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.

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.

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.

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. 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.

While subspace identification methods (SIMs) are appealing due to their simple parameterization for MIMO systems and robust numerical realizations, a comprehensive statistical analysis of SIMs remains an open problem, especially in the non-asymptotic regime. In this work, we provide a finite sample analysis for a class of SIMs, which reveals that the convergence rates for estimating Markov parameters and system matrices are O(1 / SN), in line with classical asymptotic results. Based on the observation that the model format in classical SIMs is non-causal because of a projection step, we choose a parsimonious SIM that bypasses the projection step and strictly enforces a causal model to facilitate the analysis, where a bank of ARX models are estimated in parallel. Leveraging recent results from a finite sample analysis of an individual ARX model, we obtain a union error bound for an array of ARX models and proceed to derive error bounds for system matrices using robustness results for the singular value decomposition.

In the presence of uncertainty, the optimum obtained based on a nominal identified model can neither provide any performance guarantee nor ensure that critical constraints are satisfied, which is crucial for e.g., bioprocess applications characterized by a high degree of complexity combined with costly experiments. Hence, uncertainty should be considered in the optimization and, furthermore, experiments designed to reduce the uncertainty most important for optimization. Herein, we propose a general framework that combines model-based robust optimization with optimal experiment design. The proposed framework can take advantage of prior knowledge in the form of a mechanistic model structure, and the importance of this is demonstrated by comparing it to more standard black-box models typically employed in learning. Through optimal experiment design, we repeatedly reduce the uncertainty most relevant for optimization so as to maximize the potential for improving the worst-case performance by balancing between exploration and exploitation. This makes the proposed method an efficient model-based robust optimization framework, especially in cases with limited experimental resources. The main part of the paper focuses on the case with modeling uncertainty that can be reduced with the availability of more experimental data. Towards the end of the paper, we consider extending the method to also include inherent uncertainty, such as input uncertainty and unmeasured disturbances. The effectiveness of the method is illustrated through a realistic simulation case study of medium optimization of Chinese hamster ovary cell cultivation in continuous monoclonal antibody production, where the metabolic network consists of 23 extracellular metabolites and 126 reactions.

Soft sensing is a way to indirectly obtain information of signals for which direct sensing is difficult or prohibitively expensive. It may not a priori be evident which sensors provide useful information about the target signal, and various operating conditions often necessitate different models. In this paper, we provide a systematic method to construct a soft sensor that can deal with these issues. We propose a single estimation criterion, where the objectives are encoded in terms of model fit, model sparsity (reducing the number of different models), and model parameter coefficient sparsity (to exclude irrelevant sensors). The proposed method is tested on real-world scenarios involving prototype vehicles, demonstrating its effectiveness.