Non-unity Feedback Research Articles

Necessary and Sufficient Conditions for the Existence of Decoupling Controllers in the Generalized Plant Model

Necessary and sufficient conditions for the existence of diagonal, block-diagonal, and triangular decoupling controllers in linear multivariable systems for the most general setting are presented. The plant model in this study is sufficiently general to accommodate non-square plant and non-unity feedback cases with one-degree-of-freedom (1DOF) or two-degree-of-freedom (2DOF) controller configuration. The existence condition is described in terms of rank conditions on the coefficient matrices in partial fraction expansions.

Verifying and Simulating of the Novel Analytical Method of the Steady-State Error for Nonunity Feedback Control Systems

Traditionally, the first step to analyze the steady-state error of nonunity feedback control system is to convert the system into an equivalent unity feedback control system, and calculate the steady-state error by using the concept of type number for unity feedback control system. In 2011, Mou is based on the concept of old type number and the definition of new steady-state error functione(t)=(1/KH)r(t)-y(t) by Kuo to offer the method for calculating the steady-state error of nonunity feedback control system. In this paper, three typical examples will be analyzed by the calculation of step response and by the simulation of Matlab. Therefore, we can prove that the definition of new steady-state error function e(t)=(1/KH)r(t)-y(t) by Kuo and the concept of old type number are useful to solve the steady-state error of nonunity feedback control system.

H 2 design of one-degree-of-freedom decoupling controllers for square plants

Decoupling design of one-degree-of-freedom controllers is treated for the generalised plant model within the H 2 framework. The class of all realisable closed loop transfer matrices is parametrised under a mild assumption on non-unity feedback sensors. A set of compact assumptions is presented to guarantee the existence of the optimal H 2 controller. Together with the optimal one, the class of all H 2 controllers that yield finite H 2 cost is obtained. The optimal closed transfer matrix and its associated controller are shown to be strictly proper under the reasonable order assumptions on the generalised plant.

Wiener?Hopf design of optimal decoupling one-degree-of-freedom controllers for plants with rectangular transfer matrices

This paper is a sequel to an earlier one which treated the design of optimal decoupling one-degree-of-freedom stabilizing multivariable controllers for plants with square transfer matrices. Here designs for plants with rectangular transfer matrices are given which allow for feedforward compensation and more generality in the specification of the desired closed-loop transfer matrix. As in the earlier work, all controllers are placed in the forward path of the feedback loop and non-unity feedback is permitted. The criterion for optimality is a quadratic-cost functional that penalizes both tracking error and saturation. Explicit formulas are derived which give the set of all those controllers that yield finite cost, as well as the ones that are optimal. It is shown that these controllers are strictly-proper under conditions usually prevailing in practice. The solution for plants with rectangular transfer matrices is expressed in terms of both Schur and Kronecker matrix products. When the plant transfer matrix is square, the solution reduces to the one obtained in the earlier work and involves only Schur matrix products.

Wiener-Hopf design of optimal decoupling one-degree-of-freedom controllers

This reasonably self-contained paper is directed to the design of a multivariable optimal one-degree-of-freedom feedback loop which incorporates a decoupling controller in the forward path. The criterion for optimality is a quadratic-cost functional that penalizes both tracking error and saturation. The controllers on which optimization is based are general enough to allow for non-unity feedback and rectangular plant transfer matrices possessing normal row rank. Nevertheless, for the sake of brevity and clarity, attention is focused mainly on the square case. Earlier treatments of the problem have employed multi-degree-of-freedom controllers. The solution we present for the one-degree-of-freedom case is considerably more difficult to obtain, especially when saturation is taken into account. Explicit formulas are derived for the set of all decoupling controllers yielding finite cost, as well as those that are optimal. It is shown that these controllers are strictly-proper under conditions usually prevailing in practice. Four fully worked examples serve to illustrate many important numerical aspects of the theory and all major proofs are transferred to the Appendix.

Error coefficient and nonunity feedback revisited

A review of the concepts of type, error coefficient, and steady-state error is presented from the point of view of usefulness in applications and relevance in an introductory control systems course. The approach employed makes a clear distinction between system error and loop error. It is shown that categorizing feedback control systems as unity or nonunity is of limited practical value. Error constants are seen to be valid for both nonunity and unity feedback but are useful only in "simple" systems. A system classification scheme based on this definition of "simplicity" is suggested.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Unity and non-unity feedback: a comparison†

Abstract The paper compares the unity feedback with the non-unity feedback servo problem from the point of view of sensitivity and the minimum order of the realization. It is proved that the non-unity feedback solution has smaller sensitivity ‘on the band of interest’ and that the ‘dimension’ (defined in a precise sense) of the non-unity feedback solution is never smaller than that of the unity feedback solution.

The Linear Servomechanism with Non-Unity Feedback

The paper studies the robust linear time-invariant servomechanism problem with non-unity feedback, using the transfer function set-up and the polynomial matrix factorization method. It is shown that in the scalar case, unity feedback is indeed not necessary for a robust solution with respect to "arbitrary" perturbations of both plant and loop stabilizer. In the multivariable problem however unity feedback becomes necessary when every mode of the signal to be tracked is present in each of its components and when there exists no sensor such that the loop is stabilizable by a stable loop-stabilizer.

On Error Evaluation and Non-Unity Feedback

The concept of error as applied to feedback systems is considered. Error constants and coefficients are developed for application to systems with non-unity feedback. Non-unity scalar feedback factors are considered as well as feedback functions of s.

Comments on "Linear systems and nonunity feedback"

In the above paper generalized error coefficients were defined with respect to an equivalent transfer function for nonunity feedback systems. The inadequacy of the accepted definition of error in a particular situation and the distinction between the actuating error and system error in view of the nonunity feedback are brought to the fore in this correspondence. The most general definition of error is also pointed out.

Linear systems and nonunity feedback

A straightforward extension of the error coefficient concept to the evaluation of linear systems with nonunity feedback is presented. Examples show the effects of several types of nonunity feedback.

Necessary and sufficient conditions stability for&amp;lt;tex&amp;gt;n&amp;lt;/tex&amp;gt;-input,&amp;lt;tex&amp;gt;n&amp;lt;/tex&amp;gt;-output convolution feedback systems with a finite number of unstable poles

We consider n-input n-output convolution feedback systems characterized by y = G * e and e = u - y, where the open-loop transfer function ? contains a finite number of unstable poles and the expansion ?a(s) + ?i=0 ? Gi exp(-sti). The latter is such that, i) ?a is the Laplace transform of L1 functions, ii) t0 = 0, iii) the delay-times ti, for i = 1, 2, ..., are positive and iv) the coefficient matrices Gi form an absolutely convergent series, i.e. ?i=0 ? |Gi| 0, and the second is required to prevent the closed-loop transfer function from being unbounded in some small neighborhood of each open-loop unstable pole. The latter condition has a nice interpretation in terms of McMillan degree theory. The modification of the theorem for the discrete-time case is immediate. The same is true if, instead of unity feedback, constant nonunity feedback represented by a nonsingular matrix is used. This allows us, through the use of the small gain theorem or the contraction mapping theorem, to consider non-linear systems, derived from the original one by introducing a non-linear time-varying gain in the forward loop.

Comment on ‘Some Observations concerning Response Calculations for Non-unity Feedback Systems’

Comment on ‘Some Observations concerning Response Calculations for Non-unity Feedback Systems’

Some Observations concerning Response Calculations for Non-unity Feedback Systems

Some Observations concerning Response Calculations for Non-unity Feedback Systems

A deterministic study of delta modulation

Delta modulation has often been viewed as a type of PCM and has been analyzed by probabilistic methods. This paper views delta modulation as a hybrid PDM/PAM system. A nonlinear feedback model is constructed and subjected to digital computation on an IBM 650. Computation is enabled by trapezoidal convolution, an approximate Z -transform method for solving dynamical systems. Responses to various steps and ramp inputs for unity and nonunity feedback are considered. Dead space is also introduced into the nonlinearity in the forward link. A series of computer runs is condensed and observations are reported. A typical observation is that a trade-off exists between average error and noise and that both may not be minimized by any one of two versions of the delta modulator.