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.

This paper describes a robust nonlinear control system design procedure inspired by the nonlinear control ideas of Horowitz's Quantitative Feedback Theory.(1) The central concept is the identification of a family of linear time-invariant (LTI) plants that is equivalent to an uncertain nonlinear (and/or time varying) plant in the sense that an LTI controller feasible for this linear plant family is also feasible for the original nonlinear plant. We identify two conditions for evaluating an equivalent linear family (the equivalence condition and the continuity condition) and show that when these two conditions are satisfied an LTI controller that provides satisfactory robust control of an equivalent linear plant family also provides satisfactory robust control for the related uncertain nonlinear plant, independent of the robust design technique used. We then use these two conditions to analyse the validity of the nonlinear QFT design technique published earlier. Our results suggest that nonlinear QFT can be an attractive approach to nonlinear robust control but its validity (in the sense that the linear design solves the nonlinear control problem) can be demonstrated only if additional conditions and contraints not previously reported are satisfied. (C) 1998 John Wiley & Sons, Ltd.

The frequency domain design of robust feedback control systems involves the fitting of a designable complex function of frequency to specification/uncertainty derived constraints. At present there are two basically different approaches to these problems: the several gain vs. frequency approaches and the two gain-phase vs. frequency approaches. Unfortunately, the commonalities, advantages and disadvantages of these approaches are not widely appreciated. In this paper the authors develop a common framework for examining these alternatives and use this framework to reveal some of their similarities, strengths and weaknesses.

This paper describes a robust nonlinear control system design procedure based on the construction of a family of linear plants that isequivalentto an uncertain nonlinear (and/or time varying) plant in the sense that a linear controller feasible for this linear family is also feasible for the original nonlinear plant. The results presented are inspired by the nonlinear control ideas of Horowitz’s Quantitative Feedback Theory and are used to provide a new theoretical foundation for this earlier work.

This paper addresses the existence of loop gain-phase shaping (LGPS) solutions for the design of robust digital control systems for SISO, minimum-phase, continuous-time processes with parametric uncertainty. We develop the frequency response properties of LGPS for discrete-time systems using the DELTA-transform, a transform method that applies to both continuous-time and discrete-time systems. A theorem is presented which demonstrates that for reasonable specifications there always exists a sampling period such that the robust digital control problem has a solution. Finally, we offer a procedure for estimating the maximum feasible sampling period for LGPS solutions to robust digital control problems.

A solution to the robust multi-axis coordinated motion problem is presented using the loop gain-phase shaping (LGPS) technique of quantitative feedback theory (QFT). Robust contouring specifications are translated into closed loop gain and phase robustness specifications by using a relaxed performance criterion. A two axis example is presented to demonstrated the concepts.

The authors study the linearization problem of nonlinear plants using a stable feedback structure. This problem is solved in an operator framework for those nonlinear systems having a known inverse model with some particular characteristics for the stability of its nonlinear part. For this purpose some results are used about stability of feedback systems given by the factorization approach to the study of nonlinear systems.<>

The use of root-locus-based robust design techniques in the design of a digital controller for a high-performance variable-amplitude fatigue test is described. Evaluation of the resulting controller on a prototype laboratory materials testing system, designed to simulate the mechanical stress loads experienced by certain aircraft components, shows that it meets design performance goals of 0.25% specimen load accuracy while providing robustness to specimen compliance over a 10 to 1 range. This performance significantly exceeds that obtained with traditional hand-tuned controllers.< >

The steps in loop gain-phase shaping design, an approach that offers significant advantages in the design of robust single-input-single-output control systems, are summarized. A family of CAD tools that implement the design procedure is described. several examples are presented showing the use of these tools and demonstrating some of the performance advantages available with loop gain-phase shaping as compared to classical loop shaping and its derivatives.< >

This paper discusses the extension of the loop gain-phase shaping (LGPS) techniques of QFT to SISO robust controller design for mixed uncertainty situations: that is, processes having both parametric and non-parametric uncertainty. The mixed uncertainty problem leads to a set of frequency dependent high frequency stability boundaries replacing the traditional (single) universal high frequency boundary considered in [1]. In the presence of these new frequency dependent, high frequency boundaries the existence of a LGPS solution becomes problematical - even in the case of minimum phase stable plants. We will outline the modified LGPS problem, describe new CAD software for its formalation and solution and present a simple test that can be used to indicate the existence of solutions to the fitting problem. An example demonstrates the application of these tools and techniques in the design of a robust controller for a motion control systems with uncertain resonances.

An algorithm for fast computation of a class of parametric rational functions whose coefficients are assumed to be affine functions of some underlying parameters is described. The algorithm has been implemented in Microsoft Quick BASIC 4.0. Execution times are minimal.<>

The steps in loop gain-phase shaping (LGPS) and a family of CAD tools that make LGPS a realistic design procedure are described. Several examples demonstrating the use of these tools and the performance advantages of LGPS over classical loop shaping and its derivatives are presented. It is reported that the LGPS approach uses phase information explicitly, providing an increased range of applicability, reduced bandwidth requirements, more accurate control of performance robustness, and convenient control of relative stability robustness

An algorithm for fast computation of interval rational functions is described.

Two algorithms are described that are useful in the computer-aided design implementation of the robust controller design scheme proposed by Horowitz (1963) and Horowitz and Sidi (1972).

A notation is developed to describe operations on two-dimensional arrays of data and develop a compact description of the Gaussian random matrix as an intuitively obvious extension of the Gaussian random vector. The application of these concepts is then illustrated in some typical problems encountered in the processing of two-dimensional signals.

A change of notation is suggested to clarify results reported in [1]. It is noted that the proposed cost function is generally not linear. Thus, the 0-1 integer programming formulation is inappropriate without additional assumptions.

Splitting methods are examined for the iterative solution of the classical linear least-squares problem in Hilbert space. Conditions for convergence of the class of iterations studied generalize existing conditions in the literature. Throughout, the emphasis is on an organization-theoretic interpretation of the algorithm, thereby clarifying certain questions of decentralization of information and ...

Certain results paralleling those of Bailey and Ramapriyan [1] for linear dynamic systems are presented for nonlinear systems. For "weakly perturbed" systems certain suboptimal controls are shown to be stabilizing and upper and lower bounds on the suboptimality are investigated.

This paper describes a digital control laboratory facility which teaches the central concepts of digital control without requiring a background in digital systems and/or machine language programming. The heart of the laboratory is a flexible hardware digital controller replacing the traditional analog control components in a D. C. servo system. The digital controller can be adjusted to provide versions of the standard PID control algorithm with adjustable word length and computation delay. Through a series of graded experiments this hardware can provide: (1) an introduction to digital control concepts and components, (2) an introduction to digital signal processing, (3) experience with basic DDC algorithms, (4) experience with problems encountered in the choice of sampling rate and digital word length, and (5) experience in working with and trouble-shooting an operating digital control system. Since one controller can be provided for each laboratory station the student has considerable freedom to learn by doing at his own rate.

Coordination by augmentation as proposed by Isaksén and Payne [2] is compared with a simple decentralized control structure in which there is no coordination because couplings are simply ignored. A simple comparison test is derived and it is shown that there are nontrivial situations in which coordination by augmentation actually degrades performance.