e R >>+ Sigma >>> Dc(s) >>>>> G(s) >>>>>>Y Y goes to Sigma and it is (-) sign (

Question

                          e

R >>+   Sigma >>>   Dc(s) >>>>> G(s) >>>>>>Y

Y goes to Sigma and it is (-) sign ( feedback system)   ( Sorry for Don't use picture)

(a) Suppose
G(s) =
2500K

s(s + 25)
.
Design a lead compensator so that the phase margin of the system is more than 45; the steady state error due to a ramp should be less than or equal to 0.01. Verify your answer with the Bode plot and the time response via the Simulink.

(b) Using the plant transfer function from part (a), design a lead compensator so that the overshoot is less than 25% and the 2% settling time less than 0.1 sec. Verify your answer with the step response (MATLAB).

(c) Suppose
G(s) =
                K

s(0.1s + 1)(0.2s + 1)

and let the performance specications now be Kv = lims0 sG(s) = 100 and the phase margin greater than equal to 40. Is the lead compensation eective for this system? Find a lag compensator, and plot the root locus of the compensated system.

(d) Using G(s) in part (c), design a lag compensator such that the peak overshoot is less than 20% and the Kv = lims0 sG(s) = 100.

(e) Repeat part (c) using a lead-lag compensator.

(f) Find the root locus of the compensated system in part (e), and compare your ndings with those from part (c).

Explanation / Answer

2. Phase lag and lead compensator

pos=input('Type %OS '); % Input desired percent overshoot.

Tp=input('Type peak time '); % Input desired peak time.

Kv=input('Type value of Kv '); % Input desired Kv.

numg=[1]; % Define numerator of G(s).

deng=poly([0 -1 -4]); % Define denominator of G(s).

G=tf(numg,deng); % Create G(s) without K.

s=tf([1 0],1); % Create transfer function, 's'.

sG=s*G; % Create sG(s).

sG=minreal(sG); % Cancel common factors.

K=dcgain(Kv/sG); % Solve for K.

'G(s)' % Display label.

G=tf(K*numg,deng); % Put K into G(s).

G=zpk(G) % Convert G(s) to factored form and

% display.

z=(-log(pos/100))/(sqrt(pi^2+log(pos/100)^2));

% Calculate required damping ratio.

Pmreq=atan(2*z/(sqrt(-2*z^2+sqrt(1+4*z^4))))*(180/pi);

% Calculate required phase margin.

wn=pi/(Tp*sqrt(1-z^2)); % Calculate required natural

% frequency.

wBW=wn*sqrt((1-2*z^2)+sqrt(4*z^4-4*z^2+2));

% Determine required bandwidth.

wpm=0.8*wBW; % Choose new phase-margin frequency.

[M,P]=bode(G,wpm); % Get Bode data.

Pmreqc=Pmreq-(180+P)+5; % Find phase contribution required

% from lead compensator with

% additional 5 degrees.

beta=(1-sin(Pmreqc*pi/180))/(1+sin(Pmreqc*pi/180));

% Find beta.

% Design lag compensator zero, pole, and gain.

zclag=wpm/10; % Calculate zero of lag compensator.

pclag=zclag*beta; % Calculate pole of lag compensator.

Kclag=beta; % Calculate gain of lag compensator.

'Lag compensator, Glag(s)' % Display label.

Glag=tf(Kclag*[1 zclag],[1 pclag]); % Create lag compensator.

Glag=zpk(Glag) % Convert Glag(s) to factored form

% and display.

% Design lead compensator zero, pole, and gain.

zclead=wpm*sqrt(beta); % Calculate zero of lead compensator.

pclead=zclead/beta; % Calculate pole of lead compensator.

Kclead=1/beta; % Calculate gain of lead compensator.

'Lead compensator' % Display label.

Glead=tf(Kclead*[1 zclead],[1 pclead]);

% Create lead compensator.

Glead=zpk(Glead) % Convert Glead(s) to factored form

% and display.

'Lag-Lead Compensated Ge(s)' % Display label.

Ge=G*Glag*Glead % Create compensated system,

% Ge(s)=G(s)Glag(s)Glead(s).

sGe=s*Ge; % Create sGe(s).

sGe=minreal(sGe); % Cancel common factors.

Kv=dcgain(sGe) % Calculate Kv

T=feedback(Ge,1); % Find T(s).

step(T) % Generate closed-loop, lag-lead-

% compensated step response.

title('Lag-Lead-Compensated Step Response')

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