含有有色噪声的非线性分数阶系统自适应扩展卡尔曼滤波器

研究了含有未知参数的情况下，分别含有分数阶有色过程噪声和有色测量噪声的连续时间非线性分数阶系统状态估计问题.采用Grünwald-Letnikov （G-L）差分方法和1阶泰勒展开公式，对描述连续时间非线性分数阶系统的状态方程进行离散化和线性化.构造由状态量、未知参数和分数阶有色噪声的增广向量，设计自适应分数阶扩展卡尔曼滤波算法实现对有色噪声情况下的连续时间非线性分数阶系统的状态和参数的估计.最后，通过分析两个仿真实例，验证了提出算法的有效性.     

Information filtering for time-delay systems

This paper presents the information filter, which is developed in filtered forms, for time-delay systems which include continuous-time case and discrete-time case. By introducing delay-free observation, the optimal filtering problems for time-delay systems can be converted into multiple steps prediction problems for delay-free systems. Then, the recursive forms for optimal filter are given via innovation approach in Hilbert space. Furthermore, the information filter in Riccati recursions can be derived by using the matrix inversion Lemma to optimal filtering formulas. Finally, the distributed Kalman filter for time-delay systems is designed based on information filter developed.

     

Exact finite-dimensional nonlinear filters

A new nonlinear filter is derived for continuous-time processes with discrete-time measurements. The filter is exact, and it can be implemented in real time with a computational complexity that is comparable to the Kalman filter. This new filter includes both the Kalman filter and the discrete-time version of the Benes filter as special cases. Moreover, the new theory can handle a large class of nonlinear estimation problems that cannot be solved using the Kalman or discrete-time Benes filters. A simple approximation technique is suggested for practical applications in which the dynamics do not satisfy the required conditions exactly. This approximation is analogous to the so-called "extended Kalman filter" [10], and it represents a generalization of the standard linearization method.     

Adaptive dynamic programming and optimal control of nonlinear nonaffine systems

In this paper, a novel optimal control design scheme is proposed for continuous-time nonaffine nonlinear dynamic systems with unknown dynamics by adaptive dynamic programming (ADP). The proposed methodology iteratively updates the control policy online by using the state and input information without identifying the system dynamics. An ADP algorithm is developed, and can be applied to a general class of nonlinear control design problems. The convergence analysis for the designed control scheme is presented, along with rigorous stability analysis for the closed-loop system. The effectiveness of this new algorithm is illustrated by two simulation examples.     

A receding horizon Kalman FIR filter for linear continuous-timesystems

A receding horizon Kalman finite-impulse response (FIR) filter is suggested for continuous-time systems, combining the Kalman filter with the receding horizon strategy. In the suggested filter, the horizon initial state is assumed to be unknown. It can always be obtained irrespective of unknown information on the horizon initial state. The filter may be the first stochastic FIR form for continuous-time systems that may have many good inherent properties. The suggested filter can be represented in an iterative form and also in a standard FIR form. The suggested filter turns out to be a remarkable deadbeat observer. The validity of the suggested filter is illustrated by numerical examples     

The Cramér-Rao estimation error lower bound computation for deterministic nonlinear systems

For continuous-time nonlinear deterministic system models with discrete nonlinear measurements in additive Ganssian white noise, the extended Kalman filter (EKF) convariance propagation equations linearized about the true unknown trajectory provide the Cramér-Rao lower bound to the estimation error covariance matrix. A useful application is establishing the optimum filter performance for a given nonlinear estimation problem by developing a simulation of the nonlinear system and an EKF linearized about the true trajectory.     

Asymptotic stabilisation of a class of nonlinear systems via sampled-data output feedback control

This article considers sampled-data output feedback control of a class of nonlinear systems in output feedback form. The underlying continuous-time controller is designed based on backstepping technique, employing a linear dynamic filter, and globally asymptotically stabilises the system. Rigorous analysis shows that when implemented with a sampling and zero-order hold device, the sampled-data version of the continuous-time controller semi-globally asymptotically stabilises the original system, given that the sampling period is smaller than a specific value, which depends on the initial values and nonlinearity of the system. Simulation results of a physical system are included in the end.     

Flocking for multi-agent systems with optimally rigid topology based on information weighted Kalman consensus filter

This paper investigates the leader-follower flocking problem of multi-agent systems. The leader with input noise is estimated by a proposed continuous-time information weighted Kalman consensus filter (IWKCF) for agents. A novel distributed flocking algorithm based on the IWKCF is further presented to make agents achieve flocking to the leader. It is shown that the proposed flocking algorithm based on the continuous-time IWKCF is asymptotically stable. Applying the topology optimization scheme, the communication complexity of system topologies of multi-agent systems is effectively reduced. Finally, simulations are provided to demonstrate the effectiveness of the proposed results.     

Small-gain theory for stability and control of dynamical networks: A Survey

This paper provides a personal account of the small-gain theory as a tool for stability analysis, control synthesis, and robustness analysis for interconnected uncertain systems. A milestone in modern control theory is the development of a transformative stability criterion known as the classical small-gain theorem proposed by George Zames in 1966, that surpasses Lyapunov theory in that there is no need to construct Lyapunov functions for the finite-gain stability of feedback systems. Under the small-gain framework, a feedback system composed of two finite-gain stable subsystems remains finite-gain stable if the loop gain is less than one. Despite its apparent simplicity at first sight, Zames’s small-gain theorem plays a crucial role in the development of linear robust control theory. Borrowing techniques in modern nonlinear control, especially Sontag’s notion of input-to-state stability (ISS), the first generalized, nonlinear ISS small-gain theorem proposed by one of the authors in 1994 overcomes the two shortcomings of Zames’s small-gain theorem. First, the use of nonlinear gains allows to consider strongly nonlinear, interconnected systems. Second, the role of initial conditions is made explicit so that both internal Lyapunov stability and external input-output stability can be studied in a unified framework. In this survey paper, we first review early developments in the nonlinear small-gain theory for interconnected systems of various types such as continuous-time systems, discrete-time systems, hybrid systems and time-delay systems, along with applications in robust nonlinear control. Then, we describe how to obtain a network small-gain theory for large-scale dynamical networks that are comprised of more than two interacting nonlinear systems. Constructive methods for the generation of Lyapunov functions for the total network are presented as well. Finally, this paper discusses how the network/nonlinear small-gain theory can be applied to obtain innovative solutions to quantized and event-based nonlinear control problems, that are important for the development of a complete theory of controlling cyber-physical systems subject to communications and computation constraints.     

Linearization of discrete-time systems

We characterize the equivalence of single-input single-output discrete-time nonlinear systems to linear ones, via a state-coordinate change and with or without feedback. Four cases are distinguished by allowing or disallowing feedback as well as by including the output map or not; the interdependence of these problems is analyzed. An important feature that distinguishes these discrete-time problems from the corresponding problem in continuous-time is that the state-coordinate transformation is here directly computable as a higher composition of the system and output maps. Finally, certain connections are made with the continuous-time case.     

Fractional-order Kalman filters for continuous-time linear and nonlinear fractional-order systems using Tustin generating function

This paper presents the fractional-order Kalman filters using Tustin generating function for linear and nonlinear fractional-order systems involving process noise and measurement noise. By using the Tustin generating function, the differential equation model is obtained by discretising the investigated continuous-time fractional-order system. The two kinds of fractional-order Kalman filters are given for the correlated and uncorrelated cases in terms of the process noise and measurement noise for linear fractional-order system, respectively. In addition, based on the first-order Taylor expansion formula, the extended fractional-order Kalman filter using Tustin generating function is proposed to improve the accuracy of state estimation. Finally, three examples are illustrated to verify the effectiveness of the Tustion fractional-order Kalman filters for linear and nonlinear fractional-order systems.     

Stability of nonlinear 2D systems described by the continuous-time Roesser model

This paper considers systems with two-dimensional dynamics (2D systems) described by the continuous-time nonlinear state-space Roesser model. The sufficient conditions of exponential stability in terms of vector Lyapunov functions are established. These conditions are then applied to analysis of the absolute stability of a certain class of systems comprising a linear continuous-time plant in the form of the Roesser model with a nonlinear characteristic in the feedback loop, which satisfies quadratic constraints. The absolute stability conditions are reduced to computable expressions in the form of linear matrix inequalities. The obtained results are extended to the class of continuous-time systems governed by the Roesser model with Markovian switching. The problems of absolute stability and stabilization via state- and output-feedback are solved for linear systems of the above class. The solution procedures for these problems are in the form of algorithms based on linear matrix inequalities.     

A Kalman filter type of extension to a deterministic gradient technique for parameter estimation

It is shown that a method for the identification of deterministic systems derived from the Kalman filter is related to a gradient technique of parameter estimation and that the range of problems to which the gradient method may be applied is thereby extended. Various versions of the discrete-time algorithm are compared from theoretical and computational points of view and also contrasted with the continuous-time algorithm. An important outcome is that the system containing unknown parameters and the identification algorithm may be formulated with one in discrete-time and the other in continuous-time.     

An analytic approach for computing pi-coefficients of continuous-time systems

The pi-sharing theory for continuous-time systems is discussed in this paper. An analytic approach is proposed for computing pi-coefficients of LTI systems. For the two scalar pi-coefficients, feasible regions are characterized graphically on a plane, and for the other two matrix pi-coefficients, the corresponding values are obtained by solving a Riccati equation. The results are applied in an example to determine the stability of a feedback system containing both linear and nonlinear subsystems.     

离散时间系统零动态的研究现状和未来挑战

在数字控制系统的分析与设计中, 零动态是一个被广泛关注的重要概念, 近年来取得了诸多新的理论与方法进展. 本文首先描述了离散时间系统零动态理论的研究背景和研究意义, 同时简要介绍了离散时间系统零动态理论所涉及到的3个相关问题, 如: 信号的采样与重建、连续时间系统的等价离散时间系统模型以及在离散连续时间系统过程中所需要的工具(q算子和±算子). 其次, 立足现有文献, 针对离散零动态的特点, 从线性离散时间系统和非线性离散时间系统两个方面全面而深入地介绍了近年来离散零动态研究工作的进展. 最后分析了零动态在数字控制系统分析与设计中的局限性以及出现的挑战性课题, 并指明未来工作的研究方向.     

Upper bounds for approximation of continuous-time dynamics using delayed outputs and feedforward neural networks

The problem of approximation of unknown dynamics of a continuous-time observable nonlinear system is considered using a feedforward neural network, operating over delayed sampled outputs of the system. Error bounds are derived that explicitly depend upon the sampling time interval and network architecture. The main result of this note broadens the class of nonlinear dynamical systems for which adaptive output feedback control and state estimation problems are solvable.     

Robust model predictive control for constrained continuous-time nonlinear systems

In this paper, a robust model predictive control (MPC) is designed for a class of constrained continuous-time nonlinear systems with bounded additive disturbances. The robust MPC consists of a nonlinear feedback control and a continuous-time model-based dual-mode MPC. The nonlinear feedback control guarantees the actual trajectory being contained in a tube centred at the nominal trajectory. The dual-mode MPC is designed to ensure asymptotic convergence of the nominal trajectory to zero. This paper extends current results on discrete-time model-based tube MPC and linear system model-based tube MPC to continuous-time nonlinear model-based tube MPC. The feasibility and robustness of the proposed robust MPC have been demonstrated by theoretical analysis and applications to a cart-damper springer system and a one-link robot manipulator.     

Optimal linear mean square filter for continuous-time jump linear systems

We consider a class of hybrid systems which is modeled by continuous-time linear systems with Markovian jumps in the parameters (LSMJP). Our aim is to derive the best linear mean square estimator for such systems. The approach adopted here produces a filter which bears those desirable properties of the Kalman filter: A recursive scheme suitable for computer implementation which allows some offline computation that alleviates the computational burden. Apart from the intrinsic theoretical interest of the problem in its own right and the application-oriented motivation of getting more easily implementable filters, another compelling reason why the study here is pertinent has to do with the fact that the optimal nonlinear filter for our estimation problem is not computable via a finite computation (the filter is infinite dimensional). Our filter has dimension Nn, with n denoting the dimension of the state vector and N the number of states of the Markov chain.     

Optimal state estimation for stochastic systems: an informationtheoretic approach

In this paper, we examine the problem of optimal state estimation or filtering in stochastic systems using an approach based on information theoretic measures. In this setting, the traditional minimum mean-square measure is compared with information theoretic measures, Kalman filtering theory is reexamined, and some new interpretations are offered. We show that for a linear Gaussian system, the Kalman filter is the optimal filter not only for the mean-square error measure, but for several information theoretic measures which are introduced in this work. For nonlinear systems, these same measures generally are in conflict with each other, and the feedback control policy has a dual role with regard to regulation and estimation. For linear stochastic systems with general noise processes, a lower bound on the achievable mutual information between the estimation error and the observation are derived. The properties of an optimal (probing) control law and the associated optimal filter, which achieve this lower bound, and their relationships are investigated. It is shown that for a linear stochastic system with an affine linear filter for the homogeneous system, under some reachability and observability conditions, zero mutual information between estimation error and observations can be achieved only when the system is Gaussian     

Discretization of cascaded continuous-time controllers for state and input delayed systems

This paper presents a new interpolation method that utilizes a Newton backward interpolation formula for the discretization of a continuous-time system with state and input delays. Also, a new output feedback state-matching digital redesign method, which utilizes the Newton extrapolation formula and the Chebyshev quadrature formula, enables the discretization of a cascaded continuous-time controller or filter for the continuous-time systems with delayed input and states, without employing a discrete time observer. The proposed method provides an alternative methodology for indirect digital design of a continuous-time unity-output feedback delayed system and enables us to implement accurately and effectively an existing or pre-designed cascaded analogue controller or filter via a digital controller.