1.systems and simulations (very important)

8/9/2019 1.Systems and Simulations (VERY Important)

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UNIVERSITY OF NATAL

DEPARTMENT OF ELECTRICAL ENGINEERING

SYSTEMS AND SIMULATION - DNEL3SS1

Examinations June 1996

Time: 3 hours Examiners: Dr E Boje

Total marks: 100 Dr P Gleeson

Instructions to candidates:

1) Attempt FOUR out of five questions. All questions carry equal marks.

2) Candidates are allowed a single A4 sheet of hand written notes. Both sides may be used.

3) Normalised charts and Laplace and z transform tables are attached. Semilogarithmic graph paper

is available.

Question 1 (25 marks)uA, TA, Γ A uB, TB, Γ B



Consider the following chemical reaction in which substances A

and B form a new substance AB by means of the reversible

reaction,

A + B ⇔ AB

The reaction rate of formation of AB is,

α Tm

m m

AB

A B

log( )2

(in kg/s)

where

T is the temperature in ºKelvin,

α is a constant

mA, mB, mAB are the masses (in kg) of A, B and AB

respectively

Each kg of AB formed releases q Joules of energy (i.e. exothermic reaction) (if the reaction is

reversed, the energy is absorbed).

The reactants, A and B enter a well stirred, insulated vessel of constant volume, V=constant (i.e.

volumetric outflow = volumetric inflow). The substances are diluted in water and the thermal

behaviour is that of water, with specific heat, cw

and density

ρw. For inputs A and B, for example,

Γ A is the inflow concentration in kg of A per m3

inflow

u is the inflow rate of A in [m3/s]

V=const

outflow

Question 2 (25 marks)

a) In Simulink, how would you do the following?

i) Simulate the linear system, SA B

C D=

=

−−

1 1 3

0 2 4

4 5 1

. (3)

ii) Change the initial condition of an integrator to 3.5. (2)

b) Consider the model of a non-linear pendulum shown.

( )& sin( )

&

ω θ βω

θ ω

= − − +

=

1

Jmg Tul

where,

J m= l2 is the inertia,

//////////////////////////////////

θ lT



J ml a,

β is the linear friction

θ is the angle from the vertical

ω is the angular velocity

Draw the Simulink implementation diagram if m=2kg,l

=1m, β=0.1, g=10 and Tu=1Nm unit stepapplied at t=0.2. The state variables and the time should be sent to the workspace. (15)

c) Show how you would output the Potential and Kinetic energy in the system above (5)


a) Find the state space representation of the following

linear mechanical system. Use x =

x

v

x

v

1

1

2

2

The output is y=x1, the position of mass m1.

mTu

///////////////////////////////

B1

m1

k 1

k2

x1, v1

B2

Q3bii) Write down the observer form state space description of

P ss s

s s( ) =

+ +

+ +

2

2

1

2 3 (3)


ai) Draw the Bode plot of the transfer function,

P ss

s s( )

( / )

( / ) . ( / )=

+

+ × +

3 5 1

2 2 0 4 2 12(10)

aii) Without doing any long calculations, what would the approximate response of the above system

be to an input, u(t) = 2 sin(2t)? (3)

bi) Find the unit step response (assuming zero initial conditions) of the system,



) p p ( g ) y ,

P ss

s s( )

( / )

( )( / )=

++ +3 2 1

1 3 1(5)

bii) Sketch your solution (3)

ci) What damping factor is required from a second order system to have an overshoot of 10%? (3)

cii) What would the corresponding maximum dB overshoot be on the Bode plot? (1)


ai) The system, P s ss s

s( ) = ++ +4 7

3 22 is subject to a staircase input with T=0.5. Calculate the z

domain transfer function, Pz(z). (Hint: Use a partial fraction expansion.) (7)

aii) Use the final value theorem to find the final value of the output if the input is a step. Does your

answer make sense with respect to the final value of the step response of the continuous system? (3)

b) Given P ss

s( ) = +2 3, find the w domain description for T=0.2 and comment on the similarity and

diff b t P ( ) d P ( ) (U 3 d i l l i l l ti ) (8)

UNIVERSITY OF NATAL





Total marks: 120 Dr G Bright


1) Attempt all six questions. All questions carry equal marks.


3) Normalised charts and Laplace and z transform tables are attached. Semilogarithmic graph paper

is available.

Question 1 (20 marks)liquid

C

sugar

r C



Consider the ice-cream manufacture process shown alongside.

The ice cream is made up of the following ingredients:

i) Liquid (cream, milk, water etc.) added at a rate rl [kg/hr] and

with a density ρl [kg/l].

ii) Sugar, added at a rate rs [kg/hr]. The sugar dissolvescompletely.

iii) Air is added (fully entrained - does not dissolve or escape) at

a rate of ra [litres/hr] and only adds to the volume of the ice-

cream product, not to the weight.

The tank has a cross sectional area, A, and is well mixed by the

agitator.Ice cream outflow is uout [l/hr]

a) Find a model for the process with height in the tank and density of manufactured ice-cream as

outputs. Be careful of units. Clearly state any further assumptions you find necessary. (15)

b) By considering profit as the difference between cost and income, add a state differential equation

to your system to give profit as an additional output. The liquid ingredients cost Cl [Rand/litre], the

sugar costs Cs [Rand/kg] and the air is free (i e negligible compression costs etc ) The selling price

rl, ρl, Cl rs, Cs

ice cream

outflow

air

ra

h


a) Explain the relationship between step size and global accuracy for a second order method. Make

a sketch to illustrate your discussion. (5)

b) The linear equation, &x A x= , x x( )t0 0

= has the solution, ( )x x A x( ) !t e t k t k

k

= =

=

∞

∑

A

0 0 0

Show that the modified Euler method,

x x

f x

f x

i i

i i

i i i i

t

t

t t t

+

+ +

= ++

== + +

1

1 1 2 2

1

2 1 21 1 1 2

∆

∆ ∆

ϕ

ϕ = ϕ ϕ

ϕ

ϕ ϕ

α α

β γ ( , )

( , )

withα α

β γ1 2 1 2

1

= = /



β γ 21 2 1= =is second order and not third order for this linear equation. (10)

c) Draw the Simulink implementation diagram for the following model of a system:

& ( )

& sin( )

x x x x h u

x x x u

y x x

y x u

1 2 1 2

2 1 2

1 12

22

2 1

= + += +

= += +

h(u) is available as a set of measured data,

u -4.0 -1.0 0.0 1.0 3.0 5.0

h(u) 1.213 2.124 3.234 2.456 1.235 1.123

The input will come from a function generator and the output will be sent to the workspace for

plotting. (5)


a) Find the state space representation of the following

linear mechanical system. Use x =

x

v

x

v

x

1

1

2

2

3

The output is y=x1, the position of mass m1.

f

///////////////////////////////

B1

m1

k 1

x2, v2

x1, v1

m2

k 2

B2

B3

x3



(15)

b) Give rough sketches of the Bode magnitude and phase plots and the unit step responses of second

order low-pass, high-pass and band-pass systems (for example as investigated in the laboratory) to

highlight the differences. (5)


ai) Draw the Bode plot (on semi-logarithmic graph paper) of the transfer function,

P ss e

s s( )

( / )

( / ) . ( / )

.

=+

+ × +

−3 2 1

5 2 05 5 1

005s

2(12)

aii) From the Bode plot, what would the approximate harmonic response of the above system be to

an input, u(t) = 2 sin(2t)? Confirm your answer by simple calculation. (3)


a) Find the response of the system, SA B

C D=

= − −

0 1 1

2 2 2

3 2 1

,

with initial condition x(0) = (1,0)T, and input, u(t) =

σ(t), a unit step. Sketch your solution. (12)

b) Consider the ping pong ball levitator shown alongside.

Assume that a nozzle makes a cone of air with velocity,

v h

u t

hair ( )

( )

( )= + α 2

u(t) is the opening of an air supply valve, h is the height above the2

h

nozzle

ball vair



nozzle and α is a constant. The drag force on the ball is , ( )f c A v vup d air ball= − 2. “A” is the ball

cross sectional area and cd is the drag coefficient. The model of the ball dynamics is then,

( )

dv

dt m

c Au t

h

v mg

dh

dtv

balld ball

ball

=+

−

−

=

1

2

2

( )

α

The output of interest is the total energy of the ball, Emv

mghballball= +

2

2

Find the linearisation of the system around the equilibrium point with vball = 0, hball = 0.1.Take m = 0.03kg, g = 9.8Nm/s

2, A = 1E-3m

2, cd = 0.4, α = 0.05m (8)


a) The system, P ss ss

( ).

=+ +

3

2 52is subject to a staircase input with T=0.2. Calculate the Z

domain transfer function, Pz(z). (8)

Show that aliasing will occur if the signal y t e tt( ) sin( )= − 2 is sampled with T=0.5 by finding the z

transform of the sampled sequence and then finding the inverse z transform. (7)

c) If a system has the transfer function P ss

( ) /

=+

1

3 1, and T=0.2, find Pz(w).

d) Show that the unit step signal ,u(t) = σ(t), has

aii) Use the final value theorem to find the final value of the output if the input is a step. Does your

answer make sense with respect to the final value of the step response of the continuous system? (3)



UNIVERSITY OF NATAL



Examinations: June 1998


Total marks: 110 Dr G Bright


1) Attempt ALL five questions. Questions do not carry equal marks.


3) Normalised charts and Laplace and z transform tables are attached. Semi-logarithmic graphpaper is available.


A hydroelectric pumped storage scheme moves water between an upper and a lower water



Vu

Upper reservoir

A hydroelectric pumped storage scheme moves water between an upper and a lower water

reservoir (subscripts u and l respectively). During periods of low electricity demand, power ischeap and water is pumped up to the upper reservoir. (Energy is stored as potential energy).

During peak electricity demand, water flows (backwards) through the turbines and the

potential energy is recovered as electrical energy (i.e. electricity is sold back onto the grid at a

higher price). (Electricity is usually sold in kWh but we will use MJ as a measure of power.)Develop a model for such a scheme under the following assumptions:

• The electricity price (tariff) is T(t) [Rand/MJ] and is a function of time because of

varying demand.

• The power set-point of the generator station is set at P(t) [MW = MJ/s] by the operator.

P(t) > 0 implies that the station is generating (i.e. selling electricity).

P(t) < 0 implies that electricity is being used to pump water.

• The evaporation rate from the reservoirs is given as Eu(t) and El(t) [m3 /s].

• The inflow rate into the reservoirs from rainfall is given as Ru(t) and Rl(t) [m3 /s].

• The acceleration due to gravity is g [m/s2].

• The density of water is ρ [kg/m3]

• To keep the model simple, the efficiency η = (what you get)/(what you pay for) of

pumping or generating is η = 1.

• The head, H, can be regarded as constant.

• There is a constant (overhead) cost, C [R/s]of running the station if it is generating

UNIVERSITY OF NATALDEPARTMENT OF ELECTRICAL ENGINEERING

SYSTEMS AND SIMULATION - DNEL3SS1 June 1998

PAGE 2


a) Consider the model of a dc machine driving a mechanical load,

( )

( )aa

mecha

Rik VL

1

dt

di

dt

d

)(Tik J1

dtd

−ω−=

ω=θ

ω−βω−=ω

φ

φ

where the state variables are,

ω = machine speed

θ = shaft position

i = armature current



ia = armature current

and L (inductance), J (moment of inertia), k φ (torque constant), R (resistance), β (friction

constant) and µ (load constant) are parameters. Tmech(ω) is the mechanical load which is

known from measurements and is given as a table.

ω Tmech(ω)

0 050 5

100 15

150 40

200 100

i) Set up the Simulink implementation diagram for the system with V a step from 200V

to 300V at time t=2s. The output speed should be sent to the workspace along withthe simulation time. The armature current should be displayed on a graph. (10)

ii) Explain how to set up initial values for (ω, θ, ia) (2)

iii) Explain how to run the simulation for 10 seconds with at least 100 points between

the start and end time. (3)

b) Show that the Linear Implicit Euler method is a single step method by expressing the

method in the form,

ii1i t?xx ∆+=+ (5)



PAGE 3

Q3b) In the laboratory, the following response was measured as the step response of a secondorder low-pass system. Identify the system transfer function - i.e. write down P(s). NB: The

gain is not unity. (8)

Step Response

1.5

2



Time (sec.)

A m p l i t u d e

0 0.5 1 1.5 2 2.5 3

0

0.5

1


a) Find the state space representation of the following linear electrical system. (NB x1 is thecurrent through L1 and y1 is the voltage across L1.) (12)

y2

+R2

x1 R1L3

x3



PAGE 4


a) Given3s

4)s(P

s +=, find the w domain description for T=0.2 and comment on the

similarity and difference between Ps(s) and Pw(w). (Use 3 decimal places in your

calculations.) (8)

bi) Calculate the z-transform of the signal, y(t) = e(-t)

sin(5t) sampled at T=1.0. (4)bii) By finding the inverse z-transform, show that aliasing has occurred and explain why. (3)

c) Find the first 4 terms in the inverse z-transform of 3.0z5.0z

z2)z(Y 2 ++= by means of

long division. Use the final value of the signal to confirm that you result makes sense. (5)



UNIVERSITY OF NATAL

School of Electrical and Electronic Engineering




Total marks: 115 Mr B Burton


1) You may attempt all six questions. A maximum of 100 marks will be awarded. Questions do not

carry equal marks.


3) Normalised second order charts and Laplace- and z-transform tables are attached. Semi-

logarithmic graph paper is available.





Consider the evaporating system illustrated. The system works at

standard atmospheric conditions. Water enters at a flow rate of rin

[m3

/s] and temperature Tin [°C]. An electric element is used to heatand if the temperature is at 100°C, evaporate the water. (Assume

that no evaporation occurs below 100°C.) The element power is

Pelec [watts]. The heat capacity of the water is Cw [J/kg/ °C] and the

latent heat of evaporation (i.e. water → gas) is L [J/kg].

Develop a dynamic model for the system, choosing the height of the water in the vessel as the output

variable. The vessel has a uniform cross-sectional area, A. Assume that the water density, ρ, does

not change with temperature.

Note that depending on the water temperature, you will require two different differential equations

for the same state variable. You may use an IF {condition} THEN {SDE_Option1} ELSE

{SDE_Option2} to achieve this.


rin, Tin

h

Pelec

UNIVERSITY OF NATAL


SYSTEMS AND SIMULATION - DNEL3SS1- June 1999

Page 2 of 4

Question 2 (continued)

2c) Consider the system,

+− +=211

2xxx

txx& , −= 120x

Illustrate the modified Euler by simulating 2 steps (i.e. calculate x1 and x2 by hand), with ∆T = 0.1.

Use 3 decimal digits and present all (intermediate) calculations in a tabulated form. (10)


Consider the following model of a cell growth in a bio-system. The state differential equation is

dC S V/



dC

dt

S V

S V KC

s

=+

µmax /

/ cell growth rate

inins

max uCKV / S

V / S

dt

dSρ+

+µγ −= substrate conservation

outin uudt

dV−= volume conservation

with,

growth rate, rS V

S V Ks

=+

µmax /

/ .

µmax is the maximum growth rate.

S is the substrate massV is the volume

KS is the limiting substrate concentration.

γ is the yield

C is the number of cells

uin is the substrate volumetric feed rate

ρin is the substrate concentration= [0.1 (kg substrate)/(m

3feed)]

uout is the outflow rate

uin and uout are shown below.

uin

13

uout

35

UNIVERSITY OF NATAL



Page 3 of 4


a) Find the state space representation of the following linear electrical system. The outputs are the

voltages across the inductor and resistor. The state variables are the voltages across the

capacitors and the current through the inductance.

(12)

y1

+

y2

C1 x1u1

R1

C2 x2

L x3



4bi) Given the following unit step response of a second order system (for example as investigated in

the laboratory), find the parameters, k, ωn and ζ of the transfer function,

( ) ( ) 1 / s2 / s

k )s(P

n2

n +ωζ+ω= . (8)

bii) Make a rough sketch of the corresponding Bode plot showing salient features. Semi-log paper is

not required. You should find an approximate value of the phase at ωn /2 and 2ωn using the

normalised chart). (5)

m p l i t u

d e

Step Response

2

2.5

3

3.5

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Page 4 of 4


ai) Draw the Bode plot (on semi-logarithmic graph paper) of the transfer function,

( )( )( )110 / s11 / s13 / se3)s(P

s2.0

++ +=−

(12)

aii) From the Bode plot, what would the approximate harmonic response of the above system be to

an input, u(t) = 2 sin(2t)? Confirm your answer by a simple calculation. (3)

b) Find the natural response of the system,

−−=

=

123

242

110

DC

BAS , with input, u(t) = 0 and



123

initial condition x(0) = (-1, 1)T. Sketch your solution. (10)


a) The system,3s2s

4)s(P

2s++

= is subject to a staircase input with T=0.2 seconds. Calculate the

Z domain transfer function, Pz(z). (5)

b) The system,2s

1)s(Ps += is subject to a staircase input with T=0.1 seconds. Calculate the w-

domain transfer function, Pz(w) and comment on the relationship between s- and w-domain transfer

functions. (5)

UNIVERSITY OF NATAL





Total marks: 117 Mr B Burton


1) You may attempt all six questions. A maximum of 100 marks will be awarded. Questions

do not carry equal marks.


3) Normalised second order charts and Laplace- and z-transform tables are attached. Semi-logarithmic graph paper is available.




Two tanks are arranged as shown in the diagram below. Salt and water are added into the first

tank at rates

Γ s[kg/s] and

Γ w[m

3 /s] respectively. The tanks are well mixed and the salt

dissolves completely. The flow between the two tanks depends on the pressure difference,

2211AA hhk f ρ−ρ= . (Recall that density, ρ = mass/volume, ρwater=1000kg/m3.) The

flow out of the second tank is given by 22BB hk f ρ= . Both flows are in m3 /s of the

solution in the respective tanks and you can assume that they are always positive. The tank

areas are A1 and A2 respectively. Find the model of the system. The (two) outputs required are

the density of the solution in each of the tanks.

Γ water [m3 /s]

Γ salt [k /s]

UNIVERSITY OF NATAL



Page 2 of 4


a) For fixed ∆ti+1=∆t, the linear equation, &x A x= , x x( )t0 0= has the solution,

( ) )t(!k t)t(e)t( i

0k

k k i1i xAxx

tA ∑∞

=

∆+ ∆==

A 2rd

order Runge Kutta method (not modified Euler) is,

xi+1 = xi + ∆t ϕϕ (xi, ti, ∆t)

214

1

4

3ϕϕϕϕϕϕ +=

with,

( )ii1 t,xf =ϕϕ

∆+∆+= t

2

3t,t

2

3i1i2 ϕϕϕϕ xf



For the above linear equation, show that this method is 2nd

order and not 3rd

order. (6)

b) Explain how two types of step-size control algorithms work. (You may wish to use a

combination of explicit- and modified-Euler as an example.) (8)

c) Find the observer form state space description of the following system:

( )

( ) ( )( )( )11 / s12 / s5.022 / s

13 / s2)s(P

2 ++×+

+= (6)


a) A system has the following model:

( )

)uxtan(y

)uxsin(3)xcos(2xx

xx

1

1122

21

−=−−+−=

=&

&

Find the linearised state space model about the steady state with x = (0, 0)T. (6)

b) Draw a Simulink implementation diagram of the system in Question 3a. The input comes

from a function generator and the output and simulation time must be sent to theworkspace. (5)

UNIVERSITY OF NATAL



Page 3 of 4


a) Find the state space model of the following electrical circuit. Note that the input u2 is a

current source and the output, y2is the voltage across this source. Be careful to work

systematically. (20)

Q ti 5 (23 k )

y1

+R1

L2

x2

u1

R2L3

C1 x1

x3

u2 y2




a) The Bode plot of a second order, low pass system (such as was measured in the practical

SS2) is shown below. Find the plant model and hence sketch the step response showing

salient features (tp, Mp, tr, ts, etc). (8)

P h a s e ( d e g ) ; M a g n i t u d e ( d B )

Bode Diagrams

-30

-25

-20

-15

-10

-5

0

5

From: U(1)

-120

-90

-60

-30

0

T o : Y

( 1 )

UNIVERSITY OF NATAL



Page 4 of 4


a) Find the z-transform of the following system transfer function assuming the input is a unit

staircase (i.e. zero order held). The sampling rate is T=0.2.

)5s)(1s(

2s)s(P

+++= (6)

b) Given the z-transform of a sequence, U(z) = Z{ui},

6.0z8.0z

z)z(U

2 +−=

i) Find the first 4 terms of ui by long division. (4)

ii) Find the underlying signal u(t) assuming that T=1 and no aliasing has occurred.

(4)



( )

1.systems and simulations (very important)

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