Motivation
Nonlinear dynamic systems are everywhere, but tools for the analysis/design
of nonlinear systems are poorly developed. No system is, in reality, linear
but methods for linear systems are well developed and a wealth of practical
experience with them has been accumulated. Hence, it is attractive to
adopt a divide and conquer approach whereby the analysis/design task for a nonlinear
system is decomposed into a number of linear tasks.
Conventional gainscheduling
Gainscheduling is a divide and conquer approach for the design of nonlinear
control systems which has been applied in fields ranging from aerospace to process
control. The conventional gainscheduling design approach typically involves
 linearise the nonlinear plant about a number of equilibrium points
 design a linear controller for each of the plant linearisations
 combine the linear controllers to obtain a nonlinear controller
(See survey of gainscheduling
methods).
Limitations of conventional gainscheduling
Conventional gainscheduling controllers are generally confined to near
equilibrium operation (because they are designed on the basis of the plant equilibrium
linearisations). Moreover, although gainscheduled controllers are widely
applied, the underlying theory is poorly developed.
Velocitybased gainscheduling
The velocitybased framework resolves many of the deficiencies of conventional
gainscheduling.
A linear system (the 'velocitybased linearisation') is associated with
every operating point of a nonlinear system (not just the equilibrium points).
A family of velocitybased linearisations is therefore associated with
the nonlinear system. This family embodies the entire dynamics of the
nonlinear system and so is an alternative representation. It is emphasised
that this representation is valid globally and does not involve any restriction
to the vicinity of the equilibrium points. Large transients and sustained
nonequilibrium operation can both be accommodated. This suggests the
following velocitybased gainscheduling design procedure.
 Determine the velocitybased linearisation family of the plant
 Design a linear controller for each member of the plant family.
 Realise a nonlinear controller with velocitybased linearisation
family corresponding to the linear controller family
designed at step 2.
The gainscheduled controller is not inherently confined to near equilibrium
operation or subject to any slow variation constraint. As a concrete illustration
of this lack of restrictions, the velocitybased gainscheduling approach can
be used to design a dynamic inversion controller which is valid globally and
does not involve any slow variation constraints whatsoever. This freedom
is achieved while still retaining the divide and conquer approach and continuity
with linear methods which is the principal advantage of the conventional gainscheduling
approach.
An extended summary of velocitybased
modelling and control is also available (.pdf, 55Kb. Requires Adobe Acrobat Reader 3.1 or better to view).
InputOutput Linearisation/Dynamic Inversion
The velocitybased gainscheduling approach is quite general and directly
supports the design of feedback configurations for which the closedloop dynamics
are nonlinear. Dynamic inversion corresponds to the special case where the closedloop
dynamics are linear. The velocitybased
approach to dynamic inversion is quite distinct from (and indeed in many
ways complementary to) standard inputoutput linearisation techniques based
on differential geometric methods. In particular, the velocitybased approach
 is a direct generalisation to nonlinear systems of the classical frequencydomain
polezero inversion approach. (cf. conventional methods are a generalisation
of Silverman's work on statespace inversion techniques).
 requires, in general, only a measurement or estimate of the scheduling variable,
r . Frequently, r depends on only a small number of elements of the state
and/or input vectors. Indeed, in the case of purely linear systems, there
is no scheduling variable and, consequently, plant measurements are not required
to implement the polezero inverse. (cf. the full statefeedback required
in conventional approaches, even in the linear case)
 decomposes the nonlinear design task into a number of straightforward linear
subproblems; that is, the methodology supports the divide and conquer philosophy
and maintains continuity with well established linear methods. In this sense,
it is closely related to the gainscheduling methodology. However, it is emphasised
that the velocitybased approach does not necessitate a slow variation
requirement.
LPV gainscheduling & velocitybased methods
In addition to velocitybased gainscheduling methods, a number of other approaches
have recently been developed. These are widely referred to as LPV gainscheduling
methods owing to their use of a quasiLPV/LPV representation of the plant and
controller. A considerable body of results now exists relating to the
design of controllers for plants which are in LPV or quasiLPV form. However,
the literature typically takes the existence of a plant in LPV/quasiLPV form
as its starting point, largely neglecting the critical issue of how general
nonlinear dynamics might be transformed to LPV/quasiLPV form. It
is important to emphasise the importance of this issue since the rigorous basis
of LPV methods is removed if the plant is not placed in LPV/quasiLPV form using
soundlybased techniques.
Apparently lacking practical, generally applicable methods for carrying out
such a transformation, a number of ad hoc approaches have been proposed in the
literature. Although these might lead to acceptable control designs on some
occasions, this need not be the case in general. E.g. for one such popular
method at least it is straightforward to devise counterexamples
where the control design fails (closedloop is unstable).
The velocitybased framework provides very general and soundlybased methods
for transforming systems into LPV/quasiLPV
form.
MATLAB velocitybased gainscheduling demo
Frequently Asked Questions
