Linear Robust Control (Prentice Hall Information and System Sciences)
| Date: 11 June 2011, 02:56
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A good idea of the scope of the book may be obtained from a perusal of the list
of contents. Chapter 1 introduces the idea of H1 optimization by considering a
number of simple scalar examples which are solved using Nevanlinna-Pick-Schur interpolation
theory. In this way the reader knows what H1 optimal control is about
after reading only a few pages. Chapter 2 deals with the use of singular values in
multivariable control system design. A multivariable generalization of the Nyquist
stability theorem and the interpretation of the minimum singular value of a matrix
as a measure of the distance to a singular matrix are used to establish robustness
results for linear time-invariant systems. The interpretation of the maximum singular
value as the maximum gain is then used to show how performance issues may
be addressed. Chapter 3 reviews background material on signals and systems and
introduces the small gain theorem and the bounded real lemma. The small gain
theorem states that stable systems can be connected to form a stable closed-loop if
the loop gain product is less than unity; it is the basis for the general robust stability
results. The bounded real lemma gives a condition for a linear time-invariant
system to have less than unity gain. Chapter 4 discusses linear fractional transformations
and their role in control systems. It is argued that various closed-loop and
open-loop design problems can be posed in terms of a linear fractional transformation
involving a fixed system known as the generalized plant and a to-be-designed
system known as the controller. Linear fractional transformations therefore provide
a general framework for controller synthesis theory and for computational software.
The synthesis problem we consider is to find a controller that achieves a specified
norm bound on a linear fractional transformation involving the controller and the
generalized plant. Because the established theory and sign conventions of linear
fractional transformations induce a positive sign convention on feedback problems,
we use a positive feedback sign convention throughout the book.
Chapters 5 to 8 develop the control system synthesis theory. We begin with a
brief treatment of the Linear Quadratic Guassian problem in Chapter 5. Chapters 6,
7 and 8 are the core of the book and concentrate on the synthesis of controllers that
meet H1-norm objectives. The main result is that a controller that satisfies the
objectives exists if and only if two Riccati equations have appropriate solutions.
In this case, all controllers that satisfy the objectives can be given in terms of
a linear fractional transformation involving a stable, norm bounded, but otherwise
unconstrained, parameter. The development of the LQG and H1 synthesis theories
is split into two parts. In the first, we analyze a finite-horizon version of the problem.
For this part the plant may be assumed to be time-varying. The second part tackles
the infinite-horizon extension by invoking limiting arguments. The infinite-horizon
results are only developed in a time-invariant setting—we restrict ourselves to timeinvariant
plant before taking limits. Our approach to the synthesis theory is based,
therefore, on time-domain techniques which are deeply rooted in the existing and
widely known theory of linear quadratic optimal control. The application to H1
optimization requires that we consider a quadratic objective function which is not
positive definite, but which connects precisely with the theory of linear, zero-sum
differential games with quadratic pay-off functions. This time-domain, optimalcontrol
based approach has several advantages. Firstly, the techniques are widely
known and are covered in excellent texts such as [11], [33] and [125]. Secondly,
they require almost no advanced mathematical theory. For the most part, a solid
background in linear algebra and differential equations is sufficient. Thirdly, the
main ideas and equations can be developed in a finite time horizon setting in which
stability issues do not arise. The sufficiency theory in this case is almost trivial,
amounting to little more than “completing the square”. Finally, they are applicable
to time-varying problems and are amenable to generalization to nonlinear systems.
In order to provide the reader with some insight into the alternative approaches
that have been developed, we have: (a) included two complete proofs of the bounded
real lemma, one algebraic and one based on optimal control; (b) covered the fourblock
general distance problem in some detail; (c) explored the connection with
factorization methods in several of the problems. The approach based on the fourblock
problem is given fairly detailed coverage because it is the only approach that
has yielded a complete treatment of the optimal cases and because it is able to
deal (easily) with problems involving optimization subject to the constraint that
the solution contains no more than a prespecified number of unstable poles. This
problem is of interest in frequency weighted model reduction applications which are
also covered.
Chapters 9 to 11 deal with the approximation of high-order systems by others of
lower order. This approximation process is known as model reduction. The inclusion
of model reduction is motivated by our belief that control system design cannot
be separated from the process of plant modelling. Any serious application of the
optimal synthesis methods in this book is bound to involve some model reduction. In
addition, the similarity of the mathematical techniques involved in model reduction
and H1 optimal control makes it appropriate to include this material.
Chapter 12 contains two design case studies. The first considers the design of a
controller to stabilize the vertical dynamics of the elongated plasma in a tokamak
fusion reactor and the second considers the design of a composition controller for a
high-purity distillation column.
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