In this letter, we study conditions for the existence of stabilizing solutions of two classes of modified discrete algebraic Riccati equations (MAREs) emerging in stochastic control problems. In order to do so, we first rewrite each MARE in terms of a standard ARE subject to specific constraints, which allows us to connect their solution with a set of linear-control constrained problems. With this result we also determine, for each MARE, a linear matrix inequality problem whose feasibility is guaranteed if and only if a stabilizing solution of the original MARE exists.

This paper addresses the problem of stabilizing a MIMO discrete-time LTI system over a MIMO additive noise channel. To treat this problem, we assume that the communication link between the plant output and the controller input consists of multiple additive colored mutually correlated noise channels subject to independent signal-to-noise ratio (SNR) constraints. We derive analytical conditions for which mean square stabilization (MSS) can be achieved under such constraints. We also formulate numerical methods to test these conditions when the noise is white and correlated. Moreover, for simpler plant models, a characterization of the set of power constraints compatible with MSS is obtained. Our results show that the frequency response of the spectral factor related to the channel noise affects the minimum SNR for stability depending mostly on the unstable poles and their directions. This is aggravated by the existence of non minimum phase zeros and higher relative degree of the plant. On the other hand, we detect that high correlated noise show lower SNR requirements for stability compared to independent noise channels. Numerical simulations are provided to illustrate the theoreti

The problem of state estimation of a linear, dynamical state-space system where the output is subject to quantization is challenging and important in different areas of research, such as control systems, communications, and power systems. There are a number of methods and algorithms to deal with this state estimation problem. However, there is no consensus in the control and estimation community on (1) which methods are more suitable for a particular application and why, and (2) how these methods compare in terms of accuracy, computational cost, and user friendliness. In this paper, we provide a comprehensive overview of the state-of-the-art algorithms to deal with state estimation subject to quantized measurements, and an exhaustive comparison among them. The comparison analysis is performed in terms of the accuracy of the state estimation, dimensionality issues, hyperparameter selection, user friendliness, and computational cost. We consider classical approaches and a new development in the literature to obtain the filtering and smoothing distributions of the state conditioned to quantized data. The classical approaches include the extended Kalman filter/smoother, the quantized Kalman filter/smoother, the unscented Kalman filter/smoother, and the sequential Monte Carlo sampling method, also called particle filter/smoother, with its most relevant variants. We also consider a new approach based on the Gaussian sum filter/smoother. Extensive numerical simulations—including a practical application—are presented in order to analyze the accuracy of the state estimation and the computational cost.

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 paper, 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 minimise 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 behaviour 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.

Traditionally, non-parametric impulse and frequency response functions are estimated by taking the ratio of power spectral density estimates. However, this approach may often lead to non-causal estimates. In this letter, we derive a closed form expression for the impulse response estimator by smoothed empirical transfer function estimate, which allows optimal enforcement of causality on non-parametric estimators based on spectral analysis. The new method is shown to be asymptotically unbiased and of minimum covariance in a positive semidefinite sense among a broad class of linear estimators. Numerical simulations illustrate the performance of the new estimator.