Recursive identification under scarce measurements — convergence analysis

doi:10.1016/S0005-1098(01)00236-9

Automatica

Volume 38, Issue 3, March 2002, Pages 535-544

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Abstract

In this paper, the problem of recursive identification under scarce-data operation is addressed. The control action is assumed to be updated at a fixed rate, while the output is assumed to be measured synchronously with the input update, but with an irregular availability pattern. Under these conditions the use of pseudo-linear recursive algorithms is studied. The main result is the convergence analysis for the case of regular but scarce data availability. The existence of wrong attractors is demonstrated, and a local stability condition of the identification algorithm is derived.

Introduction

In many industrial applications the control signal is updated at a fixed rate, but the output is measured with a different timing pattern and, sometimes, by various sensors with different sampling rates and reliability. In some cases the output is not available at every sampling time due to communication errors, shared or slow sensors, or the use of destructive measuring methods.

Different authors have dealt with the modeling of such systems when the measurement pattern is periodic. Their approach is based on the definition of a model that relates outputs measured at one rate with inputs updated at another rate. This allows the implementation of a standard RLS algorithm (see Albertos, Salt, & Tornero, 1996) where the regression vector is constructed with only measured variables. This is only possible if the sampling pattern is regular (periodic).

If the pattern of data availability is not regular, the multirate approach cannot be used. In that case, the regression vector cannot be filled with only measured values. It is then necessary to include estimated values of the non measured outputs on the regression vector. This results in a non-linear recursive algorithm usually called pseudo-linear because of the “linear-like” aspect.

In Isaksson (1993) the problem of identification with missing-data is addressed by means of expectation–maximization off-line algorithms. A recursive version of that algorithm is also studied in Isaksson (1994), but no convergence analysis is carried out.

This paper deals with the use of pseudo-linear recursive algorithms for estimating the parameters of the discrete transfer function of the process from scarcely sampled output measurements, assuming that the availability of data may be irregular. The convergence analysis is carried out, however, only for the regular case.

The basic pseudo-linear recursive algorithm studied in this paper was introduced in Albertos, Goodwin, and Isaksson (1992). In that work a sufficient condition for local convergence of the pseudo-linear algorithm is derived based on the passivity property of a given operator. The result is very restrictive and useless in the case of scarce measurements. As an example, if there is one measurement every three input periods, the condition never holds, but simulations show that in many cases the algorithm converges. In Adams, Albertos, Goodwin, and Isaksson (1994) a similar condition is derived for a modified version of the algorithm with worse convergence properties. The result is again very restrictive and useless. Furthermore, in those two papers, the stability of the predictor used to estimate the missing outputs is not taken into account.

The study of the pseudo-linear identification algorithm has been addressed by the authors in previous works (Albertos, Sanchis, & Sala 1995b, Albertos, Sanchis, & Sala 1997; Albertos, Sala, & Sanchis, 1995a; Sanchis, Sala, & Albertos, 1997), where the basic algorithm has been generalized by the introduction of different possible predictors. In Albertos, Sanchis, and Sala (1999) different output predictors used to obtain the estimates of the unmeasured outputs are studied, analyzing their stability properties. In this sense, this paper can be considered as an extension of those works, presenting as a new contribution a result concerning the convergence of the identification algorithm with a simple predictor based on an input–output representation.

The layout of the paper is as follows: Section 2 presents the definition of the problem, defining the pseudo-linear recursive identification algorithm. In Section 3 the existence of attractors is demonstrated for the limit case when the input updating period tends to zero. Section 4 is devoted to the convergence analysis of the algorithm, based on the properties of an associated differential equation. In Section 5 a procedure to obtain the associated differential equation is developed. Section 6 contains examples of the application of the convergence results for a second order system. Finally, the conclusions are summarized in Section 7.

Section snippets

Problem statement

Consider a SISO continuous time linear system of order n whose input is updated periodically (updating period T) by a computer with a zero-order hold. Assume that the output is measured synchronously with the input update. Also assume that there is a disturbance such that the discrete difference equation at period T can be written as $A(q)y_{k} =B(q)u_{k} +ν_{k} ⇔ y_{k} =ψ_{k}^{T} θ+ν_{k},$ where q⁻¹ is the unit delay operator, θ=[a₁⋯a_nb₁⋯b_n]^T is the parameter vector, $ψ_{k} =[−y_{k−1} ⋯−y_{k−n} u_{k−1} ⋯u_{k−n}]^{T}$ (with y_i=y(iT), u_i=u(iT)) is

Attractors

The existence of wrong attractors where the algorithm can be caught is difficult to be proved for a general system with arbitrary data pattern. Nevertheless, it can be easily proved for the limit case where the updating period T tends to zero with a periodic 1/N sampling pattern. The following theorem expresses this result.

Theorem 1

Consider a SISO continuous time linear system of ordernwhose input is updated at a constant period T and whose output is measured with a periodic 1/Nsampling pattern with no

Convergence analysis

The extension of the previous result to the case of arbitrary input updating period is not evident due to the non-linear and stochastic nature of the identification algorithm.

One technique that can be used to analyze the convergence is based on the calculation of a differential equation whose convergence properties are directly related to the properties of the recursive algorithm. This approach was first introduced by Ljung (1977), and a detailed description can be found in Ljung and Södeström

Procedure to calculate the associated differential equation

For general systems of order n with a periodic 1/N sampling pattern, the expression of Eq. (4) can be obtained following the procedure described in this section. It will be assumed that the system is described by Eq. (1) where ν_k is an independent disturbance.

Let us define the matrix M̂ as in (8) and M the corresponding matrix with the true parameters. Let us define also the vector v as in the previous section, and the vector $w=[1 0 ⋯ 0]^{T}$ . On the other hand, the true regression vector is defined

Example

To illustrate the theoretical result, the procedure developed in the previous section will be applied to a second order system. Consider a system that can be approximated by the second order model G(s)=100/(s²+2s+2). The input is assumed to be updated at constant period T. The discrete parameter vector is $θ=[a_{1} a_{2} b_{1} b_{2}]^{T}$ . Assume that there is no disturbance, and the input is an independent discrete signal of zero mean and variance σ_u². Assume that the output is measured every N=3 input periods,

Conclusions

In this paper the problem of online identification of systems with scarce measurements has been addressed. For this purpose a pseudo-linear recursive identification algorithm has been analyzed. The algorithm can be applied to regular or irregular data availability. Nevertheless, the analysis has been carried out for the case of regular (but scarce) output measurements.

The first result is Theorem 1 where the existence of wrong equilibrium points for the identification algorithm is proven for the

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Albertos, P., Sanchis, R., & Sala, A. (1997). Scarce data operating conditions: Output predictors. SYSID IFAC symposium...

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Roberto Sanchis was born in Genovés, Valencia, Spain in 1968. He received the M.Sc. Degree in Electrical Engineering in 1993 and his Ph.D. in Control Engineering in 1999 from the Polytechnic University of Valencia (UPV), Spain. He was awarded the first national prize for university graduation in 1993. During 1994 and 1995 he was a teaching assistant at the Systems and Control Engineering Department of the UPV. He has been working since 1996 at the Department of Technology of the University Jaume I of Castellón, Spain. His current position is as Associate Professor at the same department. His research interests include missing-data identification and control, variable structure systems and control applications on the ceramic industry.

Prof. Pedro Albertos was born in Valencia, Spain, in 1943. Full Professor of Automatic Control (1975), at the Department of Systems Engineering and Control, Universidad Politécnica Valencia, since 1977 and past head of the Department in the periods 1979–1995 and 1998. He is Honorary Professor at the Northwestern University, Senhyang, China (1998). Invited Professor and Researcher at the Universities of Illinois at Urbana- Champaign, USA.; Northwestern University, Senhyang, RPC; Campinas, Brazil; Newcastle, Australia; California at Santa Barbara, USA.; C.U.J.A.E., Havana, Cuba; Universidad Tecnológica de Panamá; Technical University of Santa Maria, Valparaiso, Chile; Universidad de Oriente, Venezuela; Universidad Autónoma de Occidente, Cali, Colombia; and University of British Columbia, Canada; and Lecturer at other 30 Universities and Research Centers, all around the world. His main fields of interest are Multirate Digital Control, Parameter estimation and adaptive control, Intelligent Control and their application on industrial processes. Plenary speaker in more than 10 Workshops, Symposia and Congresses and co-authoring more than 200 conference papers, some results have been also published in more than 20 journals papers. He has been coeditor of 6 international books. Chairman of many Technical Committees, Technical Sessions, Round Tables, and other meetings, he has been also member of more than 40 International Program Committees.

Since 1974, he has conducted and participated in different research projects, local, European and international, as well as educational projects (SOCRATES, TEMPUS) with many European Universities.

He has been reviewer and advisor of many institutions and technical journals such as IEEE Trans. on Automatic Control, IEEE Trans. on Control Technology, IEEE Fuzzy Sets and Systems and Automatica, being Associated editor of Control Engineering Practice. He has been the director of 14 Doctoral Thesis, and launched an international Ph.D. Program on Industrial Control and Computers in University Del Valle, in Cali, Colombia (1995), Universidad de Oriente, Puerto La Cruz, Venezuela, (1997) and with the Technological Institutes General Board in Chihuahua, México (2000).

He was elected as IFAC Vice-President for the triennium 1993–1996, President Elect for 1996–1999 and currently (1999–2002) he is the President of IFAC. As President of the Spanish association of automatic control, CEA-IFAC, he is promoting a number of new activities to stress the collaboration with the industry: courses, meetings and publications The organization of the XV IFAC World Congress, to be held in Barcelona in July 2002, is one of his main duties.

Prof. Albertos is Senior Member of IEEE, being member of the Board of Governors of the Control Systems Society during the period 1996–1997. He is fluent in French and English.

^☆: The original version of this paper was presented at the 12th IFAC Symposium on System Identification, USA, 2000. This paper was recommended for publication in revised form by Associate Editor Antonio Vicino under the direction of Editor Torsten Söderström.

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Brief PaperRecursive identification under scarce measurements — convergence analysis☆

Abstract

Introduction

Section snippets

Problem statement

Attractors

Convergence analysis

Procedure to calculate the associated differential equation

Example

Conclusions

Automatica

Dual rate adaptive control

Automatica

Brief Paper
Recursive identification under scarce measurements — convergence analysis☆