Robust optimal identification experiment design for multisine excitation

In least costly experiment design, the optimal spectrum of an identification experiment is determined in such a way that the cost of the experiment is minimized under some accuracy constraint on the identified parameter vector. Like all optimal experiment design problems, this optimization problem depends on the unknown true system, which is generally replaced by an initial estimate. One important consequence of this is that we can underestimate the actual cost of the experiment and that the accuracy of the identified model can be lower than desired. Here, based on an a-priori uncertainty set for the true system, we propose a convex optimization approach that allows to prevent these issues from happening. We do this when the to-be-determined spectrum is the one of a multisine signal. 1 Introduction We consider in this paper the problem of optimally designing the spectrum Φ u of the excitation signal u of an open-loop identification experiment. By optimal spectrum , we here mean the spectrum yielding the smallest experiment cost while guaranteeing that the accuracy of the identified parameter vector of the plant transfer function is larger than a given threshold. We thus consider the least costly experiment design framework [5], but the approach can easily be adapted to other (dual) frameworks [10,17,13]. The experiment cost J can be defined as a linear combination of the power of the exci-tation signal u and of the power of the part of the output signal induced by u. The experiment cost will therefore be a function of the spectrum Φ u , but also of the unknown true parameter vector θ 0 (we therefore denote the cost as J (θ 0 , Φ u)). Likewise, the accuracy constraint will also depend on θ 0 and on Φ u since the classical accuracy constraints are of the type P −1 (θ 0 , Φ u) ≥ R adm where P (θ 0 , Φ u) is the covariance matrix of the to-be-identified parameter vector (which depends on θ 0 and Φ u) and R adm a matrix reflecting the desired accuracy. The dependency of the optimal spectrum Φ u,opt on the unknown true parameter vector θ 0 is the so-called chicken-and-egg issue encountered in optimal experiment design. This issue is generally circumvented by replacing θ 0 b

On the calculation of the D-optimal multisine excitation power spectrum for broadband impedance spectroscopy measurements

The successful application of impedance spectroscopy in daily practice requires accurate measurements for modeling complex physiological or electrochemical phenomena in a single frequency or several frequencies at different (or simultaneous) time instants. Nowadays, two approaches are possible for frequency domain impedance spectroscopy measurements: (1) using the classical technique of frequency sweep and (2) using (non-)periodic broadband signals, i.e. multisine excitations. Both techniques share the common problem of how to design the experimental conditions, e.g. the excitation power spectrum, in order to achieve accuracy of maximum impedance model parameters from the impedance data modeling process. The original contribution of this paper is the calculation and design of the D-optimal multisine excitation power spectrum for measuring impedance systems modeled as 2R-1C equivalent electrical circuits. The extension of the results presented for more complex impedance models is also discussed. The influence of the multisine power spectrum on the accuracy of the impedance model parameters is analyzed based on the Fisher information matrix. Furthermore, the optimal measuring frequency range is given based on the properties of the covariance matrix. Finally, simulations and experimental results are provided to validate the theoretical aspects presented.Peer ReviewedPostprint (published version

Planiranje eksperimenta za robusnu identifikaciju dinamičkih sistema

Primenom principa crne kutije i teorija verovatnoće, stohastičkih procesa i matematičke statistike, uz korišćenje ulazno/izlaznih merenja razmatra se mogućnost dobijanja matematičkih modela. Okviri za dobijanje modela su opšti jer se pretpostavlja da su stohastički poremećaji negausovi. Takav model je preduslov za projektovanje široke klase industrijskih regulatora. Teorija planiranja eksperimenta ima važnu ulogu u povećanju brzine konvergencije rekurzivnih algoritama kao i u skraćenju vremena identifikacije. Povećana brzina konvergencije algoritama čini ih veoma povoljnim za praktičnu primenu. Ulazni signali za identifikaciju kreiraju se preko rekurzivne relacije za autokovarijacionu funkciju. Sinteza autokovarijacione funkcije zasnovana je na idejama iz prediktivnog upravljanja, pri čemu upravljački signal ima konačan alfabet. Praktična istraživanja pokazuju da poremećaj, u opštem slučaju, ima negausovu raspodelu. Posebno je važan slučaj kada se pojave opservacije koje su nekonzistentne u odnosu na glavninu populacije, autlajeri (outliers). Raspodele verovatnoće za taj slučaj su približno normalne (e -kontaminirane) i predmet su intenzivnog proučavanja u matematičkoj statistici. Za takav slučaj se predlažu robusni algoritmi identifikacije, pri čemu robusnost ima statistički karakter. Razmatra se primena robusnog Kalmanovog filtra u identifikaciji modela zasnovanih na grešci izlaza. Robusni prošireni Kalmanov filtar se koristi za identifikaciju opšte forme nelinearnog modela u prostoru stanja. Identifikacija procesa opisanih opštim modelom (nepoznati parametri i stanja procesa) zahteva uvođenje proširenog Masreljez-Martinovog filtra. Uvođenjem predloženih heurističkih modifikacija povećava se fleksibilnost, u smislu praktične primene kao i brzine konvergencije robusnog filtra. Prikazana je nadmoćnost predloženih robusnih algoritama u identifikaciji sistema sa vremenski promenljivim parametrima, koji zasnovani na OE klasi modela. Praktični aspekt dobijenih rezultata potvrđen je kroz eksperiment na pneumatskom cilindru koji se nalazi u laboratoriji centra za Automatsko upravljanje i fluidnu tehniku Fakulteta za mašinstvo i građevinarstvo u Kraljevu.By applying the principles of black boxes and probability theory, stochastic processes and mathematical statistics, with the use of input / output measurements, the possibility of obtaining mathematical models is considered. Frames for obtaining the model are general because it is assumed that the stochastic disturbances are non-Gaussian. Such a model is a prerequisite for the design of wide range of industrial controlers. The theory of experiment design plays an important role in increasing the speed of convergence of recursive algorithms as well as shortening the time of identification. Increased speed of convergence of algorithms makes them very favorable for practical application. The input signals for identification are created through the recursive relation for autocovariance. Design of autocovariance is based on the idea of predictive control, where the control signal has a finite alphabet. Practical studies show that disturbances, in general, have non-Gaussian distribution. Particularly important is the case when there are observations that are inconsistent with respect to the majority of population (outliers). Probability distribution for this case is approximately normal (e -contaminated) and is the subject of intensive study in mathematical statistics. In such case, the robust algorithms for identification, where robustness has a statistical nature, are proposed. It is considered the application of robust Kalman filter in identification of output error model. Robust extended Kalman filter is used for identification of the general form of the nonlinear statespace model. Identification of the processes described by general model (the unknown parameters and states of the process) requires the introduction of extended Masreliez-Martin's filter. By introducing of the proposed heuristic modification increases the flexibility in terms of practical application and the speed of convergence of the robust filter. The superiority of the proposed robust algorithms for system identification with timevarying parameters, which are based on OE models, has been shown. The practical aspects of the results have been confirmed by experiment on a pneumatic cylinder, which is located in the laboratory of the centre for automatic control and fluid technique of The Faculty of Mechanical and Civil Engineering in Kraljevo