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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 gain-scheduling
Gain-scheduling 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 gain-scheduling 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 gain-scheduling
methods).
Limitations of conventional gain-scheduling
Conventional gain-scheduling controllers are generally confined to near
equilibrium operation (because they are designed on the basis of the plant equilibrium
linearisations). Moreover, although gain-scheduled controllers are widely
applied, the underlying theory is poorly developed.
Velocity-based gain-scheduling
The velocity-based framework resolves many of the deficiencies of conventional
gain-scheduling.
A linear system (the 'velocity-based linearisation') is associated with
every operating point of a nonlinear system (not just the equilibrium points).
A family of velocity-based 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
non-equilibrium operation can both be accommodated. This suggests the
following velocity-based gain-scheduling design procedure.
- Determine the velocity-based linearisation family of the plant
- Design a linear controller for each member of the plant family.
- Realise a nonlinear controller with velocity-based linearisation
family corresponding to the linear controller family
designed at step 2.
The gain-scheduled 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 velocity-based gain-scheduling 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 gain-scheduling
approach.
An extended summary of velocity-based
modelling and control is also available (.pdf, 55Kb. Requires Adobe Acrobat Reader 3.1 or better to view).
Input-Output Linearisation/Dynamic Inversion
The velocity-based gain-scheduling approach is quite general and directly
supports the design of feedback configurations for which the closed-loop dynamics
are nonlinear. Dynamic inversion corresponds to the special case where the closed-loop
dynamics are linear. The velocity-based
approach to dynamic inversion is quite distinct from (and indeed in many
ways complementary to) standard input-output linearisation techniques based
on differential geometric methods. In particular, the velocity-based approach
- is a direct generalisation to nonlinear systems of the classical frequency-domain
pole-zero inversion approach. (cf. conventional methods are a generalisation
of Silverman's work on state-space 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 pole-zero inverse. (cf. the full state-feedback required
in conventional approaches, even in the linear case)
- decomposes the nonlinear design task into a number of straightforward linear
sub-problems; 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 gain-scheduling methodology. However, it is emphasised
that the velocity-based approach does not necessitate a slow variation
requirement.
LPV gain-scheduling & velocity-based methods
In addition to velocity-based gain-scheduling methods, a number of other approaches
have recently been developed. These are widely referred to as LPV gain-scheduling
methods owing to their use of a quasi-LPV/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 quasi-LPV form. However,
the literature typically takes the existence of a plant in LPV/quasi-LPV form
as its starting point, largely neglecting the critical issue of how general
nonlinear dynamics might be transformed to LPV/quasi-LPV 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/quasi-LPV form using
soundly-based 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 counter-examples
where the control design fails (closed-loop is unstable).
The velocity-based framework provides very general and soundly-based methods
for transforming systems into LPV/quasi-LPV
form.
MATLAB velocity-based gain-scheduling demo
Frequently Asked Questions
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