Simultaneous state estimation and parameter identification in linear fractional order systems using coloured measurement noise

In the present paper, the identification and estimation problem of a single-input–single-output (SISO) fractional order state-space system will be addressed. A SISO state-space model is considered in which parameters and also state variables should be estimated. The canonical fractional order state-space system will be transformed into a regression equation by using a linear transformation and a shift operator that are appropriate for identification. The identification method provided in this paper is based on a recursive identification algorithm that has the capability of identifying the parameters of fractional order state-space system recursively. Another subject that will be addressed in this paper is a novel fractional order Kalman filter suitable for the systems with coloured measurement noise. The promising performance of the proposed methods is verified using two stable fractional order systems.     

Parameter identification in linear distributed parameter systems

Perturbation analysis is used to identify unknown but constant parameters in a linear distributed parameter system. Noisy observations are assumed to be available at a finite number of spatial locations. A numerical example is solved to illustrate the proposed method.     

Output regulation for linear distributed parameter systems

This work extends the geometric theory of output regulation to linear distributed parameter systems with bounded input and output operator, in the case when the reference signal and disturbances are generated by a finite dimensional exogenous system. In particular it is shown that the full state feedback and error feedback regulator problems are solvable, under the standard assumptions of stabilizability and detectability, if and only if a pair of regulator equations is solvable. For linear distributed parameter systems this represents an extension of the geometric theory of output regulation developed in Francis (1997) and Isidori and Byrnes (1990). We also provide simple criteria for solvability of the regulator equations based on the eigenvalues of the exosystem and the system transfer function. Examples are given of periodic tracking, set point control, and disturbance attenuation for parabolic systems and periodic tracking for a damped hyperbolic system     

Reduced-order filters for limited state estimation from measurements containing coloured noise

The design of optimal fixed-configuration filters for estimating, without bias, a linear transformation of the state of a dynamic system in the presence of coloured measurement noise is studied. The performance index employed is the integral of error variance, and this leads to a matrix variational problem in the filter gains. It is demonstrated that the optimal time-varying filter gains may be computed by solving this variational problem using standard conjugate gradient algorithms. Consideration is also given to both algebraic and differential approximation of the optimal filter gains. The problem is formulated as one of ordinary minimization in parameters defining the approximation, and the required gradient expressions are derived for its solution using standard function minimization algorithms. An illustrative numerical example is presented.     

A method of parameter identification for linear distributed parameter systems

A method is presented for estimating unknown parameters in distributed parameter systems. The system considered is assumed to be modeled by a stochastic partial differential equation whose form is known to be linear and unknown parameters are contained in exciting terms. Unknown parameters are assumed to be a set of random constants whose a priori probabilities are known. First, the estimation process of unknown parameters is given by the Bayesian approach in the Markovian framework. The dynamics of the state estimation is also given, which is simultaneously required in the parameter identification scheme. Secondly, the computing procedure is presented, circumventing tedious calculations of the covariance function between the system state and unknown parameters. Finally, two numerical examples are shown, emphasizing that the dynamics of the observation mechanisms adopted plays an important role in both the state estimation and parameter identification.     

Stability robustness of linear normal distributed parameter systems

This paper considers the stability robustness analysis problem for linear distributed parameter systems containing known perturbation operators multiplied by uncertain parameters. The nominal system operators are assumed to be normal, but allowed to be unbounded. The perturbation operators are confined to some relative bounded set, but may be unbounded also. By using the Lyapunov stability criterion, simple bounds on uncertain parameters are derived to ensure the stability of the perturbed systems. Examples are provided to illustrate the usage of the theoretical results.     

双曲线型分布参数系统边界控制下的干扰解耦

针对干扰输入对系统输出的影响, 研究双曲线型分布参数系统在边界控制下的干扰解耦问题. 首先应用有界控制算子等价转换方法, 获取边界控制下双曲线型分布参数系统的有界拓展形式; 然后通过伽辽金近似法将有界增广系统转换成有限维系统; 接着对等价有限维系统进行干扰解耦分析, 推导出系统可干扰解耦的充分条件; 最后通过数值分析和仿真图例表明了所给出条件的有效性.

     

Controller design for distributed parameter systems with time delays in the boundary feedbacks via the backstepping method

For a distributed parameter system with an input delay in the boundary, a feedback control law is presented by means of the backstepping method. The square integrability of input signal is verified based on the target system. Then, the boundedness and invertibility of the corresponding backstepping transformation are proved under the regularity of system and the admissibility of feedback operator. Thus, the resulting closed-loop system is shown to be exponentially stable. Finally, as an application, a numerical simulation of a one-dimensional Schrödinger equation with a delay input is carried out, and the simulation results demonstrate the effectiveness of the suggested control law.     

On Recursive identification of linear and non-linear systems: case of coloured noise

Recursive identification algorithms based on least square estimation procedure have been presented for both linear and a class of nonlinear systems. The linear system is represented by an impulse response model while the non-linear system is a cascade combination of a. linear part and u zero memory non-linear part. It has been shown that even in the presence of coloured noise, for the model selected, ordinary least square estimation gives unbiased estimates of the system parameters. Results are illustrated through suitable numerical examples.     

Exponential stability of linear distributed parameter systems with time-varying delays

Exponential stability analysis via the Lyapunov-Krasovskii method is extended to linear time-delay systems in a Hilbert space. The operator acting on the delayed state is supposed to be bounded. The system delay is admitted to be unknown and time-varying with an a priori given upper bound on the delay. Sufficient delay-dependent conditions for exponential stability are derived in the form of Linear Operator Inequalities (LOIs), where the decision variables are operators in the Hilbert space. Being applied to a heat equation and to a wave equation, these conditions are reduced to standard Linear Matrix Inequalities (LMIs). The proposed method is expected to provide effective tools for stability analysis and control synthesis of distributed parameter systems.     

Parameter identification of distributed parameter systems in the presence of measurement noise

A new approach to recursive parameter identification of second-order distributed parameter systems in the presence of measurement noise under unknown initial and boundary conditions is proposed. A two-dimensional low-pass filter is introduced to pre-filter the observed data corrupted by measurement noise. The low-pass filter is designed in the continuous time-space domain and discretized by bilinear transformation. Thus a discrete estimation model of the system under study is easily constructed with filtered input-output data for recursive identification algorithms. The recursive least squares method is still efficient in the presence of low measurement noise if the filter parameters are designed so that the noise effects are reduced sufficiently. Using filtered input data as instrumental variables, a recursive instrumental variable method is also presented to obtain consistent estimates when the digital low-pass filters are not designed successfully or when the output data is corrupted by high measurement noise. Illustrative examples are given to demonstrate the applicability of the proposed methods.     

Stochastic suboptimal pointwise regulation control for linear discrete-time distributed parameter systems

Based on an examination of the concept of open-loop-optimal feedback control of Dreyfus (1964), a suboptimal stochastic pointwise regulation control scheme for linear distributed parameter systems with unknown plant and measurement noise covariances is presented. By enforcing the term coupling the control and estimation costs in the open-loop cost functional to zero, the suboptimal control scheme retains the controller-estimator in a cascade structure. An on-line algorithm is presented to adaptively select the filter parametrized by a scalar gain so as to minimize the equivalent cost functional derived from the open-loop case. Bounds on the filter gain that ensure filter tracking of the state are obtained. The control scheme is efficient and reasonably simple to use. A numerical example is presented.     

Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems

Motion planning and control of a boundary controlled quasilinear parabolic partial differential equation in one spatial variable is considered. The approach relies on a flatness property of the system; namely, that the system solution can be differentially parameterized in terms of a flat output which, in the case considered, is a boundary value. Such a parameterization allows straightforward motion planning and computation of a control law. The approach is based on power series in the spatial variable, and the convergence of these series is ensured by choosing the flat output to be a nonanalytic, smooth function of appropriate Gevrey class.     

Linear filtering for time-varying systems using measurements containing colored noise

The Kalman-Bucy filter for continuous linear dynamic systems assumes all measurements contain "white" noise, i.e. noise with correlation times short compared to times of interest in the system. It is shown here that if correlation times are not short, or if some measurements are free of noise, the optimal filter is a modification of the Kalman-Bucy filter which, in general, contains differentiators as well as integrators. It is also shown for this case that the estimate and its covariance matrix are, in general, discontinuous at the time when measurements are begun. The case of random bias errors in the measurements is shown by example to be a limiting case of colored noise.     

Analysis and identification of linear distributed systems via Chebyshev series

A double Chebyshev series is introduced to approximate functions of two independent variables and then applied to analyse and identify linear distributed systems. The solution for the coefficient matrices can be obtained directly from a Kronecker product formula. In addition, the algorithm for formulating the algebraic equations to estimate unknown parameters is derived, with the method used being algebraic and computer-oriented. Two illustrative examples are given and excellent results are obtained owing to the rapid convergence property of the Chebyshev series     

Asymptotic regulation for distributed parameter systems via zero dynamics inverse design

In this paper, we introduce a new approach, zero dynamics inverse (ZDI) design, for designing a feedback compensation scheme achieving asymptotic regulation for a linear or nonlinear distributed parameter system in the case when the value w(t) at time t of the signal w to be tracked or rejected is a measured variable. Following the nonequilibrium formulation of output regulation, we formulate the problem of asymptotic regulation by requiring zero steady‐state error together with ultimate boundedness of the state of the system and the controller(s), with a bound determined by bounds on the norms of the initial data and w. Because a controller solving this problem depends only on a bound on the norm of w not on the particular choice of w, this formulation is in sharp contrast to both exact tracking, asymptotic tracking or dynamic inversion of a completely known trajectory and to output regulation with a known exosystem. The ZDI design consists of the interconnection, via a memoryless filter, of a stabilizing feedback compensator and a cascade controller, designed in a simple, universal way from the zero dynamics of the closed‐loop feedback system. This design philosophy is illustrated with a problem of asymptotic regulation for a boundary controlled viscous Burgers' equation, for which we prove that the ZDI is input‐to‐state stable. In infinite dimensions, however, input‐to‐state stable compactness arguments are supplanted by smoothing arguments to accommodate crucial technical details, including the global existence, uniqueness, and regularity of solutions to the interconnected systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Hierarchical optimal control for distributed parameter systems via block pulse operator

This paper presents a development of the hierarchical optimal control for distributed parameter systems via block pulse operator. The proposed algorithm enables one to solve optimal control of distributed parameter systems with hierarchical structure and as a result, the resulting solutions are piecewise constant with minimal mean-square error, The algorithm is simple in form and computationally advantageous. An illustrative example shows the applicability of the proposed method in practice.     

Direct parameter identification of distributed parameter systems

A new direct approach to identifying the parameters of distributed parameter systems from noise-corrupted data is introduced. The model of the system which takes the form of a set of linear or nonlinear partial differential equations is assumed known with the exception of a set of constant parameters. Using finite-difference approximations of the spatial derivatives the original equation is transformed into a set of ordinary differential equations. The identification approach involves smoothing the measured data and estimating the temporal derivatives using a fixed interval smoother. A least-squares method is then employed to estimate the unknown parameters. Three examples that illustrate the applicability of the proposed approach are presented and discussed.     

基于小波变换的二阶线性分布参数系统预测控制

分布参数系统由于其复杂性,控制系统设计一直是控制理论的难点.寻求解决分布参数系统控制的新思路,本文提出了基于正交小波变换的线性离散时间分布参数系统预测控制概念,应用正交小波变换将二阶线性分布参数系统预测控制命题变换为集总参数系统预测控制问题,并设计预测控制器,将控制律进行反演获得原系统的具有分布特性的控制律.仿真研究表明,本文提出的分布参数系统预测控制算法取得了理想的控制效果,对系统模型失配和扰动具有较好的鲁棒性,验证了算法的有效性.