Closed-form solutions to discrete-time LQ optimal control and disturbance attenuation

This paper presents straightforward derivations of closed-form solutions to two discrete-time problems: linear-quadratic optimal control and disturbance attenuation. Then, as the main results, it shows how to reduce directly the closed-form solutions to their recursive forms. These recursive solutions are in the form of Riccati equation and, hence, establish a close relation between the closed-form solutions and the Riccati equations used in both problems.     

Event-based input and state estimation for linear discrete time-varying systems

In this paper, the joint input and state estimation problem is considered for linear discrete-time stochastic systems. An event-based transmission scheme is proposed with which the current measurement is released to the estimator only when the difference from the previously transmitted one is greater than a prescribed threshold. The purpose of this paper is to design an event-based recursive input and state estimator such that the estimation error covariances have guaranteed upper bounds at all times. The estimator gains are calculated by solving two constrained optimisation problems and the upper bounds of the estimation error covariances are obtained in form of the solution to Riccati-like difference equations. Special efforts are made on the choices of appropriate scalar parameter sequences in order to reduce the upper bounds. In the special case of linear time-invariant system, sufficient conditions are acquired under which the upper bound of the error covariance of the state estimation is asymptomatically bounded. Numerical simulations are conducted to illustrate the effectiveness of the proposed estimation algorithm.     

Design of fixed-point smoother algorithm for linear discrete time-delay systems

This paper investigates the fixed-point smoothing problems for linear discrete-time systems with multiple time-delays in the observations.The linear discrete-time systems considered have l + 1 output channels.One is instantaneous observation and the others are delayed.The fixed-point smoothers involving recursive algorithm and non-recursive algorithm are designed by using innovation analysis theory without relying on the system augmentation approach.Also, it is further shown that the design of fixed-point smoother comes down to solving l + 1 Riccati equations with the same dimensions as the original systems.     

Solving inverse laplace transform,linear and nonlinear state equations using block-pulse functions

A recursive algorithm is developed for solving the inverse Laplace transform, linear and nonlinear state equations using block-pulse functions. The relationships between the solution of the continuous-time state equation using block-pulse functions and that of the equivalent discrete-time state equation using trapezoidal rule are investigated. A complete computer program is presented for solving the differential equations of linear and nonlinear state equations using block-pulse functions.     

State estimation for discrete-time markov jump linear systems based on orthogonal projective theorem

In this paper, state estimation problem for discrete-time Markov jump linear systems is considered. Based on orthogonal projective theorem, a novel suboptimal algorithm for state estimate of discrete-time Markov jump linear systems in the sense of minimum mean square error estimate is proposed. The proposed suboptimal algorithm is recursive and finite-dimensionally computable. Computer simulations are carried out to evaluate the performance of the proposed suboptimal algorithm.     

Optimal filtering in stochastic discrete-time systems with unknowninputs

In this note we derive a recursive filtering algorithm for the linear discrete-time dynamic system with indeterminate-stochastic inputs. The algorithm is based on the minimax-optimal method of parameter estimation in the linear regression model with parameters of two different types: unknown and stochastic with partially known characteristics     

Step response of a class of second-order non-minimum phase transfer functions†

This paper investigates the problem of controlling a discrete-time linear system with jump parameters. A review of the literature is presented as well as a development of the application of dynamic programming to this class of control problems. Dynamic programming has been applied by many researchers and it was observed that no closed-form analytical solution could be constructed because of the ‘dual’ aspects of the controller. The main contribution of the present work is an algorithm, suitable for computer implementation, for the optimal dual control. The construction of the algorithm is based on transforming the dynamic programming relations into a space of sufficient statistics and using a finite-dimensional optimization procedure to obtain the optimal control as a function of the statistics. This is achieved by first developing a suitable recursive realization of a ‘filter’ which generates the sufficient statistics for the problem and then embedding this filter into the dynamic programming equations. Several examples are presented to illustrate the use of the algorithm for constructing optimal and suboptimal controllers.     

应用于离散模糊系统的最优模糊追踪器设计

针对二阶离散式模糊追踪系统,提出用于构造一种最优模糊追踪器的设计方式, 并用于追踪一类时变目标或任意模型目标.为了能让所提出的设计方法完整并达到总体最小 化,提出了一种近似线性化的离散模糊动态系统的表示法.另外,为了简化计算量,一个多 层分解方式被用于本文所提出的最优设计程序中,模拟结果显示,本文提出的最优模糊追踪 设计方式确能达到预期效果.     

Recursive algorithms for linear LMSE estimators under uncertain observations

Hadidi and Schwartz [1] proposed a recursive linear estimator algorithm for linear discrete-time systems under uncertain observations when it exists. In this note two sets of formulas for the optimal filter and one-step predictor under uncertain observations, when they exist, are derived. These formulas differ, to some extent, from those proposed in [1].     

Some new algorithms for recursive estimation in constant, linear, discrete-time systems

Certain recently developed fast algorithms for recursive estimation in constant continuous-time linear systems are extended to discrete-time systems. The main feature is the replacement of the Riccati-type difference equation that is generally used for such problems by another set of difference equations that we call of Chandrasekhar-type. The total number of operations in the new algorithm is in general less than with the Riccati-equation based Kalman filter, with significant reductions being obtained in several important special cases. The algorithms are derived via a factorization of increments of the Riccati equation variable, a method that can be extended to nonsymmetric Riccati equations as well.     

Parameter identification in a class of nonlinear systems

Two approaches are proposed for on-line identification of parameters in a class of nonlinear discrete-time systems. The system is modeled by state equations in which state and input variables enter nonlinearly in general polynomial form, while unknown parameters and random disturbances enter linearly. All states and inputs must be observed with measurement errors represented by white Gaussian noise having known covariance. System disturbances are also white and Gaussian with finite, but unknown, covariance. One method of parameter estimation is based upon a least squares approach, the second is a related stochastic approximation algorithm (SAA). Under fairly mild conditions the estimate derived from the least squares algorithm (LSA) is shown to converge in probability to the correct parameter; the SAA yields an estimate which converges in mean square and with probability 1. Examples illustrate convergence of the LSA which even in recursive form requires inversion of a matrix at each step. The SAA requires no matrix inversions, but experience with the algorithm indicates that convergence is slow relative to that of the LSA.     

Block pulse function approach for the identification of Hammerstein model non-linear systems

Two recursive algorithms based on block pulse functions are presented for identifying continuous Hammerstein models of non-linear systems with (i) a state space model and (ii) an input–output model. Since the continuous non-linear systems are transformed approximately into the corresponding difference equations via block pulse functions, these recursive estimation algorithms can easily be obtained using a derivation similar to that of the discrete-time models expressed by difference equations. Both algorithms derived here are simple and straightforward, and can easily be implemented on-line. As discussed in this paper, these algorithms can also be extended to the identification of certain continuous non-linear systems with a feedback loop or with time delays. The illustrative examples show that these recursive algorithms give satisfactory results for the identification problems of certain continuous non-linear systems.     

Extended observer form: simple existence conditions

The paper focuses on the problem of transforming the discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. The simple necessary and sufficient conditions for the existence of the extended coordinate change and the output transformation, allowing to solve the problem, are formulated in terms of certain partial derivatives, related to the input–output equation, corresponding to the state equations. Moreover, a certain algorithm for transforming the state equations into the observer form is proposed.     

An approach to automatic generation of dynamic equations of elastic joint manipulators in symbolic language

This paper describes an approach for automatic generation of the equations of motion of elastic joint manipulators in symbolic language. It is based on a vector-parametrization of the Lie groupSO(3) and uses the Lagrange's formalism to derive the dynamical equations, the final forms of which are like the equations generated by a recursive Newton-Eulerian algorithm. These characteristics together increase the computational efficiency of the algorithm and give a very good insight into the dynamical structure of the system. In addition to this, the inertia matrix is explicitly given in the final equations, which is very important for the applicability of a mathematical model in different fields of control and simulation. The suggested algorithm is therefore quite appropriate for the purposes of robot modeling, identification as well as for the applications in real time simulations and control.     

Computer generation of robot dynamics equations and the related issues

Two programs have been developed using the computer algebra system REDUCE to generate the dynamics equations of motion for robot manipulators. One of these programs is based on a Lagrange formulation and the other utilizes a recursive Newton-Euler formulation. Both programs produce equivalent scalar symbolic expressions for the generalized actuator forces, but the program based on the recursive Newton-Euler formulation is more efficient for the generation of equations. These programs have been used to generate the dynamics equations of manipulators with as many as six degrees of freedom. The efficiency of computing forces using the generated scalar symbolic expressions is compared with the efficiency of a numerical algorithm (implemented in FORTRAN '77) based on the recursive Newton-Euler formulation. Force computation by the method of symbolic equations is shown to be more efficient than the numerical recursive Newton-Euler algorithm. The technique of symbolic equations is also better adapted to multi-CPU processing.     

Recursive learning algorithms for training fuzzy recurrent models

A recurrent fuzzy neural network with external feedback, called the Dynamical-Adaptive Fuzzy Neural Network (D-AFNN), is proposed in this article for adapting discrete-time dynamical systems. The fuzzy model is based on the Takagi-Sugeno inference method with polynomial consequent functions. The D-AFNN model is trained to learn the system dynamics using a recursive training algorithm. Both the epochwise and the on-line version of the recursive algorithm are considered. The standard representation of a dynamical system, using the fuzzy recurrent model, is discussed and the computational procedure for the derivation of the output gradients is analytically described. The performance qualities of the D-AFNN model are illustrated with two examples. In the first example, the model is trained to identify a nonlinear plant, while in the second the fuzzy recurrent model implements an IIR adaptive filter for noise attenuation. © 1996 John Wiley & Sons, Inc.     

State estimation for discrete-time Markov jump linear systems with multiplicative noises and delayed mode measurements

In this paper, the state estimation problem for discrete-time Markov jump linear systems affected by multiplicative noises is considered. The available measurements for the system under consideration have two components: the first is the model measurement and the second is the output measurement, where the model measurement is affected by a fixed amount of delay. Using Bayes' rule and some results obtained in this paper, a novel suboptimal state estimation algorithm is proposed in the sense of minimum mean-square error under a lot of Gaussian hypotheses. The proposed algorithm is recursive and does not increase computational and storage load with time. Computer simulations are carried out to evaluate the performance of the proposed algorithm.     

LU分解递归算法的研究

将递归方法引入稠密线性代数的计算，能产生自动的矩阵分块，使算法适合于当今分级存储高性能计算机的结构，提高运算速度。文中对解线性代数方程组的LU分解递归算法进行了研究，给出了算法的详细推导过程。     

STWS approach for Hammerstein system of nonlinear Volterra integral equations of the second kind

In this paper, single-term Walsh series (STWS) method is applied to obtain the numerical solutions of Hammerstein systems of nonlinear Volterra integral equations of second kind (HSNVIES). Using the properties of the STWS method, HSNVIES can be easily converted into solvable recursive system of algebraic equations. Solutions obtained from the recursive system of algebraic equations are the solutions of the HSNVIES. Illustrative examples are provided with numerical solutions and the efficiency of this STWS method is also compared with the existing methods.