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The identification of continuous-time (CT) systems from discrete-time (DT) input and output signals, i.e., the sampled data, has received considerable attention for half a century. The state-of-the-art methods are parametric methods and thus subject to the typical issues of parametric methods. In the last decade, a major advance in system identification is the so-called kernel-based regularization method (KRM), which is free of the issues of parametric methods. It is interesting to test the potential of KRM on CT system identification. However, very few results have been reported, mainly because the estimators have no closed forms for general CT input signals, except for some very special cases. In this paper, we show for KRM that the estimators have closed forms when the DT input signal has the typical intersample behavior, i.e., zero-order hold or band-limited, and this paves the way for the application of KRM for CT system identification. Numerical Monte Carlo simulations show that the proposed method is more robust than the state-of-the-art methods and more accurate when the sample size is small., Comment: Submitted to IEEE CDC2024

Camera pose estimation is a fundamental problem in robotics. This paper focuses on two issues of interest: First, point and line features have complementary advantages, and it is of great value to design a uniform algorithm that can fuse them effectively; Second, with the development of modern front-end techniques, a large number of features can exist in a single image, which presents a potential for highly accurate robot pose estimation. With these observations, we propose AOPnP(L), an optimal linear-time camera-robot pose estimation algorithm from points and lines. Specifically, we represent a line with two distinct points on it and unify the noise model for point and line measurements where noises are added to 2D points in the image. By utilizing Plucker coordinates for line parameterization, we formulate a maximum likelihood (ML) problem for combined point and line measurements. To optimally solve the ML problem, AOPnP(L) adopts a two-step estimation scheme. In the first step, a consistent estimate that can converge to the true pose is devised by virtue of bias elimination. In the second step, a single Gauss-Newton iteration is executed to refine the initial estimate. AOPnP(L) features theoretical optimality in the sense that its mean squared error converges to the Cramer-Rao lower bound. Moreover, it owns a linear time complexity. These properties make it well-suited for precision-demanding and real-time robot pose estimation. Extensive experiments are conducted to validate our theoretical developments and demonstrate the superiority of AOPnP(L) in both static localization and dynamic odometry systems.

Given 2D point correspondences between an image pair, inferring the camera motion is a fundamental issue in the computer vision community. The existing works generally set out from the epipolar constraint and estimate the essential matrix, which is not optimal in the maximum likelihood (ML) sense. In this paper, we dive into the original measurement model with respect to the rotation matrix and normalized translation vector and formulate the ML problem. We then propose a two-step algorithm to solve it: In the first step, we estimate the variance of measurement noises and devise a consistent estimator based on bias elimination; In the second step, we execute a one-step Gauss-Newton iteration on manifold to refine the consistent estimate. We prove that the proposed estimate owns the same asymptotic statistical properties as the ML estimate: The first is consistency, i.e., the estimate converges to the ground truth as the point number increases; The second is asymptotic efficiency, i.e., the mean squared error of the estimate converges to the theoretical lower bound -- Cramer-Rao bound. In addition, we show that our algorithm has linear time complexity. These appealing characteristics endow our estimator with a great advantage in the case of dense point correspondences. Experiments on both synthetic data and real images demonstrate that when the point number reaches the order of hundreds, our estimator outperforms the state-of-the-art ones in terms of estimation accuracy and CPU time.

For data-driven iterative learning control (ILC) methods, both the model estimation and controller design problems are converted to parameter estimation problems for some chosen model structures. It is well-known that if the model order is not chosen carefully, models with either large variance or large bias would be resulted, which is one of the obstacles to further improve the modeling and tracking performances of data-driven ILC in practice. An emerging trend in the system identification community to deal with this issue is using regularization instead of the statistical tests, e.g., AIC, BIC, and one of the representatives is the so-called kernel-based regularization method (KRM). In this paper, we integrate KRM into data-driven ILC to handle a class of repetitive linear time-varying systems, and moreover, we show that the proposed method has ultimately bounded tracking error in the iteration domain. The numerical simulation results show that in contrast with the least squares method and some existing data-driven ILC methods, the proposed one can give faster convergence speed, better accuracy and robustness in terms of the tracking performance., Comment: 17 pages

We consider signal source localization from range-difference measurements. First, we give some readily-checked conditions on measurement noises and sensor deployment to guarantee the asymptotic identifiability of the model and show the consistency and asymptotic normality of the maximum likelihood (ML) estimator. Then, we devise an estimator that owns the same asymptotic property as the ML one. Specifically, we prove that the negative log-likelihood function converges to a function, which has a unique minimum and positive definite Hessian at the true source's position. Hence, it is promising to execute local iterations, e.g., the Gauss-Newton (GN) algorithm, following a consistent estimate. The main issue involved is obtaining a preliminary consistent estimate. To this aim, we construct a linear least-squares problem via algebraic operation and constraint relaxation and obtain a closed-form solution. We then focus on deriving and eliminating the bias of the linear least-squares estimator, which yields an asymptotically unbiased (thus consistent) estimate. Noting that the bias is a function of the noise variance, we further devise a consistent noise variance estimator that involves $3$-order polynomial rooting. Based on the preliminary consistent location estimate, a one-step GN iteration suffices to achieve the same asymptotic property as the ML estimator. Simulation results demonstrate the superiority of our proposed algorithm in the large sample case.

Input design is an important problem for system identification and has been well studied for the classical system identification, i.e., the maximum likelihood/prediction error method. For the emerging regularized system identification, the study on input design has just started, and it is often formulated as a non-convex optimization problem that minimizes a scalar measure of the Bayesian mean squared error matrix subject to certain constraints, and the state-of-art method is the so-called quadratic mapping and inverse embedding (QMIE) method, where a time domain inverse embedding (TDIE) is proposed to find the inverse of the quadratic mapping. In this paper, we report some new results on the embeddings/inverse embeddings of the QMIE method. Firstly, we present a general result on the frequency domain inverse embedding (FDIE) that is to find the inverse of the quadratic mapping described by the discrete-time Fourier transform. Then we show the relation between the TDIE and the FDIE from a graph signal processing perspective. Finally, motivated by this perspective, we further propose a graph induced embedding and its inverse, which include the previously introduced embeddings as special cases. This deepens the understanding of input design from a new viewpoint beyond the real domain and the frequency domain viewpoints.

Spatial-temporal Gaussian process regression is a popular method for spatial-temporal data modeling. Its state-of-art implementation is based on the state-space model realization of the spatial-temporal Gaussian process and its corresponding Kalman filter and smoother, and has computational complexity $\mathcal{O}(NM^3)$, where $N$ and $M$ are the number of time instants and spatial input locations, respectively, and thus can only be applied to data with large $N$ but relatively small $M$. In this paper, our primary goal is to show that by exploring the Kronecker structure of the state-space model realization of the spatial-temporal Gaussian process, it is possible to further reduce the computational complexity to $\mathcal{O}(M^3+NM^2)$ and thus the proposed implementation can be applied to data with large $N$ and moderately large $M$. The proposed implementation is illustrated over applications in weather data prediction and spatially-distributed system identification. Our secondary goal is to design a kernel for both the Colorado precipitation data and the GHCN temperature data, such that while having more efficient implementation, better prediction performance can also be achieved than the state-of-art result.

Regularized system identification is the major advance in system identification in the last decade. Although many promising results have been achieved, it is far from complete and there are still many key problems to be solved. One of them is the asymptotic theory, which is about convergence properties of the model estimators as the sample size goes to infinity. The existing related results for regularized system identification are about the almost sure convergence of various hyper-parameter estimators. A common problem of those results is that they do not contain information on the factors that affect the convergence properties of those hyper-parameter estimators, e.g., the regression matrix. In this paper, we tackle problems of this kind for the regularized finite impulse response model estimation with the empirical Bayes (EB) hyper-parameter estimator and filtered white noise input. In order to expose and find those factors, we study the convergence in distribution of the EB hyper-parameter estimator, and the asymptotic distribution of its corresponding model estimator. For illustration, we run Monte Carlo simulations to show the efficacy of our obtained theoretical results.

Most existing works on optimal filtering of linear time-invariant (LTI) stochastic systems with arbitrary unknown inputs assume perfect knowledge of the covariances of the noises in the filter design. This is impractical and raises the question of whether and under what conditions one can identify the process and measurement noise covariances (denoted as $Q$ and $R$, respectively) of systems with unknown inputs. This paper considers the identifiability of $Q$/$R$ using the correlation-based measurement difference approach. More specifically, we establish (i) necessary conditions under which $Q$ and $R$ can be uniquely jointly identified; (ii) necessary and sufficient conditions under which $Q$ can be uniquely identified, when $R$ is known; (iii) necessary conditions under which $R$ can be uniquely identified, when $Q$ is known. It will also be shown that for achieving the results mentioned above, the measurement difference approach requires some decoupling conditions for constructing a stationary time series, which are proved to be sufficient for the well-known strong detectability requirements established by Hautus., Comment: formally accepted to and going to appear in IEEE Transactions on Automatic Control. arXiv admin note: substantial text overlap with arXiv:2202.04963

State estimation is an essential part of autonomous systems. Integrating the Ultra-Wideband(UWB) technique has been shown to correct the long-term estimation drift and bypass the complexity of loop closure detection. However, few works on robotics adopt UWB as a stand-alone state estimation solution. The primary purpose of this work is to investigate planar pose estimation using only UWB range measurements and study the estimator's statistical efficiency. We prove the excellent property of a two-step scheme, which says that we can refine a consistent estimator to be asymptotically efficient by one step of Gauss-Newton iteration. Grounded on this result, we design the GN-ULS estimator and evaluate it through simulations and collected datasets. GN-ULS attains millimeter and sub-degree level accuracy on our static datasets and attains centimeter and degree level accuracy on our dynamic datasets, presenting the possibility of using only UWB for real-time state estimation., Comment: Update the content and improve consistency with the ICRA version

The Perspective-n-Point (PnP) problem has been widely studied in both computer vision and photogrammetry societies. With the development of feature extraction techniques, a large number of feature points might be available in a single shot. It is promising to devise a consistent estimator, i.e., the estimate can converge to the true camera pose as the number of points increases. To this end, we propose a consistent PnP solver, named \emph{CPnP}, with bias elimination. Specifically, linear equations are constructed from the original projection model via measurement model modification and variable elimination, based on which a closed-form least-squares solution is obtained. We then analyze and subtract the asymptotic bias of this solution, resulting in a consistent estimate. Additionally, Gauss-Newton (GN) iterations are executed to refine the consistent solution. Our proposed estimator is efficient in terms of computations -- it has $O(n)$ computational complexity. Experimental tests on both synthetic data and real images show that our proposed estimator is superior to some well-known ones for images with dense visual features, in terms of estimation precision and computing time.

We consider the range-based localization problem, which involves estimating an object's position by using $m$ sensors, hoping that as the number $m$ of sensors increases, the estimate converges to the true position with the minimum variance. We show that under some conditions on the sensor deployment and measurement noises, the LS estimator is strongly consistent and asymptotically normal. However, the LS problem is nonsmooth and nonconvex, and therefore hard to solve. We then devise realizable estimators that possess the same asymptotic properties as the LS one. These estimators are based on a two-step estimation architecture, which says that any $\sqrt{m}$-consistent estimate followed by a one-step Gauss-Newton iteration can yield a solution that possesses the same asymptotic property as the LS one. The keypoint of the two-step scheme is to construct a $\sqrt{m}$-consistent estimate in the first step. In terms of whether the variance of measurement noises is known or not, we propose the Bias-Eli estimator (which involves solving a generalized trust region subproblem) and the Noise-Est estimator (which is obtained by solving a convex problem), respectively. Both of them are proved to be $\sqrt{m}$-consistent. Moreover, we show that by discarding the constraints in the above two optimization problems, the resulting closed-form estimators (called Bias-Eli-Lin and Noise-Est-Lin) are also $\sqrt{m}$-consistent. Plenty of simulations verify the correctness of our theoretical claims, showing that the proposed two-step estimators can asymptotically achieve the Cramer-Rao lower bound.

In this paper, we give a tutorial on asymptotic properties of the Least Square (LS) and Regularized Least Squares (RLS) estimators for the finite impulse response model with filtered white noise inputs. We provide three perspectives: the almost sure convergence, the convergence in distribution and the boundedness in probability. On one hand, these properties deepen our understanding of the LS and RLS estimators. On the other hand, we can use them as tools to investigate asymptotic properties of other estimators, such as various hyper-parameter estimators.

This paper investigates two issues on identification of switched linear systems: persistence of excitation and numerical algorithms. The main contribution is a much weaker condition on the regressor to be persistently exciting that guarantees the uniqueness of the parameter sets and also provides new insights in understanding the relation among different subsystems. It is found that for uniquely determining the parameters of switched linear systems, the minimum number of samples needed derived from our condition is much smaller than that reported in the literature. The secondary contribution of the paper concerns the numerical algorithm. Though the algorithm is not new, we show that our surrogate problem, relaxed from an integer optimization to a continuous minimization, has exactly the same solution as the original integer optimization, which is effectively solved by a block-coordinate descent algorithm. Moreover, an algorithm for handling unknown number of subsystems is proposed. Several numerical examples are illustrated to support theoretical analysis.

The asymptotic optimality (a.o.) of various hyper-parameter estimators with different optimality criteria has been studied in the literature for regularized least squares regression problems. The estimators include e.g., the maximum (marginal) likelihood method, $C_p$ statistics, and generalized cross validation method, and the optimality criteria are based on e.g., the inefficiency, the expectation inefficiency and the risk. In this paper, we consider the regularized least squares regression problems with fixed number of regression parameters, choose the optimality criterion based on the risk, and study the a.o. of several cross validation (CV) based hyper-parameter estimators including the leave $k$-out CV method, generalized CV method, $r$-fold CV method and hold out CV method. We find the former three methods can be a.o. under mild assumptions, but not the last one, and we use Monte Carlo simulations to illustrate the efficacy of our findings., Comment: 36 pages, 3 figures

In this paper, we focus on the influences of the condition number of the regression matrix upon the comparison between two hyper-parameter estimation methods: the empirical Bayes (EB) and the Stein's unbiased estimator with respect to the mean square error (MSE) related to output prediction (SUREy). We firstly show that the greatest power of the condition number of the regression matrix of SUREy cost function convergence rate upper bound is always one larger than that of EB cost function convergence rate upper bound. Meanwhile, EB and SUREy hyper-parameter estimators are both proved to be asymptotically normally distributed under suitable conditions. In addition, one ridge regression case is further investigated to show that when the condition number of the regression matrix goes to infinity, the asymptotic variance of SUREy estimator tends to be larger than that of EB estimator.

In this paper, we study the recursion of measurement outcomes for open quantum networks under sequential measurements. Open quantum networks are networked quantum subsystems (e.g., qubits) with the state evolutions described by a continuous Lindblad master equation. When measurements are performed sequentially along such continuous dynamics, the quantum network states undergo random jumps and the corresponding measurement outcomes can be described by a vector of probabilistic Boolean variables. The induced recursion of the Boolean vectors forms a probabilistic Boolean network. First of all, we show that the state transition of the induced Boolean networks can be explicitly represented through realification of the master equation. Next, when the open quantum dynamics is relaxing in the sense that it possesses a unique equilibrium as a global attractor, structural properties including absorbing states, reducibility, and periodicity for the induced Boolean network are direct consequences of the relaxing property. Particularly, we show that generically, relaxing quantum dynamics leads to irreducible and aperiodic chains for the measurement outcomes. Finally, we show that for quantum consensus networks as a type of non-relaxing open quantum network dynamics, the communication classes of the measurement-induced Boolean networks are encoded in the quantum Laplacian of the underlying interaction graph., Comment: 21 pages, 3 figures

In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that $\ell_2$-regularization with proper regularization gain provides even lower mean-square error under a cost in bias. The regularization parameter is tuned by two estimators in terms of a risk characterization. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed tuning estimators are asymptotically optimal as the number of samples grows to infinity under the risk metric. Additionally, numerical examples are provided to validate the established results., Comment: 33 pages, 8 figures

In this paper, we study dynamical quantum networks which evolve according to Schr\"odinger equations but subject to sequential local or global quantum measurements. A network of qubits forms a composite quantum system whose state undergoes unitary evolution in between periodic measurements, leading to hybrid quantum dynamics with random jumps at discrete time instances along a continuous orbit. The measurements either act on the entire network of qubits, or only a subset of qubits. First of all, we reveal that this type of hybrid quantum dynamics induces probabilistic Boolean recursions representing the measurement outcomes. With global measurements, it is shown that such resulting Boolean recursions define Markov chains whose state-transitions are fully determined by the network Hamiltonian and the measurement observables. Particularly, we establish an explicit and algebraic representation of the underlying recursive random mapping driving such induced Markov chains. Next, with local measurements, the resulting probabilistic Boolean dynamics is shown to be no longer Markovian. The state transition probability at any given time becomes dependent on the entire history of the sample path, for which we establish a recursive way of computing such non-Markovian probability transitions. Finally, we adopt the classical bilinear control model for the continuous Schr\"odinger evolution, and show how the measurements affect the controllability of the quantum networks., Comment: 26 pages, 7 figures

Input design is an important issue for classical system identification methods but has not been investigated for the kernel-based regularization method (KRM) until very recently. In this paper, we consider in the time domain the input design problem of KRMs for LTI system identification. Different from the recent result, we adopt a Bayesian perspective and in particular make use of scalar measures (e.g., the $A$-optimality, $D$-optimality, and $E$-optimality) of the Bayesian mean square error matrix as the design criteria subject to power-constraint on the input. Instead to solve the optimization problem directly, we propose a two-step procedure. In the first step, by making suitable assumptions on the unknown input, we construct a quadratic map (transformation) of the input such that the transformed input design problems are convex, the number of optimization variables is independent of the number of input data, and their global minima can be found efficiently by applying well-developed convex optimization software packages. In the second step, we derive the expression of the optimal input based on the global minima found in the first step by solving the inverse image of the quadratic map. In addition, we derive analytic results for some special types of fixed kernels, which provide insights on the input design and also its dependence on the kernel structure.

The kernel-based regularization method has two core issues: kernel design and hyperparameter estimation. In this paper, we focus on the second issue and study the properties of several hyperparameter estimators including the empirical Bayes (EB) estimator, two Stein's unbiased risk estimators (SURE) and their corresponding Oracle counterparts, with an emphasis on the asymptotic properties of these hyperparameter estimators. To this goal, we first derive and then rewrite the first order optimality conditions of these hyperparameter estimators, leading to several insights on these hyperparameter estimators. Then we show that as the number of data goes to infinity, the two SUREs converge to the best hyperparameter minimizing the corresponding mean square error, respectively, while the more widely used EB estimator converges to another best hyperparameter minimizing the expectation of the EB estimation criterion. This indicates that the two SUREs are asymptotically optimal but the EB estimator is not. Surprisingly, the convergence rate of two SUREs is slower than that of the EB estimator, and moreover, unlike the two SUREs, the EB estimator is independent of the convergence rate of $\Phi^T\Phi/N$ to its limit, where $\Phi$ is the regression matrix and $N$ is the number of data. A Monte Carlo simulation is provided to demonstrate the theoretical results.