Extended root-locus technique for design of linear multivariable feedback systems

This paper presents an algebraic theory for analysis and design of linear multivariable feedback systems. The theory is developed in an algebraic setting sufficiently general to include, as special cases, continuous and discrete time systems, both lumped and distributed. Designs are implemented by construction of a controller with two vector inputs and one vector output. Use of controllers of this type is shown to generate convenient stability results, and convenient global parametrizations of all I/O maps and all disturbance-to-output maps achievable, for a given plant, by a stabilizing compensator. These parametrizations are then used to show that any such I/O map and any such disturbance-to-output map may be simultaneously realized by choice of an appropriate controller. In the special case of lumped systems, it is shown that the design theory. can be reduced to manipulations involving polynomial matrices only. The resulting design procedure is thus shown to be more efficient computationally. Finally, the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting. It is shown that the classical results on asymptotic tracking can be generalized to this setting. Additionally, sufficient conditions for robustness of asymptotic tracking, and robustness of stability are developed.

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.

An analytical method is described for the design of linear time-invariant SISO feedback systems to meet specified performance criteria in the presence of structured and unstructured uncertainty. The design framework also provides for control over the degree of the compensator.

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 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.

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 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.

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).

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.

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.

This paper presents a pole placement method for designing linear time-invariant SISO feedback systems. The idea of this proposed method is based on the fact that the internal signals of a feedback control system can be eliminated by performing row operations on a governing matrix. After choosing proper closed-loop poles, the new design procedure can be implemented by solving a Diophantine equation easily and insightfully. An example extracted from literatures is conducted to validate the effectiveness of the method in the single-input single-output (SISO) cases. Based on the analytical and operator-based design procedure of the proposed method, extending this new approach to the control of linear multi-input multi-output (MIMO) systems is possible.

This paper gives bounds for the inverse transfer function.between the jth input-output pair in an n × n multivariable feedback system when the jth loop is open and all other loops k≠ j are closed. Choice of the minimal such bounds are also studied as well as the effect of feedback gains on these bounds. These results are useful in the design of linear multi variable systems.

SOME UNIFYING CONCEPTS IN MULTIVARIABLE FEEDBACK DESIGN

A general procedure for the design of multivariable feedback systems is presented. The system is first rendered weakly coupled by a suitable choice of some controller ratios and then designed by conventional single-loop techniques. One major characteristic of the method is that various degree of coupling of the system can be obtained, dependent on the complexity of the controller ratios chosen. Furthermore, a criterion is given to decide when the system can be regarded as weakly coupled, such that single-loop design techniques can be applied. The method gives the designer more freedom in the design and provides him with many more choices for a suitable controller. The effectiveness of the method is demonstrated by an application to a 4 x 4 boiler furnace example.