Discrete Control Design

Discrete Control Design

Name: Ebuka Isiadinso Student ID: 150144135 Module: ACS224

150144135

Discrete Control Design

ACS224

Introduc)on The servo system uses a closed-loop feedback system measuring the speed and posiHon associated with the servo plant. The designed diagram below shows the connected components that power different parts of the servo system. Motor

Gearbox

Load

Servo Amplifier

Figure 1: Components of the Servo Control System

Speed Sensor

Data AcquisiHon Device



The mulH-funcHonal data acquisiHon device sends control signals to which the servo amplifier uses to power the motor (rotary servo plant). While the data acquisiHon device is capable of calculaHng signals from separate systems, its other funcHonality is to write analogue and digital amplitudes/voltages to other devices. The motor consists of internal gearbox supplying the external gears (Soliman, Saoudi and Metwally, 2015). Furthermore, the overall plant consists of several sensors such as the potenHometer, encoder and tachometer, which measures the angular posiHon and speed of the load respecHvely (De Silva, 2010). The objecHve is to design and analyse a PI (proporHonal and integral) controller and a deadbeat response controller in the discrete domain using MATLAB applicaHon. The remaining parts of this report briefly highlight the procedures used to design the two controllers, PI and deadbeat controls, with the appropriate results and findings. Lastly, the bo[om of the report outlines all related soluHons to relevant issues associated with the system's responses and behaviours. Methods The servo system has parameters which are voltage as input and speed/velocity as output. Therefore the transfer funcHon taken is in the form below; K EquaHon 1 G(s) = (τ s + 1) Where K is gain and the Hme constant of the plant is τ . Below are the procedures used to design the PI and deadbeat response controllers. Design: PI Control To maintain the rotaHonal speed, a proporHonal and integral controller is most appropriate. The transfer funcHon, in the Laplace domain, of the PI controls is in the form below;





G(s) =

K s + Ki Es + F K ⇔ Gc (s) = K p + i ⇔ p s s s









EquaHon 3

EquaHon 2

However by comparison, the closed-loop transfer funcHon is;



Gc (s)G(s) =

KK i τ s +s 2

(

(

Kp Ki

KK p +1 τ

)

s +1

)

+

KK i τ

⇔

ω n2 s + 2ζω n s + ω n 2



2

Where Kp and Ki are the PI controls, and ω n and ζ are natural frequency and damp raHo of the system. EquaHon 4 below shows how to calculate the values of Kp and Ki;



Kp =

2ζω nτ − 1 ω 2τ ! Ki = n K K



EquaHon 4

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150144135

Discrete Control Design

ACS224

Simula4on The transfer funcHons, conHnuous and discrete domain, of the simulated system is as follows; 3.1 0.05905 EquaHon 5 G(s) = ⇔ G(z) = 0.026s + 1 z − 0.981 With a natural frequency of 42.914 rad/s and a damp raHo of 0.5169, the values of the proporHonal-integral controls, Kp and Ki calculated are 0.0495 and 15.446 respecHvely. The discrete domain transfer funcHon for the PI controller, D(z) with a sample Hme of 0.5 ms is;







D(z) =

0.04953z − 0.04181 z −1





EquaHon 6

Experiment The transfer funcHons in conHnuous and discrete domain, of the experimented system is as follows; 1.64 0.1355 EquaHon 7 G(s) = ⇔ G(z) = 0.058s + 1

z − 0.9174

With a natural frequency of 17.1657 rad/s and a damp raHo of 0.5169, the values of the proporHonalintegral controls, Kp and Ki calculated are 0.0179 and 10.4210 respecHvely. The discrete domain transfer funcHon for the PI controller, D(z) with a sample Hme of 5 ms is; 0.01788z − 0.03422 EquaHon 8 D(z) = z −1 Design: Deadbeat Response Control To obtain zero error between reference point R[z] and the output Y[z] with a defined sample Hme, a deadbeat response controller is most appropriate (Nise, 2017). EquaHon 9 illustrates the calculaHon of the deadbeat controller;



M (s) = z −1 =

Dz (z)G(z) 1 z −1 ⇔ Dz (z) = 1+ Dz (z)G(z) G(z) 1− z −1





EquaHon 9

Simula4on The deadbeat response controller, from EquaHon 9, for the simulated system in discrete domain is (sample Hme is 0.1 s); z − 0.02136 EquaHon 10 Dz (z) = 3.034(z − 1) Experiment The deadbeat response controller, from EquaHon 9, for the experimented system in discrete domain is (sample Hme is 0.5 s);





Dz (z) =

z 2 − 0.0001803z 1.64z 2 − 1.64





EquaHon 11

Results Simula4on Below are the simulated performances for each different sample period - results obtained using MATLAB. According to Table 1 below, the effect of reducing sample rate on the control system will lead to more oscillaHons (disturbances) with higher overshoots. Figure 2 shows the similarity between the the system in both conHnuous and discrete domains each respecHvely with overshoot of 15.1 % and 16.7 %.

Table 1: Performance for Simulated System with PI Controller Sample )me - s

Overshoot - %

Stable / Not Stable

0.001

16.7

Stable

0.002

18.5

Stable

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150144135

Discrete Control Design

Sample )me - s

Overshoot - %

Stable / Not Stable

0.005

24.7

Stable

0.01

38.7

Stable

0.02

77.1

Stable

0.03

NA

Not Stable

Continuous Domain

Discrete Domain

1.2

1.2

1

1

0.8

0.8

Amplitude

Amplitude

ACS224

0.6

2. a - Con)nuous

0.4

Figure 2: PI control Response of Simulated System in S and Z (sample )me of 0.001 s) Domains

0.6

2. b - Discrete 0.4

0.2

0.2

0 0

0.05

0.1

0.15

0.2

0

0.25

0

Time (seconds)

0.05

0.1

0.15

0.2

0.25

0.3

Eventually, the system becomes closed-loop unstable if sample Hme increases as seen in figure 3 a - 3 b. Figure 3 shows the sample Hme increased from 0.005 s to 0.03 s. Time (seconds)

Discrete Domain with Sample time - 0.005 1.4

3. a - 0.005 s

3. b - 0.03 s

2

1

1

0.8

0

Amplitude

Amplitude

1.2

Discrete Domain with Sample time - 0.03

10 25

3

0.6

-1

0.4

-2

0.2

-3

0

Figure 3: PI control Response of Simulated System in Z Domain from sample )me of 0.005 s to 0.03 s

-4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

2

4

6

8

10

12

14

16

18

20

Experiment The tables below are the experimented performances for different sample Hme for both PI and deadbeat controllers - results obtained using MATLAB and LAB-VIEW. According to Table 2, the PI controller increases the overshoot while making the system become more unstable with more oscillaHons whenever sample period raises from 0.01 to 0.03 s. Table 3 consists of the results for hardware actua)on and feedback where the overshoot decreases as the sample Hme is further raised from 0.03 to 0.5 s. Figure 4 displays the deadbeat control responses for the actuaHon and feedback with sample Hme of 0.03 s. Time (seconds)

Time (seconds)

Table 2: Experimented System with PI Controller Sample )me - s

Overshoot - %

Stable / Not Stable

0.01

32.67

Stable

0.015

46.67

Stable

0.02

60.5

Stable

0.025

73.34

Stable

0.03

106.41

Not Stable

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150144135

Discrete Control Design

ACS224

Table 3: Experimented System with Deadbeat Controller Hardware Actua)on

Hardware Feedback

Sample Hme(s) Overshoot(%) Stable / Not Stable Sample Hme(s) Overshoot(%) Stable / Not Stable

5

0.03

146.8

Stable

0.03

24.50

Stable

0.1

38.00

Stable

0.1

18.70

Stable

0.5

10.51

Stable

0.5

9.78

Stable

PosiHon (radians) - Hardware FeedBack 4. a

8

2.5

4

0

0

-2.5

-4

-5

0.96 2.373.78 5.19 6.6 8.01 9.42

PosiHon (radians) - Hardware ActuaHon

-8

4. b

Figure 4: Deadbeat control Response of Experimented System (Actuator and Feedback) for sample )me of 0.03 s

0.962.043.12 4.2 5.286.36 7.448.52 9.6



According to Table 3 and Figure 4 above, the hardware feedback, which has a lag of one second, experienced an iniHal steady-state error of 2.47 % at sample Hme, 0.5 s when compared to the simulated results from MATLAB. While both the feedback and hardware actuaHon experienced increased overshoots with reducing sampling Hme, the hardware actuaHon does not se[le at desired point. Whereas the hardware feedback, despite the iniHal steady-state error of 2.47 %, eliminates steady-state error to se[le at set point. It is noHceable that the deadbeat is not the proper controller for this system as it results to more oscillaHons and higher overshoot with a steady-state error. Meanwhile, applying PI controls cause the system to become less stable with increasing sample Hme as shown in Figure 5 below.

6

PosiHon (radians) - Hardware FeedBack 5. a - 0.015 s

3

PosiHon (radians) - Hardware FeedBack

9

0

3

-3

-3

5. b - 0.03 s

-6 -9 0.015 3.45 6.885 10.3213.755 17.19 0.03 1.53 3.034.53 6.03 7.539.03

Figure 5: PI control Response of Experimented System (Actuator and Feedback) for sample )me from 0.015 - 0.03 s

Conclusion The overall aim of the experiment is to design and analyse two controllers namely proporHonal-integral (PI) and deadbeat controllers, both in discrete z-domain. Through the methods of designing the two control systems, it is important to note that both the plant and controllers are closed-loop stable from Equa)on 5 - 11. The results from Table 1 - 2, illustrate overshoots are increasing with increasing sampling Hme when a PI controller is applied. However, Table 3 shows different results, with overshoots decreasing with increasing sampling Hme when a deadbeat controller is applied. By comparing deadbeat simulated and experimented results, the experimented system included a lag of 1 s, causing a different between the two results. The different chosen sample Hmes, as well as data exportaHon from MATLAB, is seen to be the main reason. Unfortunately, the deadbeat controller is not a correct choice for the servo system despite maintaining stability for lower sample rates due to more oscillaHons and steady-state error. Furthermore, the effect of saturaHon in the presence of a delay (lag) will cause the system to be slower. Also, fast acHng disturbances will degrade the servo system.


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150144135

Discrete Control Design

ACS224

References De Silva, C. (2010). Mechatronics. 1st ed. Boca Raton, Fla.: CRC, pp.345-346. Nise, N. (2017). Digital Control System. In: N. Nise, ed., Control Systems Engineering, 6th ed. pp.795(#37). Soliman, E., Saoudi, M. and Metwally, H. (2015). A Controller Design for Servo Control System Using Different Techniques. 4th ed. [ebook] p.203. Available at: h[p://www.ijeit.com/Vol%204/Issue%209/ IJEIT1412201503_35.pdf [Accessed 17 May 2017].

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