3.4 Control of Linear Oscillations and Relaxations 81
The matrix A is unstable, i.e., there exists some positive eigenvalues. Although
this example seems to be very simple, a numerical solution [16] of the alge-
braic Ricatti equation is required for a reasonable structure of the quadratic
performance functional. The main problem is that the nonlinear Ricatti equa-
tion has usually more than one real solution. However, the criterion to decide
which solution is reasonable follows from the eigenvalues of the transfer matrix
D = A − BR
−1
B
T
G.
Inverted pendulum systems are classical control test rigs for verification
and practice of different control methods with wide ranging applications from
chemical engineering to robotics [17]. Of course, the applicability of the linear
regulator concept is restricted to small deviations from the nominal behavior.
It is a typical feature of linear optimal regulators that they can control the un-
derlying system only in a sufficiently close neighborhood of the equilibrium or
of another nominal state. However, the inverted pendulum or several modifi-
cations [18, 19], e.g., the rotational inverted pendulum, the two-stage inverted
pendulum, the triple-stage inverted pendulum or more general a multi-link-
pendulum, are also popular candidates for the check of several nonlinear con-
trol methods. However, the investigation of such problems in beyond the scope
of this book. For more information, we refer the reader to the comprehensive
literature [20, 21, 22, 23, 24, 25].
In principle, one can also invert the optimal regulator problem, i.e., we ask
for the performance which makes a certain controller to an optimum regulator.
The first step, of course, is now the creation of a regulator as an abstract or
real technological device. We assume that the regulator stabilizes the system.
This is not at all a trivial task, but this problem concerns the wide field
of modern engineering [1, 26, 27, 28, 29]. The knowledge of the regulator is
equivalent to the knowledge of the transfer matrix D =
A − BR
−1
B
T
G
Y
∗
.
The remaining problem consists now in finding the performance index to which
the control law of the control instrument is optimal. This problem makes sense
because the structure of Q allows us to determine the weight of the degrees
of freedom involved in the control process [30].
3.4 Control of Linear Oscillations and Relaxations
3.4.1 Integral Representation of State Dynamics
Oscillations
Oscillations are a very frequently observed type of movement. In princi-
ple, most physical models with a well-defined ground state can be approx-
imated by the so-called harmonic limit. This is, roughly spoken, the expan-
sion of the potential of the system in terms of the phase space coordinates
X = {X
1
,X
2
,...,X
N
} up to the second-order around the ground state or