Håkan Hjalmarsson

Riccardo Sven Risuleo, Marco Molinari, Giulio Bottegal et al.

This paper describes a benchmark consisting of a set of synthetic measurements relative to an office environment simulated with the software IDA-ICE. The simulated environment reproduces a laboratory at the KTH-EES Smart Building, equipped with a building management system. The data set contains records collected over a period of several days. The signals to CO$_2$ concentration, mechanical ventilation airflows, air infiltrations and occupancy. Information on door and window opening is also available. This benchmark is intended for testing data-based modeling techniques. The ultimate goal is the development of models to improve the forecast and control of environmental variables. Among the numerous challenges related to this framework, we point out the problem of occupancy estimation using information on CO$_2$ concentration. This can be seen as a blind identification problem. For benchmarking purposes, we present two different identification approaches: a baseline overparametrization method and a kernel-based method.

Niklas Everitt, Giulio Bottegal, Håkan Hjalmarsson

We present a new method of identifying a specific module in a dynamic network, possibly with feedback loops. Assuming known topology, we express the dynamics by an acyclic network composed of two blocks where the first block accounts for the relation between the known reference signals and the input to the target module, while the second block contains the target module. Using an empirical Bayes approach, we model the first block as a Gaussian vector with covariance matrix (kernel) given by the recently introduced stable spline kernel. The parameters of the target module are estimated by solving a marginal likelihood problem with a novel iterative scheme based on the Expectation-Maximization algorithm. Additionally, we extend the method to include additional measurements downstream of the target module. Using Markov Chain Monte Carlo techniques, it is shown that the same iterative scheme can solve also this formulation. Numerical experiments illustrate the effectiveness of the proposed methods.

Riccardo Sven Risuleo, Giulio Bottegal, Håkan Hjalmarsson

In this paper, we propose a novel algorithm for the identification of Hammerstein systems. Adopting a Bayesian approach, we model the impulse response of the unknown linear dynamic system as a realization of a zero-mean Gaussian process. The covariance matrix (or kernel) of this process is given by the recently introduced stable-spline kernel, which encodes information on the stability and regularity of the impulse response. The static non-linearity of the model is identified using an Empirical Bayes approach, i.e. by maximizing the output marginal likelihood, which is obtained by integrating out the unknown impulse response. The related optimization problem is solved adopting a novel iterative scheme based on the Expectation-Maximization (EM) method, where each iteration consists in a simple sequence of update rules. Numerical experiments show that the proposed method compares favorably with a standard algorithm for Hammerstein system identification.

Riccardo Sven Risuleo, Giulio Bottegal, Håkan Hjalmarsson

In this contribution, we propose a kernel-based method for the identification of linear systems from noisy and incomplete input-output datasets. We model the impulse response of the system as a Gaussian process whose covariance matrix is given by the recently introduced stable spline kernel. We adopt an empirical Bayes approach to estimate the posterior distribution of the impulse response given the data. The noiseless and missing data samples, together with the kernel hyperparameters, are estimated maximizing the joint marginal likelihood of the input and output measurements. To compute the marginal-likelihood maximizer, we build a solution scheme based on the Expectation-Maximization method. Simulations on a benchmark dataset show the effectiveness of the method.

Niklas Everitt, Giulio Bottegal, Cristian R. Rojas et al.

Substantial improvement in accuracy of identified linear time-invariant single-input multi-output (SIMO) dynamical models is possible when the disturbances affecting the output measurements are spatially correlated. Using an orthogonal representation for the modules composing the SIMO structure, in this paper we show that the variance of a parameter estimate of a module is dependent on the model structure of the other modules, and the correlation structure of the disturbances. In addition, we quantify the variance-error for the parameter estimates for finite model orders, where the effect of noise correlation structure, model structure and signal spectra are visible. From these results, we derive the noise correlation structure under which the mentioned model parameterization gives the lowest variance, when one module is identified using less parameters than the other modules.

Miguel Galrinho, Riccardo Prota, Mina Ferizbegovic et al.

For identification of systems embedded in dynamic networks, applying the prediction error method (PEM) to a correct tailor-made parametrization of the complete network provided asymptotically efficient estimates. However, the network complexity often hinders a successful application of PEM, which requires minimizing a non-convex cost function that in general becomes more difficult for more complex networks. For this reason, identification in dynamic networks often focuses in obtaining consistent estimates of particular network modules of interest. A downside of such approaches is that splitting the network in several modules for identification often costs asymptotic efficiency. In this paper, we consider the particular case of a dynamic network with the individual systems connected in a serial cascaded manner, with measurements affected by sensor noise. We propose an algorithm that estimates all the modules in the network simultaneously without requiring the minimization of a non-convex cost function. This algorithm is an extension of Weighted Null-Space Fitting (WNSF), a weighted least-squares method that provides asymptotically efficient estimates for single-input single-output systems. We illustrate the performance of the algorithm with simulation studies, which suggest that a network WNSF may also be asymptotically efficient estimates when applied to cascade networks, and discuss the possibility of extension to more general networks affected by sensor noise.

Niklas Everitt, Miguel Galrinho, Håkan Hjalmarsson

In system identification, it is often difficult to find a physical intuition to choose a noise model structure. The importance of this choice is that, for the prediction error method (PEM) to provide asymptotically efficient estimates, the model orders must be chosen according to the true system. However, if only the plant estimates are of interest and the experiment is performed in open loop, the noise model may be over-parameterized without affecting the asymptotic properties of the plant. The limitation is that, as PEM suffers in general from non-convexity, estimating an unnecessarily large number of parameters will increase the chances of getting trapped in local minima. To avoid this, a high order ARX model can first be estimated by least squares, providing non-parametric estimates of the plant and noise model. Then, model order reduction can be used to obtain a parametric model of the plant only. We review existing methods to perform this, pointing out limitations and connections between them. Then, we propose a method that connects favorable properties from the previously reviewed approaches. We show that the proposed method provides asymptotically efficient estimates of the plant with open loop data. Finally, we perform a simulation study, which suggests that the proposed method is competitive with PEM and other similar methods.

Anubhab Ghosh, Mohamed Abdalmoaty, Saikat Chatterjee et al.

Stochastic nonlinear dynamical systems are ubiquitous in modern, real-world applications. Yet, estimating the unknown parameters of stochastic, nonlinear dynamical models remains a challenging problem. The majority of existing methods employ maximum likelihood or Bayesian estimation. However, these methods suffer from some limitations, most notably the substantial computational time for inference coupled with limited flexibility in application. In this work, we propose DeepBayes estimators that leverage the power of deep recurrent neural networks in learning an estimator. The method consists of first training a recurrent neural network to minimize the mean-squared estimation error over a set of synthetically generated data using models drawn from the model set of interest. The a priori trained estimator can then be used directly for inference by evaluating the network with the estimation data. The deep recurrent neural network architectures can be trained offline and ensure significant time savings during inference. We experiment with two popular recurrent neural networks -- long short term memory network (LSTM) and gated recurrent unit (GRU). We demonstrate the applicability of our proposed method on different example models and perform detailed comparisons with state-of-the-art approaches. We also provide a study on a real-world nonlinear benchmark problem. The experimental evaluations show that the proposed approach is asymptotically as good as the Bayes estimator.

Yue Ju, Bo Wahlberg, Håkan Hjalmarsson

Regularized system identification has become a significant complement to more classical system identification. It has been numerically shown that kernel-based regularized estimators often perform better than the maximum likelihood estimator in terms of minimizing mean squared error (MSE). However, regularized estimators often require hyper-parameter estimation. This paper focuses on ridge regression and the regularized estimator by employing the empirical Bayes hyper-parameter estimator. We utilize the excess MSE to quantify the MSE difference between the empirical-Bayes-based regularized estimator and the maximum likelihood estimator for large sample sizes. We then exploit the excess MSE expressions to develop both a family of generalized Bayes estimators and a family of closed-form biased estimators. They have the same excess MSE as the empirical-Bayes-based regularized estimator but eliminate the need for hyper-parameter estimation. Moreover, we conduct numerical simulations to show that the performance of these new estimators is comparable to the empirical-Bayes-based regularized estimator, while computationally, they are more efficient.

Arun Venkitaraman, Håkan Hjalmarsson, Bo Wahlberg

We address the issue of estimating the topology and dynamics of sparse linear dynamic networks in a hyperparameter-free setting. We propose a method to estimate the network dynamics in a computationally efficient and parameter tuning-free iterative framework known as SPICE (Sparse Iterative Covariance Estimation). The estimated dynamics directly reveal the underlying topology. Our approach does not assume that the network is undirected and is applicable even with varying noise levels across the modules of the network. We also do not assume any explicit prior knowledge on the network dynamics. Numerical experiments with realistic dynamic networks illustrate the usefulness of our method.

Jack Umenberger, Mina Ferizbegovic, Thomas B. Schön et al.

This paper concerns the problem of learning control policies for an unknown linear dynamical system to minimize a quadratic cost function. We present a method, based on convex optimization, that accomplishes this task robustly: i.e., we minimize the worst-case cost, accounting for system uncertainty given the observed data. The method balances exploitation and exploration, exciting the system in such a way so as to reduce uncertainty in the model parameters to which the worst-case cost is most sensitive. Numerical simulations and application to a hardware-in-the-loop servo-mechanism demonstrate the approach, with appreciable performance and robustness gains over alternative methods observed in both.

Lirong Huang, Håkan Hjalmarsson, László Gerencsér

Optimal experiment design for parameter estimation is a research topic that has been in the interest of various studies. A key problem in optimal input design is that the optimal input depends on some unknown system parameters that are to be identified. Adaptive design is one of the fundamental routes to handle this problem. Although there exist a rich collection of results on adaptive experiment design, there are few results that address these issues for dynamic systems. This paper proposes an adaptive input design method for general single-input single-output linear-time-invariant systems.

Riccardo Sven Risuleo, Giulio Bottegal, Håkan Hjalmarsson

In this work, we present a new class of models, called uncertain-input models, that allows us to treat system-identification problems in which a linear system is subject to a partially unknown input signal. To encode prior information about the input or the linear system, we use Gaussian-process models. We estimate the model from data using the empirical Bayes approach: the input and the impulse responses of the linear system are estimated using the posterior means of the Gaussian-process models given the data, and the hyperparameters that characterize the Gaussian-process models are estimated from the marginal likelihood of the data. We propose an iterative algorithm to find the hyperparameters that relies on the EM method and results in simple update steps. In the most general formulation, neither the marginal likelihood nor the posterior distribution of the unknowns is tractable. Therefore, we propose two approximation approaches, one based on Markov-chain Monte Carlo techniques and one based on variational Bayes approximation. We also show special model structures for which the distributions are treatable exactly. Through numerical simulations, we study the application of the uncertain-input model to the identification of Hammerstein systems and cascaded linear systems. As part of the contribution of the paper, we show that this model structure encompasses many classical problems in system identification such as classical PEM, Hammerstein models, errors-in-variables problems, blind system identification, and cascaded linear systems. This allows us to build a systematic procedure to apply the algorithms proposed in this work to a wide class of classical problems.

Giulio Bottegal, Håkan Hjalmarsson, Gianluigi Pillonetto

In this paper we introduce a novel method for linear system identification with quantized output data. We model the impulse response as a zero-mean Gaussian process whose covariance (kernel) is given by the recently proposed stable spline kernel, which encodes information on regularity and exponential stability. This serves as a starting point to cast our system identification problem into a Bayesian framework. We employ Markov Chain Monte Carlo methods to provide an estimate of the system. In particular, we design two methods based on the so-called Gibbs sampler that allow also to estimate the kernel hyperparameters by marginal likelihood maximization via the expectation-maximization method. Numerical simulations show the effectiveness of the proposed scheme, as compared to the state-of-the-art kernel-based methods when these are employed in system identification with quantized data.

Miguel Galrinho, Niklas Everitt, Håkan Hjalmarsson

High-order ARX models can be used to approximate a quite general class of linear systems in a parametric model structure, and well-established methods can then be used to retrieve the true plant and noise models from the ARX polynomials. However, this commonly used approach is only valid when the plant is stable or if the unstable poles are shared with the true noise model. In this contribution, we generalize this approach to allow the unstable poles not to be shared, by introducing modifications to correctly retrieve the noise model and noise variance.

Riccardo Sven Risuleo, Giulio Bottegal, Håkan Hjalmarsson

Recent developments in system identification have brought attention to regularized kernel-based methods, where, adopting the recently introduced stable spline kernel, prior information on the unknown process is enforced. This reduces the variance of the estimates and thus makes kernel-based methods particularly attractive when few input-output data samples are available. In such cases however, the influence of the system initial conditions may have a significant impact on the output dynamics. In this paper, we specifically address this point. We propose three methods that deal with the estimation of initial conditions using different types of information. The methods consist in various mixed maximum likelihood--a posteriori estimators which estimate the initial conditions and tune the hyperparameters characterizing the stable spline kernel. To solve the related optimization problems, we resort to the expectation-maximization method, showing that the solutions can be attained by iterating among simple update steps. Numerical experiments show the advantages, in terms of accuracy in reconstructing the system impulse response, of the proposed strategies, compared to other kernel-based schemes not accounting for the effect initial conditions.

Riccardo Sven Risuleo, Giulio Bottegal, Håkan Hjalmarsson

In this paper we propose a new identification scheme for Hammerstein systems, which are dynamic systems consisting of a static nonlinearity and a linear time-invariant dynamic system in cascade. We assume that the nonlinear function can be described as a linear combination of $p$ basis functions. We reconstruct the $p$ coefficients of the nonlinearity together with the first $n$ samples of the impulse response of the linear system by estimating an $np$-dimensional overparameterized vector, which contains all the combinations of the unknown variables. To avoid high variance in these estimates, we adopt a regularized kernel-based approach and, in particular, we introduce a new kernel tailored for Hammerstein system identification. We show that the resulting scheme provides an estimate of the overparameterized vector that can be uniquely decomposed as the combination of an impulse response and $p$ coefficients of the static nonlinearity. We also show, through several numerical experiments, that the proposed method compares very favorably with two standard methods for Hammerstein system identification.

Giulio Bottegal, Gianluigi Pillonetto, Håkan Hjalmarsson

In this paper we introduce a novel method for linear system identification with quantized output data. We model the impulse response as a zero-mean Gaussian process whose covariance (kernel) is given by the recently proposed stable spline kernel, which encodes information on regularity and exponential stability. This serves as a starting point to cast our system identification problem into a Bayesian framework. We employ Markov Chain Monte Carlo (MCMC) methods to provide an estimate of the system. In particular, we show how to design a Gibbs sampler which quickly converges to the target distribution. Numerical simulations show a substantial improvement in the accuracy of the estimates over state-of-the-art kernel-based methods when employed in identification of systems with quantized data.

Giulio Bottegal, Riccardo S. Risuleo, Håkan Hjalmarsson

We propose a new method for blind system identification. Resorting to a Gaussian regression framework, we model the impulse response of the unknown linear system as a realization of a Gaussian process. The structure of the covariance matrix (or kernel) of such a process is given by the stable spline kernel, which has been recently introduced for system identification purposes and depends on an unknown hyperparameter. We assume that the input can be linearly described by few parameters. We estimate these parameters, together with the kernel hyperparameter and the noise variance, using an empirical Bayes approach. The related optimization problem is efficiently solved with a novel iterative scheme based on the Expectation-Maximization method. In particular, we show that each iteration consists of a set of simple update rules. We show, through some numerical experiments, very promising performance of the proposed method.

Giulio Bottegal, Aleksandr Y. Aravkin, Håkan Hjalmarsson et al.

Recent developments in system identification have brought attention to regularized kernel-based methods. This type of approach has been proven to compare favorably with classic parametric methods. However, current formulations are not robust with respect to outliers. In this paper, we introduce a novel method to robustify kernel-based system identification methods. To this end, we model the output measurement noise using random variables with heavy-tailed probability density functions (pdfs), focusing on the Laplacian and the Student's t distributions. Exploiting the representation of these pdfs as scale mixtures of Gaussians, we cast our system identification problem into a Gaussian process regression framework, which requires estimating a number of hyperparameters of the data size order. To overcome this difficulty, we design a new maximum a posteriori (MAP) estimator of the hyperparameters, and solve the related optimization problem with a novel iterative scheme based on the Expectation-Maximization (EM) method. In presence of outliers, tests on simulated data and on a real system show a substantial performance improvement compared to currently used kernel-based methods for linear system identification.

Cristian R. Rojas, Dimitrios Katselis, Håkan Hjalmarsson

In this article, we analyze the SPICE method developed in [1], and establish its connections with other standard sparse estimation methods such as the Lasso and the LAD-Lasso. This result positions SPICE as a computationally efficient technique for the calculation of Lasso-type estimators. Conversely, this connection is very useful for establishing the asymptotic properties of SPICE under several problem scenarios and for suggesting suitable modifications in cases where the naive version of SPICE would not work.