Riccati differential equation in optimal filtering of periodic non-stabilizable systems

This paper discusses the periodic solutions of the matrix Riccati differential equation in the optimal filtering of periodic systems. Special emphasis is given to non-stabilizable systems and the question addressed is the existence and uniqueness of a steady-state periodic non-negative definite solution of the periodic Riccati differential equation which gives rise to an asymptotically stable steady-state filter. The results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution. The convergence of the general solution of the periodic Riccati differential equation to a periodic equilibrium solution is also investigated. The results are extensions of existing time-invariant systems results to the case of periodic systems     

Disturbance decoupling in fault detection of linear periodic systems

This paper studies fault detection problems of linear discrete-time periodic systems. The aim is to design residual generators, which deliver a residual signal fully decoupled from unknown disturbances. First, a periodic parity relation based full decoupling residual generator with a periodically varying parity vector is established. Then, the relation between periodic parity vectors and periodic observer-based residual generators is investigated. It is shown that a periodic observer-based full decoupling residual generator can be obtained from a periodic full decoupling parity vector. Finally, the condition of disturbance decoupling is discussed and an example is given to illustrate the proposed approaches.     

Computation of transfer function matrices of periodic systems

We present a numerical approach to evaluate the transfer function matrices of a periodic system corresponding to lifted state-space representations as constant systems. The proposed pole-zero method determines each entry of the transfer function matrix in a minimal zeros-poles-gain representation. A basic computation is the minimal realization of special single-input single-output periodic systems, for which both balancing-related as well as orthogonal periodic Kalman forms based algorithms can be employed. The main computational ingredient to compute poles is the extended periodic real Schur form of a periodic matrix. This form also underlies the solution of periodic Lyapunov equations when computing minimal realizations via balancing-related techniques. To compute zeros and gains, numerically stable fast algorithms are proposed, which are specially tailored to particular single-input single-output periodic systems. The new method relies exclusively on reliable numerical computations and is well suited for robust software implementations. Numerical examples computed with MATLAB-based implementations show the applicability of the proposed method to high-order periodic systems.     

On Natural Continua of Periodic Solutions of the Systems with Hysteresis

Theorems on the existence of continua of periodic solutions for systems with hysteretic nonlinearities of the type of Preisah models are suggested. It is found that in the general situation, the standard criteria for existence of at least one periodic solution, which rest on the use of prior upper-bound estimates of the norm of periodic solutions and the Leray–Schauder method, ensure the existence of a one-parameter continuum of periodic solutions to these systems. The proofs are based on the use of new classes of operators that describe periodic responses of hysteretic nonlinearities to periodic inputs.     

遗传算法在周期信号傅立叶变换中的应用

为了克服周期信号进行傅立叶变换时各次谐波幅度值需要进行复杂的理论计算问题,提出一种利用遗传算法进行周期信号傅立叶变换的方法,介绍了周期信号傅立叶变换和遗传算法的基本原理;给出了使用遗传算法对周期信号进行分解适应度函数的实现方法和确定各次谐波幅值计算的方法。并提供使用遗传算法对周期信号进行分解的具体步骤。仿真实验结果表明:该方法能够满足周期信号傅立叶变换的要求,与周期信号傅立叶变换理论计算方法相比,其突出优点是算法简单,易于实现。     

Output stabilization via pole placement of discrete-time linearperiodic systems

The output stabilization problem for discrete-time linear periodic systems is solved. Both the state-feedback control law and the state-predictor are based on a suitable time-invariant state-sampled reformulation associated with a periodic system. Preliminary concepts of periodic system theory are briefly recalled. In particular, the structural properties of a linear discrete-time periodic system are properly related to those of a time-invariant system associated with it. By resorting to such a time-invariant reformulation, the output stabilization problem via pole placement is solved. The stabilizing controller is constituted by a control law and an asymptotic state predictor, both of which are shown to be periodic     

Maximum periodic solutions of the Volterra integrodifferential equations in the critical case of a pair of pure imaginary roots

Systems with aftereffect are considered, which are described by integrodifferential equations of the Volterra type in the presence of a small perturbation prescribed by the periodic (or maximum periodic) time function. In the critical case of a pair of pure imaginary roots of a characteristic equation, the question is solved as to existence in the system of maximum periodic motions (i.e., motions tending at the unlimited increase of time to periodic modes) provided that the frequency of the periodic part of disturbance coincides with the natural frequency of a linearized homogeneous system. It is shown that in the analytic case the equations of motion of the system possess a set of the maximum periodic solutions representable by power series in a small parameter specifying a value of the disturbance, and in small arbitrary initial values of uncritical variables of the problem. The conditions of the existence of such solutions are indicated, which are defined by the terms up to the third order of equations.     

Existence conditions and properties for the maximal periodic solution of periodic Riccati difference equations

The existence and properties of the maximal symmetric periodic solution of the periodic Riccati difference equation, is analysed for the optimal filtering problem of linear periodic discrete-time systems. Special emphasis is given to systems not necessarily reversible and subject only to a detectability assumption. Necessary and sufficient conditions for the existence and uniqueness of periodic non-negative definite solutions of the periodic Riccati difference equation which gives rise to a stable filter are also established. Furthermore, the convergence of non-negative definite solutions of the Riccati equation is investigated.     

Robustness of periodic trajectories

A robustness problem for periodic trajectories is considered. A nonautonomous system with a periodic solution is given. The problem is to decide whether a stable periodic solution remains in a neighborhood of the nominal periodic solution when the dynamics of the system is perturbed. The case with a structured dynamic perturbation is considered. This makes the problem a nontrivial generalization of a classical problem in the theory of dynamical systems. A solution to the robustness problem will be obtained by using a variational system obtained by linearizing the system dynamics along a trajectory, which is uncertain but within the prespecified neighborhood of the nominal trajectory. This gives rise to robustness conditions that can be solved using integral quadratic constraints for linear time periodic systems.     

Control and state observation of periodic linear multimodal systems

This paper investigates the control and state observation problem for periodic linear controllable and observable multimodal systems. For a controllable periodic linear multimodal system, this paper proposes an alternative minimum-norm dynamic state feedback control scheme for state transition matrix assignment. Also presented is a new design of state observers for an observable periodic linear multimodal system. Finally, an analysis of closed-loop behaviour of controlled periodic linear multimodal systems using a state-estimate is given     

A second order ODE is sufficient for modelling of many periodic signals

Which is the minimum order an autonomous non-linear ordinary differential equation (ODE) needs to have to be able to model a periodic signal? This question is motivated by recent research on periodic signal analysis, where non-linear ODEs are used as models. The results presented here show that a second order ODE is sufficient for a large class of periodic signals. More precisely, conditions on a periodic signal are established that imply the existence of an ODE that has the periodic signal as a solution. A criterion that characterizes the above class of periodic signals by means of the overtone contents of the signals is also presented. The reason why higher order ODEs are sometimes needed is illustrated with geometric arguments. Extensions of the theoretical analysis to cases with orders higher than two are developed using this insight.     

Performance limitations of nonlinear periodic sampled-data controllers for L/sub p/ disturbance rejection

This note presents a performance analysis of periodic nonlinear sampled-data controllers for the rejection of L/sub p/ specific and uniform disturbances. Earlier results on the performance of linear periodic controllers are extended to nonlinear controllers. For a given periodic controller, a time invariant controller is constructed which in general gives strictly better L/sub p/ disturbance rejection performance than the periodic controller.     

神经元周期放电模式的分岔

利用一种可以计算自治非线性系统周期解及周期的改进打靶法,求解了神经元电活动Rose—Hind-marsh（R-H）模型自发放电的周期解和周期;计算了周期放电的Floquet乘子并分析了周期解的分岔,如倍周期分岔,鞍-结分岔．研究结果有助于进一步理解神经放电模式转迁的动力学和生物学意义．     

一类自治脉冲微分方程的动力学研究

对一类自治脉冲微分方程的动力学性质进行了研究,给出了半平凡周期解的存在与稳定的充分条件,建立Poincare映射将周期解问题转化为不动点问题．理论分析及数值模拟表明,半平凡周期解通过跨临界分岔获得稳定的正周期-1解．数值模拟显示,随着控制参数的变化,正周期-1解通过倍周期分岔出正周期-2解,再通过一系列倍周期分岔通向混沌．     

Analysis of periodically forced bioreactors using nonlinear transfer functions

Nonlinearity is an underlying property of process systems that allows for time-average performance enhancement with periodic forcing of external parameters. One can use the well-known π-criterion to obtain a sufficient condition for optimal periodic operation, but it is valid only for weakly nonlinear systems with infinitesimal forcing amplitudes. Among such process systems, bioreactors often exhibit highly nonlinear dynamics, thus posing difficulties for a systematic analysis of their periodic operation. It becomes desirable to understand how inherent nonlinearities would contribute to performance improvement if periodic forcing is applied to such processes. This paper explores how nonlinear characteristics affect periodic process operation in the context of bioreactor systems. By taking advantage of iterated integrals and shuffle algebra, an analytical solution of a nonlinear bioprocess is approximated with the functional expansion method. The resulting Laplace-Borel transform of the nonlinear system facilitates the expression of the solution in the frequency domain based on the nonlinear transfer function. The method enables the separation of the transient dynamics and the stationary periodic behavior, facilitating the analysis of periodic operations and leading to a generalizable platform. Specifically, the optimal periodic operation problem is solved by finding the proper forcing amplitude and frequency to maximize the offset. Compared with the π-criterion, our method is proven to be globally valid for any forcing frequency and amplitude, so long as the process is globally stable.     

An algebraic method for steady-state responses of linear state equations with periodic input

An algebraic method is presented to evaluate periodic conditions for the steady-state responses of state equations with a continuous or piecewise-continuous periodic input. As a result, the analytical and experimental steady-state responses to periodic excitation may be directly obtained.     

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.     

Robust performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs

A particular class of uncertain linear discrete-time periodic systems is considered. The problem of robust stabilization of real polytopic linear discrete-time periodic systems via a periodic state-feedback control law is tackled here, along with performance optimization. Using additional slack variables and the periodic Lyapunov lemma, an extended sufficient condition of robust stabilization is proposed. Based on periodic parameter-dependent Lyapunov functions, this last condition is shown to be always less conservative than the more classic one based on the quadratic stability framework. This is illustrated on a numerical example from the literature.     

Implementation of repetitive controller for rejection of position-based periodic disturbances

Time-varying periodic motions have appeared in many industrial processes, such as cam-follower systems, in that the control system has to operate at a periodic variable velocity specified in terms of the angular position-domain. This paper employs a repetitive control for rejecting the time-varying periodic disturbances, and presents an implementation technique based on angular position which can generate a fixed number of cycles in real-time. This control technique can accommodate the period variation in a time-varying periodic signal such that variable samples per period for synchronization can be achieved by the regular fixed-time sampling rate. A technique using position information to manipulate the delayed data and an interpolation scheme to properly access data in the buffer memory is proposed. An anti-vibration control system with time-varying periodic disturbances is studied to illustrate control performance. The experimental results are given to illustrate that time-varying repetitive control can effectively eliminate steady-state errors within a few cycles.     

Periodic strong solution for the optimal filtering problem oflinear discrete-time periodic systems

The periodic Riccati difference equation (PRDE) for the optimal filtering problem of linear periodic discrete-time systems is addressed. Specifically, the author provides a number of results on the existence, uniqueness, and stability properties of symmetric periodic nonnegative-definite solutions of the periodic Riccati difference equation in the case of nonreversible and nonstabilizable periodic systems. The convergence of symmetric periodic nonnegative-definite solutions of the periodic Riccati difference equation is also analyzed. The results have been established under weaker assumptions and include both necessary and sufficient conditions. The existence and properties of symmetric periodic nonnegative-definite solutions of the PRDE are established directly from the PRDE