Comparing FOPDT and IPDT Model Based PI Controllers with Disturbance Observer ˇ ak and Mikul´asˇ Huba Peter Tap´ Slovak University of Technology in Bratislava Ilkoviˇcova 3, 812 19 Bratislava, Slovakia Email: peter.tapak,
[email protected]
Abstract—The first order process model altogether with the proportional-integral (PI) controllers represent the most popular setting for controller design applied in practice. The paper compares the performance of two disturbance observer based PI controllers, one designed using a FOPDT model combined with a loop linearization based on inverse input-output steady-state characteristic and the second using an IPDT model identified in several working points by a relay experiment. Both approaches are compared by means of a nonlinear laboratory plant model.
I. I NTRODUCTION Numerous papers investigate the transition point when the designer should choose to use more complex models - the First Order Plus Dead Time (FOPDT) representing one of the most frequently chosen starting models [1], [2], [3]. Nevertheless, already the most popular approach by Ziegler and Nichols [4], [5] based on an approximation of the process reaction curve by a tangent showed that for a relatively low ratio of the dead time and the plant time constant it is enough to use the Integral Plus Dead Time (IPDT) approximations also for dealing with stable nearly first order processes with a monotonic setpoint step response treated in this paper. However, when using the IPDT approximation for the FOPDT process, one meets several problems. The first one appears already in the plant identification. The plant feedback that is around an operating point equivalent to a load disturbance, will lead to asymmetrical behavior also in the case without an additional load and with symmetrical relay. Hence, with respect to the precision of the whole approximation, in the relay identification this oscillation asymmetry is an important issue and usually requires to look for alternative approaches to the describing function method. The second important problem in the control design for the considered plant is represented by its strongly nonlinear inputoutput steady-state characteristic that may either be tackled by an robust approach with a tuning chosen usually to get an acceptable performance in the worst situations, or by an active nonlinearity compensation based on an inverse nonlinearity. To illustrate pros and cons of both mentioned approaches, the laboratory plant model will be extended by an additional dead time with a chosen length. Such a loop will be controlled by the predictive disturbance observer based filtered PI controller [6], [7] and evaluated by the time and shape related performance measures introduced in [6].
II. D EAD T IME TO P ROCESS T IME C ONSTANT R ATIO I NFLUENCE A NALYSIS Let us consider linear FOPDT plant with transfer function G(s) =
Ks −Td s e s+a
(1)
Td being a dead time and T = 1/ |a| being a plant time constant. The influence of the dead time to process time constant ratio aTd on IPDT approximation Ks −Td s e (2) s has been firstly investigated by simulation. Fig. 1 shows the parameters of the model (2) with Ks = 0.5, a = 0.5 obtained by the IPDT relay feedback experiment based on [8] for the system (1) with dead time varying on interval Td ∈ [0.01, 10]. The method can be used for plants with unknown load disturbances without additional controller (see e.g. [9]). There is not necessary to bias the relay reference value to compensate the static disturbance, which does not have to be known in advance (see e.g. [10], [11], [12]). The proposed method uses a curve fitting approach. The FOPDT system output compared to its IPDT approximation is shown in Figs. 2-3. There is obvious that with an increasing aTd the approximation fitting decreases. Nevertheless the quality of control using either of the models will be investigated further in the paper. G(s) =
A. PI1 - controller The simplest PI1 - controller (sometime denoted as DO-PI controller) employs disturbance observer as the I-action. For an IPDT plant and the dead time neglected in the disturbance observer the simplified controller structure consisting of Paction and DO is presented in Fig. 4. Index ”1” used in its title has to be related to one saturated pulse of the control variable that can occur in accomplishing large reference signal steps. In this way it should be distinguished from the PI0 - controller reacting to a reference step by a monotonic transient of the manipulated variable. More complex structures with a static feed forward and a dead time compensation in the disturbance observer are treated in [6], [7]. To achieve in a closed loop the fastest possible transients without overshooting, a controller tuning may be derived by using the performance portrait method [7]. In following simulations the FOPDT system (1) is
IPDT Model Parameters
IPDT Model Parameters Dependancy on FOPDT Plant Dead Time to Process Time Constant Ratio 0.6
6
0.4
4
0.2
2
Fig. 4. PI1 - controller
IPDT Process Gain Real Process Gain Real Dead Time IPDT Dead time 0
0
1
30
2 3 4 Deadtime to Process Time constant Ratio
5
0
29 28 27 System output
Fig. 1. Approximation parameters dependency on dead time to process time constant ratio aTd
Model vs Measured Data Comparison Dead Time to Process Time Constant Ratio= 0.005
26 25 24 Setpoint IPDT based control FOPDT based control
23 22
40.3
21
40.25 IPDT Approximation Measured Data − FOPDT Model Output
40.2
20 1
1.05
1.1
Process Variable
40.15
1.15 1.2 Time [s]
1.25
1.3
1.35
40.1
Fig. 5. Control Performance - aTd = 0.005
40.05 40 39.95 39.9 39.85 39.8
5
5.05
5.1
5.15 Time [s]
5.2
5.25
5.3
Fig. 2. Model vs Real System Comparison - aTd = 0.005, Td = 0.01
controlled. Small, medium and large aTd was used. For very low dead time with aTd < 0.1 the performance of controllers using particular models was similar (Fig. 5). In Fig. 6 where the dead time is set to one half of the process time constant one can see that the controller using relay identification based on an IPDT model yields slower transients.
Model vs Measured Data Comparison Dead Time to Process Time Constant Ratio= 5 100
30
90
29 IPDT Approximation Measured Data − FOPDT Model Output
80
28 27
60
System output
Process Variable
70
50 40
25 24
30
23
20
22
10
21
0 140
160
180 200 Time [s]
220
240
Fig. 3. Model vs Real System Comparison - aTd = 5, Td = 10
Setpoint IPDT based control FOPDT based control
26
20 100
105
110
115
120 125 Time [s]
130
135
140
Fig. 6. Control Performance - aTd = 0.5
145
Input−output characteristic y=f(u) 80 30 70
29 28
60 50
26
−−−> y
System output
27
25 24
Setpoint IPDT based control FOPDT based control
23
Measured points of IO characteristics Linear interpolation of IO characteristics
40 30 20
22 10
21 20 1200
1300 Time [s]
1400
0
1500
Fig. 7. Control Performance - aTd = 5
III. R EAL E XPERIMENT - L IGHT
0
20
40
60
80
100
−−−> u
Fig. 8. Input-to-output characteristic of the light channel
System Output
INTENSITY
70
In this section, several experiments will be reported carried out by using the laboratory thermo-optical plant [13]. The light channel consist of the light bulb which acts as a light source, the system input is the bulb voltage[0-100 %], the output is the value from the light intensity sensor filtered with the first order low pass filter with the time constant set to 2 seconds, which corresponds to a typical FOPDT model. The input-tooutput characteristics is strongly non-linear as one can see in Fig. 8. This plant could be described by an input nonlinearity followed by a dominant first-order filter dynamics dy/dt = f (uc + di ) − ay
(3)
with y = f (uc + di ) being an invertible input nonlinearity, di and uc being an input disturbance and a plant input and y being a plant output. After linearization by an inverse nonlinearity uc = f −1 (ur ) the plant may be characterized by the transfer function Ks Ks 1 Y (s) ≈ ; = 1 ; T1 = P (s) = Ur (s) di =0 s + a a a
Light Intensity
1100
65 Setpoint FOPDT IPDT
60
55
0
1
2
3
4 5 Time [s] Control Signal
6
7
8
9
80 Bulb Power
1000
70
60
50
FOPDT IPDT 0
1
2
3
4 5 Time [s]
6
7
8
9
Fig. 9. Control performance - setpoint step response - no additional delay linearized system
(4)
(5)
After such a nonlinearity compensation, ideally, it should hold K = Ks /a = 1. But, due to different imperfections, as e.g. the fluctuations of the plant parameters due to the temperature changes, equivalent input disturbances occur that will be included within the identified value of an input disturbance di . By adding to the relatively short plant dead time an additional software dead time Td , one gets typical first order plus dead time plant. A. Identification The results from an IPDT relay feedback experiment based on [8] and the FOPDT model approximation using setpoint step responses processed by the Matlab ident toolbox are compared in Figs 9-12. For both models and loops with, or without the nonlinearity compensation, one can from the
measured data similar deviations. The first experiment was made without any additional delay in the control loop. The IPDT model parameters were Ks1 −Td1 s 0.3718 −0.3035s e = e s s The FOPDT model parameters were G1 (s) =
G2 (s) =
Ks2 −Td2 s 1.6771 −0.0812s e = e s+a s + 0.4993
(6)
(7)
The second experiment was made by using an additional delay 200 ms. 0.2752 −0.4994s Ks1 −Td1 s e = e s s The FOPDT model parameters were G1 (s) =
G2 (s) =
Ks2 −Td2 s 1.6771 −0.2861s e = e s+a s + 0.4993
(8)
(9)
IV. C ONCLUSION
System Output
Light Intensity
75 70 65 Setpoint FOPDT IPDT
60 55
0
1
2
3
4 5 Time [s] Control Signal
6
7
8
9
Bulb Power
80 FOPDT IPDT
60 40 20 0
0
1
2
3
4 5 Time [s]
6
7
8
9
Fig. 10. Control performance - setpoint step response - no additional delay - nonlinear system
System Output
ACKNOWLEDGMENT
Light Intensity
70
This work has been partially supported by the grants APVV0343-12 Computer aided robust nonlinear control design and VEGA 1/0937/14 Advanced methods for nonlinear modeling and control of mechatronic systems.
65 Setpoint FOPDT IPDT
60
55
0
5
10
15 Time [s] Control Signal
20
25
30
Bulb Power
80
70
60
50
FOPDT IPDT 0
5
10
15 Time [s]
20
25
30
Fig. 11. Control performance - setpoint step response - 200 ms additional delay - linearized system
System Output
Light Intensity
70
65 Setpoint FOPDT IPDT
60
55
0
5
10
15 Time [s] Control Signal
20
25
30
20
25
30
Bulb Power
40 FOPDT IPDT
30
20
10
0
5
10
15 Time [s]
The simulation showed expected results: the control performance is better if one uses the FOPDT model to control the FOPDT plant. However when applied to a real plant without the non-linearity compensation, the IPDT model performed better. The performance of the control loop with the IPDT model was slower in all examined cases, both in simulations and real experiments. However, this made the loop more robust, it yielded transients without an overshooting with or without the non-linearity compensation. This altogether with many closed loop identification methods can make it a safe choice for a practical application. The second important advantage of the relay experiments is that they may also be used for identification of unstable processes, where the alternative setpoint step responses may not be mostly applied. The third inteeresting point is that the IPDT based relay experiments may also be used for identification of the internal feedback parameter a.
Fig. 12. Control performance - setpoint step response - 200 ms additional delay - nonlinear system
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