Algorithm for closed-loop eigenstructure assignment by state feedback in multivariable linear systems

In this paper an algorithm is presented which greatly facilitates the complete exploitation of state feedback in the assignment of the entire closed-loop eigenstructure of controllable multi-input systems. This algorithm is a generalization of the algorithm of MacLane and Birkhoff (1968) for the computation of a basis for the null space of a matrix and is ideally suited to digital computer implementation. The algorithm readily yields the vectors which are required (Porter and D'Azzo 1978) for the simultaneous assignment of Jordan canonical forms, eigenvectors, and generalized eigenvectors to the plant matrices of closed-loop controllable multivariable linear systems. The effectiveness of the algorithm is illustrated by assigning the entire closed-loop eigenstructure of a third-order two-input discrete-time system in such a way that the resulting closed-loop system exhibits time-optimal behaviour.     

Closed-loop eigenstructure assignment by state feedback in multivariable linear systems with slow and fast modes

It is shown that the recent results of Moore (1976) and Porter and D'Azzo (1977) can be readily extended so as to provide a computationally attractive method of closed-loop eigenstructure assignment by state feedback in multivariable systems with ‘ slow ’ and ‘ fast ’ modes. The general results are illustrated by designing a state-feedback controller for a third-order system with ‘ slow ’ and ‘ fast ’ modes which assigns asymptotically the entire closed-loop eigenstructure as characterized by both ‘ slow ’ and ‘ fast ’ eigenvalues and eigenvectors.     

Symbolic realization of asymptotic time-scale and eigenstructure assignment design method in multivariable control

This paper reports on a symbolic realization of the asymptotic time-scale and eigenstructure assignment (ATEA) state feedback design technique for multivariable control. The resulting state feedback laws are parameterized in a scalar ε. Under these state feedback laws, the closed-loop system possesses a pre-specified time-scale and its eigenstructure approaches a pre-specified one, as the value of the parameter ε approaches zero. By appropriately specifying the time-scale and the eigenstructure, the feedback laws can be obtained to solve various control problems, such as the H ₂ and H _∞ suboptimal control, and almost disturbance decoupling problems. We present, in this paper, the software implementation of the ATEA design algorithm using the MATLAB symbolic programming technique. Our m-functions are capable of returning a result, which is explicitly expressed in terms of a symbolic variable epsilon, which represents ε. The controller design for a piezoelectric bimorph actuator is used to illustrate how the symbolic realization works.     

Recursive eigenstructure assignment in linear systems

In Arbel and Rath (1985) an iterative eigenvalue assignment algorithm was presented. This algorithm has the shortcomings of clustering most of the closed-loop eigenvalues on the real axis. A general method is given here for recursive eigenstructure assignment in linear systems. The method eliminates the shortcomings of the algorithm of Arbel and Rath. It is shown that the right and left eigenvectors of the closed-loop system matrix can be determined in terms of those of a small-dimension matrix A?_c . It is also shown that the results of Arbel and Tse (1980) are special cases of those of the proposed method. Moreover, the arbitrary parameters, beyond eigenvalue assignment, are shown to be embedded in the choice of a certain arbitrary invertible matrix S. Furthermore, computer-oriented steps are outlined for recursive eigenstructure assignment in large-scale systems. Numerical examples are worked out to illustrate the generality and feasibility of the proposed method.     

Automated fault diagnosis in nonlinear multivariable systems usinga learning methodology

The paper presents a robust fault diagnosis scheme for detecting and approximating state and output faults occurring in a class of nonlinear multiinput-multioutput dynamical systems. Changes in the system dynamics due to a fault are modeled as nonlinear functions of the control input and measured output variables. Both state and output faults can be modeled as slowly developing (incipient) or abrupt, with each component of the state/output fault vector being represented by a separate time profile. The robust fault diagnosis scheme utilizes on-line approximators and adaptive nonlinear filtering techniques to obtain estimates of the fault functions. Robustness with respect to modeling uncertainties, fault sensitivity and stability properties of the learning scheme are rigorously derived and the theoretical results are illustrated by a simulation example of a fourth-order satellite model.     

广义二阶动力学系统的鲁棒特征结构配置

考虑了具有摄动的广义二阶动力学系统的鲁棒特征结构配置问题, 提出了一种优化算法. 该算法的目标是使得特征结构配置的误差最小. 根据广义二阶动力学系统基于状态加微分反馈的参数化特征结构配置结果, 给出了优化指标的完全参数形式. 该算法不包含“返回”过程, 允许特征值在一定范围内参与优化, 能够给出鲁棒性较强的控制系统. 数值例子说明了算法的有效性.     

On the synthesis of multivariable systems

The general question of compensating a linear multivariable system via linear state variable feedback (lsvf) in combination with input dynamics is considered. In particular, it is shown that, since lsvf and input dynamics can both be represented by proper feedforward compensation, any design based on feedforward compensation alone can usually be realized in a more efficient manner through the combined use of "low-order" feedforward compensation (input dynamics) and lsvf. A major result of this paper is the frequency-domain synthesis algorithm outlined in Section III for "separating" any proper feedforward transfer matrix in order to achieve the more efficient compensation scheme. Specific examples which illustrate the employment of the algorithm to the questions of decoupling and exact model matching are also included.     

Coordination of nonstationary systems

In this paper on-line optimization of a complex system subject to disturbances is considered. A hierarchical control structure is described, which uses feedback from the controlled system along with simplified mathematical models. Operation of the control structure is analyzed for the case of time-varying disturbances. The effect of model-reality differences and delays caused by computation and communication is investigated. Theoretical considerations are illustrated by a simulation study.     

The eigenstructure of sampled-data systems with confluent eigenvalues

It is well known that the eigenvectors of a sampled-data system are the same as the eigenvectors of the continuous time system from which the sampled-data system has been derived provided that the eigenvalues of the continuous time system arc distinct. In this paper the eigenvectors of a sampled-data system with confluent eigenvalues arc deduced in terms of the eigenvectors of the continuous time system from which the sampled -data system has boon derived. The theory is illustrated by numerical examples.     

Canonical structures in the identification of multivariable systems

In this paper a unitary identification procedure for linear multivariable systems is described. The approach considered is based on a preliminary canonical structure identification, i.e. on the direct determination, from input-output sequences, of a set of invariant indexes completely describing the input-output structure of the system.Reduced computation time and storage, canonical representation of the identified model and easy estimation of the initial state of the system are in this way achieved. The results obtained from the application of this procedure to a simulated process and to a chemical reactor are also reported.     

Partial eigenstructure assignment and its application to largescale systems

The notion of partial eigenstructure assignment (PEA) via linear state feedback control in linear multivariable systems is introduced. This notion is a natural extension of eigenstructure assignment and partial eigenvalue assignment. Some theoretical basis for PEA is provided, and a parametric expression for feedback gain matrices achieving PEA is derived. An effective numerical algorithm for PEA tailored to large-scale systems is presented. As an extension of the algorithm, a recursive algorithm for eigenstructure assignment is presented. These algorithms possess the following desired properties: (1) compared to existing methods, the presented algorithms significantly reduce the required computation time via transforming high-dimensional matrix computations into low-dimensional matrix computations; (2) they can be implemented in a parallel fashion. The proposed algorithm for PEA is applied to modal control of large flexible space structure systems     

Dynamic controllers in linear multivariable systems

The theory of pole assignment in multivariable stationary linear systems using a controller of given order is reassessed. Novel proofs, both simpler and more general than hitherto are given, enabling efficient design algorithms to be formulated. Examples of such algorithms are worked out.     

Integrity of multivariable first-order-type systems

Recent results on the unity feedback control analysis of multivariable first-order-type systems are extended to provide necessary and sufficient conditions for integrity of the closed-loop system.