Multivariable Stochastic Systems Research Articles

Filtered generalized iterative parameter identification for equation‐error autoregressive models based on the filtering identification idea

SummaryBy using the collected batch data and the iterative search, and based on the filtering identification idea, this article investigates and proposes a filtered multi‐innovation generalized projection‐based iterative identification method, a filtered generalized gradient‐based iterative identification method, a filtered generalized least squares‐based iterative identification method, a filtered multi‐innovation generalized gradient‐based iterative identification method and a filtered multi‐innovation generalized least squares‐based iterative identification method for equation‐error autoregressive systems described by the equation‐error autoregressive models. These filtered generalized iterative identification methods can be extended to other linear and nonlinear scalar and multivariable stochastic systems with colored noises.

Non-Gaussian disturbance rejection control for multivariate stochastic systems using moment-generating function

In this paper, a non-Gaussian disturbance rejection control algorithm for a class of nonlinear multivariate stochastic systems is studied. Based on the moment-generating functions obtained from the deduced probability density functions of the output tracking errors, a new criterion representing the stochastic properties of the system is proposed, motivated by a minimum entropy design. A time-variant linear model can be established by the sampled moment-generating functions. Using this model, a control algorithm is developed that minimizes the newly developed criterion. Moreover, a stability analysis is performed for the closed-loop control system. Finally, simulation results of a numerical example demonstrate the effectiveness of the presented control algorithm. The contribution and novelty of this work can be summarized as follows: (1) a novel non-Gaussian disturbance rejection control scheme is proposed based on the minimum entropy principle, (2) the randomness of the multi-variable non-Gaussian stochastic nonlinear system is attenuated based on the new performance criterion, (3) a theoretical convergence analysis has been given for the proposed control system, and (4) a potential framework has been established for the design of a general stochastic system control.

Filtered auxiliary model recursive generalized extended parameter estimation methods for Box–Jenkins systems by means of the filtering identification idea

AbstractFor equation‐error autoregressive moving average systems, that is, Box–Jenkins systems, this paper presents a filtered auxiliary model generalized extended stochastic gradient identification method, a filtered auxiliary model multi‐innovation generalized extended stochastic gradient identification method, a filtered auxiliary model recursive generalized extended gradient identification method, a filtered auxiliary model multi‐innovation recursive generalized extended gradient identification method, a filtered auxiliary model recursive generalized extended least squares identification method, and a filtered auxiliary model multi‐innovation recursive generalized extended least squares identification method by using the filtering identification idea and the auxiliary model identification idea. The proposed filtered auxiliary model recursive generalized extended identification methods can be generalized to other linear and nonlinear multivariable stochastic systems with colored noises.

Performance analysis of least squares algorithm for multivariable stochastic systems

Performance analysis of least squares algorithm for multivariable stochastic systems

From Nonlinear Dominant System to Linear Dominant System: Virtual Equivalent System Approach for Multiple Variable Self-Tuning Control System Analysis

The stability and convergence analysis of a multivariable stochastic self-tuning system (STC) is very difficult because of its highly nonlinear structure. In this paper, based on the virtual equivalent system method, the structural nonlinear or nonlinear dominated multivariable self-tuning system is transformed into a structural linear or linear dominated system, thus simplifying the stability and convergence analysis of multivariable STC systems. For the control process of a multivariable stochastic STC system, parameter estimation is required, and there may be three cases of parameter estimation convergence, convergence to the actual value and divergence. For these three cases, this paper provides four theorems and two corollaries. Given the theorems and corollaries, it can be directly concluded that the convergence of parameter estimation is a sufficient condition for the stability and convergence of stochastic STC systems but not a necessary condition, and the four theorems and two corollaries proposed in this paper are independent of specific controller design strategies and specific parameter estimation algorithms. The virtual equivalent system theory proposed in this paper does not need specific control strategies, parameters and estimation algorithms but only needs the nature of the system itself, which can judge the stability and convergence of the self-tuning system and relax the dependence of the system stability convergence criterion on the system structure information. The virtual equivalent system method proposed in this paper is proved to be effective when the parameter estimation may have convergence, convergence to the actual value and divergence.

Constrained tracking control of stochastic multivariable nonlinear systems via Gaussian process predictions

To overcome the difficulty of propagating the stochastic uncertainties through a nonlinear model for successful online implementation of the stochastic nonlinear model predictive control (SNMPC) framework, this paper proposes the utilisation of Gaussian processes (GPs) in the context of SNMPC for the stochastic multivariable nonlinear systems to track a given trajectory in the presence of stochastic uncertainties and system constraints. Taking advantage of the GP regression which not only provides predictions, but also uncertainty quantification in the function estimation, the proposed GP-SNMPC architecture exploits the probability distribution of stochastic uncertainties and utilises the predictions provided by the GPs to formulate the model cost and constraint functions, which yields a tractable framework for handling nonlinear constrained control problems with Gaussian parametric uncertainties. By employing a cancellation strategy, the control law consists of two components, that is, tracking control law and disturbance rejection control law. Theoretical results regarding the performance bounds of the closed-loop system are derived. Numerical examples are provided to verify the effectiveness of the proposed GP-SNMPC scheme.

Stochastic physics-informed neural ordinary differential equations

Stochastic differential equations (SDEs) are used to describe a wide variety of complex stochastic dynamical systems. Learning the hidden physics within SDEs is crucial for unraveling fundamental understanding of these systems' stochastic and nonlinear behavior. We propose a flexible and scalable framework for training artificial neural networks to learn constitutive equations that represent hidden physics within SDEs. The proposed stochastic physics-informed neural ordinary differential equation framework (SPINODE) propagates stochasticity through the known structure of the SDE (i.e., the known physics) to yield a set of deterministic ODEs that describe the time evolution of statistical moments of the stochastic states. SPINODE then uses ODE solvers to predict moment trajectories. SPINODE learns neural network representations of the hidden physics by matching the predicted moments to those estimated from data. Recent advances in automatic differentiation and mini-batch gradient descent with adjoint sensitivity are leveraged to establish the unknown parameters of the neural networks. We demonstrate SPINODE on three benchmark in-silico case studies and analyze the framework's numerical robustness and stability. SPINODE provides a promising new direction for systematically unraveling the hidden physics of multivariate stochastic dynamical systems with multiplicative noise.

Constrained tracking control of stochastic multivariable nonlinear systems using unscented Kalman filter

To overcome the challenges, including propagation of the stochastic uncertainties through a nonlinear model and full determination of the states over the prediction horizon for successful online implementation of the stochastic nonlinear model predictive control (SNMPC) framework, this paper proposes the utilisation of unscented Kalman filter (UKF) in the context of SNMPC for the stochastic multivariable nonlinear systems to track a given trajectory in the presence of stochastic uncertainties and system constraints. Taking advantage of the UKF which estimates and propagates the entire conditional distribution of the states along the time horizon by taking into account errors due to external disturbances and measurement noises as well as errors introduced due to the imperfect knowledge of the states through the state estimation, the proposed UKF-SNMPC framework utilises the conditional mean and conditional covariance of the states to reformulate the model cost and constraint functions, which yields a tractable framework for handling nonlinear chance-constrained tracking control problems. By employing a cancellation strategy, the control law consists of two components, that is, a tracking control law is designed to follow the desired trajectory and an uncertainty compensation control law designed to accommodate the nonlinearities in the plant. The concept of ‘robust horizon’ is introduced as the terminal constraint to prevent the covariance from the explosion and guarantee the convergence of the UKF-SNMPC framework. Performance bounds of the closed-loop system are theoretically analysed. Comparative studies on numerical examples are carried out to verify the effectiveness of the proposed UKF-SNMPC scheme.

Backstepping-based state estimation for a class of stochastic nonlinear systems

The state estimation problem is investigated for a class of continuous-time stochastic nonlinear systems, where a novel filter design method is proposed based on backstepping design and stochastic differential equation. In particular, the structure of the filter is developed following the nonlinear system model, and then the estimation error dynamics can be described by a stochastic differential equation. Motivated by backstepping procedure, the nonlinear dynamics can be converted to an Ornstein–Uhlenbeck process via the control loop design. Thus, the estimation can be achieved once the estimation error is bounded and the variance of the error can be optimized. Since the ideal estimation error is a Brownian motion, the filter parameters can be selected following the Lyapunov stability theory and variance assignment method. Following the same framework, the multivariate stochastic systems can be handled with the block backstepping design. To validate the presented design approach, a numerical example is given as the simulation results to demonstrate the state estimation performance.

Competitive failure analysis of a stochastic degradation system based on performance characteristics fusion

With the aim of solving the reliability modeling and calculation of multivariate stochastic degradation systems, two stochastic degradation models based on the bivariate Wiener process are proposed, in which two performance characteristics are composited to one variable. Two different failure modes including the defect-based failure and the duration-based failure are considered. The explicit expressions of the system reliability are derived in the cases that the performance characteristics are not composited or the performance characteristics are composited according to the linear combination of the degradation measurements. An algorithm based on the Monte Carlo simulation is proposed to simulate the degradation process, in which the performance characteristics are composited in arbitrary forms, and the correctness of the analytical results is also verified. Finally, some numerical examples are presented to illustrate the present reliability assessment method。

Statistical Method for Investigating Transient Enhancements of Dynamical Responses due to Random Disturbances: Application to Railway Vehicle Motion

Abstract The paper presents a statistical method for determining a specific variation of random excitations that leads to large transient enhancements (peaks) of a particular dynamical response in a stochastic mechanical system. Such a variation is found by calculating the weighted mean of the excitation variations close to a small number of largest peaks of the response obtained for a single long realization of the system motion. This statistical formula is derived by using the conditional expectation with respect to the rare event of unusually large response values and the ergodic theorem; optionally, a minimal interpeak distance is introduced. A similar formula gives the specific variations of other system variables around the peaks, and it can also be generalized to investigate any multivariable stochastic dynamical system or any set of correlated random signals. This method is applied to transient enhancements of quantities related to running safety and ride comfort of a railway vehicle: the derailment coefficient and the vertical acceleration of the vehicle body, respectively, obtained in simulations of the vehicle motion along a track with random irregularities. The averaged variations of the lateral irregularities and track superelevation close to the track locations of largest peaks of the derailment coefficient show characteristic oscillations leading to enhanced wheelset hunting in a short track section before the peak occurrence. A different pattern is found for the average variation of vertical track irregularities in the vicinity of the track points where largest maxima (or minima) of the vertical body acceleration occur.

Remaining useful life prediction for multivariable stochastic degradation systems with non‐Markovian diffusion processes

AbstractMultivariable stochastic degradation system (MSDS) is quite common in industries such as blast furnace ironmaking, vehicle transportation, and aerospace manufacturing. Large‐scale complex equipments may be affected by multiple factors, resulting in not just a single deteriorating performance characteristic. It is difficult to handle unknown failure structures of practical systems by using traditional univariate degradation modeling methods. A novel health index (HI) is constructed to quantitatively analyze the health state for the overall system. Considering the interaction between internal reactions and external environments, the fractional Brownian motion (FBM), a typical non‐Markovian diffusion process, is added for the purpose of reflecting stochastic uncertainties and memory effects. Based on the wavelet estimators and the maximum likelihood estimation (MLE) algorithm, multi‐sensor observations of degradation variables are analyzed simultaneously to identify model parameters. A closed‐form distribution of system‐level remaining useful life (RUL) is obtained with a mild two‐layer approximation. Relevant case studies are then handled that adequately demonstrate the effectiveness and the practical utility of the proposed method.

Output Feedback Stabilization for MIMO Semi-linear Stochastic Systems with Transient Optimisation

This paper investigates the stabilisation problem and consider transient optimisation for a class of the multi-input-multi-output (MIMO) semi-linear stochastic systems. A control algorithm is presented via an m-block backstepping controller design where the closed-loop system has been stabilized in a probabilistic sense and the transient performance is optimisable by optimised by searching the design parameters under the given criterion. In particular, the transient randomness and the probabilistic decoupling will be investigated as case studies. Note that the presented control algorithm can be potentially extended as a framework based on the various performance criteria. To evaluate the effectiveness of this proposed control framework, a numerical example is given with simulation results. In summary, the key contributions of this paper are stated as follows: 1) one block backstepping-based output feedback control design is developed to stabilize the dynamic MIMO semi-linear stochastic systems using a linear estimator; 2) the randomness and probabilistic couplings of the system outputs have been minimized based on the optimisation of the design parameters of the controller; 3) a control framework with transient performance enhancement of multi-variable semi-linear stochastic systems has been discussed.

Modeling and application of industrial process fault detection based on pruning vine copula

Industrial processes are usually nonlinear multivariate stochastic systems. Describing the distribution characteristics of variables through a vine copula model can well define complex correlation information. However, modeling based on vine copula involves computational complexity. This study proposes a method to prune vine copula and introduces an indicator to conduct pruning process. This technique constructs a simplified model by removing the weakly correlated component from the copula structure without reducing the accuracy of model. Lastly, fault detection for industrial processes based on pruning vine copula is conducted based on a generalized local probability monitoring index. Experimental results of the monitoring process show that the method can reduce the time of modeling and improve the effect of fault detection.

Fault detection and fuzzy tolerant control for complex stochastic multivariable nonlinear systems

Abstract In this paper, the problem of fault-tolerant control (FTC) is investigated for a class of multivariable stochastic nonlinear systems in the strict-feedback form. The considered systems possess unknown nonlinear functions, unmeasured states, virtual control coefficients and actuator faults (bias and gain faults). Neural networks (NNs) are adopted to approximate the unknown nonlinear functions. Then, a state observer is constructed to solve the problem of unmeasured states. The problem of “explosion of complexity” is solved by using dynamic surface control (DSC) method. Based on the combination of Nussbaum gain function with Lyapunov function stability theory, an adaptive NNs output feedback FTC method is developed in the frame of adaptive backstepping design technique. It is shown that all signals in the closed-loop system are proved to be bounded in probability, and the system output can follow the given reference signal well. The effectiveness of the proposed approach is verified by a simulation example.

Entropy optimization based filtering for non-Gaussian stochastic systems

This paper is concerned with the entropy optimization based filter design for a class of multivariate dynamic stochastic systems with simultaneous presence of non-Gaussian process noise and measurement noise. The filter consists of time update and measurement update two steps, where the selection of the filter gain in the measurement update equation is a key issue to be addressed. Different from the classic Kalman filter theory, entropy rather than variance is employed as the filtering performance criterion due to the non-Gaussian characteristic of the estimation error. Following the establishment of the relationship between the probability density functions of random noises and estimation error, two kinds of entropy based performance indices are provided. On this basis, the corresponding optimal filter gains are obtained respectively by using the gradient optimization technique. Finally, some numerical simulations are provided to demonstrate the effectiveness of the proposed filtering algorithms.

Output Feedback Stabilization for a Class of Multi-Variable Bilinear Stochastic Systems With Stochastic Coupling Attenuation

In this technical note, stochastic coupling attenuation is investigated for a class of multi-variable bilinear stochastic systems and a novel output feedback $m$ -block backstepping controller with linear estimator is designed, where gradient descent optimization is used to tune the design parameters of the controller. It has been shown that the trajectories of the closed-loop stochastic systems are bounded in probability sense and the stochastic coupling of the system outputs can be effectively attenuated by the proposed control algorithm. Moreover, the stability of the stochastic systems is analyzed and the effectiveness of the proposed method has been demonstrated using a simulated example.

Controller design for nonlinear and non-Gaussian multivariable systems based on survival information potential criterion

In this paper, a new model-free control strategy for general nonlinear and non-Gaussian multivariable stochastic systems has been proposed. The strategy applies minimum survival information potential control (MSIPC) scheme to decrease the closed-loop randomness of the output in the information theory framework. Compared with traditional entropy measures, survival information potential (SIP) has many advantages, such as validity in a wide range of distributions, robustness, the simplicity in computation, and so on. In order to calculate the SIP, the empirical SIP formulation under scalar data case is derived directly based on ordered error sample data. By minimizing the performance index mainly consists of SIP, a new model-free control algorithm is obtained for the considered multivariable nonlinear and non-Gaussian stochastic systems. The analysis on the proposed MSIPC convergence is made and a numerical example is provided to show the effectiveness of the obtained control algorithm, where encouraging results have been obtained.

Data driven output joint probability density function control for multivariate non‐linear non‐Gaussian systems

This study presents a novel data-based joint probability density function (JPDF) control strategy for multivariate non-linear non-Gaussian stochastic systems so that the output JPDF of the system can be made to follow a desired JPDF. The output JPDF, which is usually immeasurable, is estimated according to the output sequence of the system. The multi-step-ahead cumulative performance index is constructed with respect to the control objective and is minimised based on an intelligent optimisation algorithm. By minimising this performance function, a new predictive controller design algorithm is established with more simple formulation and less computation load than existed results. Furthermore, a new approach is developed to guarantee ‘convergence in distribution’. Finally, simulations are given to demonstrate the effectiveness of the proposed control algorithm and some desired results have been obtained.

On the singular components of a copula

We analyze copulas with a nontrivial singular component by using their Markov kernel representation. In particular, we provide existence results for copulas with a prescribed singular component. The constructions not only help to deal with problems related to multivariate stochastic systems of lifetimes when joint defaults can occur with a nonzero probability, but even provide a copula maximizing the probability of joint default.