A new nonlinear filter for parameters identification in dynamic systems and application to a transmission channel

doi:10.1016/j.sigpro.2007.08.006

Signal Processing

Volume 88, Issue 2, February 2008, Pages 349-357

https://doi.org/10.1016/j.sigpro.2007.08.006 Get rights and content

Abstract

In this paper we propose general nonlinear models for off-line and on-line parameters identification in dynamic systems. These numerical filters can be applied to any nonlinear system represented by a state equation and an observation equation both nonlinear. The theory of hidden Markov models is used to derive these algorithms starting from a of Baum & Welch type method. The proposed identification algorithm has two levels: the first level is an iterative and global algorithm (IGA), it estimates iteratively the parameters from a block of data. The second level is an ergodic recursive algorithm (ERA), it estimates the parameters in an adaptive manner. The estimators defined by these algorithms converge almost surely to the true values of the model parameters studied in [A. Khoukhi, T. Aliziane, M. Souilah, Un Algorithme Multi-Niveau d’Identification d’un Canal en Communication Numérique, JESA 36(4) (2002) 519–537; M. Souilah, A. Khoukhi, T. Aliziane, A new multi-level algorithm for identification and stochastic adaptive control of industrial manipulators, Eng. Simulation 26(4) (2004) 83– 98; M. Souilah, A new strategy for identification and control of mobile robots, Eng. Simulation 28(3) (2006) 35–48]. The advantage of the proposed nonlinear filters in relation to classical autoregressive models is the fact that the nonlinearity of the model is taken into account as it is and no linearizations are made around a nominal position. The variances of the added noises are also estimated. The mathematical convergence of the algorithms IGA and ERA is an open problem in the general case. We propose to this end an interesting conjecture based on ergodic theory. These algorithms are applied to identify the parameters of a transmission channel in data communication. Some simulation results showing the convergence of these algorithms are given.

Introduction

All filtering models existing in the literature are based essentially on linearizations. Indeed, the dynamic of the model is generally nonlinear and the expression of the model state at a time n according to the precedent state (KF, EKF) or the state at several precedent times (AR, ARMA, DARMA, DARMAX, $\dots$ ) [1], [2] require linearization around a nominal position.

The nonlinear filters we propose are of two kinds: global and adaptive. The global algorithm is used in the off-line programming of dynamic systems or in training phase as we have used it in the identification of a transmission channel in data communication [3], in industrial manipulators robots [4] and in mobile robots [5]. The adaptive algorithm is a recursive one, it is used in the on-line programming, or during the execution of the task by an intelligent machine. This algorithm evaluates a parameter at a running time as a kind of convex combination of the value of this parameter at the previous time and the history of the model states at all previous states.

To derive these filters, we have used a method of Baum & Welch type. It consists of maximizing the information quantity of Kullback–Leibler instead of the maximum likelihood. We then obtain a system of nonlinear equations satisfied by the parameters of the model which we solve numerically by a Newton–Raphson method.

To prove the convergence of the adaptive algorithm ergodic recursive algorithm (ERA) for the studied models in [3], [4], [5], we have constructed in each case a new model, a new space and a new law of probability according to the study plane of Section 4.3. The convergence of the iterative and global algorithm (IGA) is an open problem in all studied models. We set forth an interesting conjecture giving an analogy with the ergodic Birkhoff theorem [6], [7].

Section snippets

The discrete nonlinear stochastic dynamic model

The general stochastic dynamic model considered here is defined by the system $\{\begin{matrix} x_{n} = f (x_{n - 1}, α, u_{n}) + v_{n}, \\ y_{n} = g (x_{n}, β) + w_{n}, \end{matrix}$ where the first equation is of state representing the dynamic of the model, the second is of observation, then $x_{n}, y_{n}$ represent, respectively, the state of the system and its observation at time $n$ . The constants $α$ , $β$ are the parameters of the model and $σ^{2}$ , $τ^{2}$ are the variances of the additive Gaussian noises $v_{n}$ , $w_{n}$ . The entry $u_{n}$ represents the control of the system. The vector of the

The iterative and global algorithm

The algorithm IGA, when $λ^{'}$ can be expressed as a function of $λ$ is known in the HMM theory, especially in engineering literature. Its mathematical convergence is an open problem even in simple cases. In the applications [3], [4], [5], the transformation $Q (λ, λ^{'})$ considered as a function of $λ^{'}$ satisfy the conditions of [8] (see Theorem 1 below) and have a unique global maximum at critical points.

It follows that, in our general case, certain minimal conditions must be satisfied by the functions f

The ergodic recursive algorithm (ERA)

The main contribution of this paper is to define the new adaptive algorithm ERA which is considered as a smoothing of IGA or a second level of this algorithm. On the other hand, we try below to give a mathematical perspective to prove its convergence by applying the ergodic theorems. This convergence is difficult to prove as it is clear from the nonlinearities of the model.

As in the case of the IGA algorithm, we distinguish between the parameters $σ$ , $τ$ and $α$ , $β$ .

The Birkhoff ergodic theorem

This theorem gives the convergence of the temporal mean in the space of states of a dynamic system to its spatial mean. This temporal mean represents for us the global algorithm IGA in the applications [3], [4], [5]. The algorithm ERA is a second level, a smoothing or a refinement of the algorithm IGA. We give below some versions of this theorem.

Theorem 2

Smordinsky [6]

Let $T : (X, B, μ) \to (X, B, μ)$ a preserving measure transformation in a measurable space, where $μ$ is $σ$ -finite and $f \in L^{1} (μ)$ . Then $(1 / N) \sum_{n = 0}^{N - 1} {foT}^{n}$ converges $μ$ -

Discussion

In the applications [3], [4], [5], we have derived explicitly the algorithm ERA from IGA. This is not the case in the present general study. The step n of the IGA algorithm is assimilated to the nth iterate of an implicit transformation preserving measures contained in the algorithm. The nonlinearities encountered in other models allow to algorithms with complex expressions for which the known versions of the Birkhoff theorem cannot be used to propose an analogy. The ergodicity must be

Application to a transmission channel

We consider a model of state in discrete time and discrete values where the states $x_{n} = (a_{n}, θ_{n})$ are related by the equations $\{\begin{matrix} a_{n} = 2 u_{n} - 1, \\ θ_{n} = θ_{n - 1} + d θ_{n} + Δ θ, \\ y_{n} = a_{n} A e^{j θ_{n}} + b_{n} . \end{matrix}$

In our case $u_{n}$ , is a symbol produced by the source, $a_{n}$ is a symbol used by the emitter (for example $u_{n} = 0$ or 1 and $a_{n} = - 1$ or $+ 1$ ), $θ_{n}$ is a sequence of random variables such that the density of probability of $θ_{0}$ is $P (θ_{0}) = 1 / 2 π$ , i.e. $θ_{0}$ is uniformly distributed on $[0, 2 π]$ . The conditional density $P (θ_{n} / θ_{n - 1})$ corresponds to $N (θ_{n - 1} + Δ θ, σ_{θ}^{2})$ if one

Conclusion

In this work, we proposed new nonlinear numerical filters for the parameters identification in dynamic systems with state representation consisting of two equations of state and observation both nonlinear. The nonlinearities of these models are taken into account as they are and no linearization is made around a nominal position. The convergence of the proposed algorithms is an open problem in the general case for which we give an interesting conjecture based on the Birkhoff ergodic theorem [10]

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Signal Processing

A new nonlinear filter for parameters identification in dynamic systems and application to a transmission channel

Abstract

Introduction

Section snippets

The discrete nonlinear stochastic dynamic model

The iterative and global algorithm

The ergodic recursive algorithm (ERA)

The Birkhoff ergodic theorem

Smordinsky [6]

Discussion

Application to a transmission channel

Conclusion

Hidden Markov Models. Estimation and Control

Adaptive Filtering Prediction and Control

Adaptive Filtering Prediction and Control

Un Algorithme Multi-Niveau d’Identification d’un Canal en Communication Numérique

JESA

A new multi-level algorithm for identification and stochastic adaptive control of industrial manipulators

Eng. Simulation

A new strategy for identification and control of mobile robots

Eng. Simulation