Optimal filtering for linear systems with state and observation delays

In this paper, the optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. As a result, the optimal estimate equation similar to the traditional Kalman–Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman–Bucy filter. The resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is however demonstrated that a finite set of the filtering equations, whose number is specified by the ratio between the current filtering horizon and the delay values, can be obtained in the particular case of equal or commensurable (τ=qh, q is natural) delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays is verified against the best Kalman–Bucy filter available for linear systems without delays and two versions of the extended Kalman–Bucy filter for time delay systems. Copyright © 2005 John Wiley & Sons, Ltd.     

H∞ state estimation for linear periodic systems

This note deals with the H_∞ state estimation problem for linear periodic systems. The question addressed is the design of an unbiased linear periodic and asymptotically stable estimator that achieves a prescribed H_∞ performance on an infinite horizon. Necessary and sufficient conditions for the existence of a periodic estimator have been derived. Asymptotic properties of the finite horizon estimation problem when the time-horizon tends to infinity are also investigated     

Optimal filtering for linear systems with state and multiple observation delays

This article solves the optimal filtering problem for linear systems with state and multiple observation delays. The optimal estimate equation similar to the traditional Kalman–Bucy one is derived, and the system of equations for determining the filter gain matrix consists of an infinite set of equations. It is then demonstrated that a finite set of the filtering equations can be obtained in case of commensurable delays. In the example, the designed optimal filter is compared to the traditional Kalman–Bucy filter.     

Fast model predictive control for linear periodic systems with state and control constraints

The design of stabilizing model predictive control laws for discrete‐time linear periodic systems with state and control constraints is considered. Two algorithms are presented. The first one is based on interpolation between several unconstrained periodic controllers. Among them, one controller is chosen for the performance while the rest are used to extend the domain of attraction. The second algorithm aims to improve the performance by combining model predictive control and interpolating control. The proposed approaches not only guarantee recursive feasibility and asymptotic stability but also are optimal for states near the origin. Copyright © 2017 John Wiley & Sons, Ltd.     

期望指标下一类线性周期系统鲁棒状态估计

针对满足多重区域指标约束的鲁棒滤波问题 ,以离散线性周期系统为研究对象 ,利用周期系统参数在某区间变化的特性 ,将周期系统满足极点或极点 /协方差指标的估计问题转化为区间系统的鲁棒估计问题。采用线性矩阵不等式 (L MIs)法进行凸优化 ,求解对应区间系统的满意估计增益 ,并设计相应鲁棒状态估计器。数值算例验证了相关的结论。     

Stabilizability and detectability of linear periodic systems

In this paper, four characterizations of stabilizability and detectability of linear periodic systems are considered. Two of them look as natural extensions of the classical definitions given for time-invariant systems. The remaining two are modal characterizations which turn out to be useful in the analysis of the periodic Lyapunov and Riccati equations. It is shown that all these notions of stabilizability (and detectability) are in fact equivalent to each other.One of the various definitions calls for the existence of the Kalman canonical decomposition of periodic systems. This issue is addressed in the Appendix.     

Structure and parametrization of periodic linear systems

In this paper, the beginnings of a structure theory for discrete-time periodic linear systems are developed. Canonical forms for state space realizations are described, structural invariants and their combinatorics are studied and geometric aspects of the parametrization of periodic systems are treated.     

Zeros of discrete-time linear periodic systems

In this note the concept of zero for linear discrete-time SISO periodic systems is discussed. The definition is based on a time-invariant MIMO equivalent representation. An input-output characterization of the zeros is provided which extends a well-known property for time-invariant systems.     

IQC-based -control of linear periodic systems

Stability analysis of linear periodically time-varying systems via integral quadratic constraints is extended to the problem of control design. A full-state feedback controller that satisfies exponential stability and -gain disturbance attenuation from an external disturbance to a controlled output is designed for linear systems with periodically time-dependent system matrices. The main result relies on dual forms of certain integral quadratic constraints. The solvability conditions for the problem are cast as a set of finite-dimensional linear matrix inequalities and thus, they are easily solvable. Moreover, the best possible disturbance attenuation level can be obtained as a convex problem.     

Adaptive state observation of linear time-varying systems with delayed measurements and unknown parameters

In this article, we address the problem of adaptive state observation of linear time-varying systems with delayed measurements and unknown parameters. Our new developments extend the results reported in our recently works. The case with known parameters has been studied by many researchers. However in this article we show that the generalized parameter estimation-based observer design provides a very simple solution for the unknown parameter case. Moreover, when this observer design technique is combined with the dynamic regressor extension and mixing estimation procedure the estimated state and parameters converge in fixed-time imposing extremely weak excitation assumptions.     

Stability of linear systems with periodic feedback and sampled-data systems

A linear multivariable time-invariant system with three periodic state feedback strategies—dynamic, static, and of the instantaneous state—is considered. For each of these feedback strategies the stability question of an appropriate closed-loop system in relation to its dependence upon the repetition frequency is examined. The proposed approach to the question is based on exponential matrix representations of the dynamics of the considered closed-loop systems. It is shown that at sufficiently large values of the repetition frequency the stability question may be decided on the basis of the stability of a certain matrix which is common for all the considered feedback strategies. It is also shown that peculiar to discrete-time control, the dead-beat stability phenomenon may appear only for a dynamic or static feedback strategy and it may be achieved when the repetition frequency is small enough. The proposed approach results in some new characterization of the stability property for multivariable sampled-data systems.     

Simultaneous observation of linear systems

This paper considers the problem of designing an observer which can observe the states for each of a given set of plants. The concept of the simultaneous observation is introduced. The simultaneous observation is achieved using coprime factorization techniques. Necessary and sufficient conditions for the existence of simultaneous observation are derived. The observer design is discussed, and an example is given     

Reachability and controllability of discrete-time linear periodic systems

In this paper, we consider discrete-time linear systems with periodic coefficients. Two necessary and sufficient conditions for complete controllability and reachability are given. The conditions are stated in terms of the reachability Gramian and hold for both reversible and nonreversible systems.     

线性离散周期系统满意估计

针对线性离散周期系统的状态估计问题,运用提升原理提取期望极点指标,同时期望估计误差系统满足稳态误差方差/H_∞混合指标,采用代数Riccati矩阵不等式法与数值递推算法对误差系统进行了上述指标的满意估计设计,并根据满意控制的基本理论将上述满意估计问题转化为线性矩阵不等式(LMI)的线性规划问题,从而运用LMI技术求解并设计了可行的满意估计,数值算例验证了相关算法.     

Satisfactory control of discrete-time linear periodic systems

In this paper satisfactory control for discrete-time linear periodic systems is studied. Based on a suitable time-invariant state sampled reformulation, periodic state feedback controller has been designed such that desired requirements of steady state covariance, H-infinity rejection bound and regional pole assignment for the periodic system are met simultaneously. By using satisfactory control theory, the problem of satisfactory periodic controller can be transformed into a linear programming problem subject to a set of linear matrix inequalities （LMIs）, and a feasible designing approach is presented via LMI technique. Numeric example validates the obtained conclusion.     

Fault detection of linear discrete-time periodic systems

In this note, an approach to the design of optimal fault detection systems for linear discrete-time periodic systems is proposed, which leads to an optimized compromise between robustness to unknown disturbances and sensitivity to faults. The needed computation mainly consists in solving a difference periodic Riccati system. The proposed approach is finally illustrated by a numerical example.     

Zero-error regulation of discrete-time linear periodic systems

In this paper the regulation problem for discrete-time linear periodic systems is investigated. A control structure is introduced which is shown to be capable of robustly zeroing the asymptotic error produced by periodically time-varying exogenous signals. The cases when the plant is time-invariant and/or the exogenous signals are constant are discussed.     

Pole-placement problem for discrete-time linear periodic systems

The pole-placement problem for the monodromy matrix of a discrete-time linear periodic system is considered. Given a partition of the complex plane into a ‘good’ part ? and a ‘bad’ part ?_b, the result obtained by Wonham (1979) is extended to the periodic case. This extension is based on the decomposition of a completely controllable periodic system into two subsystems, such that the first subsystem is completely reachable and the eigenvalues of the monodromy matrix of the second one are equal to zero.     

Pole-assignment problem for discrete-time linear periodic systems

This paper considers the pole-assignment problem for discrete-time linear periodic systems through the use of linear periodic state-variable feedback control. It is shown that if the N-periodic system with m inputs and n states is completely reachable then the problem can be reduced to the pole-assignment problem for a discrete-time linear invariant system with Nm inputs and n states.     

Monodromy eigenvalue assignment in linear periodic systems

This note addresses the problem of assigning the eigenvalues of the monodromy matrix of a linear continuous-time periodic system by periodic state feedback. A controllability criterion is given which slightly strengthens known results. An explicit expression for a periodic feedback gain which performs the pole assignment is given. When the system is controllable over one period, the whole monodromy matrix is assignable by periodic feedback. An example is given.