Plant Transfer Matrix Research Articles

A Novel Algorithm for Model-Plant Mismatch Detection for Model Predictive Controllers

For Model Predictive Controlled (MPC) applications, the quality of the plant model determines the quality of performance of the controller. Model Plant Mismatch (MPM), the discrepancies between the plant model and actual plant transfer matrix, can both improve or degrade performance, depending on the context in which performance is measured. In this paper, we do not use performance metrics or “yes-no”-type tests to merely diagnose the presence or absence of MPM in the plant matrix. Rather, we achieve the further goal of locating the exact MPM-affected elements within the plant matrix. Our proposed detection algorithm consists of two system identification experiments: the first experiment diagnoses the presence of MPM, and the second experiment pinpoints the exact MPM-affected elements. We then exercise the algorithm on artificial 3x3 and 5x5 plants suffering from sparse MPM, and demonstrate the algorithm's capability of correctly locating the MPM-affected entries.

A simple existence condition of one-degree-of-freedom block decoupling controllers

A block decoupling problem in linear multivariable systems is treated for one-degree-of-freedom controller configuration with unity output feedback. The plant transfer matrix, which may be non-square, is assumed to have unstable simple poles and zeros that may coincide. A simple existence condition of a block decoupling controller is obtained by directional interpolation approaches.

On general ℋ2 control: from frequency to time domain

This article presents a general formula for discrete-time ℋ2 control. It works with the regular and singular case of ℋ2 control, i.e. in the case of possibly non-left-invertible matrices G 12 and non-right-invertible matrices G 21, with possible unit circle invariant zeros. In the generic case, it can be simplified and adapted to work with the plant transfer matrix directly, without invoking the matrices of the parameterisation of stabilising controllers. A further result of this article is the presented necessary and sufficient conditions for state-space ℋ2 control, under only stabilisability and detectability assumptions. If the conditions are satisfied, an observer-based ℋ2 controller is constructed. The corresponding numerical algorithm consists of solving two discrete-time algebraic Riccati systems (DARSs) and two eigen-problems.

Structural Instability and Minimal Realizations

In this note, we consider the structural stability of an unstable plant, feedback stabilized by a controller. It is shown that infinitesimal parameter perturbations in the plant transfer matrix can destabilize the stable feedback control system obtained from a minimal realization of the ¿nominal plant.¿ The remedy for such structural instability consists in using a minimal realization in which the dimension of the ¿antistable part¿ is maximized over the uncertain parameter values. This requires knowledge of the ¿correct class¿ of parametrizations to which the nominal plant transfer matrix belongs.

A coprime factorisation and its application in ℋ2 control

Given a plant transfer matrix as a fraction of two polynomial matrices, we present a doubly coprime factorisation identity over the proper stable rational matrices of the plant transfer matrix, and apply it to compute a controller transfer matrix in ℋ2 optimal control of discrete-time systems. If there are no uncontrollable modes of the plant transfer matrix, the controller is given by a fraction of two polynomial matrices, where the order of the denominator is less than or equal to the McMillan degree of the plant.

Characteristic locus method robustness improvement through optimal static normalizing pre‐compensation

AbstractControlled systems designed by using the characteristic locus method can be very sensitive to small perturbations in the plant input and output at the frequencies where the plant transfer matrix is far from normal. In order to improve the closed‐loop system robustness, previous work proposes the design of a dynamic normalizing pre‐compensator followed by the design of a commutative controller for the pre‐compensated plant. The restriction on its structure and the need for approximation by a stable rational transfer matrix were among the limitations of the dynamic pre‐compensator. This paper shows that it suffices to design a static pre‐compensator that makes the pre‐compensated plant as closely as possible to a normal matrix in a frequency band containing the crossover frequency. The pre‐compensator is found by solving an optimization problem, whose solution is obtained directly by computing either a singular value decomposition of a real matrix or the spectral decomposition of a symmetric matrix, depending on whether normalization is to be achieved in one or more frequencies. Numerical examples illustrate the theoretical results of the paper. Copyright © 2009 John Wiley & Sons, Ltd.

Optimal steady-state control for linear non-right-invertible systems

The output regulation problem for an LTI plant whose outputs to be regulated are more than the control inputs is considered. For the case of constant references, a two-degree-of-freedom control scheme that minimises a quadratic cost function is presented. This cost function weights the tracking error at steady state. The methodology is based on the singular value decomposition of the plant transfer matrix at s=0. It is then shown how, by introducing some stronger assumptions, it is possible to modify the controller so as to reduce the steady-state control effort at the cost of a higher tracking error. A detailed numerical design example is finally presented.

ON ***H∞ CONTROLLERS WITH INTEGRAL ACTION: AN EXPERIMENTAL EVALUATION

This paper presents an experimental evaluation of two design methods for including integral action in ***H∞ controllers. There are two basic possibilities for doing that: (i) the specification of an integral weighting of the regulation error; or (ii) the direct inclusion of an integrator in series with the plant control input. The GS/T mixed sensitivity scheme has been employed, in order to avoid the generalized plant transfer matrix inversion. Alternative (i) is performed via an approximation that takes the generalized plant poles out of the imaginary axis (namely, the pole at origin). Such approximation is necessary to allow the Riccati equation to have a solution. An approximation procedure is proposed here, based on a linear mapping of the generalized plant model into another model close to the original one, and the inverse (approximate) mapping of the resulting controller. Alternative (ii) is implemented via an LMI (Linear Matrix Inequalities) algorithm. The experimental data suggests that the integral weighting specification leads to better results, possibly due to the lower order of the resulting controller.

On the asymptotically optimal tuning of robust controllers for systems in the CD-algebra

The authors previously (2000) showed that a low-gain controller of the form C/sub /spl epsiv//(s)=/spl Sigma//sub k=-n//sup n/ /spl epsiv/K/sub k//(s-i/spl omega//sub k/) is able to track and reject constant and sinusoidal reference and disturbance signal for a stable plant in the Callier-Desoer (CD) algebra. In this note, we investigate the optimal tuning of the matrix gains K/sub k/ of the controller C/sub /spl epsiv//(s) as the scalar gain /spl epsiv//spl darr/0. The cost function is the maximum error between the reference signal and the measured output signal over all frequencies and bounded reference and disturbance signal amplitudes. Closed forms for asymptotically globally optimal solutions are given. The optimal matrix gains K/sub k/ are expressed in terms of the values of the plant transfer matrix at the reference and disturbance signal frequencies. Thus the matrices K/sub k/ can be tuned with input-output measurements made from the open loop plant without knowledge of the plant model. Although the analysis is in the CD-algebra, to the authors' knowledge the main results are new even for finite-dimensional systems.

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.

Robust Control of LTI Square Mimo Plants Using Two Crone Control Design Approaches

A multi-SISO or a MIMO design approach is proposed for the robust control of uncertain LTI square MIMO plants, depending on the extent of the plant’s diagonal nature. For the multi-SISO approach, new multivariable frequency uncertainty domains are defined using the Gershgorin and the small gains theorems. These domains are based both on the structured frequency uncertainty domains related to the diagonal elements of the plant transfer matrix, and on the Gershgorin circles related to the corresponding column elements. Each element of the diagonal controller can be thus designed independently of the others using the SISO Crone design method. The totally MIMO approach is based on the optimization of an open-loop diagonal transfer matrix to minimize the stability margin variations of the perturbed diagonal elements of the closed-loop transfer matrix.

A normalizing precompensator for the design of effective and reliable commutative controllers

Sensitivity to parameter perturbation represents the main caution regarding the use of the characteristic locus method on the design of multivariable control systems. The method is not effective when the condition number of the plant eigenvector matrix is high or, equivalently, when the plant transfer matrix differs a great deal from normality. With the view to coping with this problem, it is proposed in this paper a precompensator structure and, in the sequel, two optimization problems are formulated and solved: the first one, for 2 x 2 systems, aims at minimizing the eigenvector matrix condition number; the second one, for the general m x m case, is intended to make the precompensated system as normal as possible by minimizing a defined measure of normality. Once the precompensated system matrix has been made close to a normal one, the characteristic locus method can then be applied effectively, leading to reliable control systems, as far as stability in face of uncertainty is concerned.

Rank-One Scaled ℋ∞ Optimization

This paper gives a solution to the scaled ℋ∞ optimization problem with constant scalings and output feedback. It is shown that, when the nominal plant transfer matrix has a special rank-one property, this optimization problem is equivalent to a sequence of convex optimization problems involving linear matrix inequalities. These results are demonstrated with a flight control example. The primary contribution of the example is a method for weight selection that is applicable to problems in which ℋ∞ optimization is used as the design tool.

A Wiener-Hopf approach to trade-offs between stability margin and performance in two- and three-degree-of-freedom multivariable control systems

Trade-offs between stability margin and performance are considered in two- and three-degree-of-freedom multivariable control systems using a Wiener-Hopf design approach. Maximum improvement in an approximate measure of stability margin is achieved at the expense of a prescribed increase in the quadratic cost functional measuring system performance. In order to attain an analytical solution to this fundamental trade-off problem, the approximate measure of stability margin chosen is also a quadratic cost function. A novel approach is introduced which allows structured perturbations in the coprime polynomial matrix fraction description of the plant transfer matrix to be taken into account. As a consequence, it is believed that the use of an approximate measure of stability margin is mitigated. Moreover, if needed, the solution obtained could serve as a very good initial one from which to search for better solutions iteratively.

Relay Tuning of Multivariable PI Controllers

A new method for identification of linear multivariable systems using relay feedback is presented. The procedure is a combination of sequential to simultaneous loop closing and relay experiments that are often used for tuning scalar systems. Once the plant transfer matrix is identified at a certain given frequency, Maciejowski technique is used to design multivariable PI controllers. A simulation example is given to illustrate both the identification and design method. Comparative results are shown to demonstrate that the new tuning method has the best set-point response performance. The results can be considered as a significant step towards autotuning of multivariable PI controllers.

Parametrization Issues of the Diagonal Stabilization Problem

In this paper parametrization issues for the problem of Diagonal Decentralized Stabilization, (DDSP), of linear, time invariant, MIMO systems, are addressed via the algebraic framework of Matrix Fractional Description, (MFD), of the system transfer matrices. It is shown that the DDSP is intimately related to plants that exhibit the property of cyclicity. The family of diagonal stabilizing controllers is parametrized via what is termed as "mode T mutually stabilizing pairs", with T taken from a family of matrices related to the plant transfer matrix. The above parametrization makes possible the determination of certain types of stabilizing controllers such as, proper, stable, reliable

A COMPLEMENT TO NIEDERLINSKI'S INDEX FOR MULTILOOP INTEGRAL CONTROL SYSTEMS

Niederlinski's theorem essentially provides a method for recognizing the lack of structural stabi-lizability in multiloop integral control systems. The purpose of this study is to complement Niederlinski's original theorem by relaxing some of the restrictions it encounters for its application. The fact that Niederlinski's index can be applied to the plants exhibiting time delays and unstable poles in their diagonal (controlled) elements is noted in this paper. It is also shown that Niederlinski's original assumption, that in order to apply the NI to the problem of integral instability all the elements of the loop transfer matrix must be strictly proper, can in fact be relaxed to the condition that all the triangular elements of the loop transfer matrix must be strictly proper. Thus even in a case where some elements of plant transfer matrix, and consequently of loop transfer matrix are not strictly proper, the NI still remains applicable if in fact the triangular elements of the loop transfer matrix are s...

Design of G-interactor and its application to direct multivariable adaptive control

In the design of multi-input multi-output (MIMO) adaptive control systems an interactor plays an important role. The derivation of the interactor requires system parameter values unless the interactor is in diagonal form. This is a considerable restriction when the interactor is applied to the adaptive control system design. In MIMO adaptive control many parameters must be estimated to achieve good tracking. Over-parametrization causes many problems, e.g. bad tracking performance and long calculation time of control input. A design of interactor, called a G-interactor, for unknown MIMO plants is given. It is applied to MIMO adaptive control systems. The G-interactor requires minimum information on the plant, such as the relative degree of each element of the plant transfer matrix. For this design a canonical form, called the G-canonical form, of a matrix A is investigated. The canonical form is also used to eliminate redundant parameters of MIMO adaptive control systems. Numerical examples show the effect...

An important property of non-minimum-phase multiple-input-multiple-output feedback systems

The non-minimum-phase (NMP( property is easily determined from the requirement that the plant input is bounded. In the single-input-single-output (SISO) system, a right-half-plane (RHP) plant zero at s = b constrains the system transfer function to have a zero at b. Also, the available feedback benefits are significantly restricted. The n × n multiple-input-multiple-output (MIMO) system is NMP if the plant determinant δhas any RHP zeros, say at plant transfer matrix and T = [tij is the closed-loop system transfer matrix. It has been thought that all n2tij (and the n2 plant disturbance response function rfj), must suffer from the NMP liability in their feedback properties. It is shown that only one row of need so suffer, with a any fixed integer in [1, n].The remaining n(n — 1) elements can be completely free of the NMP liability. A mathematically rigorous synthesis technique previously developed for MP systems is shown to be well suited for precise numerical design for such NMP MIMO plants with significan...

Direct Model Reference Adaptive Control for Linear Mul tivariable Systems

In this paper the direct adaptive Control of Multivariable discrete time minimum phase strictly proper systems is considered. The system. under consideration, is not suppose to be decouplable by state feedback. The adaptive control design presented here is based on pole-zero placement. A new parametrization of 1inear system models for use in adaptive contrl is presented. The main interest of this parametri zation lies in the fact that neither the prir knowledge of the plant transfer matrix Hermite form nor the knowledge of the transfer matrix infinite zero orders are necessary. Implementation of the scheme requires only knowledge of the plant transfer matrix largest infinite zero order and some appropriate polynomial degrees. Some simulations are presented.