On the Optimality of Linear Signaling to Deceive Kalman Filters over Finite/Infinite Horizons

Abstract

In this paper, we address the problem of obtaining optimal deceptive signaling strategies between two agents, a sender and a receiver, over an ideal channel. Different from classical (cooperative) communication settings, here, the agents select their strategies under two different cost measures. For the case when these costs are quadratic, we analyze the Stackelberg equilibrium, where the sender leads the game by committing his/her strategies beforehand. This is an infinite-dimensional optimization problem, where the sender needs to anticipate the receiver’s reaction while selecting his/her policy within the general class of stochastic kernels. The specific model we adopt for the underlying information of interest is a discrete-time Markov process generated by a vector-valued linear dynamical system, and at each instant, the information is a realization of a square integrable multivariate random vector. Over both finite and infinite horizons, we show the optimality of memoryless, “linear” signaling rules when the receiver uses a Kalman filter to estimate its information of interest. We develop algorithms that deliver the optimal signaling strategies. Numerical analysis shows that the performance of the sender degrades slightly when the receiver uses the best nonlinear estimator even when the information of interest is a Rademacher random variable rather than Gaussian.

M.O. Sayin—This research was supported in part by the U.S. Office of Naval Research (ONR) MURI grant N00014-16-1-2710, and in part by the U.S. Army Research Labs (ARL) under IoBT Grant 479432-239012-191100.

A Proof of Lemma 1

In the following, we show each property one by one: Property (i) follows since $\mathcal {M}_{m_a}(\pmb {b},\pmb {c}) = \mathcal {M}_{m_a}(\pmb {\tilde{b}}^{\perp },\pmb {c})$. Property (ii) follows since

$$\begin{aligned} \mathbb {E}\{\pmb {a}\mathcal {E}(\pmb {a}\,|\,\pmb {c})'\}&= \mathbb {E}\{\pmb {a} (\mathbb {E}\{\pmb {a}\pmb {c}'\}\mathbb {E}\{\pmb {c}\pmb {c}'\}^{\dagger }\pmb {c})'\}= \mathbb {E}\{\pmb {a}\pmb {c}'\}\mathbb {E}\{\pmb {c}\pmb {c}'\}^{\dagger } \mathbb {E}\{\pmb {a}\pmb {c}'\}'. \end{aligned}$$

(66)

Property (iii) follows since, by Property (i), we have

$$\begin{aligned} \mathbb {E}\{\mathcal {E}(\pmb {a}\,|\,\pmb {b},\pmb {c})\mathcal {E}(\pmb {a}\,|\,\pmb {c})'\} = \mathbb {E}\{\mathcal {E}(\pmb {a}\,|\,\pmb {\tilde{b}}^{\perp },\pmb {c})\mathcal {E}(\pmb {a}\,|\,\pmb {c})'\}. \end{aligned}$$

(67)

By taking a closer look at the right-hand-side, we obtain

$$\begin{aligned} \begin{bmatrix} \mathbb {E}\{\pmb {a}\pmb {c}'\} \\ \mathbb {E}\{\pmb {a}\pmb {\tilde{b}}^{\perp }\}\end{bmatrix}'\begin{bmatrix} \mathrm {cov}\{\pmb {c}\} &{} \\ &{} \mathrm {cov}\{\pmb {\tilde{b}}^{\perp }\}\end{bmatrix}^{\dagger }\begin{bmatrix} \mathbb {E}\{\pmb {c}\pmb {c}'\} \\ \mathbb {E}\{\pmb {\tilde{b}}^{\perp }\pmb {c}'\} \end{bmatrix}\mathrm {cov}\{\pmb {c}\}^{\dagger }\mathbb {E}\{\pmb {a}\pmb {c}'\}'.\nonumber \end{aligned}$$

Since $\mathbb {E}\{\pmb {\tilde{b}}^{\perp }\pmb {c}' \} = O_{m_a\times m_c}$, it is equivalent to

$$\begin{aligned} \mathbb {E}\{\pmb {a}\pmb {c}'\}\mathrm {cov}\{\pmb {c}\}^{\dagger }\mathrm {cov}\{\pmb {c}\}\mathrm {cov}\{\pmb {c}\}^{\dagger }\mathbb {E}\{\pmb {a}\pmb {c}'\}' = \mathrm {cov}\{\mathcal {E}(\pmb {a}\,|\,\pmb {c})\},\nonumber \end{aligned}$$

which follows since the pseudo inverse of a matrix is a weak inverse for the multiplicative semi-group, i.e., $M^{\dagger }MM^{\dagger } = M^{\dagger }$. Property (iv) follows since $\mathrm {cov}\{\mathcal {E}(\pmb {a}\,|\,\pmb {b},\pmb {c})\}$ is equal to $\mathrm {cov}\{\mathcal {E}(\pmb {a}\,|\,\pmb {\tilde{b}}^{\perp },\pmb {c})\}$. By taking a closer look at the right-hand-side, we obtain

$$\begin{aligned} \begin{bmatrix} \mathbb {E}\{\pmb {a}\pmb {c}'\} \\ \mathbb {E}\{\pmb {a}\pmb {\tilde{b}}^{\perp }\}\end{bmatrix}'\begin{bmatrix} \mathrm {cov}\{\pmb {c}\} &{} \\ &{} \mathrm {cov}\{\pmb {\tilde{b}}^{\perp }\}\end{bmatrix}^{\dagger }\begin{bmatrix} \mathbb {E}\{\pmb {a}\pmb {c}'\} \\ \mathbb {E}\{\pmb {a}\pmb {\tilde{b}}^{\perp }\}\end{bmatrix} = \mathrm {cov}\{\mathcal {E}(\pmb {a}\,|\,\pmb {\tilde{b}}^{\perp })\} + \mathrm {cov}\{\mathcal {E}(\pmb {a}\,|\,\pmb {c})\}. \end{aligned}$$

(68)