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On-line parameter identification is a key problem of adaptive control and also an important part of self-tuning regulator (STR) (Astrom & Wittenmark, 1994). In the widely equipped large-scale systems, distributed systems and remote systems, the plants, controllers, actuators and sensors are connected by communication channels which possess only finite communication capability due to, e.g., data loss, bandwidth constraint, and access constraint. From a heuristic analysis perspective, the existence of communication constraints has the effect of complicating what are otherwise wellunderstood control problems, including the traditional methods, such as the *H* control (Fu & Xie, 2005), and even the basic theoretic notions, such as the stability (De Persis & Mazenc, 2010).

Due to the constraints of the communication channels, it is difficult to transmit data with infinite precision. Quantization is an effective way of reducing the use of transmission resource, and then meeting the bandwidth constraint of the communication channels. However, quantization is a lossy compression method, and hence the performance of parameter identification, even the validity or effectiveness of identification may be changed by quantization, along with which the performance of adaptive control may deteriorate. This has attracted plenty of works. The problem of system identification with quantized observation was investigated in (Wang, et al, 2003, 2008, 2010), where issues of optimal identification errors, time complexity, optimal input design, and the impact of disturbances and unmodeled dynamics on identification accuracy and complexity are included.

In the light of the fundamental effect of quantization on system identification, it is necessary to pay attention to the parameter identifiability property of quantized systems. The concept of identifiability has been defined by maximal information criterion in (Durgaryan & Pashchenko, 2001): the system is parameter identifiable by maximal

© 2012 Shen and Zhang, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Shen and Zhang, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

information criterion, if the mutual information between actual output and model output is greater than zero. However, the concept of identifiability in (Durgaryan & Pashchenko, 2001) is defined in principle, based on which there is no practical result. Reference (Zhang, 2003; Zhang & Sun, 1996; Baram & Kailath, 1988) have discussed the problem of states estimability, which is related closely with parameter identifiability, for that inputoutput description of linear systems with Gauss-Markov parameters can be transformed to state space model, and then the problem of parameter identifiability can be treated as state estimability. Reference (Zhang, 2003) has proposed the definition of parameter identifiability under the criterion of minimum maximum error entropy (MMSE) estimation referring to the definition of states estimability, and also obtained some useful conclusions. Reference (Wang & Zhang, 2011) has studied the parameter identifiability of linear systems under access constraints. However, there is few work on that for quantized systems.

Identifiability of Quantized Linear Systems 259

(6)

T 1 1 ( ) = [ ( ) ( ) ( ) ( )] *m n θ k b k b ka k a k* (2)

<sup>T</sup> *yk F k* ( )= ( ) ( )+ ( ) *θ k ek* (4)

*θ*( + 1) = ( ) + ( ) *k Aθ k Bw k* (5)

<sup>T</sup> *Fk uk uk m yk yk n* ( ) = [ ( - 1) ( - ) - ( - 1) - ( - )] (3)

<sup>q</sup>( ) = ( ( )) = , for ( ) , = 1,2, , *l l y k Qyk z yk δ l N* (7)

q q ( ) = ( ( )) *<sup>k</sup> y k Dyk* (8)

*<sup>i</sup> <sup>i</sup> <sup>a</sup>* , 1 2 +1 - = < < < =+ *<sup>N</sup> aa a* are the thresholds of the quantizer.

be the parameter vector, where <sup>T</sup> ( ) denotes the operation of transposition, and let

Suppose that the parameter*θ*( ) *k* can be modeled by a Gauss-Markov process, i.e.,

where *A*, *B* are known matrices with appropriate dimensions; noise sequence { ( )} *w k* is Gaussian and white with zero-mean and covariance *Q*; initial value of the parameter *θ*(0) is Gaussian with mean *θ* and covariance *Π*(0) . Suppose that *e k*( ) , *w k*( ) and *θ*(0) are mutually uncorrelated. Hence, linear system (1) with Gauss-Markov parameters can be

> ( + 1) = ( ) + ( ) ( )= ( ) ( )+ ( ) *θ k Aθ k Bw k yk F k θ k ek*

Due to bandwidth constraint of the channel, quantization is required. The discussion in the present paper does not focus on a special quantizer, but on general *N*-level quantization

where *Q*( ) is the general quantizer, q 12 () { , , , } *<sup>N</sup> y k zz z* denote the quantizer outputs with , = 1, , *<sup>l</sup> zl N* as the reproduction values; = ( , ], = 1,2, , *l l l+1 δ aa l N* denote the quantization

The channel is assumed to be lossless. q 12 () { , , , } *<sup>N</sup> y k zz z* is transmitted and then received at the channel receiver. , = 1,2, , *<sup>i</sup> zi N* are symbols denoting the *i*th quantization interval

where ( ) *Dk* is assumed to be a one to one mapping. A common decoding method (Curry,

q q ( ( )) = E{ ( )| ( )= }, = 1,2, , *D y k yk y k z l N k l*

and not necessarily real numbers, hence, further decoding is required, as follows

\*

T

 

(Curry 1970; Gray and Neuhoff 1998) which can be described as:

then system (1) can be described as

described by (4) and (5), i.e.,

intervals, where +1

1970) is

=1 *N*

where E{.} is the operation of expectation.

This paper mainly analyzes the parameter identifiability of quantized linear systems with Gauss-Markov parameters from information theoretic point of view. The definition of parameter identifiability proposed in (Zhang, 2003) is reviewed: the linear system with Gauss-Markov parameters is identifiable, if and only if the mutual information between the actual value and estimates of parameters is greater than zero, which is extended to quantized systems by considering the intrinsic property of the system. Then the parameter identifiability of linear systems with quantized outputs is analyzed and the criterion of parameter identifiability is obtained based on the measure of mutual information. Furthermore, the convergence property of the quantized parameter identifiability Gramian is analyzed.

The rest of the paper is organized as follows: In section 2, we introduce the model that we are interested in; Section 3 discusses the existing definition of parameter identifiability, proposes our new definition, and gives analytic conclusion focusing on quantized linear systems with Gauss-Markov parameters; The convergence property of Gramian matrix of parameter identifiability for quantized systems is discussed in section 4; Section 5 and 6 are illustrative simulation and conclusion, respectively.
