Design of decoupling and tracking controllers for continuous-time transfer function matrices with multiple time delays

This paper presents an extended adjoint decoupling method together with a reference model-based sliding mode tracking method, to design a decoupling and tracking controller for continuous-time transfer function matrices with multiple (integer/fractional) time delays in both denominators and numerators. First, for obtaining the diagonally decoupled subsystems, a decoupler is designed by utilizing the extended adjoint decoupling method. Then, by using sampled data from the unit-step response of the decoupled subsystems, the conventional balanced model-reduction method is carried out to obtain the approximated delay-free/single-delay continuous-time models for the decoupled subsystems with multiple time delays. For the integral of time multiplied by absolute error (ITAE) reference model tracking, a chain observer is designed to establish the virtual estimated states for the decoupled subsystems by utilizing the obtained approximated continuous-time models. At last, we develop a sliding mode tracking controller together with a disturbance observer (DOB), to achieve reference model tracking and disturbance rejection. Illustrative examples are given to demonstrate the effectiveness of the proposed method.     

A computer program for the minimal realization of transfer function matrices

A computer program for finding the minimal realizations of transfer function matrices is described. An example is included to illustrate the procedure.     

Balanced realization of stable transfer function matrices

Simple formulas are presented to compute the internally balanced minimal realization and the singular decomposition of the Hankel operator of a given continuous-time p×m stable transfer function matrix E(s)/d(s). The proposed formulas involve the Schwarz numbers of d(s) and the singular eigenvalues-eigenmatrices of a suitable finite matrix. Similar results are also obtained for a given discrete-time transfer function matrix     

Modelling and realization of multiple time delays systems via moving models

The formulation and realization of multiple time delays systems are considered. The systems are represented by moving models established by the use of block pulse functions. A non-isothermal continuously stirred tank reactor using react ant and coolant flow rates as input variables is presented to illustrate that the method exhibits satisfactory results.     

Minimal realization of transfer function matrices via oneorthogonal transformation

The minimal realization of a given arbitrary transfer function matrix G(s) is obtained by applying one orthogonal similarity transformation to the controllable realization of G( s). The similarity transformation is derived by computing the QR or the singular value decomposition of a matrix constructed from the coefficients of G(s). It is emphasized that the procedure has not been proved to be numerically stable. Moreover, the matrix to be decomposed is larger than the matrices factorized during the step-by-step procedures given     

New algorithm for minimal balanced realization of transfer function matrix

A simple method for calculating the gramians of any stable realization using algebraic computation is described. This allows the development of a new algorithm to obtain a minimal balanced realization. The algorithm presented is shown to be computationally simpler and more efficient than previous methods and is illustrated by an example.     

Minimal realization of transfer function matrices A comparative study of different methods

A comparative study is made of three basically different methods for obtaining minimal-order realizations of linear multivariable systems in the form of state equations for specified rational transfer function matrices. Computational efficiency and suitability for practical implementation are the main criteria used for the comparison. The possibilities of direct realization in canonical forms suitable for special applications are also examined.     

The minimal order of rational transfer function realization from its z-domain samples

We present here a new method for realization of minimal-order proper rational transfer functions from z-domain samples. A criterion for determining the minimal order of the transfer function is given. The theory is developed first for strictly proper rational transfer functions and then extended to the case of transfer functions with a direct-coupling term. The theory is valid for arbitrary z-domain samples. The special case of uniform frequency response samples is shown to allow a computationally efficient derivation of the realization. Examples are given to illustrate the theory. The theory developed here has many applications, such as modelling of communication channels, identification of systems from frequency measurements and infinite impulse response filter design.     

A new algorithm for the minimal balanced realization of a transfer function matrix

A new algorithm is proposed here to obtain a minimal balanced realization directly from the transfer function matrix (TFM). This method which employs the singular-value decomposition (SVD) of an infinite block-Hankel matrix requires neither an initial minimal realization nor the solution of a Lyapunov matrix equation. The formulation is solely in terms of the coefficients of the transfer function matrix.     

A simple algorithm on minimal balanced realization for transferfunction matrices

Given a transfer function matrix, it is shown that its minimal balanced realization can be obtained directly from the singular values and the singular vectors of a constant matrix constructed from the coefficients of the least common denominator polynomial. The algorithm is based on the close relationship of controllability and the observability gramians to the singular value decomposition of the associated infinite block Hankel matrix     

Digital decoupling controller design for multiple time-delay continuous-time transfer function matrices

This paper presents an extended adjoint decoupling method to conduct the digital decoupling controller design for the continuous-time transfer function matrices with multiple (integer/fractional) time delays in both the denominator and the numerator matrix. First, based on the sampled unit-step response data of the afore-mentioned multiple time-delay system, the conventional balanced model-reduction method is utilised to construct an approximated discrete-time model of the original (known/unknown) multiple time-delay continuous-time transfer function matrix. Then, a digital decoupling controller is designed by utilising the extended adjoint decoupling method together with the conventional discrete-time root-locus method. An illustrative example is given to demonstrate the effectiveness of the proposed method.     

Parameter-dependent Lyapunov functional for systems with multiple time delays

The separation of the Lyapunov matrices and system matrices plays an important role when one use sparameter-dependent Lyapunov functional handling systems with polytopic type uncertainties. The delay-dependent robust stability problem for systems with polytopic type uncertainties is discussed by using parameter-dependent Lyapunov functional. The derivative term in the derivative of Lyapunov functional is reserved and the free weighting matrices are employed to express the relationship between the terms in the system equation such that the Lyapunov matrices are not involved in any product terms with the system matrices. In addition, the relationships between the terms in the Leibniz Newton formula are also described by some free weighting matrices and some delay-dependent stability conditions are derived. Numerical examples demonstrate that the proposed criteria are more effective than the previous results.     

Dead-time-compensator for unstable MIMO systems with multiple time delays

In this paper a new dead-time-compensator to deal with unstable time delay systems is presented. The result is an extension to multiple-input multiple-output systems with multiple and different time delays of a previous result already reported for single-input single-output systems. There are two key issues: the system instability and the presence of different time delays in each signal channel. The proposed approach is developed in three steps. First, a non-delayed output plant is predicted. This predictor is a stable dead-time-compensator coping with multiple and arbitrary delays in all the signal channels. Then, a stabilizer controller is easily designed for the resulting non-delayed plant. For this stabilized plant, the control performance is improved in order to achieve some output tracking and regulation requirements. The results are illustrated by two examples showing their applicability to unstable multiple-input multiple-output multi-delayed plants, which is the main novelty of the proposed approach.     

State analytical predictor for systems with multiple time delays

An approach for the design of a dead-time compensator for processes with time delays is presented. The proposed algorithm deals with multivariable state-space models instead of input-output models and utilizes not only a single delay but also distributed delays in the feedback loop. Both the continuous-time and discrete-time versions of the algorithm are derived. The design strategy of the controller is based on the approach of LQG design. It is incorporated into the dead-time compensation technique of the analytical predictor, which uses the process model directly to predict the effect of input variables on the process outputs. By introducing an integral action into the observation system, the steady-state observation error of the inaccessible load inputs converges to zero. This permits us to perform disturbance prediction and compensation within a single design. Theoretical analysis shows that this control strategy fully removes time delay elements from a system characteristic equation in the ideal situation of a system without model-plant mismatch and/or noise measurements. By the estimation and prediction of the unknown plant-input disturbances, the capability for rejecting input disturbances has also been improved. The control structure developed is shown to have the same closed-loop relationships as the linear quadratic feedback structure for a system without lime delays. The potential benefits of the proposed controller are demonstrated by the control problem of an industrial grinding system studied previously by Niemi et al. (1982) and Ylinen et al. (1987), which is a challenging problem for MPC-type algorithms.     

Estimation in linear discrete systems with multiple time delays

The method of orthogonal projection is used to derive the equations for optimally estimating the state of a nonstationary linear discrete system with multiple time delays. A Kalman-type filter is developed, along with the necessary recursive error and cross error covariance matrix equations. A numerical example is included.     

State space construction for continuous time transfer function matrices via Nerode equivalence

In this paper, we show that a minimal state space realization in Jordan canonical form for linear multivariable continuous-time systems described by rational transfer function matrices could be obtained in a natural and basic way by using the concept of Nerode equivalence. While the minimal state space realization is known, the contribution of this note is to provide an alternative realization procedure which is directly introduced in a simple and self-contained manner. Both scalar and multivariable cases in the continuous-time setting are discussed. The presentation also provides a concrete construction of rational vector function with preassigned poles to interpolate any 2-norm bounded analytical vector function in the open left-half of the complex plane. The basic idea of Nerode equivalence is that the state can be identified with a corresponding equivalence class of inputs. For a linear finite dimensional time-invariant continuous-time system, the zero state is identified with the kernel of certain Hankel operator. This characterization via Nerode equivalence class sheds light on the construction of state for general nonlinear input–output systems.     

多个未知时延的MISO系统的递推辨识

对于未知时延的多输入单输出（ＭＩＳＯ）系统,借助分离性原理,推导出迭代的可分离的非线性最小二乘（ＳＮＬＳ）辨识方法．为降低收敛于局部最小的可能性,利用全局优化理论,推导了全局可分离的非线性最小二乘（ＧＳＮＬＳ）辨识方法;为消除强观测噪声所引起的参数估计的偏差,将ＧＳＮＬＳ方法调整为一新颖的全局可分离的非线性多新息递推最小二乘（ＧＳＮＭＩＲＬＳ）辨识方法,仿真实验验证了算法的有效性．     

Robust analysis of nonlinear discrete systems with multiple time delays

In this paper, we present the stability criteria, sufficient conditions that guarantee the robust stability of linear structured or unstructured variation of discrete time-delay systems subjected to the given bounds of nonlinear function. Both single and composite nonlinear discrete systems with multiple time delays are considered and each of these results is expressed by a succinct scalar inequality and independent of delay time.     

Fixed-interval smoothing of linear discrete systems with multiple time delays

In this correspondence, the problem of fixed-interval smoothing of linear discrete systems with multiple time delays is considered. An algorithm is developed which in certain cases is computationally more efficient than the one obtained through the state augmentation method.