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Computer Aided Control Systems Design for a Simple Positioning Mechanism

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COMPUTER AIDED CONTROL SYSTEMS DESIGN FOR A SIMPLE POSITIONING MECHANISM

DEN211: VIBRATIONS AND CONTROL OF DYNAMIC SYSTEM

Name: Akash Patel
Student Id: 074784880

CONTENTS

INTRODUCTION 3 DESIGN OF A PD CONTROLLER FOR A SET POINT CONTROL 4 TUNING OF THE PD CONTROLLER 7 DESIGN OF A NOTCH FILTER ( CONTROLLER DERIVATION AND DESIGN) 15 TRANSFER FUNCTION THAT INCLUDES A NOTCH FILTER AND PD CONTROLLER 18 DISCUSSION AND CONCLUSION 21
REFERENCES ......................................................................................................................................................................... 22

INTRODUCTION

A set point control is one of the most vital problems in all systems, because if for a set point control, the controller doesn’t work then it would not function for other complicated controllers. Hence for a set point all requirements play an important role and hence they have to be met.
The aim of this exercise was to design a PD controller for set point control of a cart that is being steered from its original position to a desired point. However, the PD controller was to be tuned in such a way that provided objectives are met. These objectives were in terms of performance measures and they were as follows: * The 10%-90% rise time is smaller than 0.5s. * The percentage overshoot is smaller than 5%. * The settling time is not larger than 5s.
Where,
* The 10%-90% rise time is defined as the time it takes the cart to go from 0.1xd to 0.9xd. * The percentage overshoot (P.O.) is defined as the percentage ratio of the maximum deviation from xd after xd has been passed for the first time, and the value of xd. * The settling time is defined as the time it takes the cart to remain within 2% of xd.
The format of the report is as follows: The introduction will be followed by the design of a PD controller for a set point control and the tuning procedure of the PD-controller. This will then be followed by the design of the notch filter and the final system plots. Finally, this will be followed by the discussion and conclusion.

DESIGN OF A PD CONTROLLER FOR A SET POINT CONTROL

The cart system whose controllers were to be designed was a simple positioning mechanism that constituted of a motor and a drive shaft, to which a load L was fixed. For every torque produced by the motor and the moment of inertia of the shaft and load is equal to I.
Therefore
Iθ=τ
Where
θ is the angle of rotation of the shaft
Considering a mass system shown in the figure 2 below, the equation of motion of the system below has the same form of equation as the figure 1. This is because in the figure 2 the driving force is u, velocity, while the driving force in figure 1 is torque.
Hence
mx=u
This can be expressed in Laplace form ms2X(s) = U(s)

Hence the transfer function of the system is given by

G(s) =1ms2

This being the main equation of this system, this exercised was aimed to design several controllers of it.

Assuming, that the initial position of the cart can be measured the error between final position and the position at any time e(t)= xd- x (1)

The PD controller will therefore have the following form

ut=G1se= Kpet+ KD e(t) (2)

Taking Laplace transform gives

U(s) = (KP + KDs)E(s)

Therefore the transfer function of the PD controller will be

Cs= KP + KDs

and

x= G2su (3)

Figure 3: Block diagram for the closed loop system

Substituting (1) into (2)

u= G1s(xd- x) (4)

Substituting (4) into (3)

x= G2s(G1sxd- x) (5)

The Laplace transform of (5) is given by:

X(s)= G2s(G1sXd(s)- X(s)) (6)

Substituting G2s and G1s from Figure 1 into (6) gives Xs=KDs+KP1ms2Xds- Xs= KDs+KPms2Xds- Xs (7)

Simplifying (7) gives

Xs+ KDs+KPms2Xs= KDs+KPms2Xds

hence

XsXds= KDs+KPms2+KDs+KP (8)

therefore the closed loop transfer function of system 1 is given by
Gcls=X(s)Xd(s)=KDs+KPms2+KDs+KP (9)

TUNING OF THE PD CONTROLLER

The system transfer function in the previous section was required to be presented in such a way that the performance requirements were met. To do this a programmed MFILE was created that could produce an output in such a way that showed values of Kp and KD that met the requirements and at the same time produced graphs of those values.

-------------------------------------------------
The MFILE was as shown below:

for Kp = 0:0.2:1.5 % assigns values to Kp with a step of 0.2 for Kd= 0:0.01:4.4 % assigns values to KD with a step of 0.01
G= tf ([Kd Kp],[1.1 Kd Kp]);% defined tranfer function
S = stepinfo(G,'RiseTimeLimits',[0.01,0.85]); if S(1).RiseTime < 0.5 % rise time has to be less than 0.5s if S(1).Overshoot < % percentage overshott has to be less than 5% if S(1).SettlingTime <5 % settling time has to be less than 5s S=S Kp=Kp Kd=Kd ltiview('step',G); % displays the graphs end end end end
-------------------------------------------------
end

The following values for Kp and KD were selected and Kp = 0.6 and KD = 4.39 was selected to be further analysed.

Kp | Kd | Rise Time | Settling Time | Settling Min | Settling Max | Overshoot | Undershoot | peak | Peak Time | 0.2 | 4.27 | 0.4996 | 0.9311 | 0.8906 | 1.0111 | 1.1064 | 0 | 1.0111 | 2.2109 | 0.2 | 4.39 | 0.4863 | 0.9111 | 0.89 | 1.0105 | 1.0502 | 0 | 1.0105 | 2.4176 | 0.4 | 4.22 | 0.4991 | 2.9642 | 0.902 | 1.0214 | 2.1402 | 0 | 1.0214 | 1.9838 | 0.4 | 4.39 | 0.4806 | 0.8154 | 0.9003 | 1.0199 | 1.9919 | 0 | 1.0199 | 1.9031 | 0.6 | 4.22 | 0.4928 | 4.9809 | 0.9132 | 1.0306 | 3.0564 | 0 | 1.0306 | 1.7237 | 0.6 | 4.39 | 0.4751 | 4.5718 | 0.9106 | 1.0285 | 2.8464 | 0 | 1.0285 | 1.9271 |

From the table above it can be observed that for every interval of Kp, as the values of Kd increase the rise time, settling time and percentage overshoot values decrease, but the peak time increases.

Graph 1
Kp = 0.2 and KD = 4.27

Graph 2
Kp = 0.2 and KD = 4.39
Graph 3
Kp = 0.4 and KD = 4.22

Graph 4
Kp = 0.4 and KD = 4.39
Graph 5
Kp = 0.6 and KD = 4.22

Graph 6
Kp = 0.6 and KD = 4.39

Having found the values of Kp and KD a notch filter was to be designed. A notch filter is used to eliminate vibrations in the mechanism of the system show below. These vibrations could be caused due to application of large loads, where the shaft of the mechanism exhibits a considerable amount of flexibility.

Figure 1
Assumptions can be made and the system above can be considered to be interconnected as shown below by a spring and a damper.
Figure 4

Applying Newton’s second law of motion on the 0.1kg system

ΣF=mӱ=u-kex- c(v) (10)

From system 2 it can be extracted that ex=y-x (11)

Differentiating this with respect to time

v=ẏ-ẋ (12)

Substituting (11) and (12) into (10)

mӱ=u-ky-x- c(ẏ-ẋ) But m=0.1 k=20 c=5

giving

0.1ӱ=u-20y-x- 5(ẏ-ẋ) (13)

Taking the Laplace transform of this

0.1Y(s)(s2-ṡ0-s0)=U(s)-20Ys-Xs- 5(Y(s)(s-s0)-X(s)(s-s(0))

For the transfer function the initial conditions have to be 0 hence s0=ṡ0=0

Substituting the initial conditions and simplifying the above equation gives
0.1s2Y(s)=U(s)-20Ys-Xs- 5(sYs-sXs) (14) 0.1s2Ys+20Ys+ 5sYs=20Xs+5sXs+Us

Ys0.1s2+20+ 5s=X(s)(20+5s)+Us (15)

Ys=X(s)(20+5s)+Us0.1s2+20+ 5s (16)

In the same way,

Xs=Ys20+5ss2+20+ 5s (17)

Substituting (21) into (15) and (16) in to (17),

Ys0.1s2+20+ 5s=Ys20+5ss2+20+ 5s (20+5s)+Us Ys0.1s2+20+ 5ss2+20+ 5s-Ys 20+5s (20+5s)=Uss2+20+ 5s

(s2)(0.1s2+22+ 5.5s)Ys=Uss2+20+ 5s

Hs=YsUs=s2+20+ 5s(s2)(0.1s2+22+ 5.5s)

Substituting (16) into (17)
Xss2+20+ 5s=X(s)(20+5s)+Us0.1s2+20+ 5s 20+5s

Xs0.1s2+20+ 5ss2+20+ 5s-Xs 20+5s (20+5s)=Us20+ 5s

(s2)(0.1s2+22+ 5.5s)Xs=Us20+ 5s

Gs=XsUs=20+ 5s(s2)(0.1s2+22+ 5.5s)

Figure 5

The closed loop transfer function obtained by giving feedback and series commands in MATLAB for the transfer function from xd to x will be:

Gcl=2.195s4+129.8s3+983.5s2+2064s+2640.01s6+1.539s5+61.05s4+471.7s3+1481s2+2064s+264

Graph 7: The Graph below shows the step response of the transfer function Gcl.

Graph 8: The Graph below shows bode magnitude plot of the Transfer function Gcl

DESIGN OF A NOTCH FILTER CONTROLLER DERIVATION AND DESIGN

A notch filter is used to eliminate resonance to improve the performance. Notch Filters are linear systems with a transfer function typically defined as:

Fs=s2+2ζ1ωns+ωn2 s2+2ζ2ωns+ωn2 where ζ1<ζ2

But s=iω, therefore,
Fiω= iω2+ 2ζ1ωniω+ ωn2iω2+ 2ζ2ωniω+ ωn2
Fiω=ωn2-ω2+ 2ζ1ωniω ωn2-ω2+ 2ζ2ωniω Fiω = magnitude of the notch filter
Where z=A+Bi , and z=A2+ B2 ( complex number)
Therefore, F(iω) = ωn2-(ω)2+ 2ζ1ωn(iω) ωn2-(ω)2+ 2ζ2ωn(iω)
F(iω) = ωn2-(ω)2+ 2ζ1ωn(iω) ωn2-(ω)2+ 2ζ2ωn(iω)
Dividing by ωn2, gives
F(iω)= 1- ω2ωn2+ 2ζ1(iω)ωn1-(ω)2ωn2+ 2ζ2(iω)ωn
If ω2ωn2= ω0 then Fiω= 1- ω0+ 2ζ1ω012i1- ω0+ 2ζ2ω012i
Fiω= 1- ω02+ 4ζ12ω01- ω02 +(4ζ22ω0) Fiω2= 1- ω02+ (4ζ12ω0)1- ω02 + (4ζ22ω0)= 1- ω02+ (4ζ12ω0)+(4ζ22ω0)- (4ζ22ω0)1- ω02 + (4ζ22ω0)
Fiω2=1- (4ζ22ω0)- (4ζ12ω0)(1- ω0)2 + (4ζ22ω0)= 1- 4ω0ζ22- ζ12(1- ω0)2 + (4ζ22ω0)

Therefore Fiω2= 1- 4ω0ζ22- ζ12(1- ω0)2 + (4ζ22ω0)

Fiω is between 0 and infinite when limω→0ω0=0 and limω→0Fiω=1
Hence when limω→∞ω0=∞
Therefore
limω→∞Fiω=1- 4∞ζ22- ζ12(1- ∞)2 +(4ζ22ω0)
Therefore limω→∞Fiω=1= limω→0Fiω
Hence in summary, The magnitude |F(iω)| of the Notch Filters frequency response have to satisfy the properties given below:

1. Fiω2=1-4ζ22-ζ12ω1-ω2+4ζ22ω ω=ωωn2

2. limω→0|F(iω)|=limω→∞|F(iω)|=1

3. |F(iω)| has a minimum ζ1ζ2 when ω=ωn

Random set of value selected for ζ1, ζ2 and ωn were made and tabulated as show below. These values were substituted in the notch filter equation and the transfer function was then graphed using MATLAB to verify the above given properties.

| set 1 | set 2 | 1 | 0.1 | 0.2 |  | 0.9 | 1 | n | 1 | 1.5 |

The graph below shows a plot of both set of values show in the table above. It can be concluded from the graph that the plots below confirm the properties.

Graph 9:
-------------------------------------------------

TRANSFER FUNCTION THAT INCLUDES A NOTCH FILTER AND PD CONTROLLER

Having checked the properties of the notch filter, from the bode magnitude plot of the closed loop system the resonance frequency and the magnitude of resonance frequency was extracted and the values were ωr=1.29 rad/sec
Mr=0.412 dB

Ref: Graph 8

the value for the magnitude of resonance (Mr) was replaced in the equations below:

γ=10(-Mr/20), δ=10(-0.7Mr/20)

γ=10(-0.412/20) γ=0.9536 δ=10(-0.7×0.412/20) δ=0.9673 From the bode magnitude plot frequencyω0 ; defined as the frequency at which the magnitude; is 0.7Mr was obtained.

Magnitude=0.7Mr
=0.7×0.412
=0.2884

This magnitude value was found on the closed loop Bode Magnitude graph and the corresponding frequency value of ω0= 0.348rad/sec was obtained.

Therefore, given

ωn=ωr=1.29 rad/sec

ω0=ω0ωn2=0.3481.292=0.07277 rad/sec

ζ2=1-δ2(1-ω0)24ω0(δ2-γ2)=1-0.96732(1-0.07277)24×0.07277(0.96732-0.95362)=2.6870

ζ1=γζ2=0.9537×2.6870=2.56237

After calculating the values of ζ1, ζ2 and ωnfor the main mechanism, it was substituted in the notch filter function, and the notch filter equation obtained is as shown below. Fs=s2+6.6343s+1.6641s2+6.9564s+1.6641

This notch filter function was then placed in series with the transfer function Gcl to obtain a transfer function Xl, which consisted of a well defined system that made sure most of the resonances were eliminated, and the bode magnitude and the step response graph of the transfer function of the system was plotted as shown below.

Graph 10: From the graph below it can be observed that all the values meet the set objectives although the rise time increases to 0.554. However the curve is much undisturbed therefore the mechanism has improved in terms of reduced resonance and less settling time.

Graph 11: The Plot below shows the bode magnitude plot of the final transfer system

DISCUSSION AND CONCLUSION

MATLAB is a very useful tool in engineering and is used in a wide range of applications. The main advantages of using MATLAB is it helps in understanding of theoretical principles by means of enhanced graphical aids and interactive simulations, analysis of more complex systems. For example this reports looks at how MATLAB can be used in minimising resonances and vibrations of a cart in moving from one point to another, with the help of PD controllers and Notch filter, hence making its motion smoother

A PD controller has two parts; the proportional controller that reduces the rise time by making the system faster although the negative effect is that it introduce steady state error. The other component of the PD controller is the derivative component where if the derivative component is positive the percentage overshoot of the system would reduce, however it would make the system at the same time slower.

Notch Filter is a band-stop filter designed for systems where interfering signals for example in this case it’s the extra applied loads that are degrading the performance of the system. The notch filter is to be designed for this system in particular and therefore resonance values are required for this particular system. Hence from the bode-magnitude plot of the function Gcl the maximum peak response values, ωr and Mr was taken. The resonance peak was reduced by putting the notch filter in series with the closed loop system.
From graph 10 it can be observed that 1 of the 3 performance requirements were not met. The rise time was required to be less than 5s, but by observation the rise time was 5.54s. This is due to conflicting performance criteria. They are not always met, and are hence carried forward as negligible errors. However comparing Graph 10 to graph 6, it can be discussed that although the percentage overshoot increased, but the settling time reduced. Hence as soon as the system starts, in 1.62s the system settles. Also, the negotiations made by the PD controller affect the performance of the system.

REFERENCES 1. Tongue B. Principles of vibrations. Second edition, Oxford university press, 2002. 2. Timoshenko S et al. Vibrations problems in engineering. Fourth edition, 1974. 3. Computer Aided Control Systems Design for a simple positioning mechanism (lab sheet) 4. Kelly, S.G. (2000). Fundamentals of Mechanical Vibrations. 2nd. MrGraw Hill. / ISBN:0071163255…...

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