linear control systems laboratory

Islamic University Of Gaza

Faculty of Engineering

Electrical Engineering Department

Linear Control Systems

LABORATORY

Prepared By:

Eng. Adham Maher Abu Shamla

Under Supervision:

Dr. Basil Hamed

Linear Control Systems Lab EELE (3160-3161)

Page 1 of 14

Experiment #8

Time Response Design

1) Introduction:

What is the Time Response?

It is an equation or a plot that describes the behavior of a system and contains much information about it with respect to time response specification as overshooting, settling time, peak time, rise time and steady state error.

Time response is formed by the transient response and the steady state response.

Transient time response (Natural response) describes the behavior of the system in its first short time until it arrives the steady state value and this response will be our study focus.

If the input is step function the output or the response is called step time response and if the input is ramp, the response is called ramp time response … etc.

Figure 8.1: Time response plot and specification points

Time response = Transient response + Steady state response


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2) Control systems types:

2.1 Open loop transfer function …

The open loop transfer function is a transfer function that represents a system and relates the output Y(s) to the input X(s) as a ratio.

In Control Theory, the transfer function of the system that will be controlled is called a plant and sometimes denoted as G(s).

Example of an open loop system:

Consider a motor system and we want to control its speed, so if you input the system with a suitable voltage to rotate 1000 rpm, the motor should rotate with speed 1000 rpm … etc.

But if the motor at any moment face some disturbance or noise or a huge load such that its speed become lower than 1000 rpm, in this case the motor will not correct this error and will rotate with speed lower than 1000 rpm.

So the main disadvantage of an open loop system is the absence of sensitivity to disturbance and inability to correct its behavior for this disturbance.

2.2 Closed loop transfer function …

The closed loop system overcomes the disadvantage that existed in the open loop systems.

Where 𝑬(𝒔) = 𝑳 {𝒆(𝒕)} : the error signal between the input and output.

G(s)

(Plant) X(s) Y(s)

G(s)

(Plant) X(s) Y(s) +

-

E(s)

)(

)()(

sX

sYsH


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Then the transfer function H(s) represent the closed loop system and relate Y(s) and X(s) by another way.

In the previous work we considered the gain of the forward and feedback paths is 1 (unity), but here we will remain the feedback gain unity and change the forward path gain with K gain as follows:

The new transfer function of the closed loop system is:

In this case K is called Proportional Gain or Proportional Controller.

X(s) Y(s)

K X(s) Y(s) + -

E(s)

G(s)

(Plant)

)(1

)(

)(

)()(

)()()(

)(

)()(

sG

sG

sX

sYsH

sYsXsE

sE

sYsG

emForallsyst

)(1

)()(

)(1

)(

)(

)()(

)()()(

)(

)()(

sKG

sKGsH

sKG

sKG

sX

sYsH

sYsXsE

sE

sYsKG

emForallsyst


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3) The effect of proportional gain on the system specifications:

(Overshoot, settling time, rise time, peak time, steady state error)

A speed DC Motor transfer function will be our case study to study the effect of K gain on the behavior of the system.

Let: R= 2.0 % Ohms

L= 0.5 % Henrys

Km = 0.015 % torque constant

Kb = 0.015 % emf constant

Kf = 0.2 % Nms

J= 0.02 % kg.m2/s2

Then:

02.4014

5.1)(

2

SSsG

Then the Closed loop transfer function is:

)5.102.40(14

5.1

)(1

)()(

2 KSS

K

sKG

sKGsH

Now, let us assume some values of K and plot the step response and analyze the changes in the response specifications.

K= 0.1, 1, 10, 100, 1000

bmf

m

KKKJSRLS

K

sV

sWsG

))(()(

)()(


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First method: By using MatLab

At K=0.1: The DC gain of the system = 0.0037 So the final value for the unity step input is 0.0037

No overshoot

Ts = 1.5 sec

The value of Steady state error = (input) – (output) =

1-0.0037= 0.9963 df

df

g

At K=1; The DC gain of the system = 0.036

No overshoot

Ts = 1.4 sec

S.S.E = 1 – 0.036 = 0.964

%%% study the effect of K controller on the step response %%%

%%%%%%% Open Loop Transfer Function of DC motor %%%%%%%%

num=[1.5];

den=[1 14 40.02];

G=tf(num,den)

%%%%%%%% Closed Loop transfer function %%%%%%%%%

K=0.1;

H=feedback(K*G,1)

%%%%%% Step time response for closed loop system %%%%%%%

step(H)

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3 Step Response w ith K=0.1

Time (sec)

Am

plit

ude

0 0.5 1 1.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Step Response w ith K=1

Time (sec)

Am

plit

ude


Page 6 of 14


No overshoot

Ts = 0.8 sec

S.S.E = 1 – 0.27 = 0.73

At K=100;

The DC gain of the system = 0.789

OS% = 0.913-0.8=0.113=14.1%

Ts = 0.8 sec

S.S.E = 1 – 0.789 = 0.211


OS% =1.52-0.976=0.544=55.85%

Ts = 0.7 sec

S.S.E = 1 – 0.974 = 0.026

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Time (sec)

Am

plit

ude

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

System: H

Time (sec): 0.117

Amplitude: 0.527


Time (sec)

Am

plit

ude

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Time (sec)

Am

plit

ude


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Second method: By using LabView

Block diagram: step response for closed loop system with proportional gain

Front panel: step response for closed loop system with proportional gain

- Now we will varying in the value of proportional gain and observe the

effect of increasing it on the step response of the system G(s).

At K= 0.1


So the final value for the unity step input is

0.0037

No overshoot

Ts = 1.5 sec

The value of Steady state error =

(input) – (output) =

1-0.0037= 0.0063

df

df

g


Page 8 of 14

A

At K= 1


No overshoot

Ts = 1.4 sec

S.S.E = 1 – 0.036 = 0.964

At K= 10


No overshoot

Ts = 0.8 sec

S.S.E = 1 – 0.27 = 0.73

At K= 100


OS% = 0.913-0.8=0.113=14.1%

Ts = 0.8 sec

S.S.E = 1 – 0.789 = 0.211


Page 9 of 14

At K= 1000


OS% =1.52 -0.976=0.544=55.85%

Ts = 0.7 sec

S.S.E = 1 – 0.974 = 0.026

Conclusion:

The previous results say that the proportional controller has an effect on overshooting and settling time such that increasing the K gain will increase the overshooting and decrease the settling time and rise time.

In the other hand decreasing the K gain will decrease the overshooting and increase the settling time and rise time.

How can we choose the suitable value of K Controller?

There are many methods for designing the K controller as:

1. Root locus technique. 2. Frequency Response technique. 3. State Space technique.

And here we will cover the first technique only (Root Locus).

4) Root Locus technique:

What is Root Locus?

As system parameter k varies over a continuous range of values, the root locus diagram shows the trajectories of the closed-loop poles of the feedback system. Typically, the root locus method is used to tune the loop gain of a SISO control system by specifying a designed set of closed-loop pole locations.


Page 10 of 14

Then, Root Locus technique produces a plot that shows the locations of poles of a closed loop system on the S-Plane as K varies and from this plot we can choose the suitable K that meet our specification conditions.

For example below we see two plots of root locus. The lines are represent the locations of the poles as K varying from zero to pulse infinity.

Root Locus

Real Axis

Imagin

ary

Axis

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-3

-2

-1

0

1

2

3

System: sys

Gain: 3.78

Pole: -2.88 + 2.1i

Damping: 0.808

Overshoot (%): 1.34

Frequency (rad/sec): 3.57

System: sys

Gain: 9.36

Pole: -1.95

Damping: 1

Overshoot (%): 0


Root Locus

Real Axis

Imagin

ary

Axis

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1-15

-10

-5

0

5

10

15

System: sys

Gain: 9.44

Pole: 0.335 + 2.47i

Damping: -0.134

Overshoot (%): 153



Page 11 of 14

As we note in the first locus, implemented by MatLab, the plot tell us that when the gain K = 3.78 the response will have overshooting 1.34% and damping ratio 0.8 and when K = 9.36 the response will have no overshooting. Also the locus say that all values of K will not make the system unstable and will remain in the stable region.

The second locus tell us when the gain K = 9.44 the system will be unstable because the poles become in RHP, so we can conclude that the proportional controller may drive the system from the unstable mode to a stable one and vice versa.

Extra About root locus:

We knew that root locus plot the poles for a closed loop transfer function that will be controlled by Proportional controller and the system must have unity feedback path as in figure below.

But if the system does not has unity feedback as in figure below, so we should convert it to another has unity feedback by the solution below.

Root locus will take this

transfer function and use it

to plot the poles as K varies

K X(s) Y(s) + -

E(s)

G(s)

(Plant)

Note that Unity feedback

K X(s) Y(s) + -

E(s)

G(s)

(Plant)

F(s)

NON-Unity feedback


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Note that K is out of our converting because we consider that we added it after conversion.

MatLab Command :

rlocus (sys) : This command used to plot the root locus of the open loop

transfer function which will be controlled by K controller.

[Wn,Z] = damp(sys) : Compute damping factors and natural

frequencies.

sgrid : Generate an s-plane grid.

sgrid (z,wn) : Generate an s-plane grid of constant damping factors and

natural frequencies.

Lab view front panel and block diagram:

Front panel: sketch the root locus for the transfer function

K X(s

)

Y(s

) +

- E(s)

Unity feedback


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Block diagram: construct the root locus function to find the suitable value of k

Notes:

For more information about root locus technique see the help in LabView and function browser in MatLab.


Page 14 of 14

LAB WORK: 1. (Feedback Transfer function) By MatLab and LabView, for the open loop transfer function G(s),

1

1)(

2

sssG

a. Plot the step response of the open loop system. b. Plot the closed loop transfer function with unity feedback. c. Record your notes about the open and closed loop system.

d. Obtain the closed loop transfer function with feedback 1

1)(

ssF

2. (Stability Proportional Controller) By MatLab and LabView, for the open loop transfer function G(s)

)15.0(

1)(

2

sss

ssG

a. Plot the open loop step response and comment on it.

b. Plot the closed loop transfer function and comment on it.

c. Plot the root locus of this system and obtain the value range of the

proportional gain to make the system:

1. Stable.

2. Unstable.

3. Marginal stable.

linear control systems laboratory

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