THIS paper studies the input design problem in system identification.

Transcription

1 1534 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 10, OCTOBER 2005 Input Design Via LMIs Admitting Frequency-Wise Model Specifications in Confidence Regions Henrik Jansson Håkan Hjalmarsson, Member, IEEE Abstract A framework for reformulating input design problems in prediction error identification as convex optimization problems is presented. For linear time-invariant single input/single output systems, this framework unifies extends existing results on open-loop input design that are based on the finite dimensional asymptotic covariance matrix of the parameter estimates. Basic methods for parametrizing the input spectrum are provided conditions on these parametrizations that guarantee that all possible covariance matrices for the asymptotic distribution of the parameter estimates can be generated are provided. A wide range of model quality constraints can be hled. In particular, different frequency-by-frequency constraints can be used. This opens up new applications of input design in areas such as robust control. Furthermore, quality specifications can be imposed on all models in a confidence region. Thus, allowing for statements such as with at least 99% probability the model quality specifications will be satisfied. Index Terms Convex optimization, input design, robust control, system identification. NOTATION The real the complex numbers are denoted, respectively. Furthermore, denotes the linear space of realvalued vectors of length is the corresponding space of real-valued matrices with rows columns. We use analog definitions for. The complex conjugate of is denoted by, the transpose by the complex conjugate transpose by. The matrix operation extracts the largest eigenvalue of. The determinant of is given by the trace by.for positive definite positive semidefinite matrices we use the notation, respectively. is the zero matrix of dimension by is the identity matrix of dimension by. The probability that the stochastic variable is less than is denoted by the expectation of is denoted by. denotes the normal distribution. Furthermore is defined such that denotes a distributed rom variable with degrees of freedom. The shorth notation is used to denote. The time shift operator is given by, i.e.,, consequently. Furthermore, means that is defined as. Manuscript received June 8, 2004; revised May 31, Recommended by Guest Editor A. Vicino. The authors are with the Department of Signals, Sensors Systems, Kungliga Tekniska Högskolan, S Stockholm, Sweden ( henrik.jansson@s3.kth.se; hakan.hjalmarsson@s3.kth.se). Digital Object Identifier /TAC denotes the family of functions with. A state-space realization of a transfer function is a fourtuple of matrices of suitable dimensions such that. Finally, is the -norm of the stable discrete transfer function, i.e., denotes the space of complex-valued Lesbegue measurable functions defined on the unit circle such that the -norm is finite. I. INTRODUCTION THIS paper studies the input design problem in system identification. We will consider identification of time-discrete linear time-invariant single-input single-output systems within the prediction error framework, [1]. The system is modeled by is the output, the input is zero mean white noise with finite variance. The system dynamics the noise dynamics are modeled by the transfer operators, respectively. They are parametrized by a vector. Assume that the model is flexible enough to capture the dynamics of the true system, i.e., the true system is in the model set, thus, there exists a parameter vector such that denotes the true system transfer function is defined as the unique stable inversely stable monic transfer function satisfying is the spectrum of the actual additive noise acting on the output of the system is some constant that can be thought of as the true variance of. Denoting the parameter estimate based on input/output data of sample size by, the problem we will consider in this contribution is how to select the input such that certain statistical specifications on the accuracy of are satisfied. Unfortunately, it is only possible to exactly characterize the statistical properties of for some special model structures. Briefly, the model structure has to be on linear regression form with deterministic regressors. There is an emerging theory for general finite data quality assessment [2], [3] but it does not appear to be amenable to input design yet. The route most approaches to input design take, the one that we will pursue, is to base the design on quality measures that are valid asymptotically in the sample size. It is well known (1) /$ IEEE

2 JANSSON AND HJALMARSSON: INPUT DESIGN VIA LMIs ADMITTING FREQUENCY-WISE MODEL SPECIFICATIONS 1535 that the estimate based on a quadratic prediction error criterion, under some other mild assumptions, has the following asymptotic properties [1]: is the sensitivity of the onestep ahead predictor with respect to the model parameters. So when the system is in the model set, the estimate will converge to the true system the covariance of the estimation error decays as. However, it is not only the experiment length that will influence the estimation accuracy. Assuming open loop operation, the inverse of the covariance matrix,, isan affine function of the input spectrum, is given by [1] (2) been to minimize some scalar function of the covariance matrix with constraints on input /or output power. Examples of commonly used criteria are is an application dependent weighting matrix. An important discovery in this line of work was that all achievable information matrices can be obtained using an input comprising a finite number of sinusoids [11]. In the 1980s, input design attained renewed interest when the control community recognized the utility of experiment design as a way to obtain suitable models for control design. Frequency function estimates are typically more relevant for control design than the estimated parameters themselves. Under the assumption, (2) implies that [13] with (4) 1. The relation (3) shows exactly how the variance error is related to the input spectrum. Hence, inputs with different spectra will influence the estimation error in different ways. Therefore, (3) is very useful when considering input design. Furthermore, in open loop, the input spectrum is actually the only quantity that can be used to shape the asymptotic covariance. The limitation of basing the input design on asymptotic results, such as (2), is, of course, that the designs will be valid only for sufficiently large sample sizes. There is no general limit on how large has to be for the asymptotic results to be reliable. Monte Carlo simulations indicate that for typical system identification applications, the results are quite reliable for [1]. However, it is not only the sample size but also the signal to noise ratio model complexity that influence the accuracy of the asymptotic theory. Despite this shortcoming, the relevance of using asymptotic theory for input design has been exemplified extensively in the literature; see, e.g., [1], [4] [6], the references therein. In the input design example in Section VIII to follow, the asymptotic theory yields input designs with predictable performance down to. It is also worth to notice that the asymptotic theory can be valid also when the variance errors are significant, i.e., when input design is important. We refer the reader to [7, Fig. 1] for an illustration of this. Recent studies of the validity limitations of the asymptotic prediction error theory are presented in [8] [9]. The asymptotic in sample size approach to input design in relation to identification of linear dynamical systems has a long history; see, e.g., [10], [11], [5], [12], has lead to a rich collection of theoretical results. The classical approach has 1 The frequency argument will be omitted frequently in the presentation. (3) Hence, for finite data, a useful approximation of the variance of the frequency function estimate is given by The estimation error will depend on the number of data, the parameter covariance the sensitivity of the model system to parameter changes. The influence of the experimental conditions on the variance expression (4) is not easy to interpret. With (4) as starting point, a simple instructive variance expression that is asymptotic in both the model order the number of data was derived in [13]. Under the assumption of open-loop identification, it has lead to the well-known approximation for finite model order sample size. Due to its simple structure, the expression (6) has been widely used in input design; see e.g., [13] [19], [1]. This line of research has proven very successful, both in exposing the fundamentals of the input design problem, in practical applications [20]. Basing a method on asymptotic in model order variance results yields a certain robustness against the properties of the underlying system. However, recently it has been shown in, e.g., [7] that the high-order variance expression (6) is not always accurate. Hence, the validity of input designs based on such results can be questioned. In the last few years, the pendulum has swung back there has been a renewed interest in the asymptotic in sample size approach based on (2) but without resorting to any asymptotics in the model order. The contributions [21], [22], [23] are restricted to quality constraints based on -norms which is a severe limitation in certain applications such as control, e.g., stard stability certificates correspond to frequency-byfrequency constraints on the model error. Recent contributions frequency-by-frequency constraints on the model quality (5) (6)

3 1536 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 10, OCTOBER 2005 have been incorporated into experiment designs are [24] [26], [27]. In this contribution, we will present a quite general framework for input design which is based on the asymptotic theory (2). An early version has appeared as [25]. Some of the main contributions in this paper are as follows. A unification extension of existing parametrizations of the input spectrum, e.g., [5], [28], [22], which allow a wide range of input design problems to be converted into tractable finite-dimensional convex optimization programs. Several new quality measures are proposed that can cope with frequency-by-frequency specifications. For example: 1) Specifications on the asymptotic variance of the frequency function estimate. Apart from (classical) weighted -norms, frequency-by-frequency specifications on the first order approximation (5) can be used. This means that in control applications small gain results can be used to guarantee robust stability for an identified model set. There are few existing results in this spirit. Thus, the frequency-wise bounds considered here are expected to be an important contribution to the area of identification control. 2) Quality specifications on all models in confidence ellipsoids. The asymptotic properties (2) imply that when, for sufficiently large, the estimate will (approximately) lie with probability within the ellipsoid (7) given bounds on, e.g., input energy, as has been the traditional way, cf. [5], [16] [18]. Alternatively, as recently suggested in [30], [26], [27], some signal quantity, such as the input energy, can be optimized given some model quality constraints. The key to obtain tractable input design formulations from an optimization point of view is the parametrization of the input spectrum. Different parametrizations of the spectrum are presented in Section II. Corresponding parametrizations of the asymptotic covariance matrix are derived in Section III. Building on Section III, different model quality constraints are introduced in Section IV Section V. It is shown how they can be represented as linear matrix inequalities (LMIs). The corresponding parametrizations of several input output constraints are provided in Section VI. Robust input design is considered in Section VII some of the features of the derived framework are illustrated in Section VIII. The paper is concluded in Section IX. II. PARAMETRIZATION OF THE INPUT SPECTRUM A. Introduction Expression (3) for the inverse of the asymptotic covariance matrix shows that the spectrum is the only input related quantity that has any influence on. For quality measures based on this means that the input spectrum is the only quantity that can be used to negotiate quality constraints. The spectrum for the input is defined as (9) ) In this contribution, we will show how criteria such as (8) is a model quality measure, can be approximated to fit in the framework. Since depends on the experimental conditions through the confidence ellipsoid, this means that it is possible to design inputs such that the quality objective is satisfied for all models in a confidence region resulting from the identification experiment. Thus statements such as with at least 99% probability the frequency-by-frequency specifications will be satisfied will be possible. The first contribution related to this type of worst case quality measures was presented in [24], optimal input design is performed with respect to a worst case -gap [29]. The constraint (8) presented in this contribution is quite general covers also this quality measure. A variant to quality constraints such as, e.g., (8) is considered in [26] [27] a projection of a frequency function into the Nyquist plane based on the uncertainty set (7) is used. In the presented framework, for input design there is a large flexibility in the choice of objective function. One possibility is to optimize some model quality measure are the auto-correlations. By a suitable change of basis, we can rewrite (9) as the basis functions transfer functions (10) are proper stable rational. With (11) we recover (9) we will reserve the notation for the coefficients when working with complex exponential basis functions. The coefficients must be such that (12) In Section II-B, we will discuss how to ensure this. As we will see in Sections III V, many quality constraints can be transformed such that they become linear in the input spectrum. It is thus natural to parametrize the spectrum in terms of the coefficients. However, it is impractical to use an infinite number of parameters so the parametrization has to be restricted.

4 JANSSON AND HJALMARSSON: INPUT DESIGN VIA LMIs ADMITTING FREQUENCY-WISE MODEL SPECIFICATIONS 1537 One possibility is to use a finite-dimensional parametrization, i.e., (13) for some positive integer. Here one has to impose the condition (12) to ensure that indeed is a spectrum. We will denote this type of approach as a finite dimensional spectrum parametrization. From this it is clear why the reparametrization (10) is of interest: For a finite, there may be sets of basis functions that allow more efficient input designs than the stard basis (11). We will discuss this issue further later on. Instead of parametrizing the spectrum, one may equivalently work with a parametrization of the positive real part (14) This will be our preferred choice when it comes to finite dimensional spectrum parametrizations. Alternatively, one may use a partial expansion (15) which may not be a spectrum in itself, but constrained such that there exists additional coefficients such that the expansion (10) satisfies the nonnegativity condition (12). This approach, which we denote as partial correlation parametrization, thus enables one to work with infinite dimensional expansions. The positivity condition employed in the finite dimensional spectrum parametrization is in general more restrictive than the partial correlation sequence condition. Hence the former approach leads often to more conservative results than the latter one, e.g., more input power may be required to meet the design objective. On the other h, when there are frequency-by-frequency constraints on the input (or the output) spectrum of the system, a finite dimensional spectrum parametrization is typically required since the partial correlation parametrization does not define the frequency properties of the spectrum. Here, we will discuss these two approaches in more detail. B. Finite Dimensional Spectrum Parametrizations With a finite-dimensional spectrum parametrization such as (14), the parameters have to be constrained such that the positivity condition (12) holds. To hle this condition, we will use ideas from finite impulse response (FIR) filter design [31]. This approach is based on the positive real lemma which is a consequence of the Kalman Yakubovich Popov (KYP) lemma, see e.g., [32], [33]. The key idea is to postulate that the spectrum should be realizable using an th-order FIR-filter. Using this restriction on the input spectrum, finite dimensional constraints can be used to represent constraints on the input the output. This idea was first applied to input design problems in [22]. Since any spectrum can be approximated by an FIR expansion to any demed accuracy, the approach is in principle generally applicable. However, when becomes too large, computational complexity becomes an issue. We also remark that this type of parametrization has been used in parameter estimation, see e.g., [34]. Here, we will generalize the idea by imposing a finite dimensional linear parametrization of the input spectrum in which the parametrization used in [22] appears as a special case. We will employ the parametrization (14) of the positive real part of the spectrum, is a set of known proper stable rational transfer functions. This includes Laguerre functions [35], Kautz functions [36] the general orthonormal basis function expansions discussed in [37], [38]. we have the FIR case with the se- corresponding to the correlation coefficients. Notice that it is not necessary for the basis functions to be orthogonal; but, naturally, they should be linearly independent. To ensure that the spectral constraint (12) is satisfied, it may be tempting to sample the positivity condition over the frequency axis. However, this will not guarantee a realizable solution since then the positivity condition may be violated at some point between the sampled frequencies. We will instead use the following result. When quence Lemma 2.1: Let realization of a such that if only if be a controllable state space. Then, there exists (16) Proof: This is an application of the Positive Real Lemma [32]. The idea is to let be a state space realization of the parameters appear linearly in. It is easy to construct such a realization since appear linearly in. Given this realization the symmetric matrix, the constraint may be replaced with the linear matrix inequality (16). We illustrate this with an example. Example 2.1: When the input is shaped by an FIR filter the positive real part of the spectrum becomes

5 1538 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 10, OCTOBER 2005 For an FIR system, a natural choice of state space realization for the positive real part is the controllable form With this parametrization, the correlation sequence appears linearly in the inequality (16) through. Hence (16) becomes an LMI in. One may also use discrete spectra, i.e., spectra from periodic inputs. A discrete spectrum with positive spectral lines distributed over is given by For the frequencies it holds that (17) (18) For such spectra the positive real lemma simplifies to the first condition in (18). When a linear finite dimensional parametrization of the input spectrum is used, it becomes easy to construct a state space realization of the positive real part of the spectrum; see e.g., Example 2.1. Given that a state space realization of the positive real part of the spectrum is available there exist directly applicable methods to perform the spectral factorization; see [39, Ch. 9] [40]. C. Partial Correlation Parametrizations When using a partial correlation parametrization one must ensure that there exists an extension of the sequence such that the corresponding basis function expansion (10) defines a spectrum. We will first restrict attention to the case the basis functions are complex exponentials so that the parameters are the auto-correlation coefficients. The correlation extension problem 2 is then known as the trigonometric moment problem, or as the Carathéodory extension problem. It is well known that a necessary sufficient condition for the existence of such an extension is that the Toeplitz matrix is positive definite [41], [42]. Notice that the hence a convex constraint.. 2 Or, as is more common, the covariance extension problem.... is an LMI in A complete characterization of all spectra having as first expansion coefficients is given by the Schur parameters [43], [44]. For rational expansions, the so-called maximum entropy solution is to use an all-pole, or autoregressive (AR), filter of order. The solution is easily obtained via the Yule Walker equations [45]. When there are prescribed zeros, the solution can be computed via convex optimization [42]. It is also possible to use a discrete spectrum of the form (17). All partial correlation sequences can be generated by a discrete spectrum [41], [46]. However, the frequencies do not necessarily coincide with the fundamental frequencies. In [24], one solution is presented sinusoids are determined. The procedure becomes quite involved since, apart from the amplitudes, also the locations of the frequencies have to be determined. One may also attempt to realize a discrete spectrum of the form (17) with the frequencies evenly distributed. It holds that (19) Hence, if, for given, there exists a feasible solution to the constraints (18) (19), the spectrum (17) has the desired partial correlation sequence. Notice, that there is no guarantee that there exists such a solution. In general, increasing the number of spectral lines may increase the chances of a feasible solution. We now consider the more general case when the basis functions are given by (20) for some given function. Since is positive it must hold that also is positive, hence, the positive Toeplitz condition discussed previously applies to this factor as well. This means that also orthonormal basis functions such as Laguerre, Kautz, their generalization [37], [38] can be used in partial correlation parametrizations since such parametrizations can be recast as (20) with contains some pre-specified poles, i.e.,, see [37]. This type of basis functions will also appear in Examples D. Summary The framework allows for any finite dimensional linear parametrization of the input spectrum or a partial expansion thereof with rational basis functions. For finite dimensional parameterizations this implies fixed (pre-specified) poles of the input spectrum as, e.g., in the case of FIR, Laguerre, or Kautz structures or discrete spectra. For partial correlation parametrizations, the user has a certain degree of freedom in

6 JANSSON AND HJALMARSSON: INPUT DESIGN VIA LMIs ADMITTING FREQUENCY-WISE MODEL SPECIFICATIONS 1539 the realization of the final spectrum can, e.g., use all-pole (AR) or discrete spectra. The choice of whether to use a finite-dimensional spectrum parametrization or a partial correlation parametrization is governed by optimality aspects; computational aspects; signal constraint aspects; robustness aspects. The constraint on the parameters of a partial correlation parametrization is in general less restrictive than the corresponding positivity condition required for a finite dimensional spectrum parametrization. The influence of this on the solution depends on the problem formulation, e.g., the choice of quality constraint. It also depends on the number of terms in the finite dimensional parametrization since there is a tradeoff between computational complexity generality for this type of parametrization. On the other h, when there are frequency-by-frequency constraints on signal spectra, a finite dimensional spectrum parametrization is in general required. III. PARAMETRIZATIONS OF THE COVARIANCE MATRIX We described two different types of parametrizations of the input spectrum in the previous section. A key insight is that both types are based on a finite dimensional parametrization of the type (15). In this section, we will show that by using such a parametrization of the input spectrum it is possible to obtain linear finite dimensional parametrizations of the inverse covariance matrix. A. Minimal Complete Parametrizations of the Covariance Matrix In this section, we will consider how to parametrize the input spectrum such that all possible covariance matrices [defined by (2)] for a given model structure can be generated. In particular, we are interested in minimal parametrizations, i.e., those using a minimal number of parameters. Such parametrizations are of interest from a computational point of view. The starting point is the expression (3) for the inverse of the covariance matrix. Since the elements of span a linear subspace, it follows that the set of all covariance matrices can be parametrized in terms of finite dimensional parametrizations of.in the subsections, that follow we will discuss some possibilities that exist to this end. As a preparation we characterize the space spanned by the elements of. We denote this space by we will assume that it has dimension for some positive integer. We motivate this assumption with an example. Example 3.1: Consider a stable system contained in a Box Jenkins model structure ( is the pure system delay), have analog definitions to with parameters, respectively. Let let, then (21) which, assuming to be coprime, implies that (22) (23) Since, by the stability assumption, the poles of are bounded away from the unit circle, it follows that. Let denote an orthonormal basis for which is such that. 3 We can thus write (24) for some matrices. Next, we will use (24) to characterize the set of asymptotic covariance matrices defined in (2). 1) Subspace Expansions: We can write any input spectrum on the form (25) is orthogonal to. Using (24) (25) in (3) gives (26) This shows that the finite sequence completely parametrizes all possible covariance matrices. It is also clear that the expansion does not have to be orthonormal as long as the basis functions in (25) span. Notice that then (26) no longer holds; linear combinations of will replace. Example 3.2: [Example 3.1 continued] The elements in (22) can be used as the basis functions in the first term in (25) in the Box Jenkins case. 2) Oblique Expansions: It is not necessary that the basis functions are elements of the subspace. We illustrate this with an example. Example 3.3: [Example 3.1 continued] Let the spectrum be parametrized with the basis functions 3 Due to the symmetry of F (e ; )F (e ; ) this is always possible, cf. Example 3.1.

7 1540 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 10, OCTOBER 2005 is given by (23). It then follows from (3) (21) that the design. A partial correlation parametrization can e.g., not hle frequency-wise constraints on the spectrum as pointed out at the end of Section II-A. for some matrices. Hence, parametrize all possible covariance matrices. Notice that the functions in general do not belong to but that they are not orthogonal to either. The functions corresponding to are orthogonal to. This type of parametrization has been considered in [5], been employed in, e.g., [24]. A slight modification of this parametrization that is useful for Box Jenkins model structures was introduced in [28]. In the preceding examples, the basis functions depended on the true parameter in general it is necessary to know in order to guarantee that the basis functions constitute a complete parametrization. However, it is sufficient that these functions are such that their projections on span themselves. Hence, even if is unknown, the set of basis functions which do not completely parametrize the set of all covariance matrices has Lesbegue measure zero, thus, the choice of basis functions is not that critical from this point of view. B. A Parametrization Based on a Finite-Dimensional Spectrum With a finite-dimensional spectrum parametrization of the input spectrum on the form (14), it is possible to express the inverse covariance matrix as an affine function of the variables that parametrize the input spectrum. Lemma 3.1: When the input signal has the spectrum (14), the inverse covariance matrix is given by IV. QUALITY CONSTRAINTS In this section, we will further exploit the parametrizations of the input spectrum introduced in Section II in order to rewrite different measures of the model quality as linear matrix inequalities. A. Convex Representation of Quality Constraints When variance errors are the only concern it is logical to measure the model quality by different functions of. Based on the results in the previous section, it is tractable, from an optimization point of view, to consider constraints that can be recast as being convex in. There are many classical quality measures that are convex in of which are two examples. Here, we will focus particularly on two types of nontraditional constraints of interest when the intended model use is control design. In control applications, it is common to have frequency-byfrequency conditions on the error on the frequency function estimate. To illustrate this consider the weighted relative error (27) are the true system the model, respectively, is a weighting function. When is equal to the designed complementary sensitivity function, the -norm of (27) has been considered as a relevant measure of both robust stability robust performance [48] [50]; e.g., is a classical robust stability condition. When the model is obtained from an identification experiment it will lie in an uncertainty set of the type (7). Hence, a reasonable objective is therefore to design the identification experiment such that becomes small for all models in such an uncertainty set. One way to approximately achieve this objective is by restricting the variance of. The variance can, using a first order Taylor approximation, be expressed as Proof: Use the definition of the input spectrum (14), insert this into (3). C. Summary Finite linear parametrizations of can actually be obtained for both finite dimensional spectrum parametrizations partial correlation parametrizations of the input spectrum. The difference between these two types of spectrum parametrizations are the constraints that are imposed on the variables. For finite spectrum parametrizations, these constraints typically restrict the set of achievable covariance matrices. However, as has been illustrated in [47], it may not be necessary to parametrize all covariance matrices to achieve the optimal design. The choice of parametrization depends also on what type of input/output constraints that are considered in A given upper bound on the variance of gives then the following frequency-by-frequency constraint: (28) This type of constraint can be re-expressed as a linear matrix inequality in.

8 JANSSON AND HJALMARSSON: INPUT DESIGN VIA LMIs ADMITTING FREQUENCY-WISE MODEL SPECIFICATIONS 1541 Lemma 4.1: Let be positive semidefinite matrices let Then, the constraint may be written as the following constraints: (29) is one solution to is a slack variable. Proof: The positive definite matrix can always be factorized as. The factorization of leads to (30) Introduce the slack variable. Then the constraint together with (30) can be written as (31) Using the Schur complement, (31) may be written as (29) which is linear in,. Notice that (29) is convex in. It is easily verified that Lemma 4.1 applies to (28). Thus the frequency-by-frequency constraint on the weighted model variance is convex in but infinite dimensional. One approach to hle the latter problem is to sample the constraints over the -axis. However, in the next section we will present a solution that provides a finite-dimensional constraint without approximation. B. Application of the KYP-Lemma to Quality Constraints When the weighting matrix is a finite dimensional spectrum, then can be taken as the stable minimum phase spectral factor of. Furthermore, with a controllable state space realization of, the KYP-lemma can be used to rewrite (29) as a finite-dimensional constraint, cf. Lemma 2.1. Lemma 4.2: Assume that is given by a controllable statespace realization let be defined as in Lemma 4.1. It then holds that if only if there exists such that (recall the definition (16) of ) Proof: The state space realization defined in Lemma 4.2 gives. Furthermore, if is controllable so is. Thus, the Positive Real Lemma [32] can be applied. Inequality (32) is a linear matrix inequality in since they both appear linearly in. Thus, using linear parametrizations of the input spectrum, such as (13), by combining Lemma 4.1 with Lemma 4.2, we have that quality constraints of the type (28) can be recast as linear matrix inequalities in the input spectrum parameters the matrices. V. QUALITY CONSTRAINTS IN ELLIPSOIDAL REGIONS One alternative to the variance bound on suggested in Section IV.A, is to use the confidence bound in (7) directly. This gives the following alternative to the variance constraint (28): (33) This constraint implies that for all models in a confidence region associated with the (to be) identified model. This leads to the introduction of a new family of quality measures of the model. Let, be finite-dimensional stable transfer functions. Let be defined by, is some stable finite-dimensional transfer function. Let be defined analogously. Furthermore, let be a positive definite matrix. Now the following quality measure is proposed: (34) (35) The quality measure (34) is a max-norm constraint on with respect to it has to be satisfied for all models with in the ellipsoid. Uncertainty sets such as are e.g., delivered by the prediction error method then corresponds to the model parameters. We illustrate the usefulness of the measure via two examples. Example 5.1: Taking, gives that (32) Thus, the generalized quality measure includes (33). Example 5.2 [Worst Case Chordal Distance]: The square of the chordal distance [51] between can be written as (36)

9 1542 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 10, OCTOBER 2005 Taking gives Then, (34) is equivalent to the existence of a positive function such that The objective of this section is to develop tools so that constraints like (34) can be incorporated in the framework developed in Section II IV. Thus, we want to transform (34) into linear matrix inequalities. A. Reformulation as a Convex Problem To reformulate the problem we will use slight modifications of techniques used in [52] [53]. Consider the model structure (1), which is parametrized by the vector. Assume, without loss of generality, the partition, such that let be parametrized as Proof: Lemma 5.1 gives that (40) is equivalent to (41) Expression (41) is equivalent to that for a particular. Now, this must be true for all. The ellipsoid can be parametrized as (37) is the pure system delay, are row vectors of size. Lemma 5.1: Let be defined by (35) let be parametrized as in (37). Then with (38) (39) defined analogous to. Furthermore, is the least common denominator of. Proof: Using that both the numerator the denominator of have the quadratic form exploiting the parametrization of, (37), we obtain are defined by (39). The equivalence still holds when it is multiplied by (which is positive) since is real this will be equivalent to (38). The equivalence in (38) will be further exploited in the next theorem. Theorem 5.1: Let be defined by (35) let be defined as in Lemma 5.1. Assume that for all. Furthermore, assume that is not positive semidefinite for any. Hence, the condition is equivalent to for all for all such that. Such a problem can be hled by the -procedure [54] which states the following equivalence for each such that, such that. Finally to obtain the expression (40) we change the variable according to. This can be done if we can show that. The expression is true for only if for some. The assumption that guarantees that. Hence, we obtain the condition we arrive at the statement in (40). Theorem 5.1 is very interesting from an input design point of view. Then the variable in (40) will be proportional to the inverse covariance matrix of the parameters, see (7). Furthermore, the inverse covariance matrix is affine in the input spectrum, see (3). With a linear parametrization of the input spectrum, also becomes linearly parametrized. Hence, Theorem 5.1 states that the worst case (over all models in an ellipsoidal model set) max-norm performance constraint (34) can be translated into the condition (40), which is a linear matrix inequality in for each when is fix is given by (35). There are still two problems associated with this result from an input design point of view. First of all, is an unknown function of. Furthermore, this is an infinite dimensional constraint similar to the performance constraint in (29), since the constraint has to hold for all frequencies. In the next subsection we will address these issues. B. Finite-Dimensional Relaxation It was shown in Lemma 2.1 that when a frequency-by-frequency constraint can be viewed as a positiveness constraint on a spectrum, the KYP-lemma can be applied to make the constraint finite dimensional. We now introduce some conditions that will allow the KYP-lemma to be applied to (40) such that this constraint can be reduced to a finite dimensional linear matrix inequality.

10 JANSSON AND HJALMARSSON: INPUT DESIGN VIA LMIs ADMITTING FREQUENCY-WISE MODEL SPECIFICATIONS 1543 Lemma 5.2: Let be defined as in Lemma 5.1, let be defined by (40). Introduce let be defined by (42) the basis functions are proper stable linearly independent rational transfer functions. Then, there exists a finite sequence such that (43) the variables the elements of appear linearly in. Proof: Both have the structure for some matrices. Multiply with the least common denominator of the terms which make up (which is positive on the unit circle) to both sides of. This gives the equivalence in (43). The special parametrization of will, according to Lemma 5.2, imply that the condition can be replaced by a positiveness constraint on a spectrum, see (43). This fact can now be used together with Lemma 2.1 to transform the infinite dimensional constraint (34) into a linear matrix inequality in the variables the elements of. Theorem 5.2: Let be defined by (35) assume that for all for all is defined in (34). Let be defined as in Lemma 5.1. Assume that is not positive semidefinite. Let be defined as in Lemma 5.2. Then, there exists a state space realization of in (42) appears linearly in. Similarly, there exists a state-space realization of in (43) the elements of appear linearly in. Furthermore, it holds that to Lemmas Whenever this will, according to Theorem 5.1, imply (44). Theorem 5.2 is a quite powerful result for input design problems. Notice that the inequalities (45) are LMIs in the elements of. In input design from Section II we have that this quantity becomes linearly parametrized when the input spectrum is linearly parametrized. Thus, Theorem 5.2 shows that model quality constraints of the type (8) with defined by (7) defined by (35) can be relaxed to an LMI problem. The use of this theorem will be illustrated in Section VIII for the model constraint (33). VI. PARAMETRIZATION OF SIGNAL CONSTRAINTS Here, we will further exploit the linear parametrization of the input spectrum, defined in (14), to rewrite different frequency domain signal constraints. These constraints become finite dimensional affine functions of the elements of the sequence, which is the sequence that parametrizes the input spectrum. The signal constraints considered are limitations on the input /or output spectra in terms of power as well as frequency-by-frequency constraints. A. Parametrization of Power Constraints First, consider the parametrization of input output variance constraints. Lemma 6.1: The variance of, is a stable linear filter has the spectrum (14), can be expressed as with (46) The variance of, is a stable linear filter is the output corresponding to the input, when the system is operating in open loop is given by with (47) (44) if there exist such that (45) Proof: Due to the parametrization of, the constraints (45) will assure that according Proof: The affine expression (46) is an immediate consequence of the definition of the spectrum. The expression (47) is obtained using the fact that are

11 1544 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 10, OCTOBER 2005 uncorrelated due to open-loop operation, thus, the output spectrum becomes. It is also possible to derive finite linear parametrizations of the input or the output power for some partial correlation parameterizations of the input spectrum; see, e.g., [28]. B. Parametrization of Point-Wise Constraints Point-wise constraints on the input output spectra that are rational functions of can be transformed by the KYP-lemma into linear matrix inequalities just as for the input spectrum constraint, cf. Lemma 2.1. Alternatively, these may be added as sampled frequency domain constraints. Contrary to the positivity constraint on the input spectrum, this will not have any drastic impact on the solution, as a violation of these constraints in between the sampling points not will cause the input design to be nonrealizable. However, our experience is that the computational time will generally be significantly lower with sampled constraints than the straightforward implementation of the KYP-lemma. VII. ROBUSTNESS ASPECTS Solutions to most optimal input design problems depend on the true, unknown, underlying system. One common way to overcome this is to replace the true system in the design by some estimate of the system that, e.g., is obtained from an initial identification experiment. However, due to the estimation error, there is no guarantee that a design based on such an estimate will yield a solution that is satisfactory when applied to the true system. Hence, there is a need to develop methods that are robust with respect to the true system. Below we will discuss two important issues: The parametrization of the input spectrum mini-max solutions with respect to a set of initial models. A. Input Spectrum Parametrization As we have discussed in Section II, it may happen that the input design problem only depends on, for some finite positive integer, in a certain expansion of the input spectrum (10). The additional degrees of freedom, i.e.,, can then be used to increase robustness of the design. In the finite dimensional spectrum parametrization additional constraints related to robustness can be included in the original program. For the partial correlation parametrization, it is clear that different correlation extensions yield different robustness properties. Using a discrete spectrum with a minimal number of nonzero spectral lines for the correlation extension may lead to a design that is very vulnerable to errors in a priori assumptions about the system behavior. For example, if the true system order turns out to be higher than the number of nonzero spectral frequencies (over the interval ) the system will not be identifiable if a separately parametrized noise model is used. Furthermore, even if, say, an ARX-model is used so that the denominator polynomial in the system dynamics may be estimated from the noise, certain directions of the parameters associated with the system dynamics are not improved upon by the input in this situation. An all-pole realization, on the other h, will yield a spectrum solution that is nonzero every such a spectrum yields identifiability for any model order. Hence, it is important to consider robustness issues when deciding upon which correlation extension method to use. Another aspect when using a partial correlation parametrization is that, in general, the basis functions depend on the true system. Even if this often is not critical as pointed out in Section III-A.2, one may prefer to use system independent basis functions which leads to the finite-dimensional spectrum parametrization. B. Working With Sets of a Priori Models Robustness can be achieved by posing the design problem such that performance objectives constraints are satisfied for all systems within some prior model set. A recent result in this direction is [55] the application is state feedback. For the approach described in this paper it is possible to include constraints objective functions for several systems simply by adding LMIs for each separate system to the overall problem. Even though no guarantees can be given in general that the objectives are met for the true system, this approach provides improved robustness compared with if the design problem is based on a single prior model. Given an initial estimate associated covariance matrix one may for example pick models in the corresponding uncertainty set (see (7) for a definition of ). As the density of the used models inside this set is increased, a sequence of designs that converges to the worst case design over this set is obtained. In the limit such a design will guarantee that the design objective is met with the confidence level defined for, e.g., 95% probability. A related procedure is to draw samples from the corresponding normal distribution. In the numerical example given in the next section, we will compare the optimal solution, based on the knowledge of the true system, with a design the true system is replaced by normal distributed samples lying in a confidence region obtained from an initial identification experiment. VIII. NUMERICAL ILLUSTRATION We will now illustrate some of the features of the input design framework we have developed via an example related to identification for control. Example 8.1: Let the true system be given by with is zeromean white noise with variance The magnitude of the frequency function of is shown as the dashed dotted line in Fig. 1. The modeling objective is to be able to design a controller such that the resulting closed-loop system is stable the complementary sensitivity function is close to

12 JANSSON AND HJALMARSSON: INPUT DESIGN VIA LMIs ADMITTING FREQUENCY-WISE MODEL SPECIFICATIONS 1545 Fig. 1. Thick solid line: Optimal input spectrum based on (51). Dashed line: Input spectrum obtained from robust algorithm. Thin solid line: Weighting function T. Dash-dotted line: Open loop system G. Dotted line: White noise input with variance 8 (!) =0:26. The amplitude curve of is shown as the thin solid line in Fig. 1. A sufficient condition for this is that the weighted relative model error (27) is sufficiently small (in particular guarantees stability) [50]. Here, we choose the condition (48) We choose a first-order output-error model structure When we restrict the structure of to, Theorem 5.2 can be applied giving the following relaxation to (49): minimize subject to The sample size is set to samples. The objective is to find the minimum energy required, the corresponding input spectrum, such that (48) is satisfied for all models in the resulting 95% confidence region. We, therefore, use the criterion (33). We, thus, have the problem minimize subject to. (49) (50) depend linearly on that depend linearly on. The above problem is thus convex in all free variables. In order to reduce the problem to a finite-dimensional one, the spectrum is parametrized as in (14) with. This corresponds to shaping the input spectrum with an FIR filter of order 20 that the parameters in (14) correspond to the auto-correlation sequence of the input. Lemma 2.1 gives that the positivity constraint on now is equivalent to Lemma 6.1 gives that the input variance constraint can be expressed as. Thus, the input design problem (50) leads to

13 1546 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 10, OCTOBER 2005 Fig. 2. Dots: Estimated model parameters from 1000 Monte Carlo runs based on optimal design. Dashed ellipse: Estimated 95% confidence bound for the parameter estimates. Dash-dotted ellipse: Confidence bound for white noise input with 8 =0:26. Contour lines with interval are plotted for k1k k1k = 0:1 corresponds to the thick solid contour. TABLE I RESULT FROM 1000 MONTE CARLO RUNS FOR DESIGNS BASED ON DIFFERENT SAMPLE SIZES N.HERE PSQM IS THE PERCENTAGE OF THE MODELS THAT SATISFY k1k 0:1 (51) Solving (51) gives the input spectrum shown as the thick solid line in Fig. 1. The solution conforms quite well to common intuition most of the energy is distributed around the desired bwidth of the closed-loop system which is the critical frequency region for robust performance. The minimum power required is. Fig. 2 shows the parameter estimates of 1000 Monte Carlo runs for the optimal design based on (51). We see that the estimates are clustered inside the contour as desired. In fact, 96.2%, i.e., more than the desired 95%, of the estimated models satisfy the quality constraint since some of the estimates outside the confidence ellipse still are inside the level curve. Remark 2: Given a feasible solution of (51), we know that (40) is satisfied according to Theorem 5.1 this will imply that for all models in the uncertainty set. However, the restriction imposed on will lead to a conservative input design, since the representation of only corresponds to a subclass of all. In this example, the solution is not very conservative at all. For comparison purposes, confidence ellipses for the optimal design a white noise design with the same variance (cor- responds to shown as the dotted line in Fig. 1) are also shown in Fig. 2. These ellipses are all based on estimates of the covariance matrix obtained from Monte Carlo simulations. We clearly see that the approximation of (49) made in (51) performs well as the white input design clearly does not meet the objective, in this case, is uniformly worse than the optimal design. In fact, only 67% of the models satisfy the quality constraint. To obtain a white input design with a confidence ellipsoid that is completely inside the contour curve corresponding to, an input variance of is required. Hence, more than 220% higher input power is required to achieve the objective for a white input as compared to the optimal design. The sample size was in this example chosen to. The sample size acts primarily as a scaling factor for the covariance matrix. It is thus easy to modify the results for arbitrary sample size as long as the sample size is large enough for the asymptotic theory to be accurate. In Table I, there is a comparison between designs based on values of between One can see that above, the design objective is met with the desired probability. For sample sizes down to, the probability of success shrinks moderately below the required 95%. In calculating the optimal design above, knowledge of the true system was used. Both the quality constraint the covariance matrix in (49) depend on the true system. This manifests itself in (51) in that, depend on.

14 JANSSON AND HJALMARSSON: INPUT DESIGN VIA LMIs ADMITTING FREQUENCY-WISE MODEL SPECIFICATIONS 1547 Fig. 3. Thick solid line: Spectral factor of optimal input spectrum based on (51) with N =500. Dashed lines: Spectral factors of the optimal input spectra based on (51) for N =50; 100; 200; 300; 400. Thin solid line: Weighting function T. Dashed-dotted line: Open loop system G. In a practical situation, the true system is unknown. In a second design we have used an initial identification experiment with to obtain an estimate, of the true parameters an estimate of the covariance of the parameters,. With this information at h, nine additional parameter estimates were drawn from a normal distribution with mean covariance. A total of ten parameter estimates have been used as replacements for in (51). These estimates are shown as circles in Fig. 4. This leads to an input design problem with 10 quality constraints instead of one. The resulting input spectrum is shown in Fig. 1. Here, we see that more total energy is required in order to satisfy the quality constraint for these ten systems compared to the single system design considered previously. The power of the input obtained from this robustified design is. Estimates from 1000 Monte Carlo runs with this design are shown in Fig. 4. This figure also includes confidence ellipses for this design a white noise design with the same power i.e.,. The price of having 10 different replacements for the unknown in the design is in this case that the solution becomes conservative compared to the design solely based on the true system. This is evidenced by the simulations. About 99.5% of the models satisfy the constraint. However, the robustified algorithm yields a solution that is more effective compared to a white noise input with the same power. For a white noise input with, only 87% of the models satisfy the quality constraint. IX. CONCLUSION In this contribution, we have presented a quite general framework for translating input design problems in system identification into convex programs. We have assumed that variance errors is the only concern, but apart from this the framework is quite flexible. Any specification that can be recast as a convex constraint in the input spectrum can be used. This means in particular specifications that can be expressed as convex constraints in the inverse covariance matrix. We have shown that frequency-wise constraints such as, e.g., the -gap (see (36)) bounds on the weighted relative error, can be hled but there are certainly other specifications that can be incorporated as well. The possibility of frequency-by-frequency constraints makes the framework interesting for control applications. We have pointed to one approach to robustify input designs but further work is certainly needed here. Here, the approach in [55] seems promising. Another issue that warrants further investigation is, as remarked in Section VIII, that designs based on an ellipsoidal model set leads to a conservative design when the spectral structure (42) is imposed on. The design in the example in Section VIII was not particularly conservative, indicating that in this case could be well approximated by a spectrum. However, it is of great interest to study under what conditions this holds. Finally, we remark that the results can be generalized to the case of finite sample size for certain model structures to closed-loop experiment design [56].

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Scorletti, B. D. O. Anderson, Robustness analysis tool for an uncertainty set obtained by prediction error identification, Automatica, vol. 37, no. 10, pp , [54] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems Control Theory, Studies in Applied Applied Mathematics. Philadelphia, PA: SIAM, [55] M. Barenthin H. Hjalmarsson, Input design H state feedback with ellipsoidal parametric uncertainty, in Proc. 44th IEEE Conf. Decision Control, Seville, Spain, Dec [56] H. Jansson H. Hjalmarsson, Optimal experiment design in closed loop, in Proc. 16th IFAC World Congr., Prague, Czech Republic, Henrik Jansson received the M.Sc. degree in engineering physics the Ph.D. degree in automatic control, both from KTH, Stockholm, Sweden, in , respectively. His research interests include various aspects of identification for control, such as experiment design controller tuning. Håkan Hjalmarsson (M 90) was born in He received the M.S. degree in electrical engineering the Licentiate Ph.D. degrees in automatic control, all from Linköping University, Linköping, Sweden, in 1988, 1990, 1993, respectively. He has held visiting research positions at the California Institute of Technology, Pasadena, Louvain University, the University of Newcastle, Australia. He is currently a Professor in the Department of Signals, Sensors, Systems at KTH, Stockholm, Sweden. His research interests include system identification, signal processing, control, estimation in communication networks automated tuning of controllers. Dr. Hjalmarsson has served as an Associate Editor for Automatica been a Guest Editor for the European Journal of Control Control Engineering Practice. He is currently an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL. He is a Member of the IFAC Technical Committee on Modeling, Identification Signal Processing since In 2001, he received the KTH award for outsting contribution to undergraduate education.