Biomedical Engineering Reference
In-Depth Information
11.6.3.3 Parameter Optimization Methods
Parameter optimization methods for linear time-invariant (LTI) feedback design starts
with controller structures that are motivated by ideas from classical, modern, or other
techniques. Fundamentally, however, LTI refers to the basic concept that whether we
apply an input to the system now, or
T
seconds from now, the output will be identical
except for a time delay of
T
seconds. What is generally meant by controller structure,
however, is one in which the value of one or more parameters can be adjusted in any
system model. One example is a PI (proportional-plus-integral) controller structure, which
is a generic control loop feedback mechanism that is widely used in industrial control
systems and is the most commonly used feedback controller. A PID controller calculates
an 'error' value as the difference between a measured process variable and a desired
set-point. The controller attempts to minimize the error by adjusting the process control
inputs, which contain the coefficients (proportional gain) and
K
i
(integral gain), as well
as the controller transfer function parameter
K
p
+
K
i
/
s
.
The next step in a parametric method is to select a quality system performance that
will yield the most cost-effective and logical method, such as to acquire one from an
already solved LQG problem. This is advantageous to the extent that the cost yielded
by the structured controller by parameter search can then be compared to the absolute
minimum achieved by any controller, which is analytically computable.
Another economic approach is to formulate a weighted sum or maximum of various
performance indices, such as integrated square error in response to a step command,
integrated magnitude of frequency response across some band where a disturbance is
concentrated together with some indices representing actuator use. The idea of this is that
the weights define the relative importance of different aspects of system performance.
Finally, the designer may add explicit constraints on the values of the parameters, such
as bounds on closed-loop pole locations and open-loop frequency responses.
After a controller structure is determined, together with cost functions and other con-
straints, the designer may need to rationalize. Many techniques for numerical solution
of optimization problems resulting from control design problems have been proposed
in control literature [10 - 15] although it is beyond the scope of this topic to provide a
comprehensive overview of these parameter optimization methods.
11.6.3.4 Controller Design
As previously mentioned, the position of the shaft, and the applied force by the finger
to the shaft, should follow the desired force - position curve as closely as is possible. To
minimize this tracking error, the optimum controller design method was used to find the
best possible PID controller.
11.6.3.5 Solving the Problem in Simulink
The plant is a second-order linear system shown in Figure 11.14. The closed-loop system,
including the actuator and the PID controller is shown in Figure 11.15.
