On the Complexity and Approximability of Optimal Sensor Selection and Attack for Kalman Filtering

03/24/2020 ∙ by Lintao Ye, et al. ∙ Washington State University 0

Given a linear dynamical system affected by stochastic noise, we consider the problem of selecting an optimal set of sensors (at design-time) to minimize the trace of the steady state a priori or a posteriori error covariance of the Kalman filter, subject to certain selection budget constraints. We show the fundamental result that there is no polynomial-time constant-factor approximation algorithm for this problem. This contrasts with other classes of sensor selection problems studied in the literature, which typically pursue constant-factor approximations by leveraging greedy algorithms and submodularity (or supermodularity) of the cost function. Here, we provide a specific example showing that greedy algorithms can perform arbitrarily poorly for the problem of design-time sensor selection for Kalman filtering. We then study the problem of attacking (i.e., removing) a set of installed sensors, under predefined attack budget constraints, to maximize the trace of the steady state a priori or a posteriori error covariance of the Kalman filter. Again, we show that there is no polynomial-time constant-factor approximation algorithm for this problem, and show specifically that greedy algorithms can perform arbitrarily poorly.

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I Introduction

In large-scale control system design, the number of sensors or actuators that can be installed is typically limited by a design budget constraint. Moreover, system designers often need to select among a set of possible sensors and actuators, with varying qualities and costs. Consequently, a key problem is to select an appropriate set of sensors or actuators in order to achieve certain objectives. This problem has recently received much attention from researchers (e.g., [31, 23, 16, 12, 30, 27, 24, 15, 39]

). One specific instance of this problem arises in the context of linear Gauss-Markov systems, where the corresponding Kalman filter (with the chosen sensors) is used to estimate the states of the systems (e.g.,

[19], [4]). The problem then becomes how to select sensors dynamically (at run-time) or statically (at design-time) to minimize certain metrics of the corresponding Kalman filter. The former scenario is known as the sensor scheduling problem, where different sets of sensors can be chosen at different time steps (e.g., [32, 10, 11]). The latter scenario is known as the design-time sensor selection problem, where the set of sensors is chosen a priori and is not allowed to change over time (e.g., [7, 35, 38]).

Since these problems are NP-hard in general (e.g., [40]), approximation algorithms that provide solutions within a certain factor of the optimal are then proposed to tackle them. Among these approximation algorithms, greedy algorithms have been widely used (e.g, [13], [26]), since such algorithms have provable performance guarantees if the cost function is submodular or supermodular (e.g., [22], [6]).

Additionally, in many applications, the sensors that have been selected and installed on the system are susceptible to a variety of potential attacks. For instance, an adversary (attacker) can inject false data to corrupt the state estimation, which is known as the false data injection attack (e.g., [17, 20, 36]). Another type of attack is the Denial-of-Service (DoS) attack, where an attacker tries to diminish or eliminate a network’s capacity to achieve its expected objective [33], including, for example, wireless jamming (e.g., [34], [41]) and memory exhaustion through flooding (e.g., [25]). One class of DoS attacks corresponds to removing a set of installed sensors from the system, i.e., the measurements of the attacked sensors are not used. This was also studied in [14] and [28], and will be the type of attack we consider in this work.

In this paper, we consider the sensor selection problem and the sensor attack problem for Kalman filtering of discrete-time linear dynamical systems. First, we study the problem of choosing a subset of sensors to install (under given selection budget constraints) to minimize the trace of either the steady state a priori or a posteriori error covariance of the Kalman filter. We refer to these problems as the priori and posteriori Kalman filtering sensor selection (KFSS) problems, respectively. Second, we study the problem of attacking the installed sensors (by removing a subset of them, under given attack budget constraints) to maximize the trace of either the steady state a priori or a posteriori error covariance of the Kalman filter associated with the surviving sensors. These problems are denoted as the priori and posteriori Kalman filtering sensor attack (KFSA) problems, respectively.

Contributions

Our contributions are as follows. First, we show that for the priori and posteriori KFSS problems, there are no polynomial-time constant-factor approximation algorithms for these problems (unless P NP) even for the special case when the system is stable and all sensors have the same cost. In other words, there are no polynomial-time algorithms that can find a sensor selection that is always guaranteed to yield a mean square estimation error (MSEE) that is within any constant finite factor of the MSEE for the optimal selection. More importantly, our result stands in stark contrast to other sensor selection problems studied in the literature, which leveraged submodularity of their associated cost functions to provide greedy algorithms with constant-factor approximation ratios (e.g., [13], [29]). Second, we show that the same results hold for the priori and posteriori KFSA problems. Our inapproximability results directly imply that greedy algorithms cannot provide constant-factor guarantees for our problems. Our third contribution is to explicitly show how greedy algorithms can provide arbitrarily poor performance even for very small instances (with only three states) of the priori and posteriori KFSS (resp., KFSA) problems.

A portion of the results pertaining to only the priori KFSS problem appears in the conference paper [37].

Related work

The authors in [4] and [29] studied the design-time sensor selection problem for discrete-time linear time-varying systems over a finite time horizon. The objective is to minimize the estimation error with a cardinality constraint on the chosen sensors (or alternatively, minimize the number of chosen sensors while guaranteeing a certain level of performance in terms of the estimation error). The authors analyzed the performance of greedy algorithms for this problem. However, their results cannot be directly applied to the problems that we consider here, since we aim to optimize the steady state estimation error.

The papers [35] and [40] considered the same design-time sensor selection as the one we consider here. In [35], the authors expressed the problem as a semidefinite program (SDP). However, they did not provide theoretical guarantees on the performance of the proposed algorithm. The paper [40] showed that the problem is NP-hard and provided upper bounds on the performance of any algorithm for the problem; these upper bounds were functions of the system matrices. Although [40] showed via simulations that greedy algorithms performed well for several randomly generated systems, the question of whether such algorithms (or other polynomial-time algorithms) could provide constant-factor approximation ratios for the problem was left open. We resolve this question in this paper by showing that there does not exist any polynomial-time constant-factor approximation algorithm for this problem.

In [14], the authors studied the problem of attacking a given observation selection in Gaussian process regression [5] to maximize the posteriorivariance of the predictor variable. It was shown that this problem is NP-hard. Moreover, they also gave an instance of this problem such that a greedy algorithm for finding an optimal attack will perform arbitrarily poorly. In [18], the authors considered the scenario where the attacker can target a different set of sensors at each time step to maximize certain metrics of the error covariance of the Kalman filter at the final time step. Some suboptimal algorithms were provided with simulation results. Different from [14] and [18], we study the problem where the attacker removes a set of installed sensors to maximize the trace of the steady state error covariance of the Kalman filter associated with the surviving sensors, and provide fundamental limitations on achievable performance by any possible algorithm for this problem.

Notation and Terminology

The sets of integers and real numbers are denoted as and , respectively. The set of integers that are greater than (resp., greater than or equal to) is denoted as (resp., ). Similarly, we use the notations and . For any , let be the least integer greater than or equal to . For a square matrix , let , , and be its transpose, rank, rowspace and trace, respectively. We use (or ) to denote the element in the th row and th column of . A diagonal matrix is denoted as . The set of by positive definite (resp., positive semi-definite) matrices is denoted as (resp.,

). The identity matrix with dimension

is denoted as

. The zero matrix with dimension

is denoted as . In a matrix, let

denote elements of the matrix that are of no interest. For a vector

, let be the th element of and define the support of to be . The Euclidean norm of is denoted by . Define to be a row vector where the th element is and all the other elements are zero; the dimension of the vector can be inferred from the context. Define to be a column vector of dimension with all the elements equal to 1. The set of - indicator vectors of dimension is denoted as

. For a random variable

, let be its expectation. For a set , let be its cardinality.

Ii Problem Formulation

Consider the discrete-time linear system

(1)

where is the system state, is a zero-mean white Gaussian noise process with for all , and is the system dynamics matrix. The initial state is assumed to be a Gaussian random vector with mean and covariance .

Consider a set that contains sensors. Each sensor provides a measurement of the system of the form

(2)

where is the measurement matrix for sensor , and is a zero-mean white Gaussian noise process. We further define , and . Thus, the output provided by all sensors together is given by

(3)

where and . Denote . We assume that the system noise and the measurement noise are uncorrelated, i.e., , , and is independent of and , .

Ii-a The Sensor Selection Problem

Consider the scenario where there are no sensors initially deployed on the system. Instead, the system designer must select a subset of sensors from to install. Each sensor has a cost ; define the cost vector . The designer has a budget that can be spent on choosing sensors from .

After a set of sensors is selected and installed, the Kalman filter is applied to provide an optimal estimate of the states using the measurements from the installed sensors (in the sense of minimizing the MSEE). Define a vector as the indicator vector of the selected sensors, where if and only if sensor is installed. Denote as the measurement matrix of the installed sensors indicated by , i.e., , where . Similarly, denote as the measurement noise covariance matrix of the installed sensors, i.e., , where . Let (resp., ) denote the a priori (resp., a posteriori) error covariance matrix of the Kalman filter at time step , when the sensors indicated by are installed. Take the initial covariance , . We will use the following result [1].

Lemma 1

Suppose that the pair is stabilizable. For a given indicator vector , (resp., ) will converge to a finite limit (resp., ), which does not depend on the initial covariance , as if and only if the pair is detectable.

The limit satisfies the discrete algebraic Riccati equation (DARE) [1]:

(4)

The limits and are coupled as

(5)

The limit of the a posteriori error covariance matrix satisfies the following equation [3]:

(6)

Note that we can either obtain from using Eq. (6) or by substituting Eq. (5) into Eq. (6) and solving for . The inverses in Eq. (4) and Eq. (6) are interpreted as pseudo-inverses if the arguments are not invertible.

For the case when the pair is not detectable, we define and . Moreover, for any sensor selection , we note from Lemma 1 that the limit (resp., ), if it exists, does not depend on or . Thus, we can assume without loss of generality that and in the sequel. The priori and posteriori Kalman filtering sensor selection (KFSS) problems are then defined as follows.

Problem 1

(Priori and Posteriori KFSS Problems). Given a system dynamics matrix , a measurement matrix containing all of the individual sensor measurement matrices, a system noise covariance matrix , a sensor noise covariance matrix , a cost vector and a budget , the priori Kalman filtering sensor selection problem is to find the sensor selection , i.e., the indicator vector of the selected sensors, that solves

where is given by Eq. if the pair is detectable, and , if otherwise. Similarly, the posteriori Kalman filtering sensor selection problem is to find the sensor selection that solves

where is given by Eq. if the pair is detectable, and , if otherwise.

Ii-B The Sensor Attack Problem

Now consider the scenario where the set of sensors has already been installed on the system. An adversary desires to attack a subset of sensors (i.e., remove a subset of sensors from the system), where each sensor has an attack cost ; define the cost vector . We assume that the adversary has a budget , which is the total cost that can be spent on removing sensors from .

After a subset of sensors are attacked (i.e., removed), the Kalman filter is then applied to estimate the states using the measurements from the surviving sensors. We define a vector as the indicator vector of the attacked sensors, where if and only if sensor is attacked. Hence, the set of sensors that survive is . Define as the vector such that , i.e., if and only if sensor survives. Similarly to the sensor selection problem, we denote and as the measurement matrix and the measurement noise covariance matrix, respectively, corresponding to . Furthermore, let and denote the a priori error covariance matrix and the a posteriori error covariance matrix of the Kalman filter at time step , respectively. Denote and if the limits exist, according to Lemma 1. Note that Eq. (4)-(6) also hold if we substitute with .

For the case when the pair is not detectable, we define and . Recall that we have assumed without loss of generality that and . The priori and posteriori Kalman filtering sensor attack (KFSA) problems are defined as follows.

Problem 2

(Priori and Posteriori KFSA Problems). Given a system dynamics matrix , a measurement matrix , a system noise covariance matrix , a sensor noise covariance matrix , a cost vector and a budget , the priori Kalman filtering sensor attack problem is to find the sensor attack , i.e., the indicator vector of the attacked sensors, that solves

where is given by Eq. if the pair is detectable, and , if otherwise. Similarly, the posteriori Kalman filtering sensor attack problem is to find the sensor attack that solves

where is given by Eq. if the pair is detectable, and , if otherwise.

Note that although we focus on the optimal sensor selection and attack problems for Kalman filtering, due to the duality between the Kalman filter and the Linear Quadratic Regulator (LQR) [2], all of the analysis in this paper will also apply if the priori KFSS and KFSA problems are rephrased as optimal actuator selection and attack problems for LQR, respectively. We omit the details of the rephrasing in the interest of space.

Remark 1

Our goal in this paper is to show that for the priori and posteriori KFSS problems and the priori and posteriori KFSA problems, the optimal solutions cannot be approximated within any constant factor in polynomial time. To do this, it is sufficient for us to consider the special case when , , i.e., each sensor provides a scalar measurement. Moreover, the sensor selection cost vector and the sensor attack cost vector are considered to be and , respectively, i.e., the selection cost and the attack cost of each sensor are both equal to . By showing that the problems are inapproximable even for these special subclasses, we obtain that the general versions of the problems are inapproximable as well.

Iii Inapproximability of the KFSS and KFSA problems

In this section, we analyze the approximability of the KFSS and KFSA problems. We will start with a brief overview of some relevant concepts from the field of computational complexity, and then provide some preliminary lemmas that we will use in proving our results. That will lead into our characterizations of the complexity of KFSS and KFSA.

Iii-a Review of Complexity Theory

We first review the following fundamental concepts from complexity theory [8].

Definition 1

A polynomial-time algorithm for a problem is an algorithm that returns a solution to the problem in a polynomial (in the size of the problem) number of computations.

Definition 2

A decision problem is a problem whose answer is “yes” or “no”. The set P contains those decision problems that can be solved by a polynomial-time algorithm. The set NP contains those decision problems whose “yes” answers can be verified using a polynomial-time algorithm.

Definition 3

An optimization problem is a problem whose objective is to maximize or minimize a certain quantity, possibly subject to constraints.

Definition 4

A problem is NP-complete if (a) NP and (b) for any problem in NP, there exists a polynomial-time algorithm that converts (or “reduces”) any instance of to an instance of such that the answer to the constructed instance of provides the answer to the instance of . is NP-hard if it satisfies (b), but not necessarily (a).

The above definition indicates that if one had a polynomial-time algorithm for an NP-complete (or NP-hard) problem, then one could solve every problem in NP in polynomial time. Specifically, suppose we had a polynomial-time algorithm to solve an NP-hard problem . Then, given any problem in NP, one could first reduce any instance of to an instance of in polynomial time (such that the answer to the constructed instance of provides the answer to the given instance of ), and then use the polynomial-time algorithm for to obtain the answer to .

The above discussion also reveals that to show that a given problem is NP-hard, one simply needs to show that any instance of some other NP-hard (or NP-complete) problem can be reduced to an instance of in polynomial time (in such a way that the answer to the constructed instance of provides the answer to the given instance of ). For then, an algorithm for can be used to solve , and hence, to solve all problems in NP (by NP-hardness of ).

The following is a fundamental result in computational complexity theory [8].

Lemma 2

If P NP, there is no polynomial-time algorithm for any NP-complete (or NP-hard) problem.

For NP-hard optimization problems, polynomial-time approximation algorithms are of particular interest. A constant-factor approximation algorithm is defined as follows.

Definition 5

A constant-factor approximation algorithm for an optimization problem is an algorithm that always returns a solution within a constant (system-independent) factor of the optimal solution.

We will discuss the notion of a constant-factor approximation algorithm in greater depth later in this section. As described in the Introduction, the KFSS problem was shown to be NP-hard in [40] for two classes of systems and sensor costs. First, when the matrix is unstable, it was shown in [23] that the problem of selecting a subset of sensors to make the system detectable is NP-hard, which implies that KFSS is NP-hard using Lemma 1 as shown in [40]. Second, when the matrix is stable (so that all sensor selections cause the system to be detectable), [40] showed that when the sensor selection costs can be arbitrary, the knapsack problem can be encoded as a special case of the KFSS problem, thereby again showing NP-hardness of the latter problem.

In this paper, we will show that the hardness of KFSS (resp., KFSA) does not solely come from selecting (resp., attacking) sensors to make the system detectable (resp., undetectable) or the sensor selection (resp., attack) costs. To do this, we will show a stronger result that there is no polynomial-time constant-factor approximation algorithm for KFSS (resp., KFSA) even when the corresponding system dynamics matrix is stable (which guarantees the detectability of the system), and all the sensors have the same selection (resp., attack) cost. Specifically, we consider a known NP-complete problem, and show how to reduce it to certain instances of KFSS (resp., KFSA) with stable matrices in polynomial time such that hypothetical polynomial-time constant-factor approximation algorithms for the latter problems can be used to solve the known NP-complete problem. Since we know from Lemma 2 that if P NP, there does not exist a polynomial-time algorithm for any NP-complete problem, we conclude that if P NP, there is no polynomial-time constant-factor approximation algorithm for KFSS (resp., KFSA), which directly implies that the KFSS (resp., KFSA) problem is NP-hard even under the extra conditions as described above. We emphasize that our results do not imply that there is no polynomial-time constant-factor approximation algorithm for specific instances of KFSS (resp., KFSA). Rather, the result is that we cannot have such an algorithm for all instances of KFSS (resp., KFSA).

Iii-B Preliminary Results

The following results characterize properties of the KFSS and KFSA instances that we will consider when proving the inapproximability of the KFSS and KFSA problems. The proofs are provided in Appendix A.

Lemma 3

Consider a discrete-time linear system as defined in and . Suppose the system dynamics matrix is of the form with , , the system noise covariance matrix is diagonal, and the sensor noise covariance matrix . Then, the following hold for all sensor selections .

(a) For all , and satisfy

(7)

and

(8)

respectively.

(b) If such that and the th column of is zero, then .

(c) If and such that , then and .

Lemma 4

Consider a discrete-time linear system as defined in Eq. and Eq. . Suppose the system dynamics matrix is of the form , where , and the system noise covariance matrix is .

(a) Suppose the measurement matrix is of the form with sensor noise variance , where and . Then, the MSEE of state , denoted as , satisfies

(9)

where .

(b) Suppose the measurement matrix is of the form with sensor noise covariance , where . Then, the MSEE of state , denoted as , satisfies

(10)

where .

Moreover, if we view and as functions of and , denoted as and , respectively, then and are strictly increasing functions of and , with and , respectively.

Iii-C Inapproximability of the KFSS Problem

In this section, we characterize the achievable performance of algorithms for the priori and posteriori KFSS problems. For any given algorithm (resp., ) of the priori (resp., posteriori) KFSS problem, we define the following ratios:

(11)

and

(12)

where (resp., ) is the optimal solution to the priori (resp., posteriori) KFSS problem and (resp., ) is the solution to the priori (resp., posteriori) KFSS problem given by algorithm (resp., ).

In [40], the authors showed that there is an upper bound for (resp., ) for any sensor selection algorithm (resp., ), in terms of the system matrices. However, the question of whether it is possible to find an algorithm (resp., ) that is guaranteed to provide an approximation ratio (resp., ) that is independent of the system parameters has remained open up to this point. In particular, it is desirable to find polynomial-time constant-factor approximation algorithms for the priori (resp., posteriori) KFSS problem, where the ratio (resp., ) is upper-bounded by some (system-independent) constant.111Polynomial-time constant-factor approximation algorithms have been widely studied for NP-hard problems, e.g.,[8]. We provide a negative result by showing that there are no polynomial-time algorithms that can always yield a solution that is within any constant factor of the optimal (unless P NP), i.e., for all polynomial-time algorithms (resp., ) and , there are instances of the priori (resp., posteriori) KFSS problem where (resp., ).

Remark 2

Note that the “constant” in “constant-factor approximation algorithm” refers to the fact that the cost of the solution provided by the algorithm is upper-bounded by some (system-independent) constant times the cost of the optimal solution. The algorithm can, however, use the system parameters when finding the solution. For example, an optimal algorithm for the KFSS problem will be a -factor approximation, and would use the system matrices, sensor costs, and budget to find the optimal solution. Similarly, a polynomial-time -factor approximation algorithm for KFSS would use the system parameters to produce a solution whose cost is guaranteed to be no more than times the cost of the optimal solution. As indicated above, we will show that no such algorithm exists for any constant (unless P NP).

To show the inapproximability of the priori KFSS problem, we relate it to the EXACT COVER BY 3-SETS problem described below [8].

Definition 6

Given a finite set and a collection of -element subsets of , an exact cover for is a subcollection such that every element of occurs in exactly one member of .

We will use the following result [8].

Lemma 5

Given a finite set and a collection of -element subsets of , the problem of determining whether contains an exact cover for is NP-complete.

As argued in Remark 1, in order to show that the priori KFSS problem cannot be approximated within any constant factor in polynomial time, it is sufficient for us to show that certain special instances of this problem are inapproximable. Specifically, consider any instance of the problem. Using the results in Lemma 3-4, we will first construct an instance of the priori KFSS problem in polynomial time such that the difference between the solution to KFSS when the answer to is “yes” and the solution to KFSS when the answer to is “no” is large enough. Thus, we can then apply any hypothetical polynomial-time constant-factor approximation algorithm for the priori KFSS problem to the constructed priori KFSS instance and obtain the answer to the instance. Since we know from Lemma 5 that the problem is NP-complete, we obtain from Lemma 2 the following result; the detailed proof is provided in Appendix B.

Theorem 1

If P NP, then there is no polynomial-time constant-factor approximation algorithm for the priori KFSS problem.

The following result is a direct consequence of the above arguments; the proof is also provided in Appendix B.

Corollary 1

If P NP, then there is no polynomial-time constant-factor approximation algorithm for the posteriori KFSS problem.

Iii-D Inapproximability of the KFSA Problem

In this section, we analyze the achievable performance of algorithms for the priori and posteriori KFSA problems. For any given algorithm (resp., ) for the priori (resp., posteriori) KFSA problem, we define the following ratios:

(13)

and

(14)

where (resp., ) is the optimal solution to the priori (resp., posteriori) KFSA problem and (resp., ) is the solution to the priori (resp., posteriori) KFSA problem given by algorithm (resp., ). It is worth noting that using the arguments in [40], the same (system-dependent) upper bounds for and can be obtained as those for and in [40], respectively. Nevertheless, we show that (if P NP) there is again no polynomial-time constant-factor approximation algorithm for the priori (resp., posteriori) KFSA problem, i.e., for all and polynomial-time algorithms (resp., ), there are instances of the priori (resp., posteriori) KFSA problem where (resp., ). To establish this result, we relate the priori KFSA problem to the problem as described in Definition 6 and Lemma 5. Similarly to the proof of Theorem 1, given any instance of , we will construct an instance of the priori KFSA problem and show that any hypothetical polynomial-time constant-factor approximation algorithm for the priori KFSA problem can be used to solve the problem. This leads to the following result; the detailed proof is provided in Appendix C.

Theorem 2

If PNP, then there is no polynomial-time constant-factor approximation algorithm for the priori KFSA problem.

The arguments above also imply the following result whose proof is provided in Appendix C.

Corollary 2

If P NP, then there is no polynomial-time constant-factor approximation algorithm for the posteriori KFSA problem.

Iv Failure of Greedy Algorithms

Our results in Theorem 1 and Theorem 2 indicate that no polynomial-time algorithm can be guaranteed to yield a solution that is within any constant factor of the optimal solution to the priori (resp., posteriori) KFSS and KFSA problems. In particular, these results apply to the greedy algorithms that are often studied for sensor selection in the literature (e.g., [40], [14]), where sensors are iteratively selected (resp., attacked) in order to produce the greatest decrease (resp., increase) in the error covariance at each iteration. In this section we will focus on such greedy algorithms for the priori (resp., posteriori) KFSS and KFSA problems, and show explicitly how these greedy algorithms can fail to provide good solutions even for small and fixed instances with only three states; this provides additional insight into the factors that cause the KFSS and KFSA problems to be challenging.

Iv-a Failure of Greedy Algorithms for the KFSS Problem

It was shown via simulations in [40] that greedy algorithms for KFSS work well in practice (e.g., for randomly generated systems). In this section, we provide an explicit example showing that greedy algorithms for the priori and posteriori KFSS problems can perform arbitrarily poorly, even for small systems (containing only three states). We consider the greedy algorithm for the priori (resp., posteriori) KFSS problem given in Algorithm 1, for instances where all sensors have selection costs equal to , and the sensor selection budget (i.e., up to sensors can be chosen). For any such instance of the priori (resp., posteriori) KFSS problem, define (resp., ), where (resp., ) is the solution of Eq. (4) (resp., Eq. (6)) corresponding to the sensors selected by Algorithm 1.

Input: An instance of priori (resp., posteriori) KFSS
Output: A set of selected sensors

Algorithm 1 Greedy Algorithm for Problem 1
1:,
2:for  do
3:      (resp., )
4:     ,
5:end for
Example 1

Consider an instance of the priori (resp., posteriori) KFSS problem with matrices and , and , defined as

where , , and . In addition, we have the selection budget , the cost vector and the set of candidate sensors , where sensor corresponds to the th row of matrix , for .

Based on the system defined in Example 1, we have the following result whose proof is provided in Appendix D.

Theorem 3

For the instance of the priori (resp., posteriori) KFSS problem defined in Example 1, the ratios and satisfy

(15)

and

(16)

respectively.

Examining Eq. (15) (resp., Eq. (16)), we see that for the given instance of the priori (resp., posteriori) KFSS problem, we have (resp., ) as and . Thus, (resp., ) can be made arbitrarily large by choosing the parameters in the instance appropriately. To explain the result in Theorem 3

, we first note that the only nonzero eigenvalue of the diagonal

defined in Example 1 is , and so we know from Lemma that state and state of the system as defined in Example 1 each contribute at most to (resp., ) for all . Hence, in order to minimize (resp., ), we need to minimize the MSEE of state . Moreover, the measurements of state and state can be viewed as measurement noise that corrupts the measurements of state . It is then easy to observe from the form of matrix defined in Example 1 that sensor is the single best sensor among the three sensors since it provides measurements of state with less noise than sensor (and sensor does not measure state at all). Thus, the greedy algorithm for the priori (resp., posteriori) KFSS problem defined as Algorithm 1 selects sensor in its first iteration. Nonetheless, we notice from defined in Example 1 that the optimal set of two sensors that minimizes (resp., ) contains sensor and sensor , which together give us exact measurements (without measurement noise) on state (after some elementary row operations). Since the greedy algorithm selects sensor in its first iteration, no matter which sensor it selects in its second iteration, the two chosen sensors can only give a noisy measurement of state (if we view the measurements of state and state as measurement noise), and the variance of the measurement noise can be made arbitrary large if we take in defined in Example 1. Hence, the greedy algorithm fails to perform well due to its myopic choice in the first iteration.

It is also useful to note that the above behavior holds for any algorithm that outputs a sensor selection that contains sensor for the above example.

Iv-B Failure of Greedy Algorithms for the KFSA Problem

In [14], the authors showed that a simple greedy algorithm can perform arbitrarily poorly for an instance of the observation attack problem in Gaussian process regression. Here, we consider a simple greedy algorithm for the priori (resp., posteriori) KFSA problem given in Algorithm 2, for instances where all sensors have an attack cost of , and the sensor attack budget (i.e., up to sensors can be attacked). For any such instance of the priori (resp., posteriori) KFSA problem, define (resp., ), where (resp., ) is the solution to the priori (resp., posteriori) KFSA problem given by Algorithm 2. We then show that Algorithm 2 can perform arbitrarily poorly for a simple instance of the priori (resp., posteriori) KFSA problem as described below.

Input: An instance of priori (resp., posteriori) KFSA
Output: A set of targeted sensors

Algorithm 2 Greedy Algorithm for Problem 2
1:,
2:for  do
3:      (resp., )
4:     ,
5:end for
Example 2

Consider an instance of the priori (resp., posteriori) KFSA problem with matrices , , and , defined as

where , and . In addition, the attack budget is , the cost vector is , and the set of sensors has already been installed on the system, where sensor corresponds to the th row of matrix , for .

We then have the following result, whose proof is provided in Appendix D.

Theorem 4

For the instance of the priori (resp., posteriori) KFSA problem defined in Example 2, the ratios and satisfy

(17)

and

(18)

respectively.

Inspecting Eq. (17) (resp., Eq. (18)), we observe that for the given instance of the priori (resp., posteriori) KFSA problem, we have (resp., ) as and . Thus, (resp., ) can be made arbitrarily large by choosing the parameters in the instance appropriately. Here, we explain the results in Theorem 4 as follows. Using similar arguments to those before, we know from the structure of matrix defined in Example 2 that in order to maximize (resp., ), we need to maximize the MSEE of state , i.e., make the measurements of state “worse”. Again, the measurements of state and state can be viewed as measurement noise that corrupts the measurements of state . No matter which of sensor , sensor , or sensor is attacked, the resulting measurement matrix is full column rank, which yields an exact measurement of state . We also observe that if sensor is targeted, the surviving sensors can only provide measurements of state that are corrupted by measurements of states and state . Hence, the greedy algorithm for the priori (resp., posteriori) KFSA problem defined as Algorithm 2 targets sensor in its first iteration, since it is the single best sensor to attack from the four sensors. Nevertheless, sensor and sensor form the optimal set of sensors to be attacked to maximize (resp., ), since the surviving sensors provide no measurement of state . Since the greedy algorithm targets sensor in its first iteration, no matter which sensor it targets in the second step, the surviving sensors can always provide some measurements of state with noise (if we view the measurements of state and state as measurement noise), and the variance of the noise will vanish if we take in matrix defined in Example 2. Hence, the myopic behavior of the greedy algorithm makes it perform poorly.

Furthermore, it is useful to note that the above result holds for any algorithm that outputs a sensor attack that does not contain sensor or sensor for the above example.

Remark 3

Using similar arguments to those in the proof of Theorem 3 (resp., Theorem 4), we can also show that when we set (resp., ), where , the results in Eq. (15)-(16) (resp., Eq. (17)-(18)) hold if we let . This phenomenon is also observed in [4], where the approximation guarantees for the greedy algorithms provided in that paper get worse as the sensor measurement noise tends to zero.

V Conclusions

In this paper, we studied sensor selection and attack problems for (steady state) Kalman filtering of linear dynamical systems. We showed that these problems are NP-hard and have no polynomial-time constant-factor approximation algorithms, even under the assumption that the system is stable and each sensor has identical cost. To illustrate this point, we provided explicit examples showing how greedy algorithms can perform arbitrarily poorly on these problems, even when the system only has three states. Our results shed new insights into the problem of sensor selection and attack for Kalman filtering and show, in particular, that this problem is more difficult than other variants of the sensor selection problem that have submodular (or supermodular) cost functions. Future work on extending the results to Kalman filtering over finite time horizons, characterizing achievable (non-constant) approximation ratios, identifying classes of systems that admit near-optimal approximation algorithms, and investigating resilient sensor selection problems under adversarial settings would be of interest.

Vi Acknowledgments

The authors thank the anonymous reviewers for their insightful comments that helped to improve the paper.

Appendix A

Proof of Lemma 3:

Since and are diagonal, the system represents a set of scalar subsystems of the form

where is the th state of and is a zero-mean white Gaussian noise process with variance . As is stable, the pair is detectable and the pair is stabilizable for all sensor selections . Thus, the limits and exist for all and for all (based on Lemma 1), and are denoted as and , respectively.

Proof of (a): Since and are diagonal, we know from Eq. that

which implies , . Moreover, it is easy to see that , . Since , we obtain from Eq. that

which implies that since is diagonal. Hence, , . Similarly, we also have , and we obtain from Eq. (6) that

Thus, , .

Proof of (b): Assume without loss of generality that the first column of is zero, since we can simply renumber the states to make this the case without affecting the trace of the error covariance matrix. We then have of the form

Moreover, since and are diagonal, we obtain from Eq. that is of the form

where and satisfies

which implies . Furthermore, we obtain from Eq. (6) that is of the form

where and satisfies

which implies .

Proof of (c): We assume without loss of generality that . If we further perform elementary row operations on , which does not change the solution to Eq. (4) (resp., Eq. (6)), we obtain a measurement matrix