Elsevier

Automatica

Volume 47, Issue 5, May 2011, Pages 1041-1046
Automatica

Brief paper
Asymptotic statistical analysis for model-based control design strategies

https://doi.org/10.1016/j.automatica.2011.01.058Get rights and content

Abstract

In this paper, we generalize existing fundamental limitations on the accuracy of the estimation of dynamic models. In addition, we study the large sample statistical behavior of different estimation-based controller design strategies. In particular, fundamental limitations on the closed-loop performance using a controller obtained by Virtual Reference Feedback Tuning (VRFT) are studied. We also extend our results to more general estimation-based control design strategies. We present numerical examples to show the application of our results.

Introduction

Performance limitations are of great importance in areas such as control system design (Seron, Braslavsky, & Goodwin, 1997). In general, results on performance limitations have been developed assuming perfect knowledge of the process to be controlled. Control design is typically developed by using two main strategies: (i) Model-based control: where one uses a mapping from the plant model to the controller (Goodwin, Graebe, & Salgado, 2001, Chap. 2); and (ii) Direct methods: where one uses a mapping from the data collected (in an experiment) to the controller (Campi, Lecchini, & Savaresi, 2002).

In Bombois (2000), Gevers (1993) and Hjalmarsson (2005), the impact of using an estimated model for control has been studied, giving rise to the field of “identification for control”.

Recently, fundamental limitations for the accuracy of estimated Box–Jenkins (BJ) models have been developed (Rojas, Welsh, & Agüero, 2009). In this paper, we extend the main result given in Rojas et al. (2009) considering now an arbitrary parameterization operating in open/closed loop.

We also study the effect of these fundamental limitations on the Virtual Reference Feedback Tuning (VRFT) control design strategy (Campi et al., 2002, Campi and Savaresi, 2006). In VRFT, the controller is designed using identification techniques. Thus, there exists a fundamental limitation on the accuracy of the sensitivity function of the closed loop achieved in VRFT.

In addition, we study the statistical accuracy of more general model-based control design strategies. We also specialize these results to Minimum Variance Control (MVC), and H2 control design.

The layout of the paper is as follows: In Section 2 we present preliminaries and basic definitions. In Section 3 we extend the results given in Rojas et al. (2009). In Section 4 we use fundamental limitations to analyze the performance of the VRFT control strategy. In Section 5 we present extensions for other model-based control techniques. Finally, in Section 6, we draw conclusions.

Section snippets

Preliminaries

Consider a discrete-time single-input single-output (SISO) linear time-invariant (LTI) system in open-loop operation defined by yt=Go(q)ut+vt,vt=Ho(q)wt, where u and y are the plant input and output, respectively, and w is zero mean Gaussian white noise of variance σw2, independent of u. The term q is the shift operator, i.e., qxt=xt+1, and Ho is a stable minimum phase transfer function with Ho()=1. The signal u is assumed to be known, and w is unknown.

The purpose of system identification is

Fundamental limitations for SISO models

We first define the following terms Π=[GH]T,Πˆ=[GˆHˆ]T,Πo=[GoHo]T, where G=G(θ),H=H(θ),Gˆ=G(θˆ),Hˆ=H(θˆ). Gˆ and Hˆ are the estimated plant and noise models, respectively. Also, let Φχ=[ϕuϕuwϕuwσw2], where ϕuw is the cross spectrum between w and u, which is zero if the system operates in open loop. Notice that with these definitions, (2) and Parseval’s Theorem, we have (c.f. Eq. (9.54) of Ljung, 1999) Mθ=12πσw21|Ho|2[ΠTθ]Φχ[ΠTθ]H.

Theorem 1 Limitations for SISO Models

If the system in (1) is parameter identifiable for the

Virtual reference feedback tuning

VRFT is a direct controller identification approach which uses input/output data collected in open/closed loop to directly identify a controller without using a plant model. See e.g. Campi and Savaresi (2006), Campi et al. (2002), Esparza and Sala (2007), Sala (2007) and Sala and Esparza (2005).

The purpose of VRFT is to identify a controller Co from {ut,yt}t=1N which attains a closed-loop behavior (defined, for example, by the complementary sensitivity function) as close as possible to a

Extension to more general model-based controller design

In this section, we consider the two-degree-of-freedom scheme in Fig. 3. We analyze the effect of the model’s accuracy on the loop performance. The controller (F,C) is obtained from the model (Gˆ,Hˆ).

Define the function F (see Fig. 3): [FC]T=F(Π), with Π defined as in (4), and where the function F has to be differentiable with respect to Π. That is, the controller obtained by the model-based design needs to depend explicitly on G and H.

Let P=P(C,F)=FCGo1+CGo, and Q=Q(C)=Ho(1+GoC).

The

Conclusions

In this paper we have extended the fundamental limitation in estimation presented in Rojas et al. (2009). We have considered systems with different parameterizations that are operating in open and closed loops. We have also explored the implications of these fundamental limitations in System Identification for the problem of estimation-based control design. The results presented are asymptotic in the data length, but valid when the number of parameters to be estimated is finite. We have

Alicia Esparza was born in Madrid, Spain, in 1974. She received the M.Sc. Degree in Industrial Engineering in 1998 and her Ph.D. in Control Engineering in 2006 from the Technical University of Valencia (UPV), Spain. During 1998 and 1999 she was teaching assistant at the University Jaume I of Castellón, Spain. She has been teaching at the Systems Engineering and Control Department, UPV since 2000. She has participated in several local and national research projects. Her research interests

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Alicia Esparza was born in Madrid, Spain, in 1974. She received the M.Sc. Degree in Industrial Engineering in 1998 and her Ph.D. in Control Engineering in 2006 from the Technical University of Valencia (UPV), Spain. During 1998 and 1999 she was teaching assistant at the University Jaume I of Castellón, Spain. She has been teaching at the Systems Engineering and Control Department, UPV since 2000. She has participated in several local and national research projects. Her research interests include control and system identification.

Juan C. Agüero BE, ME, Ph.D. Juan-Carlos Agüero was born in Osorno, Chile. He obtained the professional title of Ingeniero civil electrónico and a Master of engineering from the Universidad Técnica Federico Santa María (Chile) in 2000, and a Ph.D. from The University of Newcastle (Australia) in 2006. He gained industrial experience from 1997 to 1998 in the copper mining industry at El Teniente, Codelco (Chile). He is currently working as a Research Academic at The University of Newcastle (Australia). His research interest include System Identification and Control.

Cristian R. Rojas was born in 1980. He received the M.S. degree in electronics engineering from the Universidad Técnica Federico Santa María, Valparaíso, Chile, in 2004, and the Ph.D. degree in electrical engineering at The University of Newcastle, NSW, Australia, in 2008. He has currently a postdoctoral position at the ACCESS Linnaeus Centre, KTH, Sweden. His research interest is in system identification.

Boris I. Godoy was born in Valparaíso, Chile, in 1975. He received his professional title of Ingeniero Civil Electrónico and the M.Sc. degree in electronics engineering from the Universidad Técnica Federico Santa María (Chile) in 2001. He received his Ph.D. in electrical engineering from The University of Newcastle (Australia) in 2008. Since then, he has been working as a Research Academic at the ARC Centre of Excellence for Complex Dynamic Systems and Control, Australia. His main research interests are in system identification, and process control.

This work has been partially supported by the project GVPRE/2008/116 financed by Generalitat Valenciana (Spain). This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Guoxiang Gu under the direction of Editor Torsten Söderström.

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