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A method for the nonparametric estimation of the Frequency Response Function (FRF) was introduced in [5] and latercalled Transient Impulse Response Modeling Method (trimm). We present here a slightly improved version of the originalmethod and, more importantly, we thoroughly analyze the method in terms of bias and variance errors. This analysis leads toguidelines for the choice of the design parameters of the trimm method. Our theoretical expressions for the bias and varianceerrors are validated by simulations which, at the same time, highlight the eect of the design parameters on the performanceof the method., QC 20160516

The topic of this paper is optimal input signal design for identification of interconnected/networked dynamic systems. We consider the case when it is only possible to design some of the input signals, while the rest of the inputs are only measurable. This is most common in industrial applications, where external excitation can only be applied to some subsystems. One example is feed-forward control from measurable disturbances. The optimal input signal will be correlated with the measured signals. The main purpose of this paper is to reveal how to re-formulate the input design problem for networked systems as an input design problem for feedback control systems. We can then use the powerful partial correlation approach for optimal closed loop input design. This means that the corresponding networked optimal input design problem can be formulated as a semi-denite program, for which there are ecient numerical methods. We evaluate this approach using two numerical examples with important applications. The result reveals some non-trivial interesting properties of the optimal input signals., QC 20150126

This letter considers how to approximately reconstruct a cascade system from a given unstructured system estimate. Many system identification methods, including subspace methods, provide reliable but generally unstructured black-box models. The problem we consider is how to find cascade systems that are close to such black-box models. For this, we use model matching techniques and optimal weighted Hankel-norm approximation to obtain accurate low-order cascade systems. We show that it is possible to bound the reconstruction error in terms of an error tolerance parameter and weighted Hankel singular values. The suggested methods are illustrated on both a numerical example and a real double tank system with experimental data., QC 20140618

In this paper we propose a method for applications oriented input design for linear systems in open-loop under time-domain constraints on the amplitude of input and output signals. The method guarantees a desired control performance for the estimated model in minimum time, by imposing some lower bound on the information matrix. The problem is formulated as a time-domain optimization problem, which is non-convex. This is addressed through an alternating method, where we separate the problem into two steps and at each step we optimize the cost function with respect to one of two variables. We alternate between these two steps until convergence. A time recursive input design algorithm is performed, which enables us to use the algorithm with control. Therefore, a receding horizon framework is used to solve each optimization problem. Finally, we illustrate the method with a numerical example which shows the good ability of the proposed approach in generating an optimal input signal., QC 20151214

In this paper we extend a recent method for generating an input signal with a desired auto-correlation function while satisfying both input and output constraints for the system it is to be applied to. This is an important problem in system identication, firstly, the properties of the identified model is highly dependent on the used excitation signal during the experiment and secondly, on real processes, due to actuator saturation and safety considerations, it is important to constraint the input and output from the process. The proposed method corresponds to a nonlinear model predictive control problem and here we extend it to consider general input horizons and also to the multiple-input multiple-output case. In general this corresponds to solving a multivariate polynomial of order four in each time step. Here two dierent methods for solving this problem are considered: one based on convex relaxation and one based on a cyclic algorithm. The performance of the algorithm is successfully veried by simulations and the implications of dierent lengths of the input horizon is discussed., QC 20140107

This paper considers the problem of realizing an input signal with a desired autocorrelation sequence satisfying both input and output constraints for the system it is to be applied to. This is a important problem in system identification. Firstly, the properties of the identified model are highly dependent on the used excitation signal during the experiment and secondly, on real processes, due to actuator saturation and safety considerations, it is important to constrain the inputs and outputs of the process. The proposed method is formulated as a nonlinear model predictive control problem. In general this corresponds to solving a non-convex optimization problem. Here we show how this can be solved in one particular case. For this special case convergence is established for generation of pseudo-white noise. The performance of the algorithm is successfully verified by simulations for a few different autocorrelation sequences, with and without input and output constraints., QC 20130710

Generating signals with a prespecified autocorrelation, available from some input design, while satisfying constraints on input and output signals of the systems which the signal is to be applied on is an important problem in system identification. This paper extends a recently proposed method for such signal generation. The method is modified to be robust to uncertainties in the system model. An adaptive formulation is also given which allows for improving the input design online. The adaptive method is compared to the true optimal input design on a simulation example. The robustness properties of the method are illustrated on an experimental setup., QC 20130710

This paper considers optimal input design when the intended use of the identified model is to construct a feed-forward controller based on measurable disturbances. The objective is to find a minimum power excitation signal to be used in a system identification experiment, such that the corresponding model-based feed-forward controller guarantees, with a given probability, that the variance of the output signal is within given specifications. To start with, some low order model problems are analytically solved and fundamental properties of the optimal input signal solution are presented. The optimal input signal contains feed-forward control and depends on the noise model and transfer function of the system in a specific way. Next, we show how to apply the partial correlation approach to closed loop optimal experiment design to the general feed-forward problem. A framework for optimal input signal design for feed-forward control is presented and numerically evaluated on a temperature control problem., QC 20140603

This paper analyzes two recent methods for the nonparametric estimation of the Frequency Response Function (FRF) from input-output data using Prediction Error identification. Such FRF estimate can be the main goal of the identification exercise, or it can be a tool for the computation of a nonparametric estimate of the noise spectrum. We show that the choice of the method depends on the signal to noise ratio and on the objective. The method that delivers the best FRF estimate may not deliver the best estimate of the noise spectrum. Our theoretical analysis is illustrated by simulations., QC 20121121

Recently, Hägg, Hjalmarsson and Wahlberg proposed a novel non-parametric method that directly estimates the frequency response at N equidistant frequencies when N measurements are available. The specific feature of the method is that together with these estimates, the transient, or, equivalently, the leakage, is explicitly estimated. The estimates are obtained by solving a least-squares problem. The method involves three design variables, the number of estimated transient terms, a number of auxiliary impulse response coefficients (that also are estimated), and the size of a frequency window. At present there is no analysis of how these design variables affect the properties of the method, which we will call TRIMM (TRansient Impulse response Modeling Method). In this contribution we provide bias and variance analysis for two extreme cases of the window size. We show that at one extreme value, the method coincides with the Empirical Transfer Function Estimate, and at the other extreme it is close to directly estimating a FIR model. This indicates that TRIMM provides an intermediate between non-parametric and parametric estimation. The results allows us to quantify bias and variance errors at the two extreme cases under study, and gives insight into how to choose the design variables in a systematic way., QC 20121122

This paper analyzes two recent methods for the nonparametric estimation of the Frequency Response Function (FRF) from input-output data using Prediction Error identification. Such FRF estimate can be the main goal of the identification exercise, or it can be a tool for the computation of a nonparametric estimate of the noise spectrum. We show that the choice of the method depends on the signal to noise ratio and on the objective. The method that delivers the best FRF estimate may not deliver the best estimate of the noise spectrum. Our theoretical analysis is illustrated by simulations.

Traditionally, the frequency response function has been estimated directly by dividing the discrete Fourier transforms of the output and the input of the system. This approach suffers from leakage errors and noise sensitivity. Lately these errors have been studied in detail. The main observation is that the error has a smooth frequency characteristic that is highly structured. The recently proposed local polynomial method uses this smoothness, and tries to estimate the frequency response function along with a smooth approximation of the error term. In this paper we propose a method, closely related to the local polynomial method, but instead of using the smoothness of the error we explore the structure even further. The proposed approach to estimate the frequency response function seems promising, as illustrated by simulations and comparison with current state of the art methods., QC 20120119

We consider identification of systems with a parallel serial (cascade) structure with multiple-input and multiple-output signals. The statistical properties of estimated models are studied with respect to input signals and possible sensor locations. The quality of the estimates are analyzed by means of the asymptotic covariance matrix of the estimated parameters. This is an extension of previous work on identification of cascaded linear systems. The key result concerns systems where the sub-systems have common dynamics. An interesting observation is that for this case the variance for the parameters belonging to the unmeasured subsystem always is larger than for the other sub-systems. This is not true for general parameters. The variance results can be used for optimal input and sensor location design. The results are illustrated by some simple FIR examples and numerical evaluations., QC 20120123

In identification it is important to take a priori structural information into account in many applications, something that is difficult when using subspace methods. Here will study how to incorporate a special structure, a cascade structure with two subsystems. Two new methods are derived for estimating system with this structure. The problem when using subspace identification on cascade structured system is that the states from the first subsystem are mixed with states from the second subsystem via a unknown similarity transform. The first indirect method finds a similarity transform that takes the system back to a form such that the subsystems can be recovered. The second method uses the fact that the structure of the extended observability matrix is known for cascade systems. However, it only works when both subsystems have order one. In practice this is still a common case. The results of the two methods seem promising, as illustrated by applying the methods to a real process, the double tank process. The performance is comparable with state of the art methods. Finally the problem of optimal input design for cascade systems are introduced, and illustrated by a simple example., QC 20240110