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121 pages, year of publication: 1999

price: 40.00 €

Periodic Feedback for Linear Systems and Optimal Control of Bilinear Systems

Let us consider a single input T-periodic system

dx/dt=A(t)x+B(t)u

where x(0)=x_{0} and x in R^{n}. We assume that it is controllable
on [0,T] and that the system matrices depend analytically on time.
The main result of the thesis is to show that it is possible to find a
T-periodic feedback of the form

u=K^{t}(t)x

such that any matrix M in M_{n}(R) which we call the monodromy
matrix, can be chosen such that the monodromy operator
Phi(T,0) of the closed-loop system satisfies

Phi(T,0)=exp(MT)

In this way, we perform monodromy matrix assignment which generalizes
to periodic systems the notion of pole placement for linear autonomous
systems. We give a constructive way of synthesizing the periodic
feedback via an auxiliary optimal control problem with
fixed end point conditions.

The research is structured in the following way:

We provide first a motivating example for studying such a
problem. The stabilization of a periodic orbit which undergoes
possibly a period doubling bifurcation is studied. After suitable coordinate transformations, the
linearization around the periodic solution leads to a periodic linear
control system which will be generically controllable. A classical
stabilization approach via the periodic Riccati equation is
proposed. In case the mode corresponding to the period doubling is linearly
uncontrollable, the bifurcation cannot be eliminated by linear
feedback but a center manifold approach shows that the stability of
the citical orbit can be guaranteed in general by nonlinear
feedback.

Next, we show that it is possible to construct a periodic feedback
which answers the problem of monodromy matrix assignment. A consequence
is that the limitations for Jordan block assignment in the autonomous
case disappear in a periodic context.

Then, we show how to construct the periodic gain via an
optimal control problem. We have existence of a solution, normality
and appearance of two analogues to the Riccati equation which arises
in the L-Q optimization problem. These are two equations in the Lax form. They
simplify the analytical treatment and help characterizing the boundary
conditions of the optimal control problem.

This leads to the next step. We provide a theoretical justification for the two Lax
equations obtained in the previous chapter. They describe the
dynamics on coadjoint orbits
for two group actions which appear naturally when studying affine control systems made of
right-invariant and
left-invariant vector fields on Lie groups. The analoque to the
solution to the Riccati equation can be identified with the momentum
mappings associated to the corresponding hamiltonian actions. This helps studying complete
integrability of the hamiltonian dynamics corresponding to the optimal
control problem under study. Such a dynamics is completely integrable
in general in a suitable sense.

Finally, we present a framework for comparing our optimal control
problem with the standard L-Q problem. The advantage is that we
place the poles whereas the L-Q method
stabilizes only. We finally
show that the limitations concerning ouput pole placement disappear in
a periodic context: with a SISO controllable and observable system, we
can generically place any monodromy matrix.

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