Survey of gain-scheduling analysis and design

D. J. Leith, W. E. Leithead
2000 International Journal of Control  
The gain-scheduling approach is perhaps one of the most popular nonlinear control design approaches which has been widely and successfully applied in fields ranging from aerospace to process control. Despite the wide application of gain-scheduling controllers and a diverse academic literature relating to gain-scheduling extending back nearly thirty years, there is a notable lack of a formal review of the literature. Moreover, whilst much of the classical gain-scheduling theory originates from
more » ... e 1960s, there has recently been a considerable increase in interest in gain-scheduling in the literature with many new results obtained. An extended review of the gainscheduling literature therefore seems both timely and appropriate. The scope of this paper includes the main theoretical results and design procedures relating to continuous gain-scheduling (in the sense of decomposition of nonlinear design into linear sub-problems) control with the aim of providing both a critical overview and a useful entry point into the relevant literature. Introduction Gain-scheduling is perhaps one of the most popular approaches to nonlinear control design and has been widely and successfully applied in fields ranging from aerospace to process control. Although a wide variety of control methods are often described as "gain-scheduling" approaches, these are usually linked by a divide and conquer type of design procedure whereby the nonlinear control design task is decomposed into a number of linear sub-problems. This divide and conquer approach is the source of much of the popularity of gainscheduling methods since it enables well established linear design methods to be applied to nonlinear problems. (Whilst the analysis and design of nonlinear systems remains relatively difficult, techniques for the analysis and design of linear time-invariant systems are rather better developed). However, it is also emphasised that the benefits of continuity with linear methods often extend beyond purely technical considerations; for example, safety certification requirements are often based on linear methods and the development of new certification procedures using nonlinear approaches may well be prohibitive. Of course, the question must be asked as to whether the basic premise of such design approaches is in fact reasonable; that is, whether a wide class of nonlinear design tasks can genuinely be decomposed into linear sub-problems. While few results are available which relate directly to this fundamental issue, and it is well known that certain classes of problem present greater difficulty than others for gain-scheduling methods, the general usefulness of such methods is nevertheless well established both in practice and from a theoretical viewpoint. Despite the wide application of gain-scheduling controllers and a diverse academic literature relating to gainscheduling extending back nearly thirty years, there is a notable lack of a formal review of the literature. Moreover, whilst much of the classical gain-scheduling theory originates from the 1960s, there has recently been a considerable increase in interest in gain-scheduling in the literature with many new results obtained. A review of the gain-scheduling literature therefore seems both timely and appropriate. It is, unfortunately, impossible to cover the great wealth of gain-scheduling literature within the available space and the scope of this paper is thus necessarily limited. Firstly, no attempt is made to review the vast literature detailing specific applications of gain-scheduling methods. Secondly, in order to retain a reasonable focus to the paper, consideration is confined to methods based on the decomposition of the nonlinear design task into linear sub-problems; accordingly, the many interesting approaches based on decomposing the design task into simpler nonlinear sub-problems (including those involving decomposition into affine sub-problems) are not discussed. Thirdly, attention is restricted to continuous scheduling methods and no attempt is made to review the very extensive literature on hybrid/switched systems. The scope of the paper is thus restricted to the main theoretical results and design procedures relating to continuous gain-scheduling (in the sense of decomposition of nonlinear design into linear sub-problems) control and the aim is to provide a critical overview and useful entry point into the relevant literature. The subject matter covered clearly remains considerable and it has sometimes been difficult to achieve a satisfactory balance between the requirement to provide a concise critical overview of the field while covering the subject in reasonable detail. In addition, although substantial effort has been expended in striving to present as balanced perspective as possible on alternative methodologies, the reader should be aware that some degree of subjective judgement is surely inevitable. Any such deficiencies do not, of course, necessarily reflect the opinions of others. The paper is organised as follows. The theoretical results relating the dynamic characteristics of a nonlinear system to those of a family of linear systems are reviewed in section 2. The classical gain-scheduling design procedure is discussed in section 3 followed by a number of recent divide and conquer approaches which attempt to address a number of deficiencies of classical methods. LPV gain-scheduling approaches, which have recently been the subject of considerable research activity but are less strongly based on divide and conquer ideas, are reviewed in section 4 and the outlook is briefly discussed in section 5. The notation used is standard (see Appendix A). Linearisation theory Gain-scheduling design typically employs a divide and conquer approach whereby the nonlinear design task is decomposed into a number of linear sub-tasks. Such a decomposition depends on establishing a relationship between a nonlinear system and a family of linear systems. The main theoretical results which, for a broad class of nonlinear systems, relate the dynamic characteristics of a member of the class to those of an associated family of linear systems are reviewed in this section. These results fall into two main sub-classes. First, stability results which establish a relationship between the stability of a nonlinear system and the stability of an associated linear system. Second, approximation results which establish a direct relationship between the solution to a nonlinear system and the solution to associated linear systems. It is important to distinguish between these classes of result. The former are typically much more limited than the latter, being confined to specifying conditions under which boundedness of the solution to a particular linear system implies boundedness of the solution to the nonlinear system for an appropriate class of inputs and initial conditions. Notice that under such conditions the solutions are bounded but may otherwise be quite dissimilar. Reflecting this distinction, the discussion in the following sections often separately considers results relating both to stability and approximation. The section is organised as follows. Perhaps the most widespread approach for associating a linear system with a nonlinear one, namely series expansion linearisation theory, is first reviewed The literature relating to series expansion linearisations is, of course, extensive yet, unfortunately, also very fragmented and it is necessary to consolidate many separate results in writing this review. The approach taken here is, therefore, to provide an overview of the available body of theory in section 2.1 while referring the reader to Appendix B for detailed references. The series expansion linearisation is only valid in the vicinity of a specific trajectory or equilibrium point, and so there is considerable incentive to develop techniques which relax this restriction. Approaches which aim to increase the allowable operating envelope by utilising a family of linearisations (rather than just a single linearisation) are reviewed in sections 2.2-2.3. strengthen the classical frozen-input stability results discussed in section 2.2. Specifically, BIBO stability of the nonlinear system (1) is guaranteed provided the members of its velocity-based linearisation family are uniformly stable, unboundedness of the state x implies that w is unbounded (assuming the input r is bounded) and the class of inputs and initial conditions is restricted to limit the rate of evolution of the nonlinear system to be sufficiently slow (Leith & Leithead 1998b) . In addition, provided that the rate of evolution is sufficiently slow, the nonlinear system inherits the stability robustness of the members of the velocity-based linearisation family (Leith & Leithead 1998b) (with the usual trade-off between robustness and the restrictiveness of the slow variation condition required).. This velocity-based result involves no restriction to near equilibrium operation other than that implicit in the slow variation requirement; for example, for some systems where the slow variation condition is automatically satisfied the class of allowable inputs and initial conditions is unrestricted and the stability analysis is global. Divide & Conquer Gain-Scheduling Design Gain-scheduling design approaches conventionally construct a nonlinear controller, with certain required dynamic properties, by combining, in some sense, the members of an appropriate family of linear time-invariant controllers. Design approaches may be broadly classified according to the linear family utilised. Classical gainscheduling design approaches, based on the series expansion linearisation of a system about its equilibrium points, are discussed in section 3.1. Recent, and closely related, approaches based on neural/fuzzy modelling and off-equilibrium linearisations are considered, respectively, in sections 3.2 and 3.3. Classical gain-scheduling design Consider the nonlinear plant with dynamics, (1). The classical gain-scheduling design approach is based on the family of equilibrium linearisations of the plant and may be applied directly to a broad range of nonlinear systems. The design procedure typically involves the following steps
doi:10.1080/002071700411304 fatcat:oxv67zumpjecliasoaewuoeivm