Elsevier

Automatica

Volume 43, Issue 5, May 2007, Pages 758-767
Automatica

Identification for control: Optimal input intended to identify a minimum variance controller

https://doi.org/10.1016/j.automatica.2006.11.003Get rights and content

Abstract

It is well known that the quality of the parameters identified during an identification experiment depends on the applied excitation signal. Prediction error identification using full order parametric models delivers an ellipsoidal region in which the true parameters lie with some prescribed probability level. This ellipsoidal region is determined by the covariance matrix of the parameters. Input design strategies aim at the minimization of some measure of this covariance matrix. We show that it is possible to optimize the input in an identification experiment with respect to a performance cost function of a closed-loop system involving explicitly the dependence of the designed controller on the identified model. In the present contribution we focus on finding the optimal input for the estimation of the parameters of a minimum variance controller, without the intermediate step of first minimizing some measure of the model parameter accuracy. We do this in conjunction with using covariance formulas which are not asymptotic in the model order, which is rather new in the domain of optimal input design. The identification procedure is performed in closed-loop. Besides optimizing the input power spectrum for the identification experiment, we also address the question of optimality of the controller. It is a wide belief that the minimum variance controller should be the optimal choice, since we perform an experiment for designing a minimum variance controller. However, we show that this may not always be the case, but rather depends on the model structure.

Introduction

It is desirable to have an accurate process model for the design of a controller. The input signal for the identification experiment determines the nature and accuracy of the system characteristics that are identified and used for control design. Therefore, the choice of this signal is crucial for the quality and performance of the designed controller. However, there are often constraints on the length of observation time or on the magnitude of disturbance to the system produced by the input signal. This limits the choice of available inputs and makes a careful selection necessary, focussing on the demands prescribed by the intended application of the controller.

Input design problems first arose in statistics and became relevant for engineering in the 1960s. A survey of early work in this area is given in Mehra (1974). This work focussed mainly on determination of parameters and prediction, while control design was let aside. The design criteria are scalar measures of the information matrix M, or the average per data sample information matrix M¯, which were chosen rather ad hoc. Some of these cost functions are:

  • A-optimality: trM¯-1 (and as a more general case tr(WM¯-1), where W0 is a positive semidefinite weighting matrix; this is called L-optimality). This measure minimizes the average variance of the parameters.

  • E-optimality: λmax(M¯-1) where λmax denotes the maximum eigenvalue.

  • D-optimality: log|M¯-1|. This measure minimizes the volume of the confidence ellipsoid defined by the covariance matrix of the parameters. An important advantage of D-optimality is that it is invariant under linear transformations of the parameter vector, whereas A- and E-optimality are not.

The limitation to these and comparably simple cost functions was dictated by a lack of efficient optimization algorithms. All mentioned functions depend analytically on the entries of M¯ and may be efficiently minimized using, e.g. Kiefer–Wolfowitz theory (Kiefer, 1974), which was already available in the early 1970s.

For the case of a moving average with white output noise, it is demonstrated in Levin (1960) that under an input power constraint, D-optimality is achieved with an open-loop white noise input. In Ng, Goodwin, and Payne (1977) it was shown that for an autoregressive model with an output power constraint, D-optimality is achieved using a minimum feedback controller together with a white probing signal. An extension of these results to a more general model was tackled in Ng, Goodwin, and Söderström (1977). It was shown that under an output variance constraint D-optimality is achieved for a certain input under minimum variance control (MVC).

In the 1980s, inspired by the appearance of asymptotic covariance formulas for transfer function estimates (see Ljung, 1985), input design criteria became focussed on the intended model application. In Ljung (1985) it was shown that cost functions related to simulation, prediction and control design purposes can in general be written in terms of L-optimality with a weighting matrix dependent on the intended application. In Gevers and Ljung (1986) this was used for input design in several settings. The design was carried out simultaneously over the reference input power spectrum and the controller to be applied during identification. It was shown that for the estimation of the plant dynamics with input power constraints open-loop identification was optimal, while for output power constraints a MVC should be used during the identification. For MVC design the MVC should be used during the identification in both cases. In Ljung and Yuan (1985), input design was carried out minimizing performance degradation due to variance and bias errors simultaneously. In Forssell and Ljung (2000) mixed constraints on the input and output were considered, and an analytic solution was found to the input design problem if only the plant dynamics were to be identified. Several special cases were also considered. All this work was based on the covariance formula in Ljung (1985) which is asymptotic in both the number of data and the model order.

In practical applications the number of data is often sufficiently large, but the assumption of the model order tending to infinity is not very realistic. Therefore it is advisable to use a covariance formula which is asymptotic only in the number of data, but valid for finite model order. Such formulas were known for decades (Ljung, 1999), but input design problems with cost functions involving them were too difficult to solve. This changed with the appearance of powerful convex optimization methods. In Hildebrand and Gevers (2003), a more complicated input design criterion based on the more exact covariance formula was treated whose minimization was performed by the ellipsoid method (Boyd, El Ghaoui, Feron, & Balakrishnan, 1994). This is a particularly robust method which requires only information about the gradient of the cost function and about supporting hyperplanes to the feasible set of search parameters, and can hence minimize a large variety of cost functions. Sometimes input design problems or their relaxed versions can be solved by semidefinite programming (see e.g. Bombois, Scorletti, Gevers, Hildebrand, & van den Hof, 2004). A summary of such methods is given in Jansson (2004).

With these methods at hand, we are able to tackle problems where the cost criterion for the input design problem is defined directly in terms of the expected performance degradation deriving from the variance error. We present an optimal input design for a closed-loop experiment that directly minimizes a widely used closed-loop cost function, namely the output variance of the achieved closed-loop system. This is the problem considered in Gevers and Ljung (1986), but here we use the more accurate covariance formula in Ljung (1999) that is asymptotic only in the number of data. This constitutes the novelty of our contribution. We solve this problem for a minimum phase ARMAX model structure with unit delay with a fixed controller in the loop during the identification. We thus optimize not the entirety of experimental conditions, namely the controller and the external input, but only the external input, while the controller is prespecified, but not necessarily optimal. The approach can be applied also to other model structures or for bigger delays.

We show that the considered design problem is equivalent to a classical L-optimality problem, so that the cost function depends on the experimental conditions via the average per data sample information matrix M¯ of the experiment. Standard optimization methods designed to find the optimal average information matrix, i.e. the one that minimizes the cost function, allow as well to find an input power spectrum that produces this information matrix (see e.g. Hildebrand & Gevers, 2003). From this power spectrum one can directly design an optimal multisine input for the identification experiment.

In Gevers and Ljung (1986), the optimization of the same cost function was carried out not only with respect to the input power spectrum, but simultaneously also over the controller used during the identification. This problem was solved analytically, but at the cost of using a simpler covariance formula which is asymptotic in both model order and the number of data. In particular, in Gevers and Ljung (1986) the optimality of the MVC itself was established, and it was shown that a reference input does not have any influence on the expected performance degradation of the achieved loop. We investigate whether these results still hold if a more accurate parameter covariance formula is used, and partly confirm them. While the MVC is not optimal in general, for certain model structures the results of Gevers and Ljung (1986) carry over one to one. Under an output power constraint there exist also cases where the MVC is still optimal, but the independence of the performance degradation vis-à-vis the reference input is lost. We also show that the MVC is asymptotically optimal for small and large output power constraints.

The paper is structured as follows. In the next section, we define the input design problem in a closed-loop setting in general. In Section 3, we consider an ARMAX model structure and introduce the parametrizations of the involved transfer functions and the controller. In Section 4, we apply the prediction error framework to compute the cost function to be minimized by the input design procedure. Section 5 is devoted to the computation of the weighting matrix of the obtained L-optimality problem. In the next section we solve this problem. A simulation example that demonstrates the benefits of using a designed input as opposed to a white noise reference input of the same energy is presented in Section 7. In Section 8, we address the question of optimality of the MVC. Finally, some conclusions are drawn in the last section.

Section snippets

General setting

Let us consider the scheme depicted in Fig. 1. We want to minimize the closed-loop output varianceJ(G0,H0,K)=E¯yt2=σ22π-ππ|Hcl|2dω=σ22π-ππ|H0S|2dωby choosing the controller K. Here S=1/1+G0K is the sensitivity function and Hcl is the closed-loop transfer function from the noise e to the output y. The optimal controller Kopt minimizing this cost function is dependent on the transfer functions G0,H0, Kopt(G0,H0)=argminKJ(G0,H0,K).Consider the situation where the system (G0,H0) is unknown and is

Model parametrizations

Let us assume that the true system dynamics can be described by an ARMAX model structure with delay 1,yt++anayt-na=b1ut-1++bnbut-nb+et++cncet-nc.et is normally distributed white noise with variance σ2. The vector of parameters is given by ρ=(a1,,ana,b1,,bnb,c1,,cnc)T,and the true system is given by some parameter vector ρ0. Define A(z,ρ)=1+a1z-1++anaz-na,B(z,ρ)=b1z-1++bnbz-nb,C(z,ρ)=1+c1z-1++cncz-nc,where z-1 is the backward shift operator.

Assume that the closed-loop with some current

ARMAX predictor in closed-loop

The closed-loop for a model G(z,ρ),H(z,ρ) is therefore given by yt=Gcl(z,ρ)rt+Hcl(z,ρ)et, Gcl(z,ρ)=G(z,ρ)1+G(z,ρ)K(z),Hcl(z,ρ)=H(z,ρ)1+G(z,ρ)K(z).For an ARMAX model structure, the closed-loop system transfer functions are Gcl(z,ρ)=Bcl/Acl,Hcl(z,ρ)=Ccl/Acl with Acl=AD+BP,Bcl=BD,Ccl=CD. The standard ARMAX predictor is given by the expression CDy^(t|ρ)=BDrt+(CD-AD-BP)yt.Applying standard techniques (Ljung, 1999), we get for the parameter covariance Cov(ρ^N) the asymptotic expression1N12π-ππ1σ2|Hcl

Hessian calculus

While the covariance matrix in the scalar product (2) is given by the asymptotic expression (1), we still need to compute the weighting matrix 2J/ρ2(ρ0). This will be accomplished in this section.

The MVC for a minimum phase ARMAX structure is given by ut=-K(z,ρ)yt=-(P(z,ρ)/D(z,ρ))ut with P(z,ρ)=C(z,ρ)-A(z,ρ), D(z,ρ)=B(z,ρ). The achieved closed-loop transfer function from et to the output yt is then given byHclach(z,ρ)=C0(z)D(z,ρ)A0(z)D(z,ρ)+B0(z)P(z,ρ)=C0(z)B(z,ρ)A0(z)B(z,ρ)+B0(z)(C(z,ρ)-A(z,ρ

Solution of the input design problem

We derived an L-optimality cost function to be minimized by choice of the reference input rt. The positive semidefinite weighting matrix (3) is calculated from prior information about the plant and is supposed to be fixed and known for the purpose of input design. Naturally, the performance degradation will decrease if the reference input power increases, which suggests to choose a large reference input. On the other hand, a large reference input may provoke non-linearities in the behaviour of

Simulations

In this section, we demonstrate the advantages of our input design method on a simulation example. Consider the ARMAX model structure G(z,ρ)=B(z,ρ)A(z,ρ)=b1z-1+b2z-21+a1z-1+a2z-2,H(z,ρ)=C(z,ρ)A(z,ρ)=1+c1z-1+c2z-21+a1z-1+a2z-2with the true parameter vector value ρ0=(a1*,a2*,b1*,b2*,c1*,c2*)=(0.15,0.4872,0.5,0.2,0.3,0.5).The open-loop system is governed by the dynamics y=G(z,ρ0)u+H(z,ρ0)e, where y is the plant output, u is the input and e is Gaussian white noise with variance σ2.

We intend to

Optimality of the MVC

So far we optimized only the reference input spectrum for the identification experiment, while the controller K present in the loop during the identification was fixed. However, the choice of the controller also influences the parameter covariance matrix, and one can consider the problem of minimizing the expected performance degradation (2) with respect to both the input power spectrum Φr and the controller K. This task was accomplished in Gevers and Ljung (1986), but for the construction of

Conclusions

We considered an optimal input design problem for a closed-loop identification experiment intended for controller design. The reference input for the identification, more precisely the reference input power spectrum, is optimized with respect to a closed-loop cost function measuring directly the performance of the achieved loop. The cost function depends on the to-be-designed controller, which is naturally not available before the experiment. We hence have to take into account the dependence of

Acknowledgements

The authors would like to thank Michel Gevers and Brian Anderson for many interesting discussions on the subject, and especially Michel Gevers for a careful reading of the manuscript and many suggestions for improving the paper.

Roland Hildebrand was born in Leipzig, East Germany, in 1975. He received his diploma from the Free University of Berlin, Germany, and the Ph.D. degree in mathematics from Moscow State University, Russia. From 2001 to 2003 he was post-doc at the Université Catholique de Louvain, Louvain-la-Neuve, Belgium. Since 2003 he is at the Laboratoire de Modélisation et Calcul, Grenoble, France. Currently, he holds a research position from the CNRS.

References (19)

  • U. Forssell et al.

    Some results on optimal experiment design

    Automatica

    (2000)
  • M. Gevers et al.

    Optimal experiment designs with respect to the intended model application

    Automatica

    (1986)
  • T.S. Ng et al.

    Optimal experiment design for linear systems with input-output constraints

    Automatica

    (1977)
  • A. Ben-Tal et al.

    Lectures on modern convex optimization—analysis, algorithms, and engineering applications

    (2001)
  • Bombois, X., Scorletti, G., Gevers, M., Hildebrand, R., & van den Hof, P. (2004). Cheapest open-loop identification for...
  • S. Boyd et al.

    Linear matrix inequalities in system and control theory

    (1994)
  • R. Hildebrand et al.

    Identification for control: Optimal input design with respect to a worst-case ν-gap cost function

    SIAM Journal of Control Optimization

    (2003)
  • Jansson, H. (2004). Experiment design with applications in identification for control. Ph.D. Thesis, Royal Institute of...
  • S. Karlin et al.

    Tchebycheff systems: With applications in analysis and statistics

    (1966)
There are more references available in the full text version of this article.

Cited by (21)

  • Neural networks in virtual reference tuning

    2011, Engineering Applications of Artificial Intelligence
    Citation Excerpt :

    The term identification for control (Gevers, 1993; Van den Hof and Schrama, 1995) arose to deal with several issues concerning identification as a tool for control design: which is the frequency range of interest, which is the optimal complexity of the model, which is the optimal experiment (open/closed loop, online/off-line, selection of plant input), etc. This is also a field of active research at present (see, for example Bombois et al., 2006; Gevers, 2005; Hjalmarsson, 2005; Hildebrand and Solari, 2007). However, most of the above contributions are focused on linear systems, so they may fail when the issue is to obtain a model for a nonlinear plant.

  • Conditions when minimum variance control is the optimal experiment for identifying a minimum variance controller

    2011, Automatica
    Citation Excerpt :

    This was first established under an output power constraint for models of large orders (Forssell & Ljung, 2000; Gevers & Ljung, 1986; Hjalmarsson & Gevers, 1996). More recently, an important relaxation of the conditions was derived in Hildebrand and Solari (2007) where it was shown that this also holds for ARMAX models of finite order (subject to some degree and factorization conditions) under general input–output power constraints. Notice that the optimality of the minimum variance controller does not hold for all situations.

  • Closed-loop optimal input design: The partial correlation approach

    2009, IFAC Proceedings Volumes (IFAC-PapersOnline)
View all citing articles on Scopus

Roland Hildebrand was born in Leipzig, East Germany, in 1975. He received his diploma from the Free University of Berlin, Germany, and the Ph.D. degree in mathematics from Moscow State University, Russia. From 2001 to 2003 he was post-doc at the Université Catholique de Louvain, Louvain-la-Neuve, Belgium. Since 2003 he is at the Laboratoire de Modélisation et Calcul, Grenoble, France. Currently, he holds a research position from the CNRS.

Gabriel Elías Solari was born in Argentina, where he received the Ingeniero Electrónico degree from the Universidad Nacional de Rosario. From 1997 to 2001 he was with SIDERCA, a seamless steel tube company.

He obtained a fellowship from the Belgian Government and completed his M.Sc. and his Ph.D. degrees in Applied Sciences in 2003 and 2005, respectively, both at the CESAME (Centre for Systems Engineering and Applied Mechanics) at the Université Catholique de Louvain in Louvain-la-Neuve, Belgium. Since then he has been working at TenarisDalmine, a big seamless tube manufacturer based at Dalmine, Italy, as a senior control engineer specialized in casting plants and arc electric furnaces.

His research interests include process automation, practical applications of control theory, layout and logistic optimization, filter design, improvement of PID controllers, identification and electromagnetic phenomena.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Brett Ninness under the direction of Editor Torsten Söderström. This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with its authors.

View full text