When estimating a single subsystem (module) in a linear dynamic network with a prediction error method, a data-informativity condition needs to be satisfied for arriving at a consistent module estimate. This concerns a condition on input signals in the constructed, possibly MIMO (multiple input multiple output) predictor model being persistently exciting, which is typically guaranteed if the input spectrum is positive definite for a sufficient number of frequencies. Generically, the condition can be formulated as a path-based condition on the graph of the network model. The current condition has two elements of possible conservatism: (a) rather than focussing on the full MIMO model, one would like to be able to focus on consistently estimating the target module only, and (b) structural information, such as structural zero elements in the interconnection structure or known subsystems, should be taken into account. In this paper relaxed conditions for data-informativity are derived addressing these two issues, leading to relaxed path-based conditions on the network graph. This leads to experimental conditions that are less strict, i.e. require a smaller number of external excitation signals. Additionally, the new expressions for data-informativity in identification are shown to be closely related to earlier derived conditions for (generic) single module identifiability.

Subspace identification methods (SIMs) are known for their simple parameterization for MIMO systems and robust numerical properties. However, a comprehensive statistical analysis of SIMs remains an open problem. Following a three-step procedure generally used in SIMs, this work presents a finite sample analysis for open-loop SIMs. In Step 1 we begin with a parsimonious SIM. Leveraging a recent analysis of an individual ARX model, we obtain a union error bound for a Hankel-like matrix constructed from a bank of ARX models. Step 2 involves model reduction via weighted singular value decomposition (SVD), where we use robustness results for SVD to obtain error bounds on extended controllability and observability matrices, respectively. The final Step 3 focuses on deriving error bounds for system matrices, where two different realization algorithms, the MOESP type and the CVA type, are studied. Our results not only agree with classical asymptotic results, but also show how much data is needed to guarantee a desired error bound with high probability. The proposed method generalizes related finite sample analyses and applies broadly to many variants of SIMs.

The maximum absolute correlation between regressors, which is called mutual coherence, plays an essential role in sparse estimation. A regressor matrix whose columns are highly correlated may result from optimal input design, since there is no constraint on the mutual coherence, making it difficult to handle sparse estimation. This article aims to tackle this issue for fixed denominator models, which include Laguerre, Kautz, and generalized orthonormal basis function expansion models, for example. The article proposes an optimal input design method where the achieved Fisher information matrix (FIM) is fitted to the desired Fisher matrix, together with a coordinate transformation designed to make the regressors in the transformed coordinates have low mutual coherence. The method can be used together with any sparse estimation method and any desired Fisher matrix. A numerical study shows its potential for alleviating the problem of model order selection when used in conjunction with, for example, classical methods such as the Akaike information criterion.

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 letter 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.

We tackle the problem of finite-time regret minimization in linear quadratic adaptive control. Regret minimization is a scientific field in both adaptive control and reinforcement learning research communities which studies the so-called trade-off between exploration and exploitation. Even though a large focus has been on linear quadratic adaptive control with theoretical finite-time bound guarantees on the expected regret growth rate, most of the proposed optimal exploration strategies do not take into account the scaling constant associated with the growth rate. Moreover, the exploration strategies are limited to white noise excitation. Using tools from experiment design, we propose a computationally tractable solution for the design of the external excitation chosen as a white noise filtered by a finite impulse response filter which is adapted on-line. In a numerical example it is shown that this approach results in a lower regret in comparison with available strategies.

The probability density function of the input plays a crucial role in the process of identifying nonlinear systems, with a finite representation commonly employed in the process. However, input design for nonlinear models is a challenging task because it usually involves optimizing a problem with a large number of free variables, which is computationally heavy. The first contribution of this paper is to demonstrate that the majority of these free variables are zero. Consequently, there is no necessity to optimize all of them. The second contribution is to identify the non-zero variables within this set of free variables associated with input design. To address this, we propose an alternating minimization approach. In the first step, we compute the per-sample Fisher Information Matrix (FIM). Then, in the second phase, we estimate the positions of the non-zero elements within the vector of free variables using the previously derived per-sample FIM. Additionally, in the later phase, we calculate the Lagrangian multipliers in our optimization problem using the Karush–Kuhn–Tucker conditions and derive an upper bound for the hyperparameter, which promotes sparsity. This bound ensures the maximum required number of non-zero variables to represent the per-sample FIM. Following this, the original input design problem is streamlined to optimize the cost function only with respect to the non-zero elements, resulting in a significant reduction in computational time. To assess the effectiveness of our proposed method, we conduct a comprehensive numerical performance evaluation by comparing it to state-of-the-art input design algorithms.

Appropriate excitation conditions are essential for reliably identifying sparse models. In regression problems, these conditions are often characterized by properties of the regressor matrix, with mutual coherence, the maximum correlation between regressors, playing a central role in enabling sparse recovery. However, obtaining sparse estimates is rarely the sole objective, as estimation accuracy is also important. When a model is used in an application, e.g. control design, acceptable performance with high probability can be (approximately) ensured by requiring the confidence ellipsoid to be contained in a certain ellipsoid which depends on the performance specifications in the application. However, it is well known that experiments fulfilling such requirements using minimal excitation energy (least-costly experiments) tend to generate highly correlated regressors, in conflict with the requirements for sparsity. Adhering to this setting and focusing on the popular orthogonal matching pursuit algorithm (OMP), we derive conditions for simultaneously ensuring sparsity and having the confidence ellipsoid contained in a pre-specified ellipsoid. We extend this result to a recently proposed two stage sparse estimation method where a linear transformation is used in a pre-processing step before OMP to reduce mutual coherence. A final contribution is to show that our theoretical results are of importance in optimal input design for sparse models. Specifically, we show that the choice of hyperparameters in a recently proposed input design method can be guided by our contributions and we show explicitly how this can be done in this two-stage method.

The dual control problem, first introduced by Feldbaum in the 1960s, is recognized as encapsulating the "exploration versus exploitation"dilemma, central to online learning and control. Numerous heuristic-based exploration methods have been developed to facilitate active learning. However, the theoretically optimal solution provided by dynamic programming (DP) remains computationally intractable for most problems due to the curse of dimensionality. In this paper, we revisit the DP framework within the context of regret minimization for a simple real-time optimization problem, aiming to identify valuable insights and uncover new avenues for simplified DP-based exploration strategies. By deriving the two-horizon DP solution in our simple setting, we observe that the optimal input is obtained by solving a closed-form optimization problem composed of two distinct components representing exploration and exploitation separately, clearly highlighting their inherent trade-off. For longer horizons, receding and cyclic horizon control based on the iterative application of the two-horizon DP provide possible approximations, reducing computational complexity while yielding useful suboptimal control policies. A key advantage of the DP-based exploration method is its ability to automatically adjust the exploration based on the current exploitation and system uncertainty. The proposed method is studied numerically through comparative evaluations against classical heuristic exploration methods from the literature.

Data-driven control, the task of designing a controller based on process data, finds application in a wide range of disciplines and the topic has been intensively studied over more than half a century. The main purpose of this contribution is to elucidate on the commonalities between data-driven control and parameter estimation. In particular, we discuss the bias-variance trade-off, i.e. rather than aiming for the optimal controller one should aim for a constrained version, that may be characterized by tunable parameters, corresponding to hyperparameters in parameter estimation. As a result we shift attention from indirect vs direct data driven control by highlighting the important role played by (complete) minimal sufficient statistics. To keep technicalities at a minimum, still capturing the essential features of the problem, we consider the problem of minimizing the expected control cost for a quadratic open loop control problem applied to a finite impulse response system. In a Gaussian white noise setting, the maximum-likelihood parameter estimate constitutes a complete minimal sufficient statistic which allows us to focus on controllers that are functions of this model estimate without loss of statistical accuracy. We make a systematic study of three different controller structures and two different tuning techniques and illustrate their behaviours numerically.

Markov parameters play a key role in system identification. There exists many algorithms where these parameters are estimated using least-squares in a first, pre-processing, step, including subspace identification and multi-step least-squares algorithms, such as Weighted Null-Space Fitting. Recently, there has been an increasing interest in non-asymptotic analysis of estimation algorithms. In this contribution we identify the Markov parameters using weighted least-squares and present non-asymptotic analysis for such estimator. To cover both stable and unstable systems, multiple trajectories are collected. We show that with the optimal weighting matrix, weighted least-squares gives a tighter error bound than ordinary least-squares for the case of non-uniformly distributed measurement errors. Moreover, as the optimal weighting matrix depends on the system's true parameters, we introduce two methods to consistently estimate the optimal weighting matrix, where the convergence rate of these estimates is also provided. Numerical experiments demonstrate improvements of weighted least-squares over ordinary least-squares in finite sample settings.