Page 985 - The Mechatronics Handbook

0066_frame_Ch33.fm Page 9 Wednesday, January 9, 2002 8:00 PM

PID, PI, PD with Filtering
The conventional control structures used in this chapter are PI (proportional-integral), PID (proportional-
integral-derivative), and PD (proportional-derivative) with filtering coefficient.
The transfer function of the PI controller has the following expression:

Us() 1 T i s + 1
.

H PI = ----------- = .  1 + ---------- = . ------------------- (33.11)
e s() K R  T i s K R T i s
.
.
K R is the proportional factor, and T i is the time constant of the integrative component.
The transfer function of the PID controller is described by the following equation:
.
.
2
.
Us() 1 T i T d s + T i s + 1
. 
H PID = ----------- = .  1 + ---------- + T d s = . ------------------------------------------------ (33.12)
e s() K R  T i s  K R T i s
.
.
K R and T i have the same significance as previously mentioned, and T d is the time constant of derivative
component.
The transfer function corresponding to PD with filtering has the expression:
1 +
.
T d s
----------- =
H PDF s() = Us() . --------------------------- (33.13)
e s() K R 1 + a T d s
.
.
where the coefficient α could have values in the range 0.1–0.125.
Generally speaking, PID controllers are commonly used in industrial control systems and, therefore,
are well established. Nevertheless, the results obtained using a PID controller for complex control loops
are not very satisfactory because it could be costly and time consuming to retune such regulators. PI
controller is enough in situations where derivative action is not frequently used.
Observer
The theory of observers, started with the work of Luenberger and Ackermann, is fairly complete and
comprehensive. For the proposed axis, an (n − m − 1) order structure of the observer is adopted, where
n = 5 represents the order of the system and m = 1 is the number of outputs [25]. The model of the
servodrive is described by five state variables: two of them for the second-order model of the servovalve
and the other three for the third-order servoactuator. The use of a linear observer as a parallel model
reconstitutes the state-variables of the installation and delivers them to the controller. Two possibilities
could be followed: with partial and with global reconstruction. The solution chosen was with partial
reconstruction [12]. The complete system consists of the installation with nonlinearities, a parallel
second-order model for the servovalve, a third-order linear servoactuator, a correction matrix for the
observer, and a controller with five loops for the five state variables [23,25,26].
The block diagram of an electrohydraulic axis controlled with a third-order observer is shown in
T
Fig. 33.11, where A, b, c are the characteristic matrices of the linear system (electrohydraulic axis), k is
the correction matrix, and R represents the matrix corresponding to the controller.
The simulation diagram of the electrohydraulic axis, controlled by a third-order observer, as it is
depicted in SIMULINK, is illustrated in Fig. 33.12.
The algorithm used to compute the matrix k and r consists of the following steps:

i. achievement of the MM for servovalve and servoactuator;
ii. setting the state-variables of the process;
iii. obtaining the controller upon the dynamics of the closed-loop system;
iv. computing the correction matrix by using the desired poles for the observer [12].

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