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Fig. 5.1. Principle of a closed-loop control
state noise or observation noise, then the stability of the controlled system is
again a crucial issue.
In the previous paragraphs, we just mentioned control laws that depend
on the current state or on the current observation only. Such a control law
is called a static control law. Actually, all the past information can be used
for implementing the current control. Such a control law is called a dynamic
control law. In practice, however, the complexity of the computation of the
control law is must obey stringent time constraints. The computation must
be performed during the sampling period of the controller to implement the
control in real time.
5.1.2 Controllability
The purpose of a control law cannot always be achieved. Controllability is
the property of the system whereby it can reach any target if it is provided
with an appropriate control law. Note that the simplest dynamic controlled
systems, such as linear systems, are not necessarily controllable when the state
dimensionality is larger than 1.
Consider for instance the following linear system:
x ( k +1)= x ( k )+ 1
0
u ( k ) .
Its order is 2 and the control is scalar. It is not possible to change the sec-
ond component of the state with the scalar control. On the other hand, it is
very easy to show that the following linear system enjoys the controllability
property,
x ( k +1)= 11
01
x ( k )+ 1
0
u ( k ) .
Controllability can readily be expressed for a linear system: in order to reach
any state, it is necessary and su cient to be able to reach the state zero from
any initial state [Kwakernaak et al. 1972].
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