Design and validation of linear robust controllers for nonlinear plants

This paper describes a robust nonlinear control system design procedure inspired by the nonlinear control ideas of Horowitz's Quantitative Feedback Theory. 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. © 1998 John Wiley & Sons, Ltd.     

Flight controller design for uncertain and nonlinear plants with pilot compensation

Nonlinear quantitative feedback theory (QFT) is used to design a flight control system for the nonlinear model of the YF-16 aircraft (A/C) with C* as the controlled output. The resulting closed loop stability augmentation system (SAS), P_e(S), becomes part of the outer loop containing the pilot. The Neal-Smith pilot model for a compensatory tracking task is used to develop a technique which allows the designer to synthesize compensation in the outer loop, which includes a free compensator F_p(S). The latter is chosen to minimize pilot workload, increase system bandwidth, and improve handling qualities ratings as per the Neal–Smith criteria, for the tracking task. The available pilot compensation abilities are then available for further increasing of system bandwidth to improve overall capabilities. This approach can be used at the early stages of flight control design, thus saving time and money over the current practice. Simulations in the time and frequency domains demonstrate that the desired performance is attained.     

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.     

ROBUST LTV FEEDBACK SYNTHESIS FOR SISO NON-LINEAR PLANTS

Given a feedback system with uncertain nonlinear plant, it is required that the plant's output, to a set of command inputs, will satisfy certain specifications, i.e., will be bounded between a maximum response β(t) and a minimum response α(t). A rigorous synthesis technique to solve this problem using an LTV controller is developed. A design example is included, and it is shown that the LTV design has much lower bandwidth as compared to LTI designs. All design steps utilize the well-established QFT technique for LTI SISO uncertain systems. The methodology also suits rejection of disturbances at the plant's input or output, and for output specifications for nonzero initial conditions. © 1997 by John Wiley & Sons, Ltd.     

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.     

Design and experimental evaluation of a robust force controller for an electro-hydraulic actuator via quantitative feedback theory

This paper presents the design and experimental evaluation of an explicit force controller for a hydraulic actuator in the presence of significant system uncertainties and nonlinearities. The nonlinear version of quantitative feedback theory (QFT) is employed to design a robust time-invariant controller. Two approaches are developed to identify linear time-invariant equivalent model that can precisely represent the nonlinear plant, operating over a wide range. The first approach is based on experimental input–output measurements, obtained directly from the actual system. The second approach is model-based, and utilizes the general nonlinear mathematical model of a hydraulic actuator interacting with an uncertain environment. Given the equivalent models, a controller is then designed to satisfy a priori specified tracking and stability specifications. The controller enjoys the simplicity of fixed-gain controllers while exhibiting robustness. Experimental tests are performed on a hydraulic actuator equipped with a low-cost proportional valve. The results show that the compensated system is not sensitive to the variation of parameters such as environmental stiffness or supply pressure and can equally work well for various set-point forces.     

The input amplitude saturation problem in QFT: A survey

In this work the input amplitude saturation problem is analysed in the Quantitative Feedback Theory (QFT) framework. This paper reviews previous works in the literature dealing with the input amplitude saturation problem in the presence of an uncertain plant in the frequency domain using QFT. The objective of this paper is to compare the different available approaches and summarize the design process for each case so that this paper can be used as a tutorial; there are six main approaches to this problem. Two of these approaches use the classical two degrees of freedom control scheme for QFT; in both of these, the design constraints of a linear QFT compensator are added in the loop shaping stage: they are added in the first approach to avoid excitation of the actuator saturation and in the second one to guarantee global stability. The other three techniques are considered as anti-windup (AW) approaches. Starting from a base design in QFT with two degrees of freedom, the first AW approach introduces a third degree of freedom that guarantees the stability of the system, allowing for base designs for high performance. The other two AW approaches also introduce a third degree of freedom, but they take simple stability considerations into account and focus on the performance of the system. The last solution consists of using a reference governor technique, which guarantees the computation of a reference signal for an inner control loop that is shaped using QFT in such a way that robust stability will be guaranteed. The reference governor technique is a time domain approach that implies the resolution of an optimization problem. The rest of the approaches are frequency domain techniques based on a loop shaping method in the traditional QFT sense.     

Non-linear QFT synthesis by local linearization

Existing non-linear quantitive feedback theory (QFT) techniques are based on some form of linearization of the non-linear plant, where the quality of the controller is a function of the equivalent linearized plant. We propose a new approach to non-linear QFT, based on local linearization of the non-linear plant about closed-loop acceptable outputs. Here, acceptable outputs appear as uncertain parameters in an equivalent linear family, being a generalization of the more traditional linearization about equilibrium points. In some applications, this technique may give better results and is an alternative when the other existing techniques fail. A comparison of this technique to previous ones using an uncertain Van der Pol plant considered previously in the QFT literature, and a pH control system, is presented.     

LTI approximation of nonlinear systems via signal distribution theory

L₂ and L₁ optimal linear time-invariant (LTI) approximation of discrete-time nonlinear systems, such as nonlinear finite impulse response (NFIR) systems, is studied via a signal distribution theory motivated approach. The use of a signal distribution theoretic framework facilitates the formulation and analysis of many system modelling problems, including system identification problems. Specifically, a very explicit solution to the L₂ (least squares) LTI approximation problem for NFIR systems is obtained in this manner. Furthermore, the L₁ (least absolute deviations) LTI approximation problem for NFIR systems is essentially reduced to a linear programming problem. Active LTI modelling emphasizes model quality based on the intended use of the models in linear controller design. Robust stability and LTI approximation concepts are studied here in a nonlinear systems context. Numerical examples are given illustrating the performance of the least squares (LS) method and the least absolute deviations (LAD) method with LTI models against nonlinear unmodelled dynamics.     

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.     

Switching controller design for linear time invariant plant with a single I/O delay

ABSTRACT

To deal with the problem of conflicting requirements that cannot be satisfied by only a single LTI controller, this paper focuses on the design of a switching controller, which includes several stabilising linear time-invariant (LTI) controllers designed independently with different control performance criteria, for a specific LTI plant with a single I/O delay. The switching controller design procedure is divided into three steps. First, using simple loop shifting arguments, the design problem is reducible to an equivalent delay-free one. Second, traditional LTI controller synthesis methods could be considered independently for the delay-free plant. Third, based on a quadratically stable state space realisation method, a Youla parameter including a switching strategy with these controllers is designed to guarantee different requirements. Finally, a numerical example is provided to demonstrate the validity and efficiency of the approach.     

CAutoCSD – Evolutionary Search and Optimisation Enabled Computer Automated Control System Design

This paper attempts to set a unified scene for various linear time-invariant (LTI) control system design schemes, by transforming the existing concept of "computer-aided control system design" (CACSD) to novel "computer-automated control system design" (CAutoCSD). The first step towards this goal is to accommodate, under practical constraints, various design objectives that are desirable in both time and frequency domains. Such performance-prioritised unification is aimed at relieving practising engineers from having to select a particular control scheme and from sacrificing certain performance goals resulting from pre-commitment to such schemes. With recent progress in evolutionary computing based extra-numeric, multi-criterion search and optimisation techniques, such unification of LTI control schemes becomes feasible, analytical and practical, and the resultant designs can be creative. The techniques developed are applied to, and illustrated by, three design problems. The unified approach automatically provides an integrator for zero-steady state error in velocity control of a DC motor, and meets multiple objectives in the design of an LTI controller for a non-minimum phase plant and offers a high-performance LTI controller network for a non-linear chemical process.     

A quantitative inherent reconfiguration theory for a class of systems†

In aircraft flight control, most control surfaces are in pairs (elevators, ailerons, canards etc.), with each pair normally controlled as a single unit. If a surface fails, the usual approach is to attempt explicit identification and switch-in of compensation prepared for that contingency. In this paper each surface is separately controlled, permitting ‘inherent reconfiguration’, wherein the design is a priori made such that despite one or several simultaneous surface failures, the system still satisfies the original performance tolerance (of course over a smaller dynamic range), with the same original fixed compensation. Inherent reconfiguration is a natural extension of quantitative feedback theory (QFT), wherein the system design is tuned to the plant uncertainty set 𝒫 ={P}, and to the acceptable system output set,. In QFT one designs a priori so that the system output is in  for all P in 𝒫 Surface failures simply enlarge the set 𝒫. The transparency of QFT enables the designer to readily see the extra ‘cost of feedback’ for this enlargement of & by inclusion of surface failures.     

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.     

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.     

A computational method for simultaneous LQ optimal control designvia piecewise constant output feedback

The paper is concerned with simultaneous linear-quadratic (LQ) optimal control design for a set of LTI systems via piecewise constant output feedback. First, the discrete-time simultaneous LQ optimal control design problem is reduced to solving a set of coupled matrix inequalities and an iterative LMI algorithm is presented to compute the feedback gain. Then, simultaneous stabilization and simultaneous LQ optimal control design of a set of LTI continuous-time systems are considered via periodic piecewise constant feedback gain. It is shown that the design of a periodic piecewise constant feedback gain simultaneously minimizing a set of given continuous-time performance indexes can be reduced to that of a constant feedback gain minimizing a set of equivalent discrete-time performance indexes. Explicit formulas for computing the equivalent discrete-time systems and performance indexes are derived. Examples are used to demonstrate the effectiveness of the proposed method.     

Interval set‐membership estimation for continuous linear systems

This article proposes a mixed interval set‐membership estimation (ISME) method for continuous linear time‐invariant (LTI) systems by combining the positive system theory and the set theory. The proposed ISME method gives a new mixed interval‐set estimation framework for continuous LTI systems, whose benefit consists in that it has potential to achieve a balance of computational complexity and robust state estimation conservatism with respect to the interval observer (IO) and the set‐valued observer (SVO) for continuous LTI systems. Particularly, the proposed ISME method first uses a coordinate transformation such that the original system is transformed into an equivalent system. Second, the equivalent system is partitioned into two subsystems, where the first subsystem has a Meztler and Hurwitz subsystem matrix and then an IO is designed for the first subsystem based on the positive system theory. Because it is not guaranteed that the second subsystem also has a Meztler and Hurwitz subsystem matrix, a zonotopic SVO is further designed for the second subsystem based on the set theory. Consequently, an integration of the two steps above provides the whole SE results for the original system. At the end of this article, an example is used to illustrate the effectiveness of the proposed ISME method.     

Feedback based adaptive compensation of control system sensor uncertainties

In this paper, the problem of adaptively compensating sensor uncertainties is addressed in a feedback based framework. In this study, sensor characteristics are modeled as parametrizable uncertain functions and a compensator is constructed to adaptively cancel the effects of sensor uncertainties, to generate an adaptive estimate of the plant output. Such an estimated output is used for the feedback control law. Adaptive control schemes using a model reference approach with sensor uncertainty compensation are developed for LTI plants with either known or unknown plant dynamics. A new feedback controller structure is developed for the case when the plant dynamics is unknown, to handle the plant and sensor uncertainties. Simulation results are presented to show that the proposed adaptive sensor uncertainty compensation designs significantly improve system tracking performance.     

A combined QFT/H ∞ design technique for TDOF uncertain feedback systems

A new way of incorporating QFT principles into H X -control design techniques for solving the two-degrees of freedom feedback problem with highly uncertain plants is developed. The proposed practical design approach consists of two stages. In the first stage, the robustness problem, due to plant uncertainties, is solved by H X -norm optimization. In this stage, the controller inside the loop (the first degree of freedom) is designed, with the ultimate goal of minimizing the cost of feedback. Minimization of the sensor white noise amplification at the input to the plant is also performed using QFT principles. In the second stage of the design, the prefilter outside the loop (the second degree of freedom), is used to achieve the tracking specifications by conventional classical control theory, as practiced by the QFT design procedure. The combined QFT/H X design procedure for single input-single output (SISO) feedback systems is directly applicable to multi input-multi output (MIMO) feedback uncertain systems. The efficiency of the proposed technique is demonstrated with SISO and MIMO design examples for higly uncertain plants.