Algebraic theory of linear multivariable feedback systems

The root-locus-approximation procedure is extended to provide for the complex-plane analysis of equations which contain a term quadratic in the variable parameter in addition to the usual linear term. Such an extension will have many applications; in the paper, a discussion is given concerning the design of linear time-invariant multivariable feedback systems. It is shown that, particularly for 2-input/2-output problems, where the algebra involved in obtaining the equations to be analysed remains fairly simple, much useful information is available by straightforward hand calculation. Brief suggestions are given for further generalisations of the techniques (inclusion of a cubic gain term etc.)

In this paper a basis is developed for the systematic design of linear multivariable discrete-time tracking systems for plants with inaccessible states, which thus extends the results for discrete-time plants with accessible states previously obtained by Bradshaw and Porter (1975). The design method is illustrated by the presentation of the results of computer simulation studies.

A comprehensive survey is given of recent work on the use of matrix-valued functions of a complex variable for the analysis and design of linear multivariable feedback systems. Several emerging design techniques are outlined and a simple illustrative design example given showing how such methods may be used in practice.

In this paper a basis is developed for the systematic design of linear multivariable continuous-time tracking systems for plants with inaccessible states which thus extends the results for continuous-time plants with accessible states previously obtained by Porter and Bradshaw (1974). The design method is illustrated by the presentation of the results of computer simulation studies.

A new designing method of multivariable linear time-varying tracking systems with stochastic disturbances for polynomial command inputs is proposed. It has been shown that it is possible to choose the matrices of gain elements placed in feedbacks and parallel paths of the closed-loop system so that the output y(t) tracks the polynomial command input v(t) with (E expectation operator).

A design methodology for linear multivariable feedback systems is formulated which utilizes modern algebraic control theory and does not require manipulation of polynomial or rational matrices. The methodology uses direct measurements of the plant in obtaining designs.

Artificial neural networks, which hold considerable potential for recognising and classifying spatial and temporal patterns, have been used as an efficient method for automatic traffic surveillance, which is an important research topic of intelligent transport systems. An important element in the performance of the neural networks is the composition of the input and output vectors, as well as the network architecture. However, there has been little research on the performance in relation to the attributes of the input and output vectors. In this research, various input vector properties were applied to the backpropagation model, which is the most popular neural network model, to see how the general performance would be affected by the different types of input vector. Experiments were performed with the inclusion of various grey levels, image sizes, edge detection images and combinations of edge and pixel grey information as the input vectors. The experimental results showed that the network performance, in terms of computing cost for training and prediction accuracy, was highly dependent on the characteristics of the input vectors. Two combined input vectors, the grey scale pixels and edge detection image, produced better prediction performance than either the grey values on the pixel or edges alone.

We present several extensions of the classical Bode integral relations to multivariable systems. The main contributions are an extended gain-phase integral formula and an extended sensitivity integral relation. These results are useful for analyzing fundamental design limitations that arise in linear multivariable feedback systems. In particular, they provide important insights into feedback properties which are unique to multivariable systems and which cannot be inferred from the previously known integral relations applicable to single-input/single-output systems. Both results show that design limitations in multivariable systems depend strongly upon certain directionality properties.

In this paper, the class of linear multivariable continuous-time plants which are amenable to high-gain error-actuated control is characterized in terms of the relatively prime polynomial matrix factors of the transfer function matrices of such plants. It is shown that this characterization greatly facilitates the design of high-gain error-actuated controllers, and the resulting synthesis technique is illustrated by designing such a controller for a third-order plant with two inputs and two outputs.

An interactor plays a basic role in the design problems of multivariable control systems, such as the model matching control, decoupling control, adaptive control, and disturbance decoupling control. For example, it is well known that a linear multivariable system is decouplable by the static state feedback if and only if it has a diagonal interactor. But, in general, the interactor is a lower left triangular matrix, which makes the design problems of multivariable control systems complicated. From this point of view, the precompensator such that the total system has a diagonal interactor is important for multivariable control systems to simplify their design procedure. This paper concerns the systematic design method of the precompensator for a diagonal interactor using Markov parameters of the plant. It will also be proved that this compensator is always obtained by the proposed design algorithm. Furthermore, this argument will be a basic idea when this technique is applied to the nonlinear multivariable systems. © 2008 Wiley Periodicals, Inc. Electr Eng Jpn, 163(1): 65–72, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20599

In the transform analysis of linear systems the two classical approaches use complex functions to study gain as a function of frequency (the Nyquist—Bode approach)and to study frequency as a function of gain (the Evans root locus approach)The central feature of any transform approach to the analysis of linear multivariable systems is the use of transfer-function matrices, and it. is the primary purpose of this paper to explain how the ideas of studying gain as a function of frequency, and frequency as a function of gain, can be extended to the multivariable case by associating with transfer-function matrices characteristic gain functions and characteristic frequency functions The characteristic gain function and characteristic frequency function are algebraic functions, each of which is defined on an appropriate Riemann surface. Considering the Riemann surface domain for the characteristic frequency function, a stability criterion for linear multivariable feedback systems is established which doe...

The paper is concerned with the derivation of useful results that complement the linearisation-based analysis and design of multivariable systems with measurement nonlinearity. The approach is based on the closed-loop linearised system descriptions and certain nonlinearity related parameters representing sector and/or incremental sector properties, and also enables the consideration of nonlinearities with discontinuities at the origin. Nonlinear-system stability and performance is assessed on the basis of the stability and performance of the corresponding linearised system and certain easily computable sufficient conditions representing BIBO and asymptotic stability conditions, as well as absolute and derivative performance bounds. The results depend on useful linearised system descriptions such as the L∞ operator norm, thus offering the advantages of meaningful assessments and overall computational simplicity.

Robust LQG optimal controller design for multivariable discrete saturating systems with noise spectral uncertainties and nonlinear time-varying unmodeled dynamics

A systematic design method of assigning a required time-scale structure to a given multivariable system is developed. The time-scale structure assignment is done by high-gain feedback of either state or output. The method decomposes a given multivariable system into several single-input, single-output (SISO) subsystems each of which is designed separately. A standard method of design for SISO system is developed, and this forms a building block for multivariable system design. The state-feedback design depends only on the stage sequence of the given system, while the output feedback design depends only on the set of integers which specifies the order of infinite zeros. In this way the design is made robust with respect to certain parameter uncertainties. >

A systematic design method for assigning a required time-scale structure to a given multivariable system is developed. The time-scale structure assignment is carried out using high-gain feedback of either the state or the output. The method developed decomposes a given multivariable system into several single-input single-output subsystems, each of which is designed separately. A standard method of design for single-input single-output systems is developed, and this forms a building block for the multivariable system design. The state feedback design depends only on the stage sequence of the given system, while the output feedback design depends only on the set of integers that specifies the order of infinite zeros. In this way, our design is made robust with respect to certain parameter uncertainties. High-gain parameter(s) can be adjusted on line, and this provides a tool for the fine tuning of the desired performance characteristics of a control system.