IEEE SIGNAL PROCESSING LETTERS, VOL. 18, NO. 6, JUNE 2011
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Optimal Filtering Over Uncertain Wireless Communication Channels Xiao Ma, Seddik M. Djouadi, and Charalambos D. Charalambous
Abstract—In this letter, filtering over wireless communication channels subject to packet losses is considered. The packet losses are assumed to follow a Bernoulli distribution. The latter is interpreted as a special case of a Markov process for which hybrid filtering theory is shown to provide an exact solution. Unlike existing works where only the linear optimal state estimate is derived as a linear function of measurements, this paper shows that the optimal state estimate is in fact a nonlinear function of measurements. In addition, the optimal estimator is derived explicitly. Illustrative examples compare the performance of the linear optimal estimator and the optimal estimator, and show that the latter offer superior performance, in particular for unstable systems. Index Terms—Bernoulli, linear optimal estimator, optimal filtering, packet loss.
I. INTRODUCTION
N
ETWORKS are spreading in a lot of modern technologies and extend the range of information communications to large distances besides resulting in significant capital saving. In recent years, the combination of network and control has been a major topic in the control community, especially, in the context where the systems are allocated in different places. Roughly speaking, the study of networking and communications together with control systems is a major direction for modern research. [1]–[6][7]. Several investigations have been conducted on the networks’ impairments, where optimal control and estimation algorithms have been derived under certain situations. In [1] a linear quadratic Gaussian (LQG) controller taking the factors of bounded delays in communication links is proposed. In [2] and [3], the authors consider uncertain observations. In [4] and [6], Kalman filter and LQG controller with known packet loss arrivals are developed. In [5], stability conditions with Markov packet losses are derived. Reference [8] considers an optimal estimator of a packet loss model that assumes there is no packet arrival information between the plant and the estimator. However, only the linear optimal estimator is considered where the state estimate is assumed to be a linear function of
Manuscript received January 30, 2011; revised March 18, 2011; accepted March 20, 2011. Date of publication April 05, 2011; date of current version April 21, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Hsiao-Chun Wu. X. Ma and S. M. Djouadi are with the Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN 37996 USA (e-mail:
[email protected];
[email protected]). C. D. Charalambous is with ECE Department, University of Cyprus, 1678 Nicosia, Cyprus (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2011.2138134
Fig. 1. Estimation over packet loss link.
measurements. In this work, we deal with the optimal filter without any prior assumption on the form of the filter. As discussed later, the problem is formulated such that the measurement is the output of the system subject to packet dropouts and Gaussian noises as represented in Fig. 1. In this case, measurements may be non-Gaussian due to the existence of the Bernoulli packet losses, and the linear optimal estimator is not optimal. To obtain the optimal filter, we rely on a change of the probability measure from the ideal probability space where measurements are Gaussian to the current probability space (probability space of the problem) and derive the probability density of the state. II. OPTIMAL FILTER OVER BERNOULLI PACKET LOSS CHANNELS A. Problem Formulation We start with some mathematical preliminaries. Let be a probability space upon which and are independent Gaussian white sequences with zero mean and unit variance, For convenience, we assume a scalar system, however we will indicate how to extend the result to the general case later on. let be a scalar process, we consider the following linear discrete-time system with stochastic and statistically independent parameters in the output equation and observation noise: (1) where is the state of the system at time , is the measurement received at the estimator; The scalars and are system parameters. is the initial value of processes . Let be the complete filtration generated by and be the complete filtration generated by . is assumed to be a -Markov process: with state-space where are 2-canonical unit vectors, and is uniformly distributed and independent of the other processes. is an -martingale increment and is a 2 2 matrix with and . Here we consider an i.i.d Bernoulli packet loss model, we have the relationship: and the transition probability matrix is
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in this case. In our model, we set when represents the packet loss and ( is a scalar) when represents the packet arrival at that time. Assume is the probability of the packet arrival. The complete filtration generated by is labeled by . The optimal state estimate is defined as the minimizer of the following cost function:
where denotes the expectation with respect to probability measure . Then, define as the unnormalized conditional density and normalized density as follows [9]:
(2)
(7)
is the optimal estimate of the state. where Note that from (1) we see that the measurement is the sum of a Markov process and Gaussian noise which renders measurements not Gaussian. B. Optimal Filter Over Bernoulli Packet Losses Link In this subsection, we rely on hybrid filtering theory [9] (see [11] for continuous time systems) to derive the optimal filter. The main idea is to use a change of measure method, which transforms the filtering problem from the original probability space to a new probability space where the measurement and state are both Gaussian. The optimal filter under the original probability space is then obtained by using conditional Bayes’ theorem. Initially, assume that all processes are defined on a new probability space , and under : 1) is a sequence of independent and identically distributed real Gaussian random variables with density function ; 2) is a sequence of independent and identically distributed real Gaussian random variables with density function ; For , let and define
(6)
Then still from a version of Bayes’ theorem, we have (8) A recurrence relationship is obtained between the unnormalized probability densities and :
(9) Then the unnormalized probability density is computed as follows: Lemma 1: Suppose is Gaussian for , 2. Then is a finite sum of Gaussian densities and
where
(3) (4) Then the process is an -martingale with respect to the filtration , [10]. Define a probability measure on that is absolutely continuous with respect to and the Radon–Nikodym derivative on with restriction to is given by: . Then, on and under , are i.i.d. sequences random processes, such that
(5) Let be a “test function.” From Bayes’ theorem, for -adapted sequence :
The integers
are defined by the following equation:
with the integers being the quotient and rest of the division of by . Proof: The proof of this lemma follows from the arguments in [9] directly, and is omitted here. The next lemma computes the state estimate and error covariance through (8) and the results are in terms of , and .
MA et al.: OPTIMAL FILTERING OVER UNCERTAIN WIRELESS COMMUNICATION CHANNELS
Lemma 2: The optimal state estimate are
and error covariance
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Similarly for covariance, we have
where
is computed as
(10) where is computed below and both of and are non linear functions of measurements. Proof: As , we have
Extension to General Case: One can easily extend the result to matrix case as in [10] and [12]. Assume (1) to be a vector version problem where the parameters , and are matrices of compatible dimensions, then can be represented as
where denotes the transpose, and
Then, the normalized density can be computed as
Thus, the optimal state estimate is computed as
The state estimate and covariance for matrix case can be derived similarly as in Lemma 2. Note that from the expression (10) of the state estimate, we can conclude that the optimal filter is not a linear function but rather an exponential function of measurements. Since is increasing, the computation of this optimal filter require exponentially increasing memory. This is the price that one has to pay to obtain the optimal filter. However, two important features can be deduced from computing the optimal filter. The first feature is that the optimal filter can be used as a benchmark for suboptimal filters to determine how far they are from optimality. The second more important feature, is that the optimal filter shows superior performance for unstable system compared with the linear optimal filter discussed in the next section.
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Fig. 2. Root mean square error between the optimal filter (solid curve) and the linear optimal estimator (dash curve) when .
Fig. 3. Root mean square error between the optimal filter (solid curve) and the . linear optimal estimator (dash curve) when
V. CONCLUSION
III. LINEAR OPTIMAL ESTIMATOR In this section, we review the linear optimal estimator for this problem. If the state estimate is assumed to be a linear function of the measurement, then, the linear optimal estimator for (1) can be written as [8]
In this letter, optimal estimation under Bernoulli i.i.d packet loss link without packet arrival information is studied. The optimal filter is shown to be nonlinear, and is derived using hybrid filtering theory by viewing the Bernoulli i.i.d packet loss as a special Markov process. A comparison with the linear optimal estimator, which assumes that the state estimate is a linear combination of measurements is presented. The simulation results show that the exact filter outperforms the linear optimal estimator. REFERENCES
(11) In the next section, this filter is implemented together with the optimal filter and the results are compared. IV. NUMERICAL EXAMPLES: COMPARISON BETWEEN THE OPTIMAL FILTER AND LINEAR OPTIMAL ESTIMATOR To illustrate the performance of the filters, we consider a , , linear system with packet arrival probability , , , and
Simulations are performed for 30 runs from to by Monte Carlo Simulation. There are packet losses at 4, 6, 13. The performance of our estimator and the linear optimal estimator are compared. It is seen from Fig. 2 that both filters estimate the true system state very well while the root mean square error (RMSE) for the optimal filter is a bit smaller. To further compare the performance, the state parameter is increased from 0.9 to 1.14, that is, now the system is unstable. The result of RMSEs is shown in Fig. 3. The Figure clearly shows that the performance of the optimal filter is much better. The reason is when the system is unstable, the linear optimal estimator diverges [13], as a result performance degrades rapidly.
[1] J. Nilsson, “Real-Time Control Systems With Delays,” Ph.D. dissertation, Dept. Automatic Control, Lund Institute of Technology, Lund, Sweden, 1998. [2] K. J. Rajasekaran, N. Satyanarayana, and M. D. Srinath, “Optimum linear estimation of stochastic signals in the presence of multiplicative noise,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-7, pp. 462–468, 1971. [3] M. Hadidi and S. Schwartz, “Linear recursive state estimators under uncertain observations,” IEEE Trans. Inf. Theory, vol. IT-24, pp. 944–948, 1979. [4] B. Sinopoli, L. Schenato, M. Franceschetti, K. Poola, M. Jordan, and S. Satry, “Kalman filtering with intermittent observations,” IEEE Trans. Automat. Contr., vol. 49, no. 9, pp. 458–463, 2004. [5] M. Huang and S. Dey, “Stability of Kalman filtering with Markovian packet losses,” Automatica, vol. 43, pp. 598–607, 2007. [6] V. Gupta, B. Hassbi, and R. M. Murray, “Optimal LQG control across packet-dropping links,” Syst. Contr. Lett., vol. 56, pp. 439–446, 2007. [7] L. Xiao, M. Jonathan, H. Hindi, S. Boyd, and A. Goldsmith, “Joint optimization of communication rates and linear systems,” IEEE Trans. Automat. Contr., vol. 48, no. 1, pp. 148–153, 2003. [8] E. Nahi, “Optimal recursive estimation with uncertain observation,” IEEE Trans. Inf. Theory, vol. IT-15, p. 457, 1969. [9] R. J. Elliott, F. Dufour, and D. D. Sworder, “Exact hybrid filters in discrete time,” IEEE Trans. Automat. Contr., vol. 41, no. 12, pp. 1807–1810, 1996. [10] L. Aggoun and R. Elliott, Measure Theory and Filtering: Introduction With Applications. Cambridge, U.K.: Cambridge Univ. Press, 2004, pp. 169–177. [11] J. L. Hibey and C. D. Charalambous, “Conditional densities for continuous-time nonlinear hybrid systems with applications to fault detection,” IEEE Trans. Automat. Contr., vol. 44, no. 11, pp. 2164–2169, Nov. 1999. [12] R. J. Elliott, F. Dufour, and W. P. Malcolm, “State and mode estimation for discrete-time jump Markov systems,” SIAM J. Contr. Optim., vol. 44, no. 3, pp. 1081–1104, 2005. [13] W. L. De Koning, “Optimal estimation of linear discrete-time systems with stochastic parameters,” Automatica, vol. 20, no. 1, pp. 113–115, 1984.
