Linear Multivariable Feedback Systems Research Articles

Logarithmic integrals, interpolation bounds, and performance limitations in MIMO feedback systems

We study performance limitation issues found in linear multivariable feedback systems. Our main contributions include Bode and Poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions. These results characterize and quantify explicitly how open-loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance. The role of time delay is also studied in this context. Most notably, we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros, and in particular, on how pole and zero directions are aligned. The latter is characterized by angles measuring the mutual orientation between zero and pole directions, and it is shown to play a crucial role in multivariable system design.

Properties of single input, two output feedback systems

We describe properties of linear multivariable feedback systems for which the plant has a single control input and two measured outputs. Particular attention is paid to 'algebraic' design tradeoffs that occur between different feedback properties at the same frequency. To describe these tradeoffs, we use concepts of plant and controller direction and alignment. We show that if plant/controller alignment is poor, then closed loop response to noise and disturbances will be large unless the loop gain is sufficiently small. Moreover, the closed loop system will exhibit large interactions. We also describe a dual set of results that are applicable to feedback systems for which the plant has two control inputs and a single measured output.

Multivariable gain-phase and sensitivity integral relations and design trade-offs

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.

Sensitivity integral relations and design trade-offs in linear multivariable feedback systems

The purpose of this paper is to develop integral relations regarding the singular values of the sensitivity function in linear multivariable feedback systems. The main utility of these integrals is that they can be used to quantify the fundamental limitations in feedback design which arise due to system characteristics such as open-loop unstable poles and nonminimum phase zeros and to such fundamental design requirements as stability and bandwidth constraints. We present extensions to both the classical Bode sensitivity integral relation and Poisson integral formula. These extended integral relations exhibit important insights toward trade-offs that must be performed between sensitivity reduction and sensitivity increase due to the aforementioned system characteristics and design constraints. Most importantly, these results display new phenomena concerning design limitations in multivariable systems which have no analog in single-input single-output systems. >

Uncertainty and interaction of linear multivariable feedback systems

The study of robustness properties of multivariable systems in the presence of uncertainty has received a lot of attention in the recent literature. However, despite the wealth of results that are now available on robust stability/tracking and tolerance to uncertainty, no algorithms have been proposed for the assessment of the interactive properties of multivariable systems. This forms the main aim of the present paper.

On practical stability of linear multivariable feedback systems with time-delays

In this paper, practical stability properties of linear multivariable feedback systems with time-delays are studied. The control schemes considered are conventional feedback control and Smith Predictor control. Depending upon the known perturbation structures, tight conditions are given which guarantee practical stability of the control system.

New measures for robustness in linear multivariable feedback systems

AbstractNew measures for robustness are proposed for linear multivariable control systems. A new theorem is presented to account for the controllability of systems that contain an integrating element, in the presence of additive uncertainties. The theorem leads to the definition of a sufficient condition for robust integral controllability: robustness margin (RM). The ratio of robustness margins for competing control or processs structures can be used to discriminate between these structures. These measures are useful, especially in design stages, since their calculation depends only on steady state information. Applications to several published distillation systems show the merits of these robustness measures.

Robust multivariable feedback design

A primary role of feedback design is to enable a system to cope with the inevitable uncertainly associated with its description. Although it is often the case that we have at hand a practical description of both the size and structure of the expected uncertainty, simple multivariable design strategies which effectively incorporate this information have been slow to appear. A rigorous development of the uncertainly problem for linear multivariable feedback systems is presented utilizing the characteristic locus framework, and then a simple geometric alignment property is exploited in order to formulate a systematic design strategy, focusing on a PI implementation for the resulting controllers

On the design of linear multivariable feedback systems via constrained nondifferentiable optimization in H/sup infinity / spaces

The design discussed is of linear, lumped, time-invariant, multivariable feedback systems, subject to various frequency and time-domain performance specifications. The approach is based on the use of stabilizing controller parametrizations which result in the formulation of feedback system design problems as convex, nondifferentiable optimization problems. These problems are solvable by recently developed nondifferentiable optimization algorithms for the constrained minimization of regular, uniformly locally Lipschitz continuous functions in R/sup N/.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Integrity Against Arbitary Feedback-loop Failure in Linear Multivariable Control Systems

This paper is concerned with the integrity problem in linear multivariable feedback systems. System integrity is defined as the property such that the closed-loop system remains stable against arbitrary feedback-loop failures. By introducing a class of matrices, called U-motrix, necessary and sufficient conditions for integrity are derived with a stable plant. The results show that integrity is ensured if and only if all principal minors of the return difference matrix or the sensitivity matrix are units of the set of proper stable rational functions, as well as the controller being stable. Moreover, the set of all proper controllers ensuring integrity is also characterized. Based on this characterization, a method of synthesizing a controller which ensures integrity, and attains desirable input-output responses with decoupling, is proposed.

The LQG/LTR procedure for multivariable feedback control design

This paper provides a tutorial overview of the LQG/LTR design procedure for linear multivariable feedback systems. LQG/LTR is interpreted as the solution of a specific weighted H2-tradeoff between transfer functions in the frequency domain. Properties of this solution are examined for both minimum-phase and nonminimum-phase systems. This leads to a formal weight augmentation procedure for the minimum-phase case which permits essentially arbitrary specification of system sensitivity functions in terms of the weights. While such arbitrary specifications are not possible for nonminimum-phase problems, a direct relationship between weights and sensitivities is developed for nonminimum-phase SISO and certain nonminimum-phase MIMO cases which guides the weight selection process.

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Necessary and sufficient stability conditions for the case of dependent additive perturbations

In this paper we examine the stability properties of linear multivariable feedback systems in the presence of structured additive perturbations characterized by dependencies occurring in the perturbation matrix. Such dependencies may take the form of dependent phases between elements of element-by-element bounded perturbations and/or repeated blocks in block-structured perturbations. By using non-diagonal and possibly complex transformation matrices, it is shown how stability conditions that are both necessary and sufficient may be obtained for this class of perturbation.

Normality, scaling and eigenvalue inclusion regions

One approach to establishing eigenvalue inclusion regions for robust stability analysis of uncertain linear multivariable feedback systems involves replacing the nominal plant matrix by an approximating normal one. This approach incurs additional uncertainty but has the advantage that the resulting regions are insensitive to small perturbations. These regions are obviously conservative, and it is shown in this paper how this conservatism may be reduced by utilizing the eigenvalue preserving operation of similarity scaling

Singular-value and eigenvalue techniques in the analysis of systems with structured perturbations

A new range of techniques is introduced for the purpose of exploiting information about the structure of the perturbation associated with uncertain linear multi-variable feedback systems. The class of perturbation considered is wide and ignores only the phase information of the individual elements of the perturbation matrix; it encompasses diagonal, block-diagonal and other forms of block-structured perturbations. The techniques developed provide a means of assessing stability margins and also lead to an extension of the Nyquist criterion.

A CAD methodology for linear multivariable feedback systems based on algebraic theory

A CAD methodology for linear multivariable feedback systems based on algebraic theory

Design of multivariable feedback systems with simple unstable plant

This paper proposes a design methodology for distributed linear multivariable feedback systems with simple unstable plants (a simple unstable plant has either first- or second-order unstable poles). The methodology developed provides a global characterization of all realizable compensators which stabilize a given simple unstable plant. A design example is given to show that this methodology can be used to generate, in an appropriate computer-aided design environment, controllers which are optimal with respect to designer-specified criteria. Additionally, it is shown that the nature of the design methodology gives geometric insight into the dynamics of the process whereby an unstable plant is stabilized.

Algebraic theory 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.

An Analytic Stability Test for Linear Multivariable Feedback Systems with a Time Delay

The purpose of this paper is to apply an algebraic test such as the Routh algoritha, Hurwitz's determinant or any alike method to a linear multivariable feedback system with a time delay. A closed-loop characteristics function of such a system is truncated according to Kouché's theorem to determine the range of adjustable parameter k of a controller for which the truncated system and the system with a time Celay are both to be stable. An example is conosidered to illustrate the development.

Analysis and design of linear multivariable feedback systems in the presence of additive perturbations

The paper introduces the concept of normal approximations to the analysis of uncertain linear multivariable feedback systems, in order to derive a new criterion for the stability of plants subject to additive perturbations. This criterion permits a complete assessment of stability margins in that it provides a measure of tolerance to uncertainty as well as providing a measure of gain/phase margins. The proposed approach leads to a systematic design procedure which minimizes the effects of uncertainty by attaining the condition of normality or which improves the stability margins by robust gain/phase sharing. The results presented in this paper are insensitive to small perturbations on the nominal plant model and provide the means for a consideration of information concerning the structure of uncertainty.