Neural Identification of Controlled Dynamical Systems and Recurrent Networks - Neural Networks: Methodology and Applications - page 232

Hopfield networks) are described. Finally, we show how to use that type of
networks for the identification of controlled dynamical system.

4.1 Formal Definition and Examples of Discrete-Time
Controlled Dynamical Systems

4.1.1 Formal Definition of a Controlled Dynamical System by
State Equation

Since all the applications of neural networks to control are implemented on
computers, the present chapter, and the next, will be essentially devoted to
discrete-time dynamical systems. The sampling techniques of analog signals
that are delivered by physical devices will not be addressed.
The mathematical model of a dynamical system is defined by a set E ,
called the state space of the system, and an evolution equation, which de-
scribes completely the evolution of the state of the system in state space
from the initial state conditions. In most problems, the evolution is said to be
autonomous: the evolution law is not time-dependent. We will stick to that
hypothesis in order to alleviate the notations. In control problems, the state
of the system at time t +∆ t does not depend on the state of the system at
time t only: it also depends on an external signal at time t , which is called
input or control of the system. In such a case, the system is termed controlled,
in contrast to autonomous. The set of controls will be referred to as F .Using
classical notations, we will write

• x ( t ) ∈ E for the state of the system at time t ,

• u ( t ) ∈ F for the value of the control at time t .

Thus, in order to define the whole state trajectory of a controlled system
from time 0 to time τ , one needs the initial state x (0) of the system and
the control trajectory [ u ( t )] t∈ [0 ,τ ] . The control system is designed in order
to build a control trajectory that is as close as possible to a reference state
trajectory, or that minimizes the cost of the trajectory with respect to a given
cost function.
Notice that if a closed loop control law is implemented, i.e., if the control
system computes the control as a function of the current state (or the past
state trajectory of the system, or the past results of measurements performed
on the system), then the whole system (controlled dynamical system+control
system) is an autonomous dynamical system. The design of closed-loop control
law and their neural implementation will be the main topic of the next chapter.
As mentioned above, we focus here on discrete-time dynamical systems.
A discrete-time dynamical system can be derived from a continuous-time dy-
namical system by sampling the state trajectory of the system. As previously
in Chap. 2, the sampling period is denoted by T and we write time k for time