Title:

Time  and frequency  domain approximation techniques for optimal control

Approximation techniques for optimising control
functions using both time and frequency domain methods are
interest. The use of models of highorder systems with a
loworder statespace subsystem in parallel with a
convolutiondescribed subsystem is considered. An optimisation
algorithm for the parallel model, using the Dynamic Programming
approach, is developed and computed results are given
for an example. Algorithms for obtaining feedback gains
for the parallel model with a quadratic cost is developed,
using a variational approach. Similar algorithms are
also presented for the same model when the quadratic cost
includes a sensitivity function term. The convergence of
these algorithms is proved.
A continued fraction expansion (CFE) technique is proposed
for modelling multivariable transfer functions to give
good approximations to the initial transient response while
yielding the correct steadystate response for a class of in
puts. Another approximation technique, called Modified Pads
Approximation (MPA) which differs from the wellknown Pads
approximation, is also presented, for modelling multivariable
transfer functions. It is shown that, under certain conditions,
the CFB and MPA techniques yield the same model.
A technique is developed for obtaining approximations
to the transfer function of a system such that the steadystate
responses of the system and its loworder model are the same
for a class of inputs. The technique is based on modelling
the system's impulse response by a linear combination of
orthonormal functions. For multivariable systems, loworder
models of the transfer function matrices can be obtained by
applying the technique to each element of the system's
weighting function matrix. A minimal realisation of the
resulting loworder model of the transfer function matrix
can then be simply obtained.
New lower bounds for the minimum cost for convolutiondescribed
linear dynamical systems with quadratic cost
functions are presented. Besides its application as Stopping
conditions for iterative optimisation algorithms to decide
when a control sufficiently close to the optimal control has
been achieved, it can also be used to check the accuracy
of a simplified model to a system.
The CPE and MPA approximation techniques developed
before are used to approximate the optimal control for
regulator problems formulated in the frequency domain. It
is done by performing a sequence of simple spectral factorisations
which each require considerably less effort than the
complete spectral factorisation which is necessary for the
determination of the optimal control in the frequency domain.
It should be particularly useful for highorder linear
multivariable systems. A frequency domain algorithm is
proposed which iterates to give a suboptimal control with
increasingly detailed structure.
