Multivariable quantitative feedback design for tracking error specifications

The use of tracking error specifications in quantitative feedback theory (QFT) design is discussed for multi-input, multi-output (MIMO) systems. These specifications bound the closed loop transfer function within a disk around some nominal (model) performance while preserving the QFT approach that allows treatment of highly structured (and unstructured) uncertainty. Because the specifications capture phase information, the level of over-design in certain MIMO QFT designs is reduced. The method presented allows independent, two-degree-of-freedom design.     

Efficient algorithm for computing QFT bounds

This article presents an efficient algorithm for computing quantitative feedback theory (QFT) bounds for frequency-domain specifications from plant templates which are approximated by a finite number of points. To develop the algorithm, an efficient procedure is developed for testing, at a given frequency, whether or not a complex point lies in the QFT bound. This test procedure is then utilised along with a pivoting procedure to trace out, with a prescribed accuracy or resolution, the boundary of the QFT bound. The developed algorithm for computing QFT bounds has the advantages that it is efficient and can compute QFT bounds with multi-valued boundaries. A numerical example is given to show the computational superiority of the proposed algorithm.     

Non‐diagonal MIMO QFT controller design reformulation

This paper presents a reformulation of the full‐matrix quantitative feedback theory (QFT) robust control methodology for multiple‐input–multiple‐output (MIMO) plants with uncertainty. The new methodology includes a generalization of previous non‐diagonal MIMO QFT techniques; avoiding former hypotheses of diagonal dominance; simplifying the calculations for the off‐diagonal elements, and then the method itself; reformulating the classical matrix definition of MIMO specifications by designing a new set of loop‐by‐loop QFT bounds on the Nichols Chart, which establish necessary and sufficient conditions; giving explicit expressions to share the load among the loops of the MIMO system to achieve the matrix specifications; and all for stability, reference tracking, disturbance rejection at plant input and output, and noise attenuation problems. The new methodology is applied to the design of a MIMO controller for a spacecraft flying in formation in a low Earth orbit. Copyright © 2008 John Wiley & Sons, Ltd.     

Analytical formulation to compute QFT bounds: The envelope method

This paper describes an analytical formulation to compute quantitative feedback theory (QFT) bounds in one‐degree‐of‐freedom feedback control problems. The new approach is based on envelope curves and shows that a QFT control specification can be expressed as a family of circumferences. Then, the controller bound is defined by the envelope curve of this family and can be obtained as an analytical function. This offers the possibility of studying the QFT bounds in an analytical way with several useful properties. Gridding methods are avoided, resulting in a lower computational effort procedure. The new formulation improves the accuracy of previous methods and allows the designer to calculate multivalued bounds. Copyright © 2009 John Wiley & Sons, Ltd.     

GENERATION OF EXACT QFT BOUNDS FOR PLANTS WITH AFFINELY DEPENDENT UNCERTAINTIES

This paper presents an efficient method for the generation of exact QFT bounds for robust sensitivity reduction and gain‐phase margin specifications for plants with affinely dependent uncertainties. It is shown that, for a plant with m affinely dependent uncertainties, the exact QFT bounds for robust sensitivity reduction and gain‐phase margin specifications at a given frequency and controller phase can be computed by solving m^2m‐1 bivariate polynomial inequalities corresponding to the edges of the parameter domain box. Moreover, the solution set for each bivariate polynomial inequality can be computed by solving for the real roots of one fourth‐order and six second‐order polynomials. This avoids the unfavorable trade‐off between the computational burden and the accuracy of QFT bounds that has arisen in the application of many existing QFT bound generation algorithms. Numerical examples are given to illustrate the proposed method and its computational superiority.     

Quantitative feedback theory for multiple-input-multiple output feedback systems with control input failures†

In quantitative feedback theory (QFT), plant parameter and disturbance uncertainties are the reasons for using feedback. The system design is tuned to quantitative statements of these parameters and of the performance tolerances. Available design freedom is used to minimize the cost of feedback which is in the bandwidths of the loop transfer functions. This paper extends QFT to 2 × 2 linear time invariant (LTI) multiple-input-multiple-output plants, in which total failure of some control inputs is possible. Maximum possible achievement of the performance specifications is determined, with single fixed LTI compensation networks. A detailed design example is included.     

On characterizing sensitivity-based and traditional formulations for quantitative feedback theory

Recent developments in quantitative feedback theory include the 'new formulation' approach in which a robust performance and robust stability problem, similar to Horowitz's traditional QFT formulation, is developed in terms of sensitivity function bounds. The motivation for this approach was to provide the basis for a more rigorous treatment of nonminimum phase systems and/or plants characterized by mixed parametric and non-parametric uncertainty models. However, it has been found in practice that the sensitivity-based formulation exhibits some unique behaviour, i.e. in terms of the open loop design bounds obtained for various choices of nominal plant. Experience has shown that these bounds will dominate (i.e. are more conservative than) the corresponding traditional QFT bounds for the same problem; it has also been observed that the degree to which this occurs varies with choice of the nominal plant. Further, it has been found that the choice of nominal, in certain cases, can lead to a problem which is infeasible with respect to Bode sensitivity (i.e. requiring S(jomega) < 1 as omega infinity), while the traditional QFT problem remains feasible. Heretofore, this behaviour has not been fully explained. In this paper, these issues are characterized in the simplest possible setting, focusing primarily on the behaviour at zero phase angle. A 'modified' sensitivity-based QFT formulation is proposed here in which limitations on the choice of nominal plant are clearly delineated; this formulation results in open loop design bounds which are equivalent to the traditional QFT problem at zero phase angle, while over-bounding them elsewhere. The modified formulation is also shown to meet the same necessary condition for Bode feasibility as traditional QFT. In conclusion, these issues are demonstrated by means of a basic example.     

Generation of QFT bounds for robust tracking specifications for plants with affinely dependent uncertainties

This paper presents an efficient algorithm for the generation of QFT bounds for robust tracking specifications for plants with affinely dependent uncertainties. For a plant with m affinely dependent uncertainties, it is shown that whether a point in the Nichols chart lies in the QFT bound for a robust tracking specification at a given frequency can be easily tested by computing the maxima and minima of m2^m?1 univariate functions corresponding to the edges of the parameter domain box. This test procedure is then utilized along with a pivoting procedure to trace out the boundary of the QFT bound with a prescribed accuracy or resolution. The developed algorithm has the advantages that (1) it is efficient in the sense that it requires less floating point operations than other existing algorithms in the literature; (2) it can avoid the unfavorable trade‐off between the computational burden and the accuracy of the computed QFT bounds that has arisen in the application of many existing QFT‐bound generation algorithms; (3) the maximum allowable error of the computed QFT bound can be prespecified; and (4) it can compute QFT bounds with multi‐valued boundaries. Numerical examples are given to illustrate the proposed algorithm and its computational superiority. Copyright © 2010 John Wiley & Sons, Ltd.     

Feedback design for robust tracking and robust stiffness in flight control actuators using a modified QFT technique

The problem of dynamic stiffness of hydraulic servomechanisms has often been recognized as a significant performance issue in a variety of applications, the most notable of which includes flight control actuation. When a hydraulic actuator such as this is operated in position control, an aerodynamic flutter load on the control surface manifests itself as a force disturbance on the system. Although this would appear to be a standard disturbance rejection problem, the disturbance does not enter the system as in the classical sense (i.e. at the plant output) and hence, this problem must be considered in a modified formulation. A hydraulic servomechanism is said to be 'stiff' if it exhibits acceptable rejection of force disturbances within the control bandwidth. In this paper, an approach to feedback design for robust tracking and robust disturbance rejection is developed via the quantitative feedback theory (QFT) technique. As a result, it is shown that reasonable tracking and disturbance rejection specifications can be met by means of a fixed (i.e. non-adaptive), single loop controller. The methodology employed in this development is the sensitivity-based QFT formulation. As a result, robust tracking and robust disturbance rejection specifications are mapped into equivalent bounds on the (parametrically uncertain) sensitivity function; hence, the frequency ranges in which tracking or disturbance rejection specifications dominate become immediately obvious. In this paper, a realistic non-linear differential equation model of the hydraulic servomechanism is developed, the linear parametric frequency response properties of the open loop system are analysed, and the aforementioned QFT design procedure is carried out. Analysis of the closed loop system characteristics shows that the tracking and disturbance rejection specifications are indeed met.     

Computation of QFT bounds for robust tracking specifications

An algorithm is proposed for generation of QFT controller bounds to achieve robust tracking specifications. The proposed algorithm uses quadratic constraints and interval plant templates to compute the bounds, and presents several improvements over existing QFT tracking bound generation algorithms. The proposed algorithm (1) guarantees robustness against template inaccuracies, (2) guarantees robustness against phase discretization, (3) provides a posteriori error estimates, (4) is computationally efficient, achieving a reduction in flops and execution time, typically by 1-2 orders of magnitude. The algorithm is demonstrated on an aircraft example having five uncertain parameters.     

Output feedback robust MPC using general polyhedral and ellipsoidal true state bounds for LPV model with bounded disturbance

This paper considers the dynamic output feedback robust model predictive control (MPC) for a system with both polytopic model parametric uncertainty and bounded disturbance. For this topic, the techniques for handling the unknown true state are crucial, and the strict guarantee of the input/output/state constraints requires replacing the true state by its bounds in the optimisation problems. Previously, in the separate works, we (i) gave the general polyhedral bound; (ii) proposed the general ellipsoidal bound; (iii) applied some special polyhedral bounds to tighten the ellipsoidal bound since the latter is crucial for guaranteeing recursive feasibility. In this paper, (i)–(iii) are unified, and the up-to-date least conservative treatment of the true state bound is given, so the control performance can be greatly improved. The contribution mainly lies in overcoming the difficulties in developing technical details for the unification. A numerical example is given to illustrate the effectiveness of the new method.     

Interactive parameter space design for robust performance of MISO control systems

This paper presents a method for the design of nonconservative low-order controllers achieving robust performance in the case of multi-input single-output parallel structure plants subject to unstructured uncertainty. The first step is the analytical generation of gain-phase controller bounds, as in quantitative feedback theory (QFT). Then, to avoid the difficult step of QFT loop shaping, which often produces high-order controllers, these bounds are translated into the controller parameter space where the iterative design of low fixed order controllers takes place. This, as well as the design transparency offered by this technique, constitutes appreciable advantages over the other popular robust performance design method of /spl mu/-synthesis. Other important features are the fact that no extra conservatism is introduced by the method presented and the fact that the method is directly compatible with a sequential loop closing strategy. Finally, the direct search optimization of any additional secondary criteria is possible.     

A QFT Procedure for Generating Design Frequencies and Bounds of Guaranteed Accuracy

In this paper, a QFT procedure is presented to systematically determine the following (i) the set of design frequency intervals from a given design frequency range, (ii) the controller bounds of prescribed accuracy at each design frequency interval, and (iii) the controller phase intervals for efficient bound generation at each design frequency interval. The procedure is given for the robust gain-phase margin specifications, based on several new results derived in the paper in the interval analysis framework. The procedure is demonstrated on a significant practical problem concerning the longitudinal motion of an aircraft.     

Quantitative feedback theory—reply to criticisms

Quantitative feedback theory (QFT) consists of a steadily growing body of design techniques for achieving prespecified system performance tolerances, despite prespecified large plant parameter and disturbance uncertainties. Since 1959, QFT has been extended to SISO and MIMO, linear and non-linear, time-invariant and time-varying, output feedback and internal variable feedback, lumped and distributed plants. Design examples in all the above classes have been described in great detail.

In contrast, modern control theory almost completely ignored the uncertainty issue in feedback theory until about five years ago. There has since been much activity in this subject, which it denotes as the robustness problem. Despite this activity, hardly a single detailed design example involving large plant parameter uncertainty has been described. Nevertheless, researchers in robustness have ignored QFT. This conspiracy of silence has recently been broken with a list of criticisms by Doyle. These provide a very welcome means of explanation and elaboration of important QFT properties, including some new results.     

On quantitative feedback design for robust position control of hydraulic actuators

This paper discusses several practical issues related to the design of robust position controllers for hydraulic actuators by quantitative feedback theory (QFT). Important properties of the hydraulic actuator behavior, for control system design, are identified by calculating a family of equivalent frequency responses from acceptable nonlinear input–output data. The role of this modeling approach towards reducing over-design by decreasing the sizes of the QFT plant templates is described. The relationship between the geometry of the QFT bounds and the complexity of the robust feedback law is examined through the development of two low-order controllers having characteristics suitable for different applications. Experimental test results demonstrate the extent that each QFT controller is able to maintain robustness against variations in the hydraulic system dynamics that occur due to changing load conditions as well as uncertainties in the hydraulic supply pressure, valve spool gain, and actuator damping.     

Improvements on the computation of boundaries in QFT

Quantitative feedback theory (QFT) is an engineering design technique of uncertain feedback systems that uses frequency domain specifications. A key step in QFT is the mapping of these specifications into regions of the Nichols plane, whose borders are usually referred to as boundaries. Boundaries computation is a key design step, thus a precise and efficient computation is critical for both obtaining low bandwidth feedback compensators and simplification of the design process. In this work, the problem of boundaries computation is analysed, introducing a new algorithm based on the computation of level curves of a three‐dimensional surface. Besides magnitude boundaries, associated with some specification over the magnitude of a closed‐loop transfer function, phase boundaries are also considered. In addition, comparison with previous published algorithms is done in terms of precision and computational efficiency. Copyright © 2006 John Wiley & Sons, Ltd.     

定量反馈理论发展综述

定量反馈理论是一种基于频域的鲁棒控制理论,可以用于具有高度不确定性的单变量线性/非线性系统、多变量线性/非线性系统控制器设计.本文概述了定量反馈理论的基本原理、设计过程以及特点.总结了近年来QFT在提高系统性能、鲁棒稳定性、自动设计以及应用等方面的最新研究进展,并且给出了一些已有的理论应用成果.最后讨论了进一步的研究方向.     

融合GPC和QFT对不确定性系统的鲁棒控制

针对广义预测控制(GPC)与定量反馈理论(QFT)的特点,提出了把两种算法融合的鲁棒控制算法。该方法是在对QFT进行修改的基础上,采用双回路控制。内回路采用QFT控制器实现对系统不确定性的控制;外回路采用GPC控制器,实现对系统的各种性能要求并且提高鲁棒性。该方法可以充分利用两种控制理论的优点。最后的仿真结果显示,融合的算法比单独采用其中的任何一种控制算法所取得的控制效果都好。     

在GIMC结构下基于定量反馈理论的连续广义预测鲁棒稳定控制

The continuous-time generalized predictive control (CGPC) and the quantitative feedback theory (QFT) are used together to control the plant with high uncertainty. QFT conquers the plant uncertainty and stabilizes the system in the inner loop without affecting the nominal performance based on the generalized internal model control (GIMC) structure. CGPC is used to obtain the necessary control performance in the outer loop. According to several given sufficient conditions, the available tuning parameters of CGPC are selected to make the system robustly stable. Finally, an example is given to show how to use this technique; and it is shown that this combined approach gets better performance than if only one of them is used.