Closed-Loop Control Learning - Neural Networks: Methodology and Applications - page 296

been chosen for training, is the real interval [ / 5]. It corresponds to a

moderate nonlinearity. The sampling frequency is 50 Hz.
Controller training was performed through the straightforward inversion of
the process model. The optimization algorithm, which was chosen to perform
training, is the BFGS algorithm (see Chap. 2). The state is supposed to be
completely observed. The task is to stabilize the unstable equilibrium. The
cost function takes into account the difference between the current angle and
the reference angle and the difference between the angular velocity and zero.
Those two quadratic deviations must be appropriately weighted in the cost
function; that choice has an impact on the e ciency of the controller.
The controller that was computed as described as above is tested for its
capacity to stabilize the system from an initial position equal to half the max-
imal deviation of the angle range, which was used for training. The operation
of that controller is satisfactory: it stabilizes the system quickly. Then, in order
to test the robustness of the control law, the control is disturbed by a mul-
tiplicative noise of the form (1 + κε ) , where is a numerical white noise and
κ is the noise factor of the control. Generally, one investigates the robustness
of the control law with respect to the external disturbances (state noise and
measurement noise). However, it is important, in practice, to guarantee that
the control law is robust with respect to itself because the control law may
be implemented with errors (numerical roundoff errors, electro-mechanical er-
rors for the servomotors, etc.). The e ciency of the controller depends on the
choice of the cost function, as shown in the following figures.
In the first experiment, the weight of the velocity deviation in the cost
function is larger to the angle deviation weight. Figure 5.3 shows a typical
trajectory of such a controlled system. The system is stabilized only if the
noise factor is smaller than 0.5. When the noise factor is larger, generally
the trajectory leaves its viability domain during the experiment (20 seconds).
The velocity is stabilized around the reference as shown on Fig. 5.3. The
stabilization of the position is slower and equilibrium is not reached within
the allotted time.
In the second experiment, the weight of the velocity deviation is smaller
than the weight of the angle deviation. The controller is more robust with
respect to the control noise ( κ = 3) as shown in Fig. 5.4.
If the weight of the angle deviation is chosen smaller than the weight of the
velocity deviation in the cost function, the system is ill-stabilized as soon as
a control noise exists in the controller. Thus, straightforward model inversion
method assumes that empirical knowledge of the system is available in order
to choose a relevant cost function. The robustness of the optimal controller
relies heavily on that choice.
To summarize, straightforward model inversion method is a simple solu-
tion. However, a deep knowledge of the system to be controlled may be needed
to implement it e ciently. It is necessary to check its robustness versus various
disturbances and modeling errors. Quite often, improvements of that method

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