Optimal Experiment Design with Diffuse Prior Information Cristian R. Rojas Graham C. Goodwin James S. Welsh Arie Feuer Abstract— In system identification one always aims to learn as much as possible about a system from a given observation period. This has led to on-going interest in the problem of optimal experiment design. Not surprisingly, the more one knows about a system the more focused the experiment can be. Indeed, many procedures for ‘optimal’ experiment design depend, paradoxically, on exact knowledge of the system pa- rameters. This has motivated recent research on, so called, ‘robust’ experiment design where one assumes only partial prior knowledge of the system. Here we go further and study the question of optimal experiment design when the a-priori information about the system is diffuse. We show that band- limited ‘1/f ’ noise is optimal for a particular choice of cost function. I. I NTRODUCTION In system identification, there is always a strong incentive to learn as much about a system as possible from a given observation period. This has motivated substantial interest in the topic of optimal experiment design. Indeed, there exists a body of work on this topic, both in the statistics literature [5, 14, 7] and in the engineering literature [17, 10, 27]. Much of the existing literature is based on designing the experiment to optimize some scalar function of the Fisher Information Matrix [10, pg. 6]. However, a fundamental dif- ficulty is that when the system response depends non-linearly on the parameters, the Information Matrix depends, inter- alia, on the true system parameters. Moreover, we note that models for dynamical systems (even if linear) typically have the characteristic that their response depends non-linearly on the parameters. Hence, the information matrix for models of dynamical systems generally depends upon the true system parameters. This means that experiment designs which are based on the Fisher Information Matrix will, in principle, depend upon knowledge of the true system parameters. This is paradoxical since the ‘optimal experiment’ then depends on the very thing that the experiment is aimed at estimating [13, pg. 427]. The above reasoning has motivated the study of, so called, ‘robust’ optimal experiment designs with respect to uncertainty on a priori information. In this vein, various approaches have been proposed, e.g. (i) Iterative design where one alternates between param- eter estimation and experiment design based on the C. R. Rojas, G. C. Goodwin and J. S. Welsh are with the School of Electrical Engineering & Computer Science, The University of Newcastle, NSW, Australia 2308. cristian.rojas@studentmail.newcastle.edu.au, james.welsh@newcastle.edu.au, graham.goodwin@newcastle.edu.au A. Feuer is with the Department of Electrical Engineering, Technion, Haifa 32000, Israel. feuer@ee.technion.ac.il current estimates [4, 18, 25]. (ii) Bayesian design where one optimizes some function of the expected information matrix, with the expectation taken over some a-priori distribution of the parame- ters [1, 3, 6]. (iii) Min-Max design in which one optimizes the worst case over a bounded set of a-priori given parameter values [20, 8, 21]. The latter designs mentioned above are closely related to game theory. Indeed, game-theoretical ideas have been used to characterize the optimal robust (in the min-max sense) experiment. For example, several papers have studied different types of one-parameter robust experiment design problems [21, 11]. It has been shown for these problems that the optimal min-max experiment has many interesting properties, e.g. it exists, it is unique, it has compact support in the frequency domain and it is characterized by a line spectrum. For multi-parameter problems, one usually needs to use gridding strategies to carry out the robust designs numerically [21, 25]. A surprising observation from recent work on min-max optimal experiment design is that band-limited ‘1/f ’ noise is actually quite close to optimal for particular problems. Indeed, ‘1/f ’ noise has been shown to have a performance which is within a factor of 2 from the performance of robust optimal designs for first-order and resonant systems [21, 11]. It is important to note, however, that the proof of near optimality depends on a particular property of these systems which allows one to scale the parameters with respect to frequency. Here we ask a more general question: Say we are just beginning to experiment on a system and thus have very little (i.e. diffuse) prior knowledge about it. What would be a ‘good’ initial experiment to use to estimate the system? In this case we consider as diffuse prior information that the interesting part of the frequency response of the system lies in an interval [a, b]. This implies that we are seeking an experiment which is ‘good’ over a very broad class of possible systems. In this paper, we propose a possible solution to this problem, being that the experiment should consist of bandlimited ‘1/f ’ noise. The paper is structured as follows. In Section II we discuss the problem of measuring the ‘goodness’ of an experiment by using a system independent criterion. Section III gives some desirable properties that such a measure would be expected to possess. In Section IV we consider a typical input constraint generally used in experiment design. Section V shows a preliminary result for choosing a suitable cost function which satisfies the properties developed in Section