The complexity of model classes, and smoothing noisy data

We consider the problem of smoothing a sequence of noisy observations using a fixed class of models. Via a deterministic analysis, we obtain necessary and sufficient conditions on the noise sequence and model class that ensure that a class of natural estimators gives near-optimal smoothing. In the case of i.i.d. random noise, we show that the accuracy of these estimators depends on a measure of complexity of the model class involving covering numbers. Our formulation and results are quite general and are related to a number of problems in learning, prediction, and estimation. As a special case, we consider an application to output smoothing for certain classes of linear and nonlinear systems. The performance of output smoothing is given in terms of natural complexity parameters of the model class, such as bounds on the order of linear systems, the -norm of the impulse response of stable linear systems, or the memory of a Lipschitz nonlinear system satisfying a fading memory condition.     

On noncausal weighted least squares identification of nonstationary stochastic systems

In this paper, we consider the problem of noncausal identification of nonstationary, linear stochastic systems, i.e., identification based on prerecorded input/output data. We show how several competing weighted (windowed) least squares parameter smoothers, differing in memory settings, can be combined together to yield a better and more reliable smoothing algorithm. The resulting parallel estimation scheme automatically adjusts its smoothing bandwidth to the unknown, and possibly time-varying, rate of nonstationarity of the identified system. We optimize the window shape for a certain class of parameter variations and we derive computationally attractive recursive smoothing algorithms for such an optimized case.     

MPC for continuous piecewise-affine systems

A large class of hybrid systems can be described by a max–min-plus-scaling (MMPS) model (i.e., using the operations maximization, minimization, addition and scalar multiplication). First, we show that continuous piecewise-affine systems are equivalent to MMPS systems. Next, we consider model predictive control (MPC) for these systems. In general, this leads to nonlinear, nonconvex optimization problems. We present a new MPC method for MMPS systems that is based on canonical forms for MMPS functions. In case the MPC constraints are linear constraints in the inputs only, this results in a sequence of linear optimization problems such that the MPC control can often be computed in a much more efficient way than by just applying nonlinear optimization as was done in previous work.     

Coprime factorization over a class of nonlinear systems

In this paper, we consider the problem of generalizing elements of linear coprime factorization theory to a nonlinear context. The idea is to work with a suitably wide class of nonlinear systems to cover many practical situations, yet not cope with so broad a class as to disallow useful generalizations to the linear results. In particular, we work with nonlinear systems characterized in terms of (possibly time-varying) state-dependent matrices A(x), B(x), C(x), D(x) and an initial state x₀. (This class clearly does contain the class of finite-dimensional linear (time-varying) systems.) We achieve first right coprime factorizations for idealized situations. To achieve stable left factorizations we specialize to the case where the matrices are output-dependent. Alternatively, we work with systems, perhaps augmented by a direct feedthrough term, where the input is reconstructible from the output. For nonlinear feedback control systems, with plant and controller having stable left factorizations, then under appropriate regularity-conditions earlier results have allowed the generation of the class of stabilizing controllers for a system in terms of an arbitrary stable system (parameter). Plant uncertainties, including unknown initial conditions are modelled by means of a Yula–Kucera-type parametrization approach developed for nonlinear systems. Certain robust stabilization results are also shown, and simulations demonstrate the regulation of nonlinear plants using the techniques developed. All the results are presented in such a way that specialization for the case of linear systems is immediate.     

Noise covariance identification for time-varying and nonlinear systems

Kalman-based state estimators assume a priori knowledge of the covariance matrices of the process and observation noise. However, in most practical situations, noise statistics and initial conditions are often unknown and need to be estimated from measurement data. This paper presents an auto-covariance least-squares-based algorithm for noise and initial state error covariance estimation of large-scale linear time-varying (LTV) and nonlinear systems. Compared to existing auto-covariance least-squares based-algorithms, our method does not involve any approximations for LTV systems, has fewer parameters to determine and is more memory/computationally efficient for large-scale systems. For nonlinear systems, our algorithm uses full information estimation/moving horizon estimation instead of the extended Kalman filter, so that the stability and accuracy of noise covariance estimation for nonlinear systems can be guaranteed or improved, respectively.     

On robust synchronization of nonlinear systems with application to grid integration of renewable energy sources

Many systems in the natural and physical world often work in unison with similar other systems. This process of simultaneous operation is known as synchronization. In the past few decades, owing to this phenomenon’s importance, extensive research efforts have been made. However, many of the existing results consider the systems are identical and/or linear time-invariant, while practical systems are often nonlinear and nonidentical for various reasons. This observation motivated several recent studies on the synchronization of nonidentical (i.e., heterogeneous) nonlinear systems. This paper summarizes some recent results on the synchronization of heterogeneous nonlinear systems, as developed in the thesis (Ahmed 2016). First, the results on the synchronization of a particular class of robustly stable nonlinear systems are presented. Then, these results are applied to an example model known as Brockett oscillator. Finally, using the Brockett oscillator as a common dynamics, output oscillatory synchronization results are given for heterogeneous nonlinear systems of relative degree 2 or higher. An application example of Brockett oscillator for power-grid synchronization is also presented. Some outlooks are provided regarding future research directions.     

带有色观测噪声系统最优和自校正状态估计器

基于ＡＲＭＡ新息模型，通过计算白噪声估值器和输出预报器，提出了带有色观测噪声系统的一种新的最优和自校正状态估计器，可统一处理滤波、平滑和预报问题，可处理未知的非平衡有色观测噪声、不稳定系统和状态转移阵奇异的系统。一个雷达跟踪系统的仿真例子说明了其有效性。     

Stochastic system identification with noisy input-outputmeasurements using polyspectra

Two new classes of parametric, frequency domain approaches are proposed for estimation of the parameters of scalar, linear “errors-in-variables” models, i.e., linear systems where measurements of both input and output of the system are noise contaminated. The first approach consists of linear estimators where using the bispectrum or the integrated polyspectrum of the input and the cross-bispectrum or the integrated cross-polyspectrum, the system transfer function is first estimated at a number of frequencies exceeding one-half the number of unknown parameters. The estimated transfer function is then used to estimate the unknown parameters using an overdetermined linear system of equations. In the second class of approaches, quadratic transfer function matching criteria are optimized by using the results of the linear estimators as initial guesses. Both classes of the parameter estimators are shown to be consistent in any measurement noise that has symmetric probability density function when the bispectral approaches are used. The proposed parameter estimators are shown to be consistent in Gaussian measurement noise when trispectral approaches are used     

Adaptive stabilization of a class of time-varying systems with an uncertain delay

Handling delays and uncertain parameters in control systems is an interesting and challenging class of problems. In this paper, we consider the problem of “bounded-input bounded-output stabilizing” a class of single-input single-output, linear time-varying plant models with a time-delay margin as large as desired and a considerable amount of uncertainty in the input matrix of the state-space model. The proposed controller, while periodic and mildly nonlinear, is of low complexity; it tolerates slow variations in the delay and the elements of the input matrix as well as occasional jumps in these parameters, and guarantees that the effect of the initial condition decays exponentially to zero, even in the presence of noise.     

自校正Kalman滤波、预报、去卷、平滑新方法

本文用现代时间序列分析方法，提出了基于白噪声估值器和输出预报器解决线性离散定常系统稳态最优和自校正Ｋａｌｍａｎ滤波，预报，平滑问题新方法，此方法可能应用于跟踪系统、信号处理、通讯系统等领域，仿真例子说明了新方法的有效性。     

A result on output regulation of lower triangular systems with unknown high‐frequency gain sign

This paper concerns global robust output regulation of a class of nonlinear lower triangular systems with an unknown high‐frequency gain as well as an unknown exosystem. A novel class of internal model candidates is integrated with the output regulation framework. As a result, stabilization of the augmented system can be performed without parameter estimators. However, the new internal models bring challenges to the stabilization of the augmented system. To overcome these challenges, we propose a new recursive controller design procedure and use it to develop a Nussbaum‐gain‐based controller. This work extends the existing results on nonlinear output regulation of lower triangular systems to the case where both the high‐frequency gain sign and the exosystem are unknown. Copyright © 2017 John Wiley & Sons, Ltd.     

Optimal smoothing of non-linear dynamic systems via Monte Carlo Markov chains

We consider the smoothing problem of estimating a sequence of state vectors given a nonlinear state space model with additive white Gaussian noise, and measurements of the system output. The system output may also be nonlinearly related to the system state. Often, obtaining the minimum variance state estimates conditioned on output data is not analytically intractable. To tackle this difficulty, a Markov chain Monte Carlo technique is presented. The proposal density for this method efficiently draws samples from the Laplace approximation of the posterior distribution of the state sequence given the measurement sequence. This proposal density is combined with the Metropolis-Hastings algorithm to generate realizations of the state sequence that converges to the proper posterior distribution. The minimum variance estimate and confidence intervals are approximated using these realizations. Simulations of a fed-batch bioreactor model are used to demonstrate that the proposed method can obtain significantly better estimates than the iterated Kalman-Bucy smoother.     

Suboptimal stochastic linear feedback control of linear systems with state- and control-dependent noise: The incomplete information case

The stochastic regulation problem for linear systems with state- and control-dependent noise and a noisy linear output equation is considered. The optimal quadratic cost output-feedback control law in a class of linear controllers is found. This problem was first addressed in the early 1970s and solved, in the complete information case, by Wonham. In this paper we give the solution of the problem in the incomplete information case, that is, for a linear output equation corrupted by Gaussian noise. Moreover, a different method is used here, giving the solution in a more direct way even in the complete information case.     

Recursive smoothing algorithms for the estimation of signals from uncertain observations via mixture approximations

Considering discrete-time systems with uncertain observations when the signal model is unknown, but only covariance information is available, and the signal and the observation additive noise are correlated and jointly Gaussian, we present recursive algorithms for suboptimal fixed-point and fixed-interval smoothing estimators. To derive the algorithms, we employ a technique consisting in approximating the conditional distributions of the signal given the observations by Gaussian distributions, taking successive approximations of the mixtures of normal distributions. The expectation of these approximations provides us with the suboptimal estimators. In a numerical simulation example, the performance of the proposed estimators is compared with that of linear ones, via the sample mean square values of the corresponding estimation errors.     

一类仿射非线性系统的概率密度函数形状控制

针对一类仿射非线性有界动态随机系统,提出一种最优概率密度函数（PDF）跟踪控制算法,使得系统的输出PDF跟踪给定的PDF．首先利用线性B样条解耦得到仿射非线性状态方程和PDF逼近方程,使PDF跟踪转化为状态方程输出权值的跟踪;然后采用线性时变序列逼近方法将非线性系统转化为线性时变系统,通过对线性系统的迭代运算得到非线性系统的最优跟踪控制器,从而实现最优PDF跟踪．理论分析和仿真实验均表明了所提出算法的有效性．     

Obtaining initial parameter estimates for nonlinear systems using multicriteria associative memories

Parameter estimation problems for nonlinear systems are typically formulated as nonlinear optimization problems. For such problems, one has the usual difficulty that standard successive approximation schemes require good initial estimates for the parameter vector. This paper develops a simple multicriteria associative memory (MAM) procedure for obtaining useful initial parameter estimates for nonlinear systems. An easily calculated one-parameter family of associative memory matrices is developed; see Equation (25). Each memory matrix is efficient with respect to two criteria: accurate recovery of parameter-output training case associations; and small matrix norm to guard against noise arising from imprecise calculations and observations. For illustration, the MAM procedure is used to obtain initial parameter estimates for a well-known nonlinear economic model, the Solow-Swan growth model. Surprisingly accurate initial parameter estimates are obtained over broad ranges of the family of MAM memory matrices, even when observations are corrupted by i.i.d. or correlated noise.     

A note on observability tests for general polynomial and simple Wiener–Hammerstein systems

We propose an algorithm, based on symbolic computation packages, for testing observability conditions of general polynomial systems, which were formulated in Sontag, SIAM J. Control Optim. 17 (1979) 139–151. Computational complexity of the observability test can be reduced and the test simplified for classes of polynomial systems. We illustrate this by considering the class of simple Wiener–Hammerstein systems, which consist of a series of two linear dynamic blocks between which a static nonlinearity is “sandwiched”. We consider the case when the nonlinearity is a monomial . Simple necessary and sufficient conditions for observability are given and they resemble, but are subtly different from, the well known result on observability for the series connection of two linear systems.     

Adaptive noise smoothing filter for images with signal-dependent noise

In this paper, we consider the restoration of images with signal-dependent noise. The filter is noise smoothing and adapts to local changes in image statistics based on a nonstationary mean, nonstationary variance (NMNV) image model. For images degraded by a class of uncorrelated, signal-dependent noise without blur, the adaptive noise smoothing filter becomes a point processor and is similar to Lee's local statistics algorithm [16]. The filter is able to adapt itself to the nonstationary local image statistics in the presence of different types of signal-dependent noise. For multiplicative noise, the adaptive noise smoothing filter is a systematic derivation of Lee's algorithm with some extensions that allow different estimators for the local image variance. The advantage of the derivation is its easy extension to deal with various types of signal-dependent noise. Film-grain and Poisson signal-dependent restoration problems are also considered as examples. All the nonstationary image statistical parameters needed for the filter can be estimated from the noisy image and no a priori information about the original image is required.