Category A PRACTICAL. APPROACH TO. ROBUSTNESS. ANALYSIS

The book is organized as follows:

■ Chapter 1 (Part 1): the ц framework is presented. The general case of a closed loop system subject to both parametric uncertainties and neglected dynamics is considered. The first step is to transform this uncertain closed loop into a standard interconnection structure. The s. s.v. is then introduced as a tool for studying the robustness proper­ties of this interconnection structure, and thus equivalently those of the original uncertain closed loop system. Beyond its mathematical definition, a physical interpretation of the s. s.v. is especially given.

■ Chapter 2 (Part 1): the airplane and missile examples are explained in details. The plant model is described, as well as the method for synthesizing the control law. The way to introduce uncertainties in the model is also presented.

■ Chapters 3 and 4 (Part 2): the issue is to transform the ori­ginal uncertain closed loop into the standard interconnection struc­ture. Chapter 3 is devoted to the problem of parametric uncertainties entering the open loop plant model, since this appears as the key is­sue to be solved. The idea is to realize this parametrically uncertain system as an LFT transfer, in which the uncertainties appear as an internal feedback. To a large extent, this difficult problem remains open from a theoretical point of view. A simple solution is presented here for the case of physical systems. As an illustration of the tech­niques of chapter 3, the standard interconnection structures for the airplane and missile problems are obtained in chapter 4.

■ Chapter 5 (Part 3): different ways to compute (bounds of) the clas­sical s. s.v. n are presented. Are considered the particular case of real parametric uncertainties and the general case of mixed uncertainties (i. e. a model perturbation simultaneously containing real parametric uncertainties and neglected dynamics).

■ Chapter 6 (Part 3): the aim is twofold. On the one hand, the H tools developed in chapter 5 are evaluated on the aeronautical ex­amples. On the other hand, the resolution of the first physical prob­lem, which is considered in this book, is detailed (point (1) of the previous section).

■ Chapter 7 (Part 4): through the presentation of some of the phys­ically motivated problems, which are solved in this book, this chapter illustrates in a rather qualitative way that the skewed s. s.v. v can solve a large set of engineering problems. It is emphasized that many /r problems, which are encountered in practice, appear to be skewed /л problems.

■ Chapter 8 (Part 4): because of the practical importance of the skewed s. s.v. v (see chapter 7), it is interesting to develop specific tools for computing skewed fi bounds: this is done in chapter 8.

■ Chapter 9 (Part 4): the aim is twofold. The skewed v tools de­veloped in chapter 8 are evaluated on the missile example. On the other hand, the resolution of the physical problem, which corresponds to point (2) of the previous section, is detailed. Note that the skewed ц tools of chapter 8 will also be used in chapters 11 and 12.

■ Chapters 10, 11 and 12 (Part 5): these chapters present the resolution of the problems, which correspond to points (3) to (5) of the previous section.

This book is organized in a particular way, from the simplest topics to

the most technical ones. The first part introduces the /r framework and

presents the applicative examples. The second part focuses on the way

to transform a parametrically uncertain plant into an LFT form. The third part focuses on the application of classical p tools. All these three parts are expected to be readable by a large audience, with the excep­tion of chapter 5, which presents methods for computing bounds of the s. s.v.. The reading of this chapter can be nevertheless avoided, since a summary is done at the end of this chapter.

The fourth part is devoted to skewed p problems. To a large extent, this part is here again expected to be readable by a large audience, ex­cept chapter 8 which presents computational methods: a summary is nevertheless done at the end of this chapter.

The last part is the most technical. Unlike what is done before, the theoretical and practical results are presented inside a same chapter. Moreover, when compared to the problems of parts 3 and 4, the prob­lems of part 5 are more sophisticated. Nevertheless, part 5 presents new solutions to difficult engineering problems.

1. GLOSSARY

LFT: Linear Fractional Transformation

LMI: Linear Matrix Inequality

LP: Linear Programming

LTI: Linear Time Invariant

MIMO: Multi Inputs Multi Outputs

m. s.m.: multiloop stability margin

SIDF: Sinusoidal Input Describing Function

SISO: Single Input Single Output

s. s.v.: structured singular value ц

Three applicative examples are treated. The first one is a longitudinal missile autopilot. The linearization of a nonlinear missile model at a trim point is considered, with parametric uncertainties in the stability

derivatives and unmodeled high frequency bending modes.

The second applicative example is the lateral flight control system of a civil transport aircraft. Depending on the problem to be solved, we consider either the single rigid aerodynamic model (with parametric uncertainties in the stability derivatives), or a more complete model in­cluding a flexible structural model (with damping ratio about 1 % for the bending modes).

The third application is a telescope mock-up used to study high ac­curacy pointing systems. The mock-up, which is composed of a two axis gimbal system mounted on Bendix flexural pivots, is representative of very flexible plants (with damping ratio about 0.1 % for the bending modes).

The following problems will be solved in parts 3, 4 and 5:

1. Application of classical p tools to the robustness analysis of a rigid aircraft or missile (chapter 6): the idea is to evaluate existing meth­ods for computing bounds of the s. s.v. on realistic examples. The two examples are complementary: because of the small number of para­metric uncertainties in the missile model, exponential time methods for computing bounds of the s. s.v. p can be applied. Conversely, since there is a large number of parametric uncertainties in the air­craft model, only polynomial time methods can be applied.

2. Advanced robustness analysis of the missile (chapter 9): the robust stability and performance properties of the autopilot are studied in the presence of uncertain stability derivatives and unmodeled bending modes. This example also emphasizes the usefulness of the skewed p tools for some classes of practical problems.

3. Computation of the robustness margin in the special case of flexible structures (chapter 10): the s. s.v. p(p) is to be computed as a func­tion of ui. The robustness margin is then obtained as the inverse of the maximal s. s.v. over the frequency range. As said above, p(p) is usually computed at each point p of a frequency gridding, and the ro­bustness margin is deduced as the inverse of the maximal s. s.v. over this gridding. In some special cases such as the control of flexible structures, the p plot (corresponding to the value of the s. s.v. p(u>) as a function of ш) may present narrow and high peaks, so that it is possible to miss the critical frequency (i. e. the frequency which corresponds to the maximal s. s.v. over the frequency range), even when using a very fine frequency gridding. If this critical frequency is missed, the robustness margin is overevaluated, i. e. the result is too optimistic. Chapter 10 proposes a method which computes an estim­ate of the s. s.v. p(p) as a function of w and which gives a reliable value of the robustness margin. The method is first applied to the flexible transport aircraft, and then to the telescope mock-up.

4. Computation of a robust delay margin (chapter 11): the issue is to analyze the robustness properties of a closed loop in the face of clas­sical model uncertainties (uncertain parameters and neglected dy­namics) and uncertain time delays. This difficult problem presents a great practical interest. Indeed, when embedding control laws on a real-time computer, delays are to be considered simultaneously at the plant inputs (because of the time needed to compute the value of the plant input signal as a function of the plant output signal) and outputs (because of the sensors measuring the plant output signal). The method is applied to the missile.

5. Detection of limit-cycles in a nonlinear parametrically uncertain closed loop (chapter 12): as an extension of the famous Lur’e problem, con­sider the interconnection of a Linear Time Invariant (LTI) system (subject to LTI parametric uncertainties) with autonomous separable nonlinearities (e. g. saturations). The first issue is to detect the pres­ence of a limit-cycle inside this closed loop with a necessary condition of oscillation. The second issue is to guarantee the absence of limit – cycles despite parametric uncertainties, with a sufficient condition of non-oscillation. The ц and skewed /.t tools provide a solution to this interesting nonlinear analysis problem.

In many physical situations, a plant model is often provided with a qualitative or quantitative measure of associated model uncertainties. On the one hand, the validity of the model is guaranteed only inside a frequency band, so that nearly nothing can be said about the behavior of the real plant at high frequencies. On the other hand, if the model is derived on the basis of physical equations, it can be parameterized as a function of a few physical parameters, which are usually not perfectly known in practice.

This is e. g. the case in aeronautical systems: as an example, the aero­dynamic model of an airplane is derived from the flight mechanics equa­tions. When synthesizing the aircraft control law, it is then necessary to take into account uncertainties in the values of the stability derivatives, which correspond to the physical coefficients of the aerodynamic model.

Moreover, this airplane model does not perfectly represent the beha­vior of the real aircraft. As a simple example, the flight control system or the autopilot are usually synthesized just using the aerodynamic model, thus without accounting for the flexible mechanical structure: the cor­responding dynamics are indeed considered as high frequency neglected dynamics, with respect to the dynamics of the rigid model[1].

Summarizing, a model never perfectly represents the real plant to be controlled, and it is necessary to deal with associated model uncertain­ties. These correspond, either to uncertainties in the physical parameters of the plant (and more generally model perturbations inside the control bandwidth), or to high frequency unmodeled or neglected dynamics (un­
certainties beyond the control bandwidth).

However, many control design procedures only use the nominal model of the plant, and treat uncertainties in an incomplete or heuristic way. H synthesis schemes exist, which account for the available information on the nature and structure of these uncertainties: even if a great deal of work has been devoted to this subject, these design methods remain difficult to use, and it is not easy to control the order and the structure of the resulting controller. Note moreover that a properly designed control law can be robust in the face of uncertainties, even if these ones were not explicitly taken into account during the design process. Engineers often use their physical knowledge of the plant to design in an heuristic way control laws, which appear a posteriori sufficiently robust.

In this context, the issue is rather to validate a control law by analyz­ing its robust stability and performance properties. Various methods are available for solving this problem, depending on the nature and struc­ture of the uncertainties. We focus here on the structured singular value (s. s.v.) approach. The first reason is that the s. s.v. /xprovides a general framework to robustness analysis problems. As a second justification of this choice, the ц approach has been successfully applied to industrial problems.

Two general issues arise when applying this method, which have mo­tivated a great deal of work since the beginning of the Eighties. The first one is to put a specific control problem into a standard form, which is called an LFT (Linear Fractional Transformation). When applying the Ц tools to this standard LFT form, the second problem concerns the computational requirement, which must remain reasonable even for large dimension problems.

More precisely, the s. s.v. ц(ш) is to be computed at each point w of a frequency gridding, and /x lower and upper bounds are computed instead of the exact value of /x 2. Methods for computing these /x bounds can be divided into two large categories, namely the exponential and polynomial time ones: the computational requirement of exponential (resp. polyno­mial) time methods increases exponentially (resp. only polynomially) with the size of the problem.