QUESTION:

1) Consider the system shown below which represents the altitude rate control

for a certain aircraft.

(a) Design a compensator so that the dominant poles are at -2 +/- 2j.

(b) Sketch the Bode plot for your design, and select the compensation so that the

crossover frequency is at least2* 2^0.5 rad/sec and PM >=50deg.

(c) Sketch the root locus for your design, and find the velocity constant when

crossover frequency > 2*2^0.5 and damping ratio >=0.5.

SOLUTION USING MATLAB

>> %Our aim is to design a compensator D(s) that satisfies the conditions given in >> %question

>>%Let me assume D(s)=K, i.e, some constant. Let K=1. Let me check out if the >> %D(s)=k satisfies the required condition.i.e, it should pass through dominant >> %poles at 2+j2 and its conjugate

>> numf=[2 0.1]

numf =

2.0000 0.1000

>> denf=[1 0.1 4 0]

denf =

1.0000 0.1000 4.0000 0

>> fun=tf(numf,denf)

Transfer function:

2 s + 0.1

-------------------

s^3 + 0.1 s^2 + 4 s

>> %In the above lines, by selecting K=1, I have found the transfer function of the >> %uncompensated system.

>>%Now, let me find root locus of the uncompensated system and check if at all >>%there is a need for a compensator.

>> rlocus(fun)

>>%From the root locus obtained as in fig1, it can be observed that the it never >>%passes through 2+j2 and its conjugate, unlike the requirement in the question. >>%Hence, compensator is required

>>%So now, I design a lead compensator with D(s)=K(s+z)/(s+p)

>>%We need to find z,p,K that satisfy the required condition

>>%First, let me find angle of departure of dominant pole at -2+2j

>> desired_pole1=-2+2*i

desired_pole1 =

-2.0000 + 2.0000i

>> req_angle1=180/pi*[angle(polyval(numf,desired_pole1)/polyval(denf,desired_pole1))-pi]

req_angle1 =

-116.6996

>>%Thus, angle of departure of pole at -2+2j=-116.6996 deg

>>%Now, let me find angle of departure of dominant pole at -2-2j

>> desired_pole2=-2-2*i

desired_pole2 =

-2.0000 - 2.0000i

>> req_angle2=180/pi*[angle(polyval(numf,desired_pole2)/polyval(denf,desired_pole2))-pi]

req_angle2 =

-243.3004

>>%Thus, angle of departure of pole at -2-2j=116.6996

>>%Now, the compensator zero and pole must be placed such that

>>% i)root locus passes through dominant poles

>>% ii)difference of angles made by compensator zero and compensator pole >>%with respect to a given dominant pole equals angle of departure of the given >>%dominant pole

>>%We can place the zero and pole of compensator in MATLAB using RLTOOL >>%function

>> rltool(fun)

>>%The root locus was obtained as in fig1.

>>%The compensator pole was placed at -9.89 and compensator zero was adjusted >> %and placed at -0.261 such that root locus passed through the dominant poles

>>%The value of gain K was found out from CONTROLS AND ESTIMATION >> %TOOLS MANAGER (fig2).

>>%From fig2, K=0.35577*3.8/0.1=13.5166

>>%Thus, compensator design values are z=-0/261, p=-9.89, K=13.5166

>>%Question 1(a) is solved

>>%Hence, the root locus as in fig3 was obtained

>>%Now, the system transfer function H(s)=D(s)*G(s) is obtained

>> num=[27.0332 8.183 0.3528]

num =

27.0332 8.1830 0.3528

den=[1 9.99 4.989 39.56 0]

den =

1.0000 9.9900 4.9890 39.5600 0

>> sys=tf(num,den)

Transfer function:

27.03 s^2 + 8.183 s + 0.3528

------------------------------------

s^4 + 9.99 s^3 + 4.989 s^2 + 39.56 s

>>%Root locus of the overall system transfer function was found out just to verify >> %if it was passing through the dominant poles

>> rlocus(sys)

>>%The root locus is found to be passing through dominant poles as required as >> %shown in fig4

>>%Next, Bode plot is plotted and phase and gain margin were found out.

>> bode(sys)

>> margin(sys)

>>%The obtained bode plot is shown in fig5

>>%From bode plot(fig5), crossover frequency=3.66rad/s and phase

>> % margin=67.2deg

>>%Thus, the required condition for crossover frequency and phase margin are

>> %satisfied.

>>%Question 1(b) is solved

>>%Next, for designed values of damping ratio and crossover frequency, transfer >> %function is found out.

>> %From fig2, damping ratio=1. From fig5, crossover frequency=3.66rad/s.

>> numk=[13.4]

numk =

13.4000

>> denk=[1 3.66 13.4]

denk =

1.0000 3.6600 13.4000

>> sysk=tf(numk,denk)

Transfer function:

13.4

-------------------

s^2 + 3.66 s + 13.4

>> sysk=minreal(sysk)

Transfer function:

13.4

-------------------

s^2 + 3.66 s + 13.4

>> kv=dcgain(sysk)

kv =

1.0000

>>% Velocity constant=1. Root locus is shown in fig4.

>>%Question 1(c) is solved.

>>%Thus, the given compensator is designed successfully.