A First Course in FUZZY and NEURAL CONTROL

2.1. INTRODUCTORY EXAMPLES: PENDULUM PROBLEMS 19

like a brake. In technical terms, we need to adddampingto the system. The
modified control law is now

u(t)=−αφ(t)−βφ ̇(t) (2.8)

forα> 1 andβ> 0. Because of the second term inu(t), these types of control
laws are calledproportional-derivative(feedback) control laws, or simply PD
control.
To determine if Equation (2.8) is a good guess, as before, we look at the
characteristic equation
z^2 +βz+α−1=0 (2.9)

of the closed-loop, second-order linear differential equation

φ®(t)+βφ ̇(t)+(α−1)φ(t)=0 (2.10)

Forα> 1 andβ> 0 , the roots of Equation (2.9) are

z=

−β±

q

β^2 −4(α−1)

2

and hence both have negative real parts. Basic theorems in classical control
theory then imply that all solutions of Equation (2.2) with this control law will
converge to zero astgets large. In other words, the PD control laws will do the
job. In practice, suitable choices ofαandβare needed to implement a good
controller. Besidesαandβ,weneedthevalueφ ̇(t), in addition toφ(t),in
order to implementu(t)by Equation (2.8).
Suppose we can only measureφ(t)but notφ ̇(t), that is, our measurement
of the state

x(t)=

μ

φ(t)

φ ̇(t)

∂

is of the form
y(t)=Cx(t) (2.11)

for some known matrixC.Here,C=

°

10

¢

.

Equation (2.11) is called themeasurement(oroutput)equation,that,
in general, is part of the specification of a control problem (together with (2.4)
in the state-space representation).
In a case such as the above, a linear feedback control law that depends only
on the allowed measurements is of the form

u(t)=KCx(t)

Of course, to implementu(t), we need to estimate the components ofx(t)that
are not directly measured, for exampleφ ̇(t), by some procedures. A control law
obtained this way is called adynamic controller.
At this point, it should be mentioned thatuandyare referred to asinput
andoutput, respectively. Approximating a system from input-output observed
data is calledsystem identification.