UTILIZING PRIOR KNOWLEDGE IN ROBUST OPTIMAL EXPERIMENT DESIGN. EE & CS, The University of Newcastle, Australia EE, Technion, Israel.

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1 UTILIZING PRIOR KNOWLEDGE IN ROBUST OPTIMAL EXPERIMENT DESIGN Graham C. Goodwin James S. Welsh Arie Feuer Milan Depich EE & CS, The University of Newcastle, Australia 38. EE, Technion, Israel. Abstract: In this paper we propose a new approach to robust optimal experiment design. The ey departure from earlier wor is that we specifically account for the fact that, prior to the experiment, we possess only partial nowledge of the system. We also give a detailed analysis of the solution for a simple case and propose a concave optimization algorithm that can be applied more generally. Keywords: Experiment Design, Optimal Input Design.. INTRODUCTION It is well nown that the choice of experimental conditions has a strong influence on the accuracy of models obtained from system identification experiments. This has motivated substantial research on experiment design over almost a century. Early results appear in the statistics literature Wald, 943; Cox, 958; Kempthorne, 95; Kiefer and Wolfowitz, 96; Karlin and Studden, 966; Federov, 97; Whittle, 973; Wynn, 97). This wor was later adapted to the problem of identification of dynamic systems Levadi, 966; Gagliardi, 967; Goodwin and Payne, 973; G.C. Goodwin and Murdoch, 973; G.C. Goodwin and Payne, 973; Arimoto and Kimura, 973; Mehra, 974). Some of the later wor is summarised in Goodwin and Payne, 977; Zarrop, 979). Our focus here will be on experiment design for dynamic systems. Early wor on this problem focused predominately on frequency domain designs. More recently, there has been substantial effort devoted to the inter-relationship between identification and control together with the associated issue of input signal design Hilderbrand and Gevers, 3; Hjalmarsson, 5). However, a major drawbac of all the above wor is that, generically, the optimal test signal for dynamic system identification is a function of the unnown system i.e. it depends on the very thing that the experiment is aimed at finding). Indeed, quoting Hjalmarsson, 5): It should be noted that, as usual in experiment design, in order to compute the optimal design the true system has to be nown. Methods that are robust with respect to uncertainty about the system is a wide open research field. The goal of the current paper is to propose a methodology for addressing this problem. In particular, we formulate a robust optimal experiment design criterion. We also demonstrate the use of this criterion for a simple case. The layout of the remainder of the paper is as follows: In section we give a general formulation of the robust optimal experiment design problem. In section 3, we focus on a simple one parameter) problem so as to give insight into the problem. In section 4 we convert the problem to an approximate matrix formulation and discuss asymptotic behavior when the prior nowledge is diffuse. In section 5 we describe an algorithm for computing the robust optimal experiment and give a numerical example. In section 6 we describe the extension to multi-parameter systems. Finally, in section 7 we draw conclusions.

2 . GENERAL FORMULATION Our focus in the current paper is on how to design the experimental conditions so that the information obtained from a particular experiment is maximized in some specific sense. To motivate our approach we consider a single input single output linear discrete time system of the form:- y t G q)u t + G q)w t ) G q), G q) are rational transfer functions in the forward shift operator q, G ) and {w t } is Gaussian white noise of variance Σ. We recall that the log lielihood function for data Y given parameters β, is given by log py β) N log π N log Σ Σ N ε t, t ) ε t G q) [y t G q)u t ]. 3) Fisher s information matrix is obtained by taing the following expectation Goodwin and Payne, 977) [ ) ) ] T log py β) log py β) M E Y β β β 4) log py β) N ε t ε t β Σ β t [ ] 5) Σ N N ε t Σ β Σ t { ε t β G q) G β q)ε t + G } β q)u t 6) We assume an open loop experiment so that w t and u t are uncorrelated. We also assume that G q), G q) and Σ have no common parameters. We let β [ T, γ T, Σ ] T denotes the parameters in G q) and γ denotes the parameters in G q). Taing expectations, as in 4), we have that M can be partitioned as [ ] M M 7) M M is the part of the information matrix related to and M is independent of the input. Then M Σ N t εt ) εt ) T, 8) ε t G q) G q)u t. 9) Notice that M depends on the full parameter vector β. Assuming N is large, it is more convenient to wor with the scaled average information matrix for the parameter, i.e. Mβ, φ u ) lim N N M Σ. ) Utilizing Parseval s Theorem, we finally have that Mβ, φ u ) π {[ G e jω ] ) G R e jω ) π [ G e jω ] T } ) φ u e jω )dw ) φ u ) is the discrete input spectral density. It is also possible to do a parallel development Goodwin and Payne, 977) for continuous time models. In the latter case, ) is replaced by {[ ] G Mβ, φ u ) R G jω) jω) ] } T [ G jω) φ u ω)dw ) G and G are continuous time transfer functions assumed independently parameterized) and φ u ω) is the continuous time input spectral density. Since M is a matrix, we will need a scalar measure of M for the purpose of experiment design. In the nominal case treated in the literature i.e. when β is assumed nown), several measures of the size of M have been proposed. Examples are i) D - optimality Goodwin and Payne, 977) J d β, φ u ) det Mβ, φ u ) 3) ii) Experiment design for robust control Hjalmarsson, 5; Hilderbrand and Gevers, 3) { [ ] } J cc β, φ u ) sup fβ, ω) M f c β, ω) 4) ω fβ, ω) is a frequency dependent vector related to the ν-gap Hilderbrand and Gevers, 3). Thus nominal experiment design is aimed at choosing φ u ) to maximize a function of the type shown in 3), 4). Note, however, that the optimal input spectrum depends, inter-alia on the unnown parameter vector β. To address this paradox, we propose an alternative robust optimal experiment design procedure. We assume that we do not have complete a-prior nowledge of the true parameter value β. Instead, we assume that the parameters can tae any value in a compact set Θ. We propose that φ u ) be chosen as: φ u arg max φ u S min β Θ J Mβ, φ u ), β ) 5) J is any suitable scalar measure of M. Note that we also allow J to depend explicitly on β. This is standard in nominal experiment design, see for example 4). Also, we can use the explicit dependence on β to associate different weighting to Mβ, φ u ) depending on the nature of the prior nowledge regarding β. We also need to constrain the allowable set of input signals. A typical constraint used in experiment design is that the input energy is constrained i.e. we define

3 { S φ u ) : } φ u ω)dω 6).5 3. A SIMPLE EXAMPLE. To give insight into the robust optimal experiment design problem, we consider a simple continuous time problem G s) and G s) s + 7) For the model 7), it follows that M φ u, ) M, ω) φ u ω) dω 8) M, ω) is the single frequency information matrix given by M, ω) G, ω) 4 + ). 9) 3. Nominal Optimal Experiment Design. Nominal experiment design assumes that an initial estimate of is available. Based on this information, the function φ u ) is chosen so as to maximize some scalar function of M φ u, ) subject to a constraint on the output or the input power. One interesting observation is that equations 8), 9) define a convex combination of the set of all single frequency matrices. This leads to several useful facts, e.g. any value of M φ u, ) for φ u S can be generated by a finite number of frequencies. Also, in the scalar parameter case of equations 8), 9) it can actually be shown that we need only use a single frequency input for optimal experiment design Goodwin and Payne, 977), namely, φ u ω) δ ω ω ). Moreover, by differentiation it is readily seen that the optimal input frequency is ω ) This is an intuitively pleasing result, i.e. one places the test signal at the nominal) 3dB brea point. However, equation ) reinforces the fundamental difficulty in nominal experiment design, namely the optimal experiment depends on the very thing that the experiment is aimed at estimating. To gauge how important the dependence on is, we note that M, ω) in our example decays at the rate of 4dB per decade as a function of both and ω. Hence, given the prior estimate of the parameter,, say we choose ω for the input signal frequency. Also, say that the true parameter lies in the range Cost.5..5 Fig.. M, φ u ) as a function of for nominal input dots), /f noise solid). )., then min Θ M, ω) is approximately / th of the nominal value! This seems to suggest that nominal experiment design is limited to those cases a good prior estimate is available. A plot of M β, φ u ) versus is given in Figure. The reason for multiplying by is that M is a variance measure and thus [ M ] gives relative mean square) errors. 3. Robust Optimal Experiment Design We next turn to the robust experiment design described in section. For the scalar parameter problem all measures of M are equivalent and we thus use J M, φ u ), ) M, φ u ) ) Thus, our robust optimal experiment design can be stated as φ u arg max min φ u S Θ + ) φ u ω) dω ) Θ { : }. 3) In subsequent sections, we will give further insights into the above design problem. We first observe the following property of φ u: Lemma. Consider the problem stated in equation ). Then the optimal energy input has all its energy inside Θ. Namely, φ u ω) dω 4) ω / Θ Proof. Let J M, φ u), ) be the optimal solution of ). Then,

4 J M, φ u), ) J M, φ u ), ) Let us now define φ u ω) φ u ω) X Θ ω)+δ ω ) + ) φ u ω) dω 5) for all φ u S ω / Θ φ u ω) dω 6) X Θ ω) is the indicator function for the set Θ. Then we readily observe that φ u ω) dω and φ u ω), so that φ u S. Furthermore, since ω ) + 4 and φ u ω) then ω Θ + ) φ u ω) dω + ) φ u ω) dω ) φ u ω) dω ω / Θ φ u ω) dω so that by 5) we must have ω / Θ φ u ω) dω, which completes the proof. 4. APPROXIMATE ANALYSIS To gain insight into the robust experiment design problem as formulated in the last section, we next approximate the integral in equation ) by a Riemann sum. Specifically, utilising Lemma, we choose a grid of N points ω m m for m N so that ω, ω N N. Then J m ω Θ N n n m N n m + m n m ) φ u ω) dω ) φ u ω n ) ω n+ ω n ) + A m,n E n 7) A m,n n m n m ) > 8) + and E n ω n+ ω n ) φ u ω n ). 9) Note that the matrix A {A m,n } is symmetric and has positive entries. We can now state the following discrete alternative to the optimization problem in equation ): E arg max min e T m AE ) 3) E S d m<n S d { E R N : T E, E n }, E [ E E E N ] T, em is the mth column of the N dimensional identity matrix and is an N dimensional vector of ones. We next show that if there exists a feasible Ẽ S d such that all entries of AẼ are equal, then Ẽ is optimal. Lemma 3. Let Ẽ be defined by Then, if Ẽ S d we have Ẽ A T A. 3) E Ẽ. 3) Proof 4. Let E be the optimal solution for 3) and Ẽ S d. Then we have ) T E Ẽ 33) and, since e T maẽ for all m < N, T A e T m AE Namely e T ma min m<n e T m AE ) e T m AẼ T A ) E Ẽ for all m < N. 34) On the other hand, using the symmetry of A we obtain ) ) T ) Ẽ T A E Ẽ AẼ E Ẽ ) T A T E Ẽ. 35) Since by definition, all the entries of Ẽ are nonnegative we must conclude from 34) and 35) that E Ẽ which completes the proof. The discussion above holds for any choice of grid for { ωm m }. Let us now consider a special choice of a logarithmic grid: ω m m β m 36) β / ) N. 37) With this specific choice we have the following result for our original problem ), Lemma 5. The robust optimal test signal for diffuse prior information) has spectrum approximately given by φ u ω) ). 38) ω ln Proof 6. Note first that since >, ω ln ) >. Furthermore, clearly, ω ln dω ω ln ), so we have ) S. Let us now choose a large integer N

5 and the corresponding grid as defined in 37) and 38). Then, by definition φ u ω m ) φ u,m ω m ln ) and by 9) Em ω m+ ω m ) ) ω m ln β ). ln Clearly, Em > and T E β )N ) ln ) as β )N ) lim N ln ) ). On the other hand, we note that for the matrix A in 8), apart from the first and last few rows, the row sum is very nearly constant. Hence, AE ln β )A γ for some γ >. Namely, the conditions in Lemma 3 are approximately satisfied which completes the proof. Remar 7. The immediate consequence of Lemma 5 is that band limited f noise is an approximation to the robust optimal input for our example see Figure ). 5. A NUMERICAL EXAMPLE We show that the optimization problem 3) can be converted into standard linear programming LP). Let us denote [ ] x F R N+) 39) E then we can readily show that 3) is equivalent to the following optimization problem: subject to: max F CF 4) ÃF BF 4) [ ] A Ã R N N+) I B [ ] T R N+) C [ ] R N+) 4) This problem is readily recognized as a LP problem. Numerical Example We consider the scalar parameter problem described in section 3 we assume.,, N and compare i) A nominal input of frequency rad sec Note that this is the optimal input if the initial estimate of the parameter is ). ii) Band limited white noise input. iii) Band limited f noise input. Energy Fig.. Values of E for robust optimal input. iv) The robust optimal input generated by LP. Relative errors for the different experimental conditions are shown in Table. ω min Θ ) M, φ u) Single frequency at ω. Band limited white noise.64 Band limited /f noise.543 Optimal input.778 Table. Values of Cost Function for Different Input Signals We see from Table that /f noise is approximately an order of magnitude better than a white noise input in terms of the cost function 3). Furthermore, going to the true optimum gives a further 4% improvement. The optimal input energy is shown in Figure and Figure 3 shows the corresponding values of M, φ u ) as a function of. It is interesting to note from Figure 3 that M, φ u) is an almost constant function of. This should be compared with the result in Lemma 3. The latter Lemma predicts that if the input that generates M, φ u) independent of is feasible, then it is optimal. Here, the input which gives M, φ u ) constant is not feasible but we see from Figure 3 that the optimal values of M, φ u) are almost constant. 6. GENERALIZATION TO MULTI-PARAMETER PROBLEMS For the multi-parameter case we return to the general expression for M β, φ) given in ) and ). We can again convert this problems into an approximate discrete form as was done in section 4. We write Q E) m A m E m ; Θ 43) as an approximation to the integral in ). The index refers to the discretized) element of the parameter set Θ, the index m denotes the frequency and E m denotes the input energy at the mth frequency.

6 Cost Fig. 3. Variation of cost function with for optimal input. In this case, we see that Q is a matrix for each discrete parameter value. Hence, as discussed in section, we need a measure of the size of Q. Say we choose J Q, ), then the robust optimal design becomes E arg max E S d min J Q E), ) 44) Obviously, there are many choices for the scalar function J. One possible choice is J Q E), ) λ min {Q E)} 45) This cost function is motivated by the following observation: Defining the scalar function f E) minλ min {Q E)} we note that, by 43) we have f µe + µ) E ) minλ min {Q µe + µ) E )} min λ min {µq E ) + µ) Q E )} min µλ min {Q E )} + µ) λ min {Q E )}) µminλ min {Q E )} + µ) min λ min {Q E )} µf E ) + µ) f E ) which shows that f E) is a concave function of E. Hence, solving 44) becomes a standard concave maximization. This latter aspect is the subject of the journal version of this paper. 7. CONCLUSION This paper has proposed a robust optimal experiment design procedure. We have argued for a simple case, that a near optimal input is band-limited /f noise. We have also proposed an algorithm to design robust experiments in more general cases and have presented results showing the gains obtained from the use of the algorithm compared with using either the nominal optimal experiment, band limited white noise or band-limited /f noise. 8. REFERENCES Arimoto, S. and H. Kimura 973). Optimal input test signals for system identification - an information theoretic approach. Int. J. Sytems Sci. 3), Cox, D.R. 958). Planning of Experiments. Wiley, New Yor. Federov, V. 97). Theory of Optimal Experiments. Academic Press, New Yor and London. Gagliardi, R.M. 967). Input selection for parameter idenfication in discrete systems. IEEE Transactions on Automatic Control 5), G.C. Goodwin, J.C. Murdoch and R.L. Payne 973). Optimal test signal design for linear single input - single output system identification. Int. J. Control 7), G.C. Goodwin, R.L. Payne and J.C. Murdoch 973). Optimal test signal design for linear single input - single output closed loop identification. CACSD Conference, Cambridge. Goodwin, G. C. and R. L. Payne 977). Dynamic System Identification - Experiment Design and Data Analysis. Vol. 36 of Mathematics in Science and Engineering. Academic Press. Goodwin, G.C. and R.L. Payne 973). Design and characterisation of optimal test signals for linear single input - single output parameter estimation. 3rd IFAC Symposium, The Hague/Delft. Hilderbrand, R. and M. Gevers 3). Minimizing the worst-case nu-gap by optimal input design.. 3th IFAC Symposium on System Identification.. pp Hjalmarsson, H. 5). From experiment design to closed-loop control.. Automatica 43), Karlin, S. and W.J. Studden 966). Optimal experimental designs. Annals. Math. Stat. 37, Kempthorne, O. 95). Design and Analysis of Experiments. Wiley, New Yor. Kiefer, J. and J. Wolfowitz 96). The equivalence of two extremum problems. Can. J. Math., Levadi, V.S. 966). Design of input signals for parameter estimation. IEEE Transactions on Automatic Control ), 5. Mehra, R.K. 974). Optimal inputs for system identification. IEEE Transactions on Automatic Control pp. 9. Wald, A. 943). On the efficient design of statistical investigations. Annals. Math. Stat. 4, Whittle, P. 973). Some general points in the theory of optimal experimental design. J.R. Stat. Soc. ), 3 3. Wynn, H.P. 97). Results in the theory and construction of d-optimum experimental designs. J.R. Stat. Soc. ), Zarrop, M. 979). Optimal Experiment Design for Dynamic System Identification. Springer-Verlag.