Open-loop Transfer Matrix Research Articles

Multivariable Fractional-Order PID Tuning by Iterative Non-Smooth Static-Dynamic H∞ Synthesis

Abstract This paper proposes a new method for tuning the parameters of multi-input multi-output (MIMO) fractional-order PID (FOPID) controller. The aim of the proposed method is to calculate the parameters of this controller such that the rise time and steady-state errors of the feedback system are minimized without violating the predetermined stability margins. Mathematically, this problem is formulated as maximizing the spectral norm of the open-loop transfer matrix at zero frequency subject to a constraint on the H∞ -norm of the sensitivity function. This problem is nonlinear in parameters of the MIMO FOPID, which can be solved using the iterative algorithm developed in this paper based on non-smooth H∞ synthesis.

Control Design and Stability Analysis of Power Converters: The MIMO Generalized Bode Criterion

Three-phase dynamic systems and multiphase generators are frequently modeled and controlled in the synchronous reference frame. To properly model the cross-coupling terms in this reference frame, complex vector theory and transfer function matrices are commonly applied, obtaining multiple-input multiple-output (MIMO) dynamic models. The stability of MIMO systems can be assessed through the Nyquist generalized stability criterion. However, the use of the Nyquist diagram complicates the controller design. The Bode diagram is a more intuitive tool for the controller design; however, the Bode stability criterion is not applicable to MIMO systems. In this article, the MIMO generalized Bode criterion is proposed. Since this stability criterion is based on the Nyquist generalized stability criterion, it can be applied to any system. Furthermore, it is simple to use, as it only requires information contained in the open-loop transfer matrix and the Bode diagram. The proposed stability criterion thus offers an interesting tool for the controller design procedure in MIMO systems, as it is shown in this article for two common applications: the current control loop of a power converter, a 2 × 2 system, and the current control loop of two independent power converters in parallel, a 4 × 4 system.

Robust stabilisation of coprime factor plant description subject to plant perturbations

This paper presents a new procedure for the synthesis of linear multivariable system subjected to plant perturbations and external disturbances. Coprime factorisation approach is used to model the plant perturbations. External disturbances is treated as exogenous system whose unstable modes are assumed to be disjoint from the system zeros. A stable desired closed loop transfer matrix is obtained which can be expressed as the product of an open loop transfer matrix and any proper rational transfer matrix. Internal model principle is employed to realise the closed loop transfer matrix which simultaneously ensures: (1) stability robustness in the presence of plant perturbations, (2) decoupling, (3) complete and arbitrary closed loop pole placement, (4) internal stability and (5) disturbance rejection. The compensation scheme is presented in the frequency domain. This is compared with H-infinity control technique and the simulation result shows better robust performance for the frequency domain method.

Robust control design for multivariable plants with time-delays

Abstract This paper proposes an extension of CRONE control and design for unstable, multiple time-delay multivariable plants with right half-plane (RHP) zeros. The CRONE control approach developed for multivariable square plants is based on third generation scalar CRONE methodology. The aim is to find a decoupling and stabilizing controller for the open-loop transfer matrix. Fractional order transfer functions are used to define all the components of the diagonal open-loop transfer matrix, β 0 i . Optimization gives the best fractional open-loop transfer matrix, β . Finally, frequency-domain system identification is used to find a robust controller K = G −1 β . When the plant is inversed, time-delays and RHP poles or zeros can appear on the denominator of the transfer matrix, G −1 . To achieve a stable controller some time-delays, RHP zeros and unstable poles must be included within the diagonal transfer function of the open-loop transfer matrix. To assess the proposed design, the CRONE control approach is applied to a distillation example.

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.

Robust Integral Stabilization and Regulation of Uncertain Multivariable Systems

The recent concept of multivariable integral controllability (MIC) is generalized to uncertain multivariable systems. It is shown that MIC is strongly dependent on the all gain positive stability (AGPS) of the steady state open loop transfer matrix. Some sufficient conditions for AGPS are provided, from which a design scheme for inducing MIC on any given system can be determined.

The Open-Loop Transfer Matrix of an Estimator-Controller

Open-loop transfer matrices of continuous and sampled-data estimator-controllers (including those in predictor-corrector form) are derived. This paper complements an earlier paper in which open-and closed-loop transfer matrices of estimators and closed-loop transfer matrices of estimator-controllers were derived. Three different open-loop transfer matrices of each type of estimator-controller are derived by breaking the loop at different points. An example is included to illustrate the utility of such an open-loop transfer matrix.

Multivariable Control of Industrial Fractionators

The multivariable Nyquist Array method offers a concept which enables the classical singie-iiiput/s iugle-uutput Nyquist control dcoign methode to ba extended to multivariable systems. This concept is based on partial decoupling using a compensator net-work to achieve what is known as a “diagonal-dominant” structure, whereby single-loop controllers can be designed independently for the different loops. The method described here offers an approach by which a compensator network is designed automatically, a function minimization algorithm being used to obtain dominance over a specified frequency range by minimizing the ratio of moduli of the off-diagonal terms to the diagonal terms of the appropriate open-loop transfer matrix. Once a suitable compensa-tor has been found, the design of an overall control system is completed by designing single-loop controllers for the different loops separately. The applicability of the method is demonstrated, with the aid of a dynamic model of a high-vacuum distillation unit, by the design of the viscosity control configuration for that unit. Further, attention has been paid to incorporating such a control configuration into modern, distributed-control systems such is Foxboro Spectrum.

Stability enhancement for control of flexible space structures

This paper demonstrates the application of linear-quadratic-Gaussian/ loop transfer recovery (LQG/LTR) techniques to the active control of a flexible spaceborne optical support structure. LQG compensators are designed for three actuator/ sensor placements on the structure in the context of an 18-mode reduced-order structural model. The state-feedback (LQ) portion of each design meets stringent line-of-sight performance requirements in the face of two wide-band disturbances impinging on the structure. Conditions on the open-loop transfer matrix, broken at the plant input, are stated which make more robust the response of the LQG designs to the 132 high-frequency elastic modes excluded from the design model. Through loop transfer recovery, with LQ dynamics as the goal, closed-loop stability in the presence of all 150 elastic modes is maintained for two of the actuator/sensor selections. Difficulties in achieving performance robustness are noted.

Nonconservative evaluation of uniform stability margins of multivariable feedback systems

This paper discusses concepts of stability margins of multivariable feedback systems. Independent and uniform stability margins are defined. A previous conjecture that the uniform margins may be computed by using the eigenvalue magnitudes instead of the singular values in the robust stability criteria is theorized. The nonconservatism provided by this theory in the evaluation of uniform margins is discussed, along with limitations of the uniform margins. Also presented is a method of using the uniform margins to extend the region of stability beyond what can be specified by singular values. Results are demonstrated numerically in an example of a lateral attitude control system for a drone aircraft. attempt to relax the conservatism in the evaluation of stability margins of a two-input two-output lateral attitude control system of a drone aircraft, Mukhopadhyay and Newsom5 experimented with using the magnitudes of the eigenvalues instead of the singular values in the robustness criterion. By examining the Nyquist plot of the eigenvalues of the return difference matrix of the control system in their study, they conjectured that the eigenvalue-based gain (or phase) margins are limits within which the gains (or phases) of all feedback loops vary uniformly without destabilizing the feedback system while the phase angles (or gains) remain at their nominal values. In fact, since the spectral radius (maximum of the modulii of the eigenvalues) of a matrix is the greatest lower bound of all norms of that matrix, the con- jecture of Ref. 5 is the least conservative for the evaluation of the uniform stability margins by means of norm-bounded robust stability criteria. The uniform variations of multiloop gains and phases are also interesting in that the uniformity constraints give the multiloop variations a single-variabl e nature. Hence, the regions of stability in the gain and phase spaces degenerate into line segments. At each stable nominal operating point in the gain and phase spaces one such line segment may be constructed. In this fashion, the uniform gain and phase margins facilitate a nonconservativ e but discrete represen- tation of the regions of stability in multidimensional gain and phase spaces. ! In this paper, the conjecture given in Ref. 5 is proved. Uniformity in the gain and phase variations in feedback loops may be viewed as a special structure in the perturbation of an open-loop transfer matrix. Therefore, weighted £; and 4, norms3 are used in the norm-bounded stability robustness criteria to derive the formulas from which the uniform stability margins are computed. The concept and the one- dimensional characteristics of the uniform stability margins and their use in discretizing the regions of stability in multidimensional gain and phase spaces are demonstrated.

Multiple gain parameter multivariable root locus

The multivariable root locus concept is generalized here by allowing for independently variable gain parameters in the different feedback paths. Sufficient conditions are found under which the matrix of Markov parameters of the open loop transfer matrix G( s) will guarantee stability for every gain matrix in the gain space. This generalizes the recent work of Byrnes (1981) and Cook and Hall (1982), and is useful in the design of proportional plus integral controllers for multivariable feedback control systems.

Optimal systems root loci: Relation to the McMillan structure of the open-loop system

This note shows that the orders of the Butterworth patterns formed by the asymptotes of optimal closed-loop poles of a time-invariant linear feedback system, as the weight of the input in the performance criterion approaches zero, always correspond to twice the McMillan orders of the zeros at infinity of the open-loop transfer matrix. Furthermore, bounds are given, in terms of the Newton polygons, on the orders of the Butterworth patterns formed by the root loci at finite pole-zero locations. Finally, some examples are presented to show that in the latter case the branching behavior need not always correspond to the McMillan structure of the open-loop system.

Diagonal dominance for multivariable nyquist array methods using function minimization

A new method of achieving diagonal dominance for Nyquist array design methods is presented. The technique utilizes a conjugate direction function minimization algorithm to obtain dominance over a specified frequency range by minimizing the ratio of the moduli of the off-diagonal terms to the moduli of the diagonal terms of the appropriate open loop transfer matrix. The new dominance algorithm is easily implemented in either a batch or interactive computer mode and will yield dominant compensators when alternative methods fail. Several concepts new to the Nyquist array methods are also presented. The proposed method is applied to the control system design for a sixteenth order state model of the Pratt-Whitney F-100 turbofan engine with three inputs.

Multivariable system synthesis with step disturbance rejection

The question of achieving a variety of specific design goals (e.g., arbitrary pole placement, decoupling, and exact model matching) for linear multivariable systems subjected to unknown step disturbances on the entire internal state and output is constructively resolved via a synthesis procedure which simultaneously combines integral feedforward compensation with asymptotic state estimation. The technique relies on the ability to express the desired closed-loop transfer matrix of the compensated system as the product of the open-loop transfer matrix and any proper rational transfer matrix, assuming the given system has no zeros at the origin and a stable closed-loop design is sought. The method employs some recent results involving the differential operator approach to multivariable system analysis and synthesis, and the treatment is confined almost exclusively to the frequency domain.

The use of transfer matrices in the analysis of conditions of oscillation

The condition of oscillation of a feedback oscillator in terms of its open loop transfer matrix is deduced. The method of determining the open loop transfer matrix for a feedback oscillator is demonstrated by considering illustrative examples, and the frequencies of oscillation and the starting conditions of oscillation are deduced for some of these examples.

Use of matrix transformation and system eigenvalues in the design of linear multivariable control systems

A method of using matrix transformations to obtain a diagonal open-loop transfer matrix and its application to the design of linear multivariable control systems are presented. Numerous examples are used to illustrate the usefulness of the design procedure, and a justification for it is presented by a discussion of system eigenvalues.