t.}}\quad \hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} < 0 \\ \end{aligned} $$

(33)

Similarly:

$$ {\text{ G}}_{t,i} = \left\{ \begin{gathered} \frac{\beta }{\rho }e_{t}^{T} + V_{i}^{k} {,}\quad V_{i}^{k} \cdot e_{t} + \hat{x}_{t|t - 1,i} < - \frac{\beta }{\rho }e_{t}^{T} \cdot e_{t} \hfill \\ V_{i}^{k} - \frac{{\left( {\hat{x}_{t|t - 1,i} + V_{i}^{k} \cdot e_{t} } \right) \cdot e_{t}^{T} }}{{e_{t}^{T} \cdot e_{t} }}{,}\quad V_{i}^{k} \cdot e_{t} + \hat{x}_{t|t - 1,i} \ge - \frac{\beta }{\rho }e_{t}^{T} \cdot e_{t} \hfill \\ \end{gathered} \right. $$

(34)

In summary:

$$ G_{t,i}^{k + 1} = \left\{ \begin{gathered} V_{i}^{k} + \frac{\beta }{\rho }e_{t}^{T} {,}\quad \hat{x}_{t|t - 1,i} + V_{i}^{k} \cdot {e}_{t} \in \left( { - \infty , - \frac{\beta }{\rho }e_{t}^{T} \cdot e_{t} } \right) \hfill \\ V_{i}^{k} - \frac{{{\hat{x}}_{t|t - 1,i} e_{t}^{T} + V_{i}^{k} \cdot {e}_{t} \cdot e_{t}^{T} }}{{e_{t}^{T} \cdot e_{t} }}{,}\quad \hat{x}_{t|t - 1,i} + V_{i}^{k} \cdot {e}_{t} \in \left[ { - \frac{\beta }{\rho }e_{t}^{T} \cdot e_{t} ,\frac{\beta }{\rho }e_{t}^{T} \cdot e_{t} } \right] \hfill \\ V_{i}^{k} - \frac{\beta }{\rho }e_{t}^{T} {,}\quad \hat{x}_{t|t - 1,i} + V_{i}^{k} \cdot {e}_{t} \in \left( {\frac{\beta }{\rho }e_{t}^{T} \cdot e_{t} , + \infty } \right) \hfill \\ \end{gathered} \right. $$

(35)

2. When $g\left( {G_{t} } \right) = {\text{card}}\left( {\hat{x}_{t|t - 1} + G_{t} \cdot e_{t} } \right)$:

$$ \begin{aligned} \varphi (G_{t,i} ) & = \beta {\text{card}}\left( {\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} } \right) + \frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} \\ & = \left\{ {\begin{array}{*{20}l} {\frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} {,}} & {\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} = 0} \\ {\frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} + \beta ,} & {\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} \ne 0} \\ \end{array} } \right.\quad 1 \le i \le n \\ \end{aligned} $$

(36)

It can be noted that when $\hat{x}_{t|t - 1,i} + G_{t,i} \cdot e_{t} \ne 0$, $G_{t,i} = V_{i}^{k}$ minimizes the objective function, and the minimum value is $\beta$. When ${\hat{x}}_{t|t - 1,i} + G_{t,i} \cdot e_{t} = 0$, at this point, an equality constrained optimization problem is formed:

$$ \begin{gathered} \mathop {\min }\limits_{{G_{t,i} }} \, \frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} \hfill \\ s.t. \, {\hat{x}}_{t|t - 1,i} + G_{t,i} \cdot e_{t} = 0 \hfill \\ \end{gathered} $$

(37)

Its Lagrange function is:

$$ L_{2} \left( {G_{t,i} ,\lambda_{2} } \right) = \, \frac{\rho }{2}\left\| {G_{t,i} - V_{i}^{k} } \right\|_{2}^{2} + \lambda \left( {{\hat{x}}_{t|t - 1,i} + G_{t,i} \cdot e_{t} } \right) $$

(38)

By optimality conditions:

$$ \left\{ \begin{gathered} \rho \left( {G_{t,i} - V_{i}^{k} } \right) + \lambda_{2} e_{t}^{T} = 0 \, \hfill \\ {\hat{x}}_{t|t - 1,i} + G_{t,i} \cdot e_{t} = 0 \, \hfill \\ \end{gathered} \right. $$

(39)

Hence:

$$ G_{t,i} = V_{i}^{k} - \frac{{\left( {{\hat{x}}_{t|t - 1,i} + V_{i}^{k} \cdot e_{t} } \right) \cdot e_{t}^{T} }}{{e_{t}^{T} \cdot e_{t} }} $$

(40)

In summary:

$$ G_{t,i}^{k + 1} = \left\{ \begin{gathered} V_{i}^{k} - \frac{{\left( {{\hat{x}}_{t|t - 1,i} + V_{i}^{k} \cdot e_{t} } \right) \cdot e_{t}^{T} }}{{e_{t}^{T} \cdot e_{t} }}{, }\frac{\rho }{2}\left\| {\frac{{\left( {{\hat{x}}_{t|t - 1,i} + V_{i}^{k} \cdot e_{t} } \right) \cdot e_{t}^{T} }}{{e_{t}^{T} \cdot e_{t} }}} \right\|_{2}^{2} \le \beta \hfill \\ V_{i}^{k} {, }\frac{\rho }{2}\left\| {\frac{{\left( {{\hat{x}}_{t|t - 1,i} + V_{i}^{k} \cdot e_{t} } \right) \cdot e_{t}^{T} }}{{e_{t}^{T} \cdot e_{t} }}} \right\|_{2}^{2} > \beta \hfill \\ \end{gathered} \right. $$

(41)

Notice that when ${\hat{x}}_{t|t - 1,i} + V_{i}^{k} \cdot e_{t} = 0$, $G_{t,i}^{k + 1} = V_{i}^{k}$.