International Conference on Computer Aided Systems Theory

Computer Aided Systems Theory – EUROCAST 2015 pp 49-56 | Cite as

Identification of First Order Plants by Relay Feedback with Non-symmetrical Oscillations

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9520)

Abstract

The paper deals with the approximation of the systems with dominant first order dynamics by the Integrator Plus Dead Time (IPDT) model. They are attractive especially in control of unstable plants, where an open-loop identification may not be applied. This paper updates a previously published contribution based on analysis of the non-symmetrical oscillations with possible offset arising typically under relay control that has been improved to prevent computational errors in the case of a negligible disturbances, when the relay on and off times are nearly equal over one control period. The analytical results are followed by the experiments with several laboratory plant models. The obtained model parameters are used to tune disturbance observer based controllers to illustrate the performance of the proposed method in real applications.

Keywords

Closed loop identification Relay control First order model 

1 Introduction

Relay feedback test has been very popular in several commercial autotuners for decades. The research in this area was deeply analyzed in [9, 10, 11]. This paper updates an earlier contribution [7] in which the algorithm failed in situations when the disturbances were negligible and the relay on and off times were almost equal over one control period. The results of the previous paper [7] can be summarized in the following way: the proposed method can be used for plants with unknown load disturbances without additional controller (see e.g. [14]). 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. [2, 12, 13]). This paper starts with a method derivation, then the disturbance observer based PI controller is presented followed by a demonstration of the results of the identification and control using a laboratory model of a real plant.
Fig. 1.

Relay identification with nonsymmetrical plant input

Fig. 2.

Transients of basic variables of the loop in the Fig. 1

1.1 Method Derivation

The main advantage of constraining the plant approximation to the IPDT model is that both the experiment setup and the corresponding analytical formulas remain relatively simple and robust against measurement noise.
$$\begin{aligned} S(s)=\frac{K_{s}}{s}e^{-T_{d}s} \end{aligned}$$
(1)
Let us consider the control loop with a relay with the output \(u_{r}=\pm M\) and a piecewise constant input disturbance \(v=const\). Then, the actual plant input will be given as a piecewise constant signal \(u_{A}=\pm M + v\). Possible transients are shown in Fig. 2. Let us assume relay switching from the positive relay output \(u=M\) to the negative value \(u=-M\) (point 1) at the time \(t_{21i-1}\), the influence of the positive plant input \(U_{2}=(v+M)K_{s}\) will remain over interval with the length equal to the dead time value \(T_{d}\). Then, after reaching output value \(y_{21}\) at the time moment \(\tau _{21i-1}\) (point 2) due to the effective plant input \(U_{1}=(v-M)K_{s}\) the output starts to decrease. After the time interval \(t_{1}\) it reaches the reference value w (point 3). At this moment the relay switches to the positive value \(u=M\), however the plant output keeps decreasing for the time \(T_{d}\) and reaches the value \(y_{12}\) (point 4). The duration of the interval with negative relay output will be denoted as \(t^{-}\). While the output of the relay is positive the plant output starts to increase and reaches the reference value after the time \(t_{2}\) (point 5). Let us denote the total duration of the positive relay output as
$$\begin{aligned} t^{+}=t_{2}+T_{d} \end{aligned}$$
(2)
As a result of the time delay, the plant output turnover time instants \(\tau _{21i}\) are shifted with respect to the relay reversal moments \(t_{21i}\) by \(T_{d}\). Similar time shift exists among time instants \(\tau _{12i}\) and \(t_{12i}\), i.e.
$$\begin{aligned} \tau _{21i}=t_{21i}+T_{d} \end{aligned}$$
(3)
$$\begin{aligned} \tau _{12i}=t_{12i}+T_{d} \end{aligned}$$
(4)
For a single integrator one can formulate relations
$$\begin{aligned} y_{21}-w=U_{2}T_{d};t_{1}=(w-y_{21})/U_{1} \nonumber \\ y_{12}-w=U_{1}T_{d};t_{2}=(w-y_{12})/U_{2} \end{aligned}$$
(5)
Let us denote the period of one cycle as
$$\begin{aligned} P_{u}=t^{+}+t^{-}=2T_{d}+t_{1}+t_{2}=\frac{ 4T_{d} M^{2} }{M^{2}-v^{2} } \end{aligned}$$
(6)
For a known value of the relay amplitude M and a known ratio of the positive and negative relay output duration over one cycle
$$\begin{aligned} \epsilon =\frac{t^{+}}{t^{-}}=\frac{t_{2}+T_{d}}{t_{1}+T_{d}}=-\frac{v-M}{v+M} \end{aligned}$$
(7)
it is possible to express the identified disturbance as
$$\begin{aligned} v=u_{0}+v_{n} \end{aligned}$$
(8)
This may consist of a known and intentionally set offset at the relay output and of an unknown external disturbance \(v_{n}\) that may be calculated as
$$\begin{aligned} v=M\frac{1-\epsilon }{1+\epsilon } \end{aligned}$$
(9)
Then from (6) it follows
$$\begin{aligned} T_{d}=\frac{P_{u}}{4}\left[ 1-\left( \frac{v}{M}\right) ^{2}\right] =P_{u}\frac{\epsilon }{(1+\epsilon )^{2}} \end{aligned}$$
(10)
Since the identification of the plant gain \(K_s\) proposed in [7] and based on an average output value over a limit cycle does not work well for negligible disturbances and a symmetrical relay output, one may either introduce a relay offset to make the cycle assymetrical, or to derive \(K_s\) by identifying the cycle limits.
From (5) one gets
$$\begin{aligned} y_{21}-y_{12}=U_2T_{d}+w-U_1T_d-w \end{aligned}$$
(11)
Substituting \(U_{1}=(v-M)K_{s}\) and \(U_{2}=(v+M)K_{s}\) into (11) yields
$$\begin{aligned} K_s=\frac{y_{21}-y_{12}}{2T_dM} \end{aligned}$$
(12)
Substituting (10) into (12) finally yields formula for the plant gain
$$\begin{aligned} K_{s}=\frac{1}{2} \frac{(y_{21}-y_{12})(1 + \epsilon )^2}{MP_u\epsilon } \end{aligned}$$
(13)

2 \(\mathrm {PI_{1}}\) - Controller

The \(\mathrm {PI_{1}}\) - controller (sometime denoted as DO-PI controller) employs disturbance observer (DO) as the I-action. The controller structure consisting of P-action and DO is presented in Fig. 3. 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 \(\mathrm {PI_{0}}\) - controller reacting to a reference step by monotonic transient of the manipulated variable. To achieve fastest possible transients without overshooting in a closed loop with an integrator and a dead time, the closed loop pole corresponding to the fastest monotonic output transients using simple P-controller is [3]
$$\begin{aligned} \alpha _{e}=-1/(T_{d}e) \end{aligned}$$
(14)
Fig. 3.

\(\mathrm {PI_{1}}\) - controller

When using the P-controller together with the DO based I-action, some “slower” closed loop pole should be used
$$\begin{aligned} \alpha _{eI}=\alpha _{e}/c=1/(T_{d}ec);c=[1.3, 1.5] \end{aligned}$$
(15)
The gain of P-controller corresponding to the closed loop pole (15) is then
$$\begin{aligned} K_{p}=-\alpha _{eI}/K_{s} \end{aligned}$$
(16)
For the time constant of the filter used in the DO one gets
$$\begin{aligned} T_{f}=-1/\alpha _{eI} \end{aligned}$$
(17)
A preciser controller tuning may be derived by using the performance portrait method [6], or the dead-time may also be included into the DO [4].

3 Real Experiment - Fan RPM

In this section, several experiments will be reported carried out by using the laboratory thermo-optical plant [8].

3.1 Identification

The input of the plant is the fan power, values from 0 to 100 % can be used in the Simulink model. The system output is the fan rpm filtered by the first order low pass filter with time constant set to one second. Several closed loop experiments have been made to cover a large operational range. Table 1 shows the plant is non-linear, the process gain varies from 21.34 to 30.82. The parameters for each working point in Table 1 were obtained in the following way: The system started from a steady state, the relay control was applied until at least 10 oscillations were measures. The first three relay cycles were omitted, then for the each one left the identification algorithm has been applied separately. The average value of these parameters were put into Table 1. The model and the real data are compared in Fig. 4. The corresponding parameters from Table 1 have been used, the real plant and the model use the same input.
Table 1.

System parameters - fan rpm

Setpoint

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

\(K_s\)

21.34

30.81

29.03

28.06

27.29

25.73

20.75

21.26

21.68

21.77

21.82

\(T_d\)

0.49

0.29

0.36

0.42

0.48

0.50

0.47

0.48

0.49

0.49

0.49

v

-17.26

28.38

24.38

21.23

18.03

14.98

11.32

6.29

0.91

-4.74

-10.66

Fig. 4.

Real system identification results - fan rpm, model vs real data

3.2 Control

The highest values of the process gain and of the time delay from Table 1 have been used in the experiment for controller tuning. Figure 5 shows the control results for one setpoint step change from a steady state. The system output reaches monotonically the new setpoint. Due to the plant nonlinearity and long dead time, the control signal remains without an extreme point typical for the \(PI_1\) control.
Fig. 5.

Fan closed loop control performance of the controller based on the obtained model

4 Real Experiment - Temperature

4.1 Identification

A linear plant approximation corresponds to a plant with fast and slow channels [1], where the fast channel is represented by the heat radiation and the slow one corresponds to the heat conduction via body of the plant. One can expect that with the closed loop relay control, the fast channel would be dominant. Table 2 shows the identification results for various operating points. The model and the real data are compared in Fig. 6. Again the plant is non-linear, process gain varies from 0.0022 to 0.0068 and the time delay ranges from 0.3312 to 1.0533. The system parameters vary from one period to another. Since the simulation in the Fig. 6 was made using the average values of the system parameters calculated for each cycle and using the same input as the real system, one can see a difference in the amplitude of the real system and a model in Fig. 6.
Fig. 6.

Real system identification results - temperature, model vs real data

Table 2.

System parameters - temperature channel

Setpoint

40

45

50

55

60

65

70

\(K_s\)

0.0068

0.0056

0.0048

0.0041

0.0034

0.0026

0.0022

\(T_d\)

0.7263

0.9113

1.0305

1.0533

0.9805

0.7325

0.3312

v

39.1122

31.8356

24.2042

16.3182

8.1374

-1.6944

-11.4774

4.2 Control

The highest value of the process gain and time delay from Table 2 have been used in the experiment to tune the controller. Figure 7 shows the control results for setpoint step changes from a steady state. The system output transients are close to the desired monotonic ones. Due to the system nonlinearity and two dynamic modes, the control signal transients show more than one pulse. Nevertheless, when considering the simplicity of the used model, the control performance is satisfactory.
Fig. 7.

Temperature closed loop control performance of the controller based on the obtained model

5 Conclusion

The carried out experiments show that the developed identification approach yields results applicable in control of a broad range of linear and nonlinear plants with a dynamics that is considerably more complex than the approximation used. Together with new performance portrait method for the controller tuning [5] and new DO based filtered PI control [4] it considerably simplifies control of the simple plants with significantly increasing reliability of the controller tuning and achievable performance.

Furthermore, when evaluating the dependency of the identified disturbances on the setpoint (or, more precisely, on the average output value), the method may be easily extended also to identification of static plants with a linear, or a nonlinear internal feedback and thus also to precise the subsequent controller tuning, which may bring significant improvements especially in control of static systems with a longer dead time.

Notes

Acknowledgment

This work has been partially supported by the grants APVV-0343-12 Computer aided robust nonlinear control design and VEGA 1/0937/14 Advanced methods for nonlinear modeling and control of mechatronic systems.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Slovak University of Technology in BratislavaBratislavaSlovakia

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