Let P be given by the IPZ transfer function
) , 1 ( 1
2
1 sL
v e
sT s k sT
P
(3.29)
and choose the desired set point response to 1 ,
1 sL
cl
y e
M sT
(3.30) then
), 1 (
1 ) (
) 1 (
1 ) (
) 1 )(
1 (
) 1 (
1 1
1 2 1 1
2 1
2 1
2
sT T
T T k
T T sT
T T T k
T T T
T k
T
sT sT
k
sT M s
cl v
cl cl
cl v
cl cl
v
cl v
u
(3.31)
and uff can be generated as the output of the sum of a constant gain and two low pass filters. This is necessary if the DCS-systems have no component for high order filters or no possibility to implement the pure derivative. Observe that the calculations made above are independent of the feedback controller C. Under ideal conditions, the control signal ufb
remains constant during a set point change and the purpose of C is simply
CONTR MV
OUTP SP
Mu FF My y
r
uc
Figure 3.10 Implementation of the feedforward structure in Figure 3.9 in a DCS system.
To let the operator manipulate the correct set point, the objects should be grouped together and presented as one controller element in the operator station, as indicated by the dashed line.
to give good disturbance rejection and robustness. In reality, there will be a control error during a set point change due to modeling errors which the feedback loop takes care of by driving e to zero. Figure 3.10 shows how the feedforward part could be implemented in an industrial control system to make use of its anti-windup, bumpless transfer, and other functions. Observe that normally there is a pure time delay in My, which also should be implementable in the control system.
How should Tcl be chosen? Looking at the shape of the control signal gives a hint. The transfer function from set point to control signal is
), 1
(
) 1 ( ) 1 (
2
1 cl
v u
ru sT
sT sT
k M s
G
(3.32) and we have lead-network (low-pass filtered derivative) followed by a zero-pole-network. If Tcl < T2, more lead action is added to the first lead-network, and Tcl > T2 gives a low-pass filter. Observe that T1 is always greater than T2.
0 10 20 30 40 50 60 70 80 90 100
0 0.2 0.4 0.6 0.8 1.0 1.2
Process output
0 10 20 30 40 50 60 70 80 90 100
0 0.2 0.4 0.6 0.8 1
Time (s)
Control signal
Figure 3.11 Set point step response of the 2DOF-controller. Tcl = 5 (solid), Tcl = 10 (dashed), and Tcl = 20 (dash-dotted). The dotted line is the set point.
If r is a unit step, the initial value theorem gives
. )
0 (
1 2
T T k u T
cl v
c (3.33) This can also be seen in the partial fraction expansion of Mu in (3.31).
Figure 3.11 shows a simulation where the process is given by kv=0.01, T1=200, T2=10, L=2. It clearly shows how the relation between Tcl and T2
affects the shape of uc. The selection of Tcl will be a trade-off between performance and control action, and letting Tcl T2 is a good choice.
Since there is an integrator in the process, the control signal will always go to zero in absence of disturbances. However, due to the cancellation of the slow process zero in í1/T1 by (3.32), the control signal is slowly brought back to zero long after we have reached the set point.
But the control signal must behave like this to maintain the output at the
0 50 100 150 200 250 300
0 0.2 0.4 0.6 0.8 1.0 1.2
Process output
0 50 100 150 200 250 300
0 0.2 0.4 0.6 0.8 1
Time (s)
Control signal
Figure 3.12 Dashed í 2DOF-controller Tcl= 10, solid í PI with aggressive tuning, and dash-dottedí PI with robust tuning. The thin dashed curve is the set point. Note that the 2DOF-controller cancels process dynamics from rĺ y but not from load disturbance d, therefore there is a slight overshoot when the disturbance acts on the system.
desired value. The physical explanation is that the energy flow from the steam to the cylinder and paper is slowly increasing which implies that the steam consumption is slowly increasing. To compensate for this the controller must add extra steam to the cylinder long after the set point has been reached. Therefore, even though the set point has been reached the system has not reached steady-state, and the cylinder and paper temperatures are still increasing.
Figure 3.12 shows how a well-tuned PI controller behaves without the feedforward compared to the 2DOF-controller. What is meant by well tuned is obviously relative. The tuning method used for the PI controller is introduced in Chapter 5. The method has one user parameter that determines the robustness of the loop and two different settings are used in the figure, here denoted as aggressive and robust. The two settings are chosen to give both a faster and slower response compared to the 2DOF-controller. The 2DOF-controller gives a smoother performance in set point response. Since the transfer function from r to y is a first order system, the frequency response does not have a peak as in Figure 3.8, which is a nice feature of the 2DOF-controller structure. It is also an advantage that this structure is easy to implement into most commercial
0 20 40 60 80 100 120 140 160 180
0 0.5 1.0
Process output
0 20 40 60 80 100 120 140 160 180
0 0.5 1.0
Time (s)
Control signal
Figure 3.13 Robustness analysis to modeling errors of the 2DOF-controller. Perfect process model (solid), doubled time delay in process (dotted), and doubled process gain (dashed). The dash-dotted line is the set point.
DCS-systems, which is not the case for the controller structure presented in the following section.
Finally, the sensitivity to modeling errors is investigated. Figure 3.13 shows closed loop simulations where the time delay and gain in the process is increased by a factor two. The response in process output is no longer equal to the desired response in presence of modeling errors since the feedback loop is active, and the second term in (3.27) is not canceled by Mu. However, the control system proves to handle model errors well.