Filtering and identification of a state space model with linear and bilinear interactions between the states
© Al-Mazrooei et al.; licensee Springer 2012
Received: 22 April 2012
Accepted: 24 September 2012
Published: 9 October 2012
In this paper, we introduce a new bilinear model in the state space form. The evolution of this model is linear-bilinear in the state of the system. The classical Kalman filter and smoother are not applicable to this model, and therefore, we derive a new Kalman filter and smoother for our model. The new algorithm depends on a special linearization of the second-order term by making use of the best available information about the state of the system. We also derive the expectation maximization (EM) algorithm for the parameter identification of the model. A Monte Carlo simulation is included to illustrate the efficiency of the proposed algorithm. An application in which we fit a bilinear model to wind speed data taken from actual measurements is included. We compare our model with a linear fit to illustrate the superiority of the bilinear model.
Bilinear systems are a special type of nonlinear systems capable of representing a variety of important physical processes. They are used in many applications in real life such as chemistry, biology, robotics, manufacturing, engineering, and economics [1–4] where linear models are ineffective or inadequate. They have also been recently used to analyze and forecast weather conditions [5–10].
Bilinear systems have three main advantages over linear ones: Firstly, they describe a wider class of problems of practical importance. Secondly, they provide more flexible approximations to nonlinear systems than linear systems do. Thirdly, one can make use of their rich geometric and algebraic structures, which promises to be a fruitful field of research for scientists  as well as practitioners.
Bilinear models were first introduced in the control theory literature in 1960s . So far, the type of nonlinearity that is extensively treated and analyzed consists of bilinear interaction between the states of the system and the system input [1, 2, 12]. Aside from their practical importance, these systems are easier to handle because they are reducible to linear ones through the use of a certain Kronecker product. In this work, we treat the case where the nonlinearity of the system consists of bilinear interaction between the states of the system themselves. This means that our model will be able to handle evolutions according to the Lotka-Volterra models  or the Lorenz weather models [7, 8, 10], thus enabling a wider and more flexible application of such models. To the best of our knowledge, no attempt has been made to treat such systems in the general setting presented here.
The widespread use of bilinear models motivates the need to develop their parameter identification algorithms. A lot of work exists in the literature which presents methods of estimation and parameter identification of linear and nonlinear systems [13–21]. The two most widely used techniques fall under the names of least square estimation and maximum likelihood estimation, respectively.
The maximum likelihood estimation is computed through the well-known EM algorithm . It is an iterative method that tries to improve a current estimate of the system parameters by maximizing the underlying likelihood densities. The algorithm is useful in a variety of incomplete data problems, where algorithms such as the Newton-Raphson method may turn out to be more complicated. It consists of two steps called the Expectation step or the E-step and the Maximization step or the M-step; hence the name of the algorithm. This name was first coined by Dempster, Laird, and Rubin in their fundamental paper . In this paper, we develop the EM algorithm for our bilinear system. This will necessitate also the development of a Kalman filter and smoother suitable for the nonlinear system at hand. The direct development of the recursions for the nonlinear filters is very complicated if not impossible altogether. Instead, we develop our recursions based on a linearization of the quadratic term that uses the most current state estimate available.
The remainder of this article is arranged as follows. In Section 2, the bilinear state space model problem is stated along with underlying assumptions. In Section 3, we derive the bilinear Kalman filter and smoother. Section 4 estimates the unknown parameters in the bilinear state space model via the EM algorithm. Section 5 presents a simulation example that produces very satisfactory results. A real world example is given in Section 6.
2 The bilinear state space model
In this section, we introduce a bilinear state space model and describe a generalization of the Kalman filter and smoother to this model. Our model subsumes the well-known Lorentz-96 model  for weather forecast, and the Lotka-Volterra evolution equations appear in many applications in chemistry, biology and control [4, 6]. Other types of bilinear models were investigated in [2, 11], where bilinearity occurs because of the interaction between the input and states of the system.
for all ; the closed subspace of of all random vectors z which can be written as measurable functions of the elements of Y .
Lemma 1 , , .
The second equation can be shown in exactly the same way. □
3 A bilinear Kalman filter and smoother
In this section, we will develop a Kalman filter and smoother for the bilinear system (1) and (2).
3.1 A bilinear Kalman filter
In order to compute equation (4), we approximate the second-degree term by using the most current available state estimation for ; that is,
In the case of prediction, we take
In the case of filtering, we take
In the case of smoothing, we take
represents the Kalman gain.
This completes the proof. □
Also, note that the bilinear Kalman filter algorithm is a generalization of the Kalman filter for the linear case which is given in .
3.2 A bilinear Kalman smoother
where denotes the subspace spanned by .
Continuing in this manner, we get (14). □
We state the bilinear Kalman smoother in the following theorem.
which completes the proof. □
The next theorem states the bilinear lag-one recursions.
4 The bilinear EM algorithm
represents the n-variate normal density of the initial state with mean μ and the covariance matrix V.
represents the p-variate normal density with zero mean and the covariance matrix R.
represents the n-variate normal density function with zero mean and the covariance matrix Q.
The following theorem accomplishes the expectation step.
and simplifying. The middle equality follows from the fact that odd moments of Gaussian random variables vanish. □
The next step of the EM algorithm is to maximize the function with respect to θ.
which means that is maximized by separately minimizing , , . This is done by setting the partial derivative of q with respect to each parameter equal to zero (i.e., ) and solving the resulting system of equations. □
The EM algorithm for a bilinear state space model is summarized as follows.
Initialize the EM algorithm by choosing initial values of .
Calculate the incomplete-data likelihood, .
Execute the E-step by using the bilinear Kalman filter and smoother in (9)-(10) and (15)-(16), respectively.
Execute the M-step using (21)-(23) and update the estimates of θ using (M-step) to obtain .
Repeat Steps 2 to 4 until convergence.
5 Simulation results
In all simulations, the number of iterations for the EM algorithm is fixed and its value set to .
Comparison of the mean square errors
Linear MSE 
6 Application to wind speed
The first author was supported by Tayyebah University. The second and third authors would like to thank King Fahd University for the excellent research facilities they provide.
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