Linear Time-invariant Stochastic Systems Research Articles

On the Practical Exponential Stability in Mean Square of Stochastic Perturbed Systems Via a Lyapunov Approach

The Lyapunov method is one of the most effective methods to analyze the stability of stochastic differential equations (SDEs). Different authors analyzed the stability of SDEs based on Lyapunov techniques when the origin can be considered as an equilibrium point. When the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability of solutions concerning a small neighborhood of the origin. The purpose is to study the asymptotic stability of a system whose solution behavior is a small ball of state space or close to it. Thus, all state trajectories are bounded and close to a sufficiently small neighborhood of the origin. In this sense, the limited boundedness of solutions of random systems, or the chance of convergence of solutions needs to be analyzed on a ball centered on the origin. This is the so called “Practical Stability”. In this article, we mainly investigate the practical uniform exponential stability in the mean square of stochastic linear time–invariant systems. In addition, we are developing the problem of stabilization of certain classes of perturbed stochastic systems. Our crucial techniques include Lyapunov techniques and generalized Gronwall inequalities. Lastly, we provide a numerical example to illustrate our theoretical findings.

Distributionally Robust Chance Constrained Data-Enabled Predictive Control

In this article we study the problem of finite-time constrained optimal control of unknown stochastic linear time-invariant (LTI) systems, which is the key ingredient of a predictive control algorithm—albeit typically having access to a model. We propose a novel distributionally robust data-enabled predictive control (DeePC) algorithm which uses noise-corrupted input/output data to predict future trajectories and compute optimal control inputs while satisfying output chance constraints. The algorithm is based on 1) a nonparametric representation of the subspace spanning the system behavior, where past trajectories are sorted in Page or Hankel matrices; and 2) a distributionally robust optimization formulation which gives rise to strong probabilistic performance guarantees. We show that for certain objective functions, DeePC exhibits strong out-of-sample performance, and at the same time respects constraints with high probability. The algorithm provides an end-to-end approach to control design for unknown stochastic LTI systems. We illustrate the closed-loop performance of the DeePC in an aerial robotics case study.

Optimal LQG Control of Networked Systems Under Traffic-Correlated Delay and Dropout

Transmission delay and packet dropout are inevitable network-induced phenomena that severely compromise the control performance of network control systems. The real-time network traffic is a major dynamic parameter that directly influences delay and reliability of transmission channels, and thus, acts as an unavoidable source of induced coupling among all network sharing systems. In this letter, we analyze the effects of traffic-induced delay and dropout on the finite-horizon quality-of-control of an individual stochastic linear time-invariant system, where quality-of-control is measured by an expected quadratic cost function. We model delay and dropout of the channel as generic stochastic processes that are correlated with the real-time network traffic induced by the rest of network users. This approach provides a pathway to determine the required networking capabilities to achieve a guaranteed quality-of-control for systems operating over a shared-traffic network. Numerical evaluations are performed using realistic stochastic models for delay and dropout. As a special case, we consider exponential distribution for delay with its rate parameter being traffic-correlated, and traffic-correlated Markov-based packet drop model.

Semi-analytical assessment of the relative accuracy of the GNSS/INS in railway track irregularity measurements

An aided Inertial Navigation System (INS) is increasingly exploited in precise engineering surveying, such as railway track irregularity measurement, where a high relative measurement accuracy rather than absolute accuracy is emphasized. However, how to evaluate the relative measurement accuracy of the aided INS has rarely been studied. We address this problem with a semi-analytical method to analyze the relative measurement error propagation of the Global Navigation Satellite System (GNSS) and INS integrated system, specifically for the railway track irregularity measurement application. The GNSS/INS integration in this application is simplified as a linear time-invariant stochastic system driven only by white Gaussian noise, and an analytical solution for the navigation errors in the Laplace domain is obtained by analyzing the resulting steady-state Kalman filter. Then, a time series of the error is obtained through a subsequent Monte Carlo simulation based on the derived error propagation model. The proposed analysis method is then validated through data simulation and field tests. The results indicate that a 1 mm accuracy in measuring the track irregularity is achievable for the GNSS/INS integrated system. Meanwhile, the influences of the dominant inertial sensor errors on the final measurement accuracy are analyzed quantitatively and discussed comprehensively.

Delay-Sensitive Joint Optimal Control and Resource Management in Multiloop Networked Control Systems

In the operation of networked control systems (NCSs), where multiple processes share a resource-limited and time-varying cost-sensitive network, communication delay is inevitable and primarily induced by, first, intermittent sensor sampling to restrict nonurgent transmissions, and second, resource management to avoid contentions, excessive traffic, and data loss. In a heterogeneous scenario, where control systems may tolerate only specific levels of sensor-to-controller latency, delay sensitivities need to be considered in the design of control and network policies to achieve the desired performance guarantees. We propose a cross-layer optimal co-design of control, sampling, and resource management policies for an NCS consisting of multiple stochastic linear time-invariant systems which close their sensor-to-controller links over a shared network. Aligned with advanced communication technology, we assume that the network offers a range of latency-varying transmission services for given prices. The performance of the local closed-loop systems is measured by a combination of linear-quadratic Gaussian cost and a suitable communication cost, and the overall objective is to minimize a defined social cost by all three policymakers. We derive optimal control, sampling, and resource allocation policies under different cross-layer awareness models, including constant and time-varying parameters, and show that higher awareness generally leads to performance enhancement at the expense of higher computational complexity. This trade-off is shown to be a key feature to select the proper interaction structure for the codesign.

An Extended Kalman Filter for Data-Enabled Predictive Control

The literature dealing with data-driven analysis and control problems has significantly grown in the recent years. Most of the recent literature deals with linear time-invariant systems in which the uncertainty (if any) is assumed to be deterministic and bounded; relatively little attention has been devoted to stochastic linear time-invariant systems. As a first step in this direction, we propose to equip the recently introduced Data-enabled Predictive Control algorithm with a data-based Extended Kalman Filter to make use of additional available input-output data for reducing the effect of noise, without increasing the computational load of the optimization procedure.

The Kalman filter on stochastic time scales

In this paper, we discretize a stochastic linear time-invariant system to a dynamic system on a time scale. We then develop a Kalman filter to estimate the true state for the corresponding system. Here, the measurement-update and time-update equations account for the size of the time step when the time scale is generated randomly. Numerical examples are also provided.

Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay

This paper introduces stability conditions in the form of linear matrix inequalities (LMIs) for general linear retarded systems with a delay term described by Stieltjes integral. The derived LMIs provide in the unified form conditions for both discrete and distributed delays. Two Lyapunov-based methods for the asymptotic mean square stability of stochastic linear time-invariant systems are presented. The first one employs neutral type model transformation and augmented Lyapunov functionals. Differently from the existing LMI stability conditions based on neutral type transformation, the proposed conditions do not require the stability of the corresponding integral equations. Moreover, it is shown that in the simplest existing LMIs based on non-augmented Lyapunov functionals, the stability analysis of the integral equation can be omitted. The second method is based on a stochastic extension of simple Lyapunov functionals depending on the state derivative. The same two methods are further applied to delay-induced stability analysis of stochastic vector second-order systems, simplifying the recent results via neutral type transformation and leading to new conditions for stochastic systems via the second method. Numerical examples give comparison of results via different methods.

Optimal LQG Control Under Delay-Dependent Costly Information

In the design of closed-loop networked control systems (NCSs), induced transmission delay between sensors and the control station is an often-present issue which compromises control performance and may even cause instability. A very relevant scenario in which network-induced delay needs to be investigated is costly usage of communication resources. More precisely, advanced communication technologies, e.g. 5G, are capable of offering latency-varying information exchange for different prices. Therefore, induced delay becomes a decision variable. It is then the matter of decision maker's willingness to either pay the required cost to have low-latency access to the communication resource, or delay the access at a reduced price. In this article, we consider optimal price-based bi-variable decision making problem for single-loop NCS with a stochastic linear time-invariant system. Assuming that communication incurs cost such that transmission with shorter delay is more costly, a decision maker determines the switching strategy between communication links of different delays such that an optimal balance between the control performance and the communication cost is maintained. In this article, we show that, under mild assumptions on the available information for decision makers, the separation property holds between the optimal link selecting and control policies. As the cost function is decomposable, the optimal policies are efficiently computed.

Identification of a Class of Nonlinear Dynamical Networks

Identification of dynamic networks has attracted considerable interest recently. So far the main focus has been on linear time-invariant networks. Meanwhile, most real-life systems exhibit nonlinear behaviors; consider, for example, two stochastic linear time-invariant systems connected in series, each of which has a nonlinearity at its output. The estimation problem in this case is recognized to be challenging, due to the analytical intractability of both the likelihood function and the optimal one-step ahead predictors of the measured nodes. In this contribution, we introduce a relatively simple prediction error method that may be used for the estimation of nonlinear dynamical networks. The estimator is defined using a deterministic predictor that is nonlinear in the known signals. The estimation problem can be defined using closed-form analytical expressions in several non-trivial cases, and Monte Carlo approximations are not necessarily required. We show, that this is the case for some block-oriented networks with no feedback loops and where all the nonlinear modules are polynomials. Consequently, the proposed method can be applied in situations considered challenging by current approaches. The performance of the estimation method is illustrated on a numerical simulation example.

Centralized security-guaranteed filtering in multirate-sensor fusion under deception attacks

In this paper, the centralized security-guaranteed filtering problem is studied for linear time-invariant stochastic systems with multirate-sensor fusion under deception attacks. The underlying system includes a number of sensor nodes with a centralized filter, where each sensor is allowed to be sampled at different rate. A new measurement output model is proposed to characterize both the multiple rates and the deception attacks. By exploiting the lifting technique, the multi-rate sensor system is cast into a single-rate discrete-time system. With a new concept of security level, the aim of this paper is to design a filter such that the filtering error dynamics achieves the prescribed level of the security under deception attacks. By using the stochastic analysis techniques, sufficient conditions are first derived such that the filtering error system is guaranteed to have the desired security level, and then the filter gain is parameterized by using the semi-definite programme method with certain nonlinear constraints. Finally, a numerical simulation example is provided to demonstrate the feasibility of the proposed filtering scheme.

Decentralized <inline-formula> <tex-math notation="TeX">$H_{2}$</tex-math></inline-formula> Control for Multi-Channel Stochastic Systems

This technical note investigates a decentralized H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> control problem for multi-channel linear time-invariant stochastic systems governed by Itô's differential equation. After using a stochastic algebraic Riccati equation (SARE) to establish the necessary conditions for the existence of the optimal strategy set that minimizes an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm, we show that the necessary conditions can also be expressed using a linear matrix inequality (LMI). The equivalence between the solvability of the SARE and the feasibility of the LMI is proved for the first time using a Karush-Kuhn-Tucker (KKT) condition. Furthermore, the static output feedback solution was also studied. A numerical example is given to demonstrate the usefulness of the obtained features.

Adaptive control design under structured model information limitation: A cost-biased maximum-likelihood approach

Networked control strategies based on limited information about the plant model usually result in worse closed-loop performance than optimal centralized control with full plant model information. Recently, this fact has been established by utilizing the concept of competitive ratio, which is defined as the worst-case ratio of the cost of a control design with limited model information to the cost of the optimal control design with full model information. We show that an adaptive controller, inspired by a controller proposed by Campi and Kumar, with limited plant model information, asymptotically achieves the closed-loop performance of the optimal centralized controller with full model information for almost any plant. Therefore, there exists, at least, one adaptive control design strategy with limited plant model information that can achieve a competitive ratio equal to one. The plant model considered in the paper belongs to a compact set of stochastic linear time-invariant systems and the closed-loop performance measure is the ergodic mean of a quadratic function of the state and control input.

A Suggestion of Fuzzy Estimation Technique for Uncertainty Estimation of Linear Time Invariant System Based on Kalman Filter

In order to control a LTI(Linear Time Invariant) system subjected to system noise and measurement noise, first of all, it is necessary to estimate the state of system with reliability. Kalman filtering technique has been widely used to estimate the state of the stochastic LTI system with stationary noise characteristics because of its estimation ability versus algorithm simplicity. However, it often fails to estimate the state of the LTI system of which system parameter uncertainty exists partly and/or input uncertainty exists. In this paper, a new estimation technique based on Kalman filter is suggested for stochastic LTI system under parameter uncertainty and/or input uncertainty. A fuzzy estimation algorithm against uncertainties is introduced so as to compensate the state estimate filtered by Kalman filter. In order to verify the state estimation performance of the suggested technique, several simulations are accomplished.

Cramér-Rao Bounds for Estimation of Linear System Noise Covariances

The performance of Kalman filter depends directly on the noise covariances, which are usually not known and need to be estimated. Several estimation algorithms have been published in past decades, but the measure of estimation quality is missing. The Cramer-Rao bounds represent limitation of quality of parameter estimation that can be obtained from given data. In this article, The Cramer-Rao bounds for noise covariance estimation of linear time-invariant stochastic system will be derived. Two different linear system models will be considered. Further, the performance of earlier published methods will be discussed according to the Cramer-Rao bounds. The analogy between the Cramer-Rao bounds and the Riccati equa- tion will be pointed out.

Cramér-Rao bounds for estimation of linear system noise covariances

AbstractThe performance of Kalman filter depends directly on the noise covariances, which are usually not known and need to be estimated. Several estimation algorithms have been published in past decades, but the measure of estimation quality is missing. Cramér-Rao bounds represent limitation of quality of parameter estimation that can be obtained from given data. In this article, Cramér-Rao bounds for noise covariance estimation of linear time-invariant stochastic system will be derived. Two different linear system models will be considered. Further, the performance of earlier published methods will be discussed according to Cramér-Rao bounds. The analogy between Cramér-Rao bounds and Riccati equation will be pointed out.

On-line discontinuities identification in noisy signals: Application to Kalman filtering

The problem of fault detection in a stochastic linear time invariant system is dealt with, assuming that the fault is modelled by a jump in the noisy output. An on-line identification procedure is designed based on a multiscale analysis and on the application of suitable statistical tests. Once the time and the size of the discontinuity have been identified, the state estimate evolution of a Kalman filter is adequately compensated.

On the Rank Deficiency of the Least Squares Residuals in Subspace Identification

Subspace identification methods for linear time-invariant stochastic systems basically consist of two steps. In the first step, which is an orthogonal projection, followed by a Singular Value Decomposition (SVD), typically (non-steady state) Kalman filter states sequences are obtained directly from output data. In the second step, the system model matrices are identified in a least squares estimation step, which is asymptotically unbiased. For an infinite amount of data points, the measurement-process noise covariances are rank deficient, since the system is identified in innovation form. However, due to finite sample errors, this rank deficiency property is generically not satisfied when identifying on a finite number of measurement data. The required rank deficiency is achieved by replacing the least squares step by a total least squares estimate with exact row

Model reference adaptive control of the output stochastic distributions for unknown linear stochastic systems

This paper presents a model reference based adaptive control approach for the control of the output probability density function for unknown linear time-invariant stochastic systems. Different from most existing models used in stochastic control, it is assumed here that the measured control input directly affects the distribution of the system output in probability sense. As such, the purpose of control is to make the shape of the probability density function of the system output as close as possible to a prespecified one. Using the weighted integration of the measurable output probability density functions, two adaptive on-line updating rules are developed which guarantee the global stability for theclosed loop adaptive control system under certain conditions. Ithas been shown, when there is no external disturbance, that the so-formed closed loop system also realizes the perfect tracking (i.e., the probability density function of the system output approaches a class of given distributions asymptotically). A simulated example is included to illustrate the use of the developed control algorithm and desired results have been obtained.

線形確率制御系における有色システム雑音の取り扱いの影響

線形確率制御系における有色システム雑音の取り扱いの影響