The Refined Instrumental Variable method for discrete-time systems (RIV) and its variant for continuous-time systems (RIVC) are popular methods for the identification of linear systems in open-loop. The continuous-time equivalent of the transfer function estimate given by the RIV method is commonly used as an initialization point for the RIVC estimator. In this letter, we prove that these estimators share the same converging points for finite sample size when the continuous-time model has relative degree zero or one. This relation does not hold for higher relative degrees. Then, we propose a modification of the RIV method whose continuous-time equivalent is equal to the RIVC estimator for any non-negative relative degree. The implications of the theoretical results are illustrated via a simulation example.

In continuous-time system identification, refined instrumental variable methods are widely used in open and closed-loop settings. Although their robustness and performance are well documented for stable systems, these estimators are not reliable for estimating unstable continuous-time models. The main difficulty we encounter in modeling unstable systems with refined instrumental variables is that the filtered regressor and instrument vectors, as well as the filtered output, become severely ill-conditioned if the model is unstable during the iterative process. In this work, we propose a solution to this problem by including a tailor-made all-pass filter in the prefiltering step. This approach is used for obtaining an extension of the least-squares state-variable filter method, as well as extensions for the refined instrumental variable method for continuous-time systems (RIVC) and its simplified embodiment(SRIVC), that admit the identification of unstable systems and are shown to minimize the prediction error upon convergence and as the sample size goes to infinity. In addition, several implementations of these methods are proposed depending on the intersample behavior of the input (zero and first-order hold, multisine and arbitrary). The particular case when the plant has integral action is explicitly considered in this work. In an indirect system identification setting, an extension of the closed loop version of the SRIVC method is also proposed and discussed in detail. Monte Carlo simulations are used to assess the performance of our methods. 

In this paper, we analyse the consistency of the Instrumental-Variable-based State Variable Filter (IVSVF) estimator by taking into account the intersample behaviour of the input and output signals. It is found that when only sampled input and output data are available for estimation, the IVSVF estimator is not consistent for any fixed sampling period due to the interpolation error that arises from constructing the filtered output. A Bias-Eliminated IVSVF (BEIVSVF) estimator is then proposed and shown to be consistent. The theoretical results developed in the paper are also discussed from a practical standpoint. Simulations are performed to verify the performance of the proposed method as well as to support the theoretical results.

Continuous-time system identification deals with the problem of building continuous-time models of dynamical systems from sampled input and output data. There are two main approaches in this field: indirect and direct. In the indirect approach, a suitable discrete-time model is first determined, and then it is transformed into continuous-time. On the other hand, the direct approach obtains a continuous-time model from the sampled data without an intermediate discrete-time model. In both approaches there exists a dichotomy between discrete-time data and continuous-time models, which can induce robustness issues and complications in the theoretical analysis of identification methods. These difficulties are addressed in this thesis.

First, we consider the indirect approach to continuous-time system identification. For a zero-order hold sampling mechanism, this approach usually leads to an excess of model zeros when the true system has a relative degree greater than one. Inspired by the indirect prediction error method, we propose an indirect-approach estimator that guarantees stability in the model and enforces the desired number of poles and zeros in the continuous-time transfer function estimate.

The second part of this thesis concerns the asymptotic properties and extensions of direct continuous-time identification methods. We provide a comprehensive statistical analysis of the simplified refined instrumental variable method for continuous-time systems (SRIVC), which is a widely-used direct identification algorithm that applies an adaptive prefiltering to the sampled input and output data. We prove that the SRIVC estimator is generically consistent and asymptotically efficient under some mild conditions when taking into account the intersample behavior of the signals in the analysis, and we give conditions under which these statistical properties are not achieved. An extended analysis is provided for when the model is over-parameterized. Later, we propose and analyze the statistical properties of an extension of the SRIVC estimator that can deal with input signals that cannot be interpolated exactly via hold reconstructions. The standard SRIVC estimator and its extension for arbitrary inputs, together with other refined instrumental variable methods, are also investigated in closed-loop settings and are further enhanced to deal with the identification of unstable systems.

The last part of this thesis focuses on the analysis and identification of continuous-time systems subject to band-limited input excitations. The non-causal behavior of the band-limited discrete-time equivalent system is studied in detail, and the findings are later used for designing novel non-parametric and parametric identification methods for when the input is band-limited. Special treatment is given to identification with continuous-time multisine inputs. For that case, we investigate fundamental relations between prediction error methods, optimal refined instrumental variables, and interpolation and approximation of frequency response function estimates.

All of the methods and theoretical results are accompanied by extensive simulation tests that verify our findings.

Tidskontinuerlig systemidentifiering går ut på att bygga tidskontinuerliga modeller av dynamiska system genom att använda samplad data. Det finns två huvudsakliga tillvägagångssätt inom detta forskningsområde: indirekta och direkta. Med det indirekta tillvägagångssättet så hittar man först en lämplig tidsdiskret modell, för att sedan transformera den till kontinuerlig tid. Det direkta tillvägagångsättet, å andra sidan, identifierar en tidskontinuerlig modell från samplad data utan att först hitta en tidsdiskret modell som ett mellanliggande steg. Med båda tillvägagångssätten så finns en dikotomi mellan tidsdiskret data och tidskontinuerliga modeller, vilket kan skapa problem med robusthet samt komplikationer med den teoretiska analysen av identifieringsmetoder. Dessa svårigheter behandlas i denna avhandling. Vi börjar med att betrakta det indirekta tillvägagångssättet för identifiering av tidskontinuerliga system. För en samplingsmekanisk med en nollte ordningenshållkrets så leder detta tillvägagångssätt vanligtvis till ett överskott av nollställen i modellen då det sanna systemet har relativt gradtal större än ett. Vi föreslår enskattare, inspirerad av den indirekta prediktionsfelmetoden, som med det indirekta tillvägagångssättet garanterar modellens stabilitet och framtvingar önskat antal poler och nollställen i skattningen av den tidskontinuerliga överföringsfunktionen. En viktig del av denna avhandling berör direkta metoder för tidskontinuerlig systemidentifiering. Vi inkluderar en omfattande statistisk analys den förenklade raffinerade instrumentvariabelmetoden för tidskontinuerliga system (SRIVC), som är en direkt identifieringsalgoritm som används i stor utsträckning och som applicerar ett adaptivt förfilter till den samplade in- och utsignalen. Vi bevisar att SRIVCskattaren är generiskt konsistent och asymptotiskt effektiv, under vissa milda antaganden. Vi presenterar även villkor under vilka detta antagande inte är uppfyllt. En utökad analys är inkluderad för fallet då modellen är överparametriserad. Senare så föreslår och analyserar vi även de statistiska egenskaperna av en utökning av SRIVC-skattaren som kan hantera insignaler som inte kan interpoleras exakt via rekonstruktioner som använder sig av hållkretsar. Den typiska SRIVC-skattaren och dess utökning för godtyckliga insignaler, tillsammans med andra raffinerade instrumentvariabelmetoder, undersöks också i fall då systemet är återkopplat, samt förbättras ytterligare för att hantera identifiering av instabila system. Avhandlingens sista del fokuserar på analys och identifiering av tidskontinuerliga system med begränsad bandbredd av insignalen. Det icke-kausala beteendet av det ekvivalenta tidsdiskreta systemet med begränsad bandbredd undersöks i detalj, och resultaten används senare för att designa både nya icke-parametriska och parametriska identifieringsmetoder för fall då insignalen har begränsad bandbredd. Särskilt behandlas identifiering av tidskontinuerliga system då insignalen är summan av flera sinusvågor. För det fallet så undersöker vi fundamentala förhållanden mellan prediktionsfelmetoder, optimala raffinerade instrumentvariabelmetoder, samt interpolering och approximation av skattningar av frekvensfunktionen. Alla metoder och teoretiska resultat presenteras tillsammans med omfattande simulerade test som verifierar våra fynd.

The Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC) is one of the most popular direct methods for linear continuous-time system identification. Only recently, some optimal asymptotic properties of this method have been proven. Here we provide a comprehensive analysis of the convergence of the SRIVC method for over-parameterized models. The results we derive are used to discuss practical aspects of the method and to analyze the behavior of the normalized error variance norm, which is a central part in Young’s Information Criterion for model order selection. The theoretical findings are validated through simulation examples.

For many years, the Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC) has been widely used for identification. The intersample behaviour of the input plays an important role in this method, and it has been shown recently that the SRIVC estimator is not consistent if an incorrect assumption on the intersample behaviour is considered. In this paper, we present an extension of the SRIVC algorithm that is able to deal with continuous-time multisine signals, which cannot be interpolated exactly through hold reconstructions. The proposed estimator is generically consistent for any input reconstructed through zero or first-order-hold devices, and we show that it is generically consistent for continuous-time multisine inputs as well. The statistical performance of the proposed estimator is compared to the standard SRIVC estimator through extensive simulations.

In continuous-time system identification, the intersample behavior of the input signal is known to play a crucial role in the performance of estimation methods. One common input behavior assumption is that the spectrum of the input is band-limited. The sinc interpolation property of these input signals yields equivalent discrete-time representations that are non-causal. This observation, often overlooked in the literature, is exploited in this work to study non-parametric frequency response estimators of linear continuous-time systems. We study the properties of non-causal least-square estimators for continuous-time system identification, and propose a kernel-based non-causal regularized least-squares approach for estimating the band-limited equivalent impulse response. The proposed methods are tested via extensive numerical simulations.

Continuous-time system identification has primarily dealt with sampled input and output data for constructing continuous-time models. However, sampled signals can lead to inaccurate models if their intersample behavior is not addressed appropriately. In this paper, this effect is explored in detail with respect to the SRIVC and CLSRIVC estimators, which are some of the most popular methods for open and closed-loop continuous-time system identification respectively. Based on our consistency analysis, we propose an algorithm that alleviates the asymptotic bias of these methods for arbitrary input excitations and provide an alternative procedure to achieve consistent estimates for band-limited signals. Simulation examples show the effectiveness of our approach.

In this paper, we study non-asymptotic deviation bounds of the least squares estimator in Gaussian AR(n) processes. By relying on martingale concentration inequalities and a tail-bound for χ2 distributed variables, we provide a concentration bound for the sample covariance matrix of the process output. With this, we present a problem-dependent finite-time bound on the deviation probability of any fixed linear combination of the estimated parameters of the AR(n) process. We discuss extensions and limitations of our approach.

In this paper, we analyse the consistency of the Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC). It is well known that the intersample behaviour of the input signal influences the quality and accuracy of the results when estimating and simulating continuous-time models. Here, we present a comprehensive analysis on the consistency of the SRIVC estimator while taking into account the intersample behaviour of the input signal. The main result of the paper shows that, under some mild conditions, the SRIVC estimator is generically consistent. We also describe some conditions when consistency is not achieved, which is important from a practical standpoint. The theoretical results are supported by simulation examples.