20-07-2012, 04:37 PM
Loop Shaping
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A Basic Feedback Loop
In the previous chapter, we considered the use of PID feedback as a mecha-
nism for designing a feedback controller for a given process. In this chapter
we will expand our approach to include a richer repertoire of tools for shap-
ing the frequency response of the closed loop system.
One of the key ideas in this chapter is that we can design the behavior
of the closed loop system by studying the open loop transfer function. This
same approach was used in studying stability using the Nyquist criterion:
A BASIC FEEDBACK LOOP 325
The feedback loop in Figure 11.1 is influenced by three external signals,
the reference r, the load disturbance d and the measurement noise n. There
are at least three signals, , y and u that are of great interest for control,
giving nine relations between the input and the output signals. Since the
system is linear, these relations can be expressed in terms of the transfer
functions.
Performance Specifications
A key element of the control design process is how we specify the desired
performance of the system. Inevitably the design process requires a tradeoff
between different features of the closed loop system and specifications are
the mechanism by which we describe the desired outcome of those tradeoffs.
Frequency Domain Specifications
One of the main methods of specifying the performance of a system is
through the frequency response of various input/output pairs. Since spec-
ifications were originally focused on setpoint response, it was natural to
consider the transfer function from reference input to process output. For a
system with error feedback, the transfer function from reference to output is
equal to the complementary transfer function, T = PC/(1+PC).
Relations between Time and Frequency Domain Features
In Section 5.3 we described some of the typical parameters that described
the step response of a system. These included the rise time, steady state
error, and overshoot. For many applications, it is natural to provide these
time domain specifications and we can relate these to the eigenvalues of the
closed loop system, which are equivalent to the poles of the transfer function
T = PC/(1 + PC).