機密等級 : 想看就看,懶的理你
Consider the PID control system in Fig.1. 
with the perturbation 
denotes torque
constant. The plant (induction motor) 
with the perturbation 
, the tachometer
and the PID controller are of the following
form:
where 
is a conversion constant, 
is inertia of the motor and load , 
is
viscous constant, 
is constant and the
plant perturbation 
and 
are assumed to be stable but uncertain. Suppose 
and 
are bounded according to
, 
where the function 
and positive constant
are stable and known.
The robust stability result in [] reveals that if a controller
is chosen so that the nominal closed
loop system (free of 
) in Fig.1 is asymptotically
stable, and the following inequality holds,
then the closed loop system in Fig.1 is also asymptotically stable
under plant perturbation 
in ( ) , where
the 
norm in ( ) is defined as
i.e., the maximum peak of the spectral density of 
.
However, often robust stability alone is not enough in control
system. Optimal tracking performance is also appealing in many
practical control engineering applications. Therefore, the mixed
control problem is formulated as follows:
for the nominal closed loop system in Fig.1 , subject to the robust
stability constraint ( ), where 
is the
tracking error and 
is actuator signal.
That is, under the 
constraint ( ), the
summation of error energy and the control energy in ( ) must be
as small as possible. from the above analysis, the PID mixed 
control design problem involves how to specify a PID controller
to achieve the optimal tracking ( ) subject to the robust stability
constraint ( ).
In the case with uncertain disturbance 
and measurement noise 
(see Fig.1), if
the desired attenuation level 
and
are specified, then
where 
and 
are weighting functions.
Optimal Control Design
By Parseval's theorem [ ], we have
where both 
and 
for i=1,2 are Hurwitz polynomials of s with appropriate degree.
The minimization problem in the above equation can be solved
via the aid of the residue theorem. Let 
and 
;if
The value of 
can be found easily from
published tables[ }:
