As described previously, the steam and air system is combined by mid-ranging control. Generally, mid-mid-ranging refers to control problems where there is one process output and two or more manipulated inputs, see
7 8 9
Sheet moisture (%)
50 55 60 65
Dew point exhaust air (o C)
0 200 400 600 800 1000 1200
0 50 100
Time (s) Control signal supply air (%)
Figure 6.7 Results of the prestudy where it is shown that the supply air has a large impact on the sheet moisture, measured at the reel-up. The dew point of the exhaust air is also shown.
[Shinskey, 1978], [Allison and Isaksson, 1998] and [Allison and Ogawa, 2003]. A common example in the pulp and paper industry is consistency control of the pulp that is pumped from the storage tower, see Figure 6.8.
The controller has two actuators, a large dilution valve and a smaller valve for fine adjustments. The main purpose of the water through the large valve is to make the pulp pumpable. Nevertheless, it is also important to avoid letting too much water being injected at the bottom of the tower to prevent the small valve from being closed, and vice versa, since the small valve often has a smaller operating range and finer resolution. One solution is to let the small valve return to its midpoint or target value in steady state. This is known as mid-ranging. In general, the effect of the inputs differs significantly in both range and speed, and sometimes in cost. When there is a cost tied to the inputs, the optimal target value for the expensive input is often not the midpoint.
Four types of mid-ranging structures
There exist several different alternatives to implement a mid-ranging controller, and four different structures are described below. It is assumed that P1 is a fast process with a small range, and P2 is a slow process with a large range. The signals u1 and u2 are control signals for P1 and P2, respectively, and r1 is set point for y and r2 set point for u1. The last three mid-ranging structures are evaluated in [Allison and Isaksson, 1998]. The
Pulp tower
Water
QC
Figure 6.8 The consistency in the pulp tower is typically 8í12 %. Dilution to pumpable suspension takes place at the bottom of the tower by injection of dilution water. An agitator helps the mixing.
conclusion is that the MPC solution is superior compared to the other two, and the valve position controller comes in second hand.
Simple Mid-ranging
This is probably the simplest way to implement mid-ranging and it consists of only one feedback loop but two controllers in parallel with the same set point, see Figure 6.9. The controller manipulating u1 is a P controller, whereas the controller manipulating u2 is a controller with integral action. When the integral action in C removes the control error, the output u1 is the offset of the P controller. By adjusting the offset, the target value for u1 is set. The disadvantage with this solution is that P1 is limited to be controlled by a P controller. This gives a slow response to both set point changes and disturbance rejections, since the P controller gives a steady-state error which is removed by C through a much slower process P2.
Valve Position Controller
The valve position controller in Figure 6.10 is probably the most common implementation of mid-ranging found in industry today, see [Shinskey, 1978] and [Allison and Ogawa, 2003]. Controller C1 controls the output y with input u1, while C2 controls u1 with u2. In the literature describing the valve position controller, the decoupling filter CD is seldom included. An exception is [Allison and Isaksson, 1998], where it is briefly discussed without any details. The filter is not necessary for the function of the control system but it improves the performance. If the decoupling filter is removed, the only way for C2 to control u1 is through the error r1í y, consequently introducing disturbances into y. The decoupling should ideally be chosen as
C(s)
kc
Ȉ Ȉ
r1 u1
u2
y
1
P1(s) P2(s)
Ȉ
r2
Figure 6.9 The structure of the simple mid-ranging controller. The set point r2 is the bias of the P-controller. In practice, C(s) is often chosen as a PI controller
), (
) ) (
(
1 2
s P
s s P
CD (6.2)
which is realizable as long as the time delay of P1 is shorter or equal to the time delay of P2, and P1 and P2 have the same unstable zeros. In addition, the pole excess of P2 should be at least as large as the pole excess of P1. It is assumed that both P1 and P2 have stable poles or integrators. With (6.2), changes in u2 do not affect y, and the transfer function from u2 to r2í u1 is equal to íCD. If the ideal decoupling is not realizable, approximations have to be used.
This implementation does not require anti-windup protection in C1
since u1 is a controlled variable. Regardless if u1 saturates or not during an upset, controller C2 brings u1 back to r2 in steady-state. However, depending on the speed of controller C2, anti-windup protection in C1
might be beneficial, see [Haugwitz, et al, 2005].
Hybrid Mid-ranging
An approach, originally used to control pressure of a steam header, is the hybrid mid-ranging [Love, 1994]. It consists of one PI controller, two saturations, a low pass filter, and three gains, see Figure 6.11. The parameter Rm, should be chosen as
100 ,
2 1
1
k k Rm k
(6.3) where k1 and k2 are the steady-state gains of P1 and P2, respectively. The filter in combination with the direct term creates the mid-ranging function.
The transfer function from the output of the PI controller to the P2(s)
P1(s) C2(s)
C1(s)
CD(s)
Ȉ
Ȉ Ȉ Ȉ
r1 r2
u1 u2
y
1 1
Figure 6.10 The structure of the valve position controller.
summation after the filter is in effect a low pass filtered derivative. The purpose of the saturation block preceding the filter is to allow an offset in u1 when u2 is reaching its limit. By this, the controller combines mid-ranging (normal operation) with split-range (saturated u2) in a nice way.
The purpose of the other saturation block is to immediately engage control signal u2 when u1 reaches its limit.
The PI controller can, without losing the mid-ranging function, be replaced by any SISO controller.
Mid-range MPC
In [Allison and Isaksson, 1998] a mid-ranging controller is implemented in the MPC (model predictive control) approach, see Figure 6.12. By letting the minimized cost function be given by
) , (
) ( )
( ) (
)
| ˆ( ) ) (
(
1
0
2
2 1 1
0
2
1 2
1
¦
¦
»¼
« º
¬ ª
'
'
»¼
« º
¬ ª
u
p H
i R
H
i Q u k i
i k u i
k u i k r
k i k y i k k r
J (6.4)
and the weighting matrices chosen as
large , 0
0 small small ,
0 0 large
»¼
« º
¬
» ª
¼
« º
¬
ª R
Q (6.5)
PI Filter 1 Ȉ Ȉ
Ȉ
Rm
100
100 Rm
Rm
100
100 Ȉ
1
Ȉ
1
u2 u1 r1
r2
y P1(s)
P2(s)
Figure 6.11 The structure of the hybrid mid-ranging controller.
the desired behavior is achieved. Since the MPC formulation is inherently discrete-time, the cost function is given as a summation. Notation ǔ(k + i | k) denotes the i step-ahead prediction. Hp is the prediction horizon, Hu is the control horizon, and ǻ is the difference operator (ǻ = 1 í zí1, where zí1 is the backward shift operator). The main advantage with the MPC formulation is that it takes constraints into account explicitly and integrator windup is no issue. The main tuning parameters are the prediction horizon Hp, the control horizon Hu, and the weights Q and R. It is possible to omit the weight on the control increments, R, in (6.4), without losing the mid-ranging function. However, the formulation is less flexible since there in practice is only one weighting parameter (one of the parameters in Q can be set to e.g. one).