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.     

Nonconservative QFT bounds for tracking error specifications

An alternative to the traditional QFT tracking problem, in which upper and lower tolerances are imposed on the magnitude of the tracking transfer function, is to define the robust specification as a boundary on the deviation of such function from a predefined model. However, previous research exploring this approach reveals a certain overdesign and dependence on the choice of the nominal plant. This paper establishes a necessary condition on the controller from tracking error specifications. With this condition, the controller bounds introduce no overdesign, and the resulting two‐degrees‐of‐freedom design is independent of the choice of the nominal plant, similar to the traditional approach. Copyright © 2011 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.     

Stability of quantitative feedback designs and the existence of robust QFT controllers

Quantitative feedback theory (QFT) has received much criticism for a lack of clearly stated mathematical results to support its claims. Considered in this paper are two important fundamental questions: (i) whether or not a QFT design is robustly stable, and (ii) does a robust stabilizer exist. Both these are precursors for synthesizing controllers for performance robustness. Necessary and sufficient conditions are given to resolve unambiguously the question of robust stability in SISO systems, which in fact confirms that a properly executed QFT design is automatically robustly stable. This Nyquist-type stability result is based on the so-called zero exclusion condition and is applicable to a large class of problems under some simple continuity assumptions. In particular, the class of uncertain plants include those in which there are no right-half plane pole-zero cancellations over all plant uncertainties. A sufficiency condition for a robust stabilizer to exist is derived from the well-known Nevanlinna-Pick theory in classical analysis. Essentially the same condition may be used to answer the question of existence of a QFT controller for the general robust performance problem. These existence results are based on an upper bound on the nominal sensitivity function. Also considered is QFT design for a special class of interval plants in which only the poles and the DC gain are assumed uncertain. The latter problem lends itself to certain explicit computations that considerably simplify the QFT design problem.     

QFT design for uncertain non-minimum phase and unstable plants revisited

Design method for uncertain non-minimum phase and unstable plants in the quantitative feedback theory (QFT) developed by Horowitz and Sidi is revisited in this paper. It is illustrated that the existing method may not work since some design rules have not been clearly specified by several examples including non-minimum phase plants and unstable plants. Then stability of a new nominal plant is carefully examined and analysed, and an improved design method is presented. The result in this paper provides mathematical justification of the QFT design procedure for nonminimum phase and unstable plants in Horowitz and Sidi (1978) and Horowitz (1992).     

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.     

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.     

Interval QFT: a mathematical and computational enhancement of QFT

The paper presents an overview of a mathematical and computational enhancement of Horowitz's QFT design procedure. The enhancement uses methods of interval analysis and is called as interval QFT, or IQFT. IQFT addresses and solves some of the fundamental issues in QFT, concerning selection of design frequencies, selection of controller phases in bound generation, approximation of plant templates with finite plant sets, and generation of plant templates and controller bounds with reliability and to a prescribed accuracy. An example is presented to illustrate the key features of IQFT. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

QFT bounds for robust stability specifications defined on the open‐loop function

In the framework of quantitative feedback theory, this paper develops a new method to compute robust stability bounds. This is of special interest when stability is defined directly on the open‐loop function. Thus, ignorance of the plant gain and phase shift can be specifically and independently considered. Furthermore, upper and lower stability margins for both gain and phase can be chosen. However, classical quantitative feedback theory stability specifications are defined as constraining the peak magnitude of closed‐loop functions, which lack the said flexibility. Once the upper tolerance has been defined, all stability margins are determined. Moreover, confining the most restrictive stability margin may result in other excessive margins. However, the stability bounds of the new approach guard just the required distance from the open‐loop frequency response to the critical point. This allows maximization of the available feedback in the functional bandwidth and minimization of the cost of feedback beyond the crossover frequency, provided that the open‐loop frequency response is shaped to closely follow the stability bounds. It should be noted that the new bound computation algorithm performs few and simple arithmetic operations. This makes it far more efficient than traditional methods. The flight altitude control of an unmanned aerial vehicle is proposed as a practical example to show the new method's potential benefits.     

Some results in nonlinear QFT

Nonlinear QFT (quantitative feedback theory) is a technique for solving the problem of robust control of an uncertain nonlinear plant by replacing the uncertain nonlinear plant with an ‘equivalent’ family of linear plants. The problem is then finding a linear QFT controller for this family of linear plants. While this approach is clearly limited, it follows in a long tradition of linearization approaches to nonlinear control (describing functions, extended linearization, etc.) which have been found to be quite effective in a wide range of applications. In recent work, the authors have developed an alternative function space method for the derivation and validation of nonlinear QFT that has clarified and simplified several important features of this approach. In particular, single validation conditions are identified for evaluating the linear equivalent family, and as a result, the nonlinear QFT problem is reduced to a linear equivalent problem decoupled from the linear QFT formalism. In this paper, we review this earlier work and use it in the development of (1) new results on the existence of nonlinear QFT solutions to robust control problems, and (2) new techniques for the circumvention of problems encountered in the application of this approach. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis and design for ill-conditioned plants

A study is made of a special case of the robust performance problem given by Freudenberg (1989). When the weightings used to describe the uncertainty and performance specifications vary only with frequency, then it is possible to strengthen the results in the above-mentioned work by deriving both upper and lower bounds upon the structured singular value. Both sets of bounds are stated in terms of the coupling coefficients introduced by Freudenberg (1989), and essentially yield necessary and sufficient conditions for the structured singular value to be small. This information is used to suggest a strategy for compensator design to achieve robust performance despite plant ill-conditioning. Applying this strategy to an example, it can be seen how the design trade-offs quantified by the Bode gain-phase relation manifest themselves in the robust performance problem. Finally, the design is compared with one obtained using the'µ-synthesis' approach.     

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.     

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.     

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.     

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.     

Worst case identification of continuous time systems via interpolation

We consider a worst case robust control oriented identification problem recently studied by several authors. This problem is one of identification in the continuous time setting. We give a more general formulation of this problem. The available a priori information in this paper consists of a lower bound on the relative stability of the plant, a frequency dependent upper bound on a certain gain associated with the plant, and an upper bound on the noise level. The available experimental information consists of a finite number of noisy plant point frequency response samples. The objective is to identify, from the given a priori and experimental information, an uncertain model that includes a stable nominal plant model and a bound on the modeling error measured in norm. Our main contributions include both a new identification algorithm and several new ‘explicit’ lower and upper bounds on the identification error. The proposed algorithm belongs to the class of ‘interpolatory algorithms’ which are known to possess a desirable optimality property under a certain criterion. The error bounds presented improve upon the previously available ones in the aspects of both providing a more accurate estimate of the identification error as well as establishing a faster convergence rate for the proposed algorithm.     

Optimal nonparametric identification from arbitrary corrupt finitetime series

Formulates and solves a worst-case system identification problem for single-input, single-output, linear, shift-invariant, distributed parameter plants. The available a priori information in this problem consists of time-dependent upper and lower bounds on the plant impulse response and the additive output noise. The available a posteriori information consists of a corrupt finite output time series obtained in response to a known, nonzero, but otherwise arbitrary, input signal. The authors present a novel identification method for this problem. This method maps the available a priori and a posteriori information into an “uncertain model” of the plant, which comprises a nominal plant model, a bounded additive output noise, and a bounded additive model uncertainty. The upper bound on the model uncertainty is explicit and expressed in terms of both the l₁ and H_∞ system norms. The identification method and the nominal model possess certain well-defined optimality properties and are computationally simple, requiring only the solution of a single linear programming problem     

Algebraic and topological aspects of quantitative feedback theory

The current interest in robust control has called into question the applicability of the quantitative feedback theory (QFT) robust design method introduced by Horowitz. A number of issues have been raised regarding inherent restrictions of both the design method and the uncertain plant set. Using a multivariable root-locus technique extended to uncertain systems, this paper shows that the QFT assumptions are indeed not restrictive and are in fact equivalent to other well-known conditions for robust stabilizability. Because QFT is one of the very few methods to specifically address the quantitative robust performance issue, these results should lead to better methods of developing new QFT-design software, as well as improved robust control methods to satisfy a priori quantitative performance bounds.     

Multi-input,multi-output flight control system design for the YF-16 using nonlinear QFT and pilot compensation

Nonlinear quantitative feedback theory (QFT) and pilot compensation techniques are used to design a 2 × 2 flight control system for the YF-16 aircraft over a large range of plant uncertainty. The design is based on numerical input-output time histories generated with a FORTRAN implemented nonlinear simulation of the YF-16. The first step of the design process is the generation of a set of equivalent linear time-invariant (LTI) plant models to represent the actual nonlinear plant. It has been proven that the solution to the equivalent plant problem is guaranteed to solve the original nonlinear problem. Standard QFT techniques are then used in the design synthesis based on the equivalent plant models. A detailed mathematical development of the method used to develop these equivalent LTI plant models is provided. After this inner-loop design, pilot compensation is developed to reduce the pilot's workload. This outer-loop design is also based on a set of equivalent LTI plant models. This is accomplished by modelling the pilot with parameters that result in good handling qualities ratings, and developing the necessary compensation to force the desired system responses.